Pramana - journal of physics
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Raman Research Institute. Bangalore
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Indian Institute of Science, Bangalore
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B M Deb, Punjab University, Chandigarh
Rohini M Godbole, Indian Institute of Science, Banyalorc
S Kailas, Bhabha Atomic Research Centre. Bombay
R K Kaul, The Institute of Mathematical Sciences. Madras
A V Khare, Institute of Physics. Bhubaneswar
1 Padmanabhan, Inter- Univ. Centre for Astronomy and Astrophysics, Pune
R Ramaswamy, Jawaharlal Nehru University. New Delhi
A K Raychaudhuri, Indian Institute of Science, Banaalore
K C Rustagi, Centre for Advanced Technology, Indore
E V Sampathkumaran, Tata Institute of Fundamental Research, Bombay
Abhijit Sen, Institute for Plasma Research. Gandhinaaar
S K Sikka, Bhabha Atomic Research Centre. Bombay
Y Singh, Banaras Hindu University, Varanasi
Editor of Publications of the Academy
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C-MM/4CS, NAL, Banaalore
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Pramaea - journal of physics
Volume 46
1996
R Nityananda
Raman Research Institute, Bangalore
Associate Editor
H R Krishnamurthy
Indian Institute of Science, Bangalore
Editorial Board
G S Agarwal, Physical Research Laboratory, Ahmedabad
V Balakrishnan, Indian Institute of Technology, Madras
] K Bhattacharjee, Indian Assoc. for the Cultivation of Science, Calcutta
D N Bose, Indian Institute of Technology, Kharagpur
B M 'Deb, Punjab University, Chandigarh
Rohini M Godbole, Indian Institute of Science, Bangalore
S Kailas, Bhabha Atomic Research Centre, Bombay
R K Kaul, The Institute of Mathematical Sciences, Madras
A V Khare, Institute of Physics, Bhubaneswar
T Padmanabhan, Inter-Univ. Centre for Astronomy and Astrophysics, Pune
R Ramaswamy, Jawaharlal Nehru University, New Delhi
A K Raychaudhuri, Indian Institute of Science, Bangalore
K C Rustagi, Centre for Advanced Technology, Indore
E V Sampathkumaran, Tata Institute of Fundamental Research, Bombay
Abhijit Sen, Institute for Plasma Research, Gandhinagar
S K Sikka, Bhabha Atomic Research Centre, Bombay
Y Singh, Banaras Hindu University, Varanasi
Editor of Publications of the Academy
V K Gaur
C-MMACS, NAL, Bangalore
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Pramana - journal of physics
CONTENTS - VOLUME 46
(January- June 1996)
Number 1
Painleve analysis and exact solutions of two dimensional Korteweg-de
Vries-Burgers equation M P Joy
Objectification problem, CHSH inequalities for a system of two spin- 1/2
particles G Kar and S Roy
On the structure and multipole moments of axially symmetric stationary
metrics S Chaudhuri and K C Das
Effect of heavy quark symmetry on the mass difference of B-system in minimal
left right symmetric model A K Girl, L Maharana and R Mohanta
Signature inversion in the K = 4~ band in doubly-odd 152 Eu and 156 Tb
nuclei: Role of the h 9/2 proton orbital
Alpana Goel and Ashok K Jain
Conservation of channel spin in transfer reactions
V S Mathur and Anjana Acharya
Number 2
Perturbation theory of polar hard Gaussian overlap fluid mixtures
Sudhir K Gokhul and Suresh K Sinha
Structural study of aqueous solutions of tetrahydrofuran and acetone mix-
tures using dielectric relaxation technique AC Kumbharkhane,
S N Helambe, M P Lokhande, S Doraiswamy and S C Mehrotra
Composite Anderson-Newns model and density of states due to chemisorp-
tion: Quasi-chemical approximation
R Guleria, P K Ahluwalia and K C Sharma
Transient and thermally stimulated depolarization currents in pure and iodine
doped polyvinyl formal (PVF) films P K Khare
Mobile interstitial model and mobile electron model of mechano-induced
luminescence in coloured alkali halide crystals B P Chandra,
Seema Singh, Bharti Ojha and R G Shrivastava
Review
Coupled scalar field equations for nonlinear wave modulations in dispersive
media N N Rao 161
Research Articles
Self-interacting one-dimensional oscillators Mamta and Vishwamittar 203
Causal dissipative cosmology N Banerjee and Aroonkumar Beesham 213
An identity for 4-spacetimes embedded into E 5 Jose L Lopez-BoniUa
and H N N unez- Yepez 2 1 9
A q deformation of Gell-Mann-Okubo mass formula
B Bagchi and S N Biswas 223
Scaling laws for plasma transport due to ^-driven turbulence
C B Dwivedi and M Bhattacharjee 229
Brief Report
Current algebra results for the B D systems V Gupta and H S Mani 239
Number 4
Review
Waves with linear, quadratic and cubic coordinate dependence of amplitude in
crystals G N Borzdov 245
Research Articles
Dielectric behaviour of ketone-amine binary mixtures at microwave frequencies
P J Singh and K S Sharma 259
A model for the reflectivity spectra of TmTe P Nayak 271
Evidence for superconductivity in fluorinated La 2 CuO 4 at 35K: Microwave
investigations .G M Phatak, K Gangadharan,
R M Kadam, M D Sastry and U R K Rao 277
Harmonic generation studies in laser ablated YBCO thin film grown on <100>
MgO Neeraj Khare, J R Buckley,
R M Bowman, G B Donaldson and C M Pegrum 283
A Compton profile study of tantalum B K Sharma, B L Ahuja,
Usha Mittal, S Perkkid, T Paakkari and S Manninen 289
S Sivaprakasam, Ch Saradhi Babu and Ranjit Singh
Nonlinear Schrodinger equation for optical media with quintic nonlinearity
G Mohanachandran, V C Kuriakose and K Babu Joseph
Number 5
Geometric phase a la Pancharatnam
Veer Chand Rakhecha and Apoorva G Wagh
Time dependent canonical perturbation theory III: Application to a system with
nonconstant unperturbed frequencies
Mitaxi P Mehta and B R Sitaram
Cosmic strings in Bianchi II, VIII and IX spacetirnes: Integrable cases
LK Pa.tel,S D Maharaj and P G L Leach
A spherically symmetric gravitational collapse-field with radiation
P C Vaidya and L K Patel
Static and dynamic properties of heavy light mesons in infinite mass limit
D K Choudhury and Pratibha Das
Effective potentials and threshold anomaly
S V S Sastry and S K Kataria
Depopulation of Na(8s) colliding with ground state He: Study of collision
dynamics A A Khan, K K Prasad, S K Verma, V Kumar and A Kumar
Multiconfiguration Hartree-Fock calculations in Cr 5 + , Mn 6 + and Fe 7 +
S N Tiwary, P Kumar and R P Roy
Number 6
Linear periodic and quasiperiodic anisotropic layered media with arbitrary
orientation of optic axis A numerical study
V Mahalakshmi, Jolly Jose and S Dutta Gupta
Optimal barrier subdivision for Kramers' escape rate
Mulugeta Bekelc, G Ananthakrishna and N Kumar
Ratios of B and D meson decay constants with heavy quark symmetry
.A K Giri, L Maharana and R Mohama
Diquark structure in heavy quark baryons in a geometric model
Lina Paria and Afsar Abbas
Semi-empirical formulae for the A and neutron-hole oscillator frequency
M Z Rahman Khan and Nasra Neelofer
Positron scattering from hydrocarbons
Ritu Raizada and K LBaluja 431
Subject Index 451
Author Index 455
Information for Contributors
Painleve analysis and exact solutions of two dimensional
Korteweg-de Vries-Burgers equation
MPJOY
Materials Research Centre, Indian Institute of Science, Bangalore 560012, India
MS received 5 April 1995; revised 26 September 1995
Abstract. Two dimensional Korteweg-de Vries-Burgers equation is shown to be non-integr
able using Painleve analysis. Exact travelling wave solutions are obtained using an algorithmic
approach of truncating the Painleve series expansions.
Keywords. Korteweg-de Vries-Burgers equation; Painleve analysis; exact solutions.
PACSNos 02-30; 03-40
1. Introduction
Painleve analysis is a powerful tool in investigating the integrability properties o
differential equations [1,2,3]. It can be used to find Lax pairs, Baddund transform
ations, Hirota's bilinear equations, symmetries, invariants, etc., of integrable equations
It can also be used to find exact solutions of non-integrable equations in an algorithmii
way [4]. In this paper we present the Painleve analysis of two dimensional Korteweg-
de Vries-Burgers equation (2d-KdVB),
(u t + uu x + nu xxx - vu xx ) x + ffu yy = 0. (1
It is a two-dimensional generalization of KdVB equation which is used as a non-linea
wave model of fluid flow in an elastic tube with dispersion and dissipation, flow o
liquids containing small bubbles, etc. [5,6]. The 2d-KdVB serves as a model fo
propagation of shallow water waves subject to a small transverse disturbance am
influenced by viscosity. Similarity solutions of such systems are discussed in [7]
Recently travelling wave solutions of this system were derived using different method
[8,9, 10]. In all these methods they assumed solutions with travelling wave form ant
substituted such a solution in the system and determined the exact parameter values a
which they exist, if they exist.
Using the technique of truncating the Painleve series expansions at different order
we obtained exact travelling wave solutions of this equation without assuming an
particular form for the solutions a priori. Equation (1) is found to be non-Painleve typ
and due to Painleve conjecture it is non-integrable, but it has got conditional Painlev
property. When v = and a = it becomes KdV equation and when v = it become
KP equation. Both of them are integrable and have soliton solutions. When fj. = an
cr = it becomes Burgers equation which is also integrable. At o- = it is KdV]
Painleve analysis of KdVB is given in [1 1].
In the next section we present the Painleve analysis of 2d-KdVB. In 3 exact
solutions of the system are given. Last section summarises the results and conclusion.
2. Painleve analysis
In Painleve analysis we expand the solution u about a singular manifold (f)(x, y, t) in
an infinite series
J=o
where a is a negative integer determined by balancing the powers of $ of dominant
terms in the equation. is a non-characteristic manifold. Coefficients Uj are functions of
x, y and t. If solutions are single valued about the movable singular manifold, the' partial
differential equation is said to have Painleve property. There are basically three steps in
the Painleve analysis, viz, dominant behaviour analysis, finding the resonances, and
checking whether arbitrary coefficients enter at the resonance values [3].
From the dominant behaviour analysis we get a = 2. By balancing the terms of
order $ ~ 6 in the equation after substituting (2) for u(x, y, t) in (1), we obtain recurrence
relations for Uj
(j -4)0 - 5)Uj. 2 (/> x (}) t + (j-5)[uj_ 3 (j> xt + Uj_ 3tX 4> t + Uj- 3 .A]
+ u -4,v + (j~k- 2)(k - 2)U_ k u k (t) 2 x + 2(j -k- 2)u. k u k _^ x (t) x
+ (J ~ 5)[j-3^ xjeic + 3 Mj ._ ^ x (j) xx + 3w ; _ 3>:ex J + Uj_t tXXX }
+ o-{(;-5)[(;-4) Mj ._ 2 ^ 2 + u J ._ 3 ^ + 2^_ 3)y ^] + w/ _ 4 ^} = 0, (3)
where coefficients u- } with negative./ are taken to be zero. For j = we obtain
u =-12^. (4)
Using (4) in (3) collecting coefficients of u j we obtain
6)u j ^h j ((l) x ,(f) y , ( t) t ,...,u ,...,u j _ 1 \ (5)
where h } is a non-linear function. We can see that j= 1,4,5,6, are resonances at
which Uj becomes arbitrary. Resonance at - 1 corresponds to the arbitrariness of 0.
Exact solutions of 2d-KdVB
For the system to have Painleve property (PP) there should be 3 more arbitrary
coefficients u 4 , w 5 , and u 6 occurring at ; = 4, 5, and 6 respectively without any
constraint on </>. That is, at those values the compatibility relation hj = 0, should be
consistently satisfied. When hj ^ at a resonance value;, we can make u- arbitrary by
including logarithmic terms in the series or there will be a constraint on </> at that j. In
that case the arbitrariness of </> is lost and we may say that the equation has conditional
PP. Using the constraints we solve for the particular form of and it can be used to find
special solutions of such conditional PP systems.
For successive values of), from the recurrence relation (3) we obtain
7 = 4 w 4 is arbitrary with the constraint,
= ^1-2^^^ + ^^ = A. (9)
7 = 5 u 5 is arbitrary with
= 20J0J, - ^^ w - tyJJ^
- 00^, = B. (10)
We can see that (10) is satisfied identically if (9) is satisfied, because fi/0^ =
There is incompatibility at j = 6, and the recurrence relation is too lengthy and
complicated. h 6 does not vanish identically, when u 4 and u s are arbitrary. From this
analysis we see that 2d-KdVB is non-Painleve and because of Painleve conjecture it is
non-integrable.
3. Exact solutions
To find special solutions we can truncate the Painleve expansion (2) at a particular
order and find constraining equations on 0. Solving for and substituting in (2) we
obtain the corresponding special solution for system (1). The solution we obtain after
this truncation procedure is not the general solution of the system because they do not
contain sufficient number of arbitrary coefficients, but they are exact. In the truncation
procedure we impose the condition that coefficients of the higher order terms in the
expansion are zero.
(12)
= a^l +
+ 12/z</4 - Tv^^ + 44^0^0^ + 9/10*0^. (14)
= ~ O^y - ^0x0^ - 0^0^ - x xxt + 3V0 XX XX;C
- fy<t> xxx + v$ x <l> XXX x - 4 V<i> xx <i> xxxx - ^ x (t> xxxxx - (15)
A solution for of the form
= exp(- ) + ,4, (16)
where , and A are arbitrary and
= fcx + fy - cof, (17)
exists for the above equations (12-15). Here,
v _ 6v 3 5al 2
k , co
and / is arbitrary. It is to be noted here that (16) is not the most general solution of
equations (12-15), it is the simplest non-trivial solution one can obtain for 0. If we
could find other solutions for we can use them in (11) to find other solutions of the
original system (1). Here we did not assume any particular form for solutions of the
original system.
With this, the solution u given by (1 1) becomes
- 12v 2 1
2 - ( )
When A 0, the trivial solution u = constant is obtained. When A = 1 we get one of the
solutions given in [10]. Then (18) can be written as
where
S = sech[i(- )] (20)
and
T = tanh[i(- )]. (21)
Case (ii). Let us truncate the solution at the next order, i.e. u j = for all j ^ 2,
(w 7* 0,1/^0). Then
u,
" + (22)
0,
+ 2 ' (23
=
(24;
J
6fj.v(f> xxxxx + ^>^ 2 ^ xxxxxx - (26;
These equations satisfy a solution for of the form (16) with
.v / 6v 3
and / arbitrary. Here we get two waves in opposite directions. In this case, the 'solution
(22) will be (after substituting for u ,u^)
">^,, 1 7i i A ~~r Ti \~\\2 ~*~ 1 i A . r Ti E \-\ i \^n
When A = 1 we obtain the solution given by (12) in [9] and (15) in [10],
3v 2
(28)
where S and T are as defined in (20) and (21).
Case (in). Now we set Uj = for all j ^ 3 (u , u 1 , u 2 ^ 0). Then
tt n U,
u = ^ + ^ + 2 . (29)
Now the recurrence relations give
1 a((> 2 y (j) xx (j) t (t) xx
M P Joy
2v<j) x
,
T
=
+ ^xr^jcx H > i X)> /2 T^ + "5^ +
(3 1)
y 2iXx;c 2>xx
U -- T
(32)
= <2, W - + W L + U 2, Xt + U 2 U 2, XX ~ V" 2 ,xxx + W 2 . XXXX - ( 33 )
Here we see that at j = 6 the recurrence relation gives 2d-KdVB for u 2 . Hence (29) may
be considered as an auto-Backlund transformation. We can find a solution for (j) from
(30-33) in the form (16) with
v vc 5ffl 2 u
k= , co = - -
5/^ 5[i v
and / arbitrary, where,
co/c - al 2 6v 2
(34)
Here we note that /I is the value of u 2 which can be an arbitrary non-zero constant. Now
the solution for (1), given by (29) is
- 12v 2 f 1 -1 + 1
+ A.
(35)
For A = 1 we could obtain (17) of [10] or (10) of [8].
by a constant is only obtained. Therefore we do not obtain non-tnvially new travelling
wave solutions from the known solutions using (29).
The solution (36) can be obtained from (28) by using the fact that (1) is invariant
under the transformation [10]
u* = u + A, x* = x + fa, t* = t, y* = y.
The solution (36) of 2d-KdVB can be written in terms of travelling wave solutions of KP
and 2d-Burgers equations [9]. If we continue the process of truncation to higher order
we do not obtain any new solutions. From (3) we see that at j -6,7 and 8 compatibility
conditions require w 3 = and hence all other higher order coefficients should be zero
for compatibility.
4. Conclusion
We used the Painleve test as described by Weiss et al [1] to study the integrability of 2d
Korteweg-de Vries-Burgers equation and showed that it is non-integrable. We
obtained exact travelling wave solutions by using the method of truncation of Painleve
series at successive orders and all previously known travelling wave solutions as special
cases of our solution to the system. Moreover we may obtain other types of exact
solutions, if we can solve the constraining equations on </> at each order. Here we did not
assume any particular form for the solutions unlike others [8,9, 10], where they assumed
the travelling wave form for solutions and searched for them. In such methods we obtain
only special solutions of the assumed form, if they exist. The method of P-analysis is
algorithmic and it gives details on the integrability aspects of the equation also.
Acknowledgement
The author acknowledges the financial support from National Board for Higher
Mathematics, Department of Atomic Energy, India.
References
[1] J Weiss, M Tabor and G Carnevale, The Painleve property for partial differential
equations, J. Math. Phys. 24, 522-526 (1983)
[2] R Conte, Invariant Painleve analysis of partial differential equations, Phys. Lett. A140,
383-390(1989)
[3] A Ramani, B Grammaticos and T Bountis, The Painleve property and singularity analysis
of integrable and non-integrable systems, Phys. Rep. 180, 159-245 (1989)
[4] F Cariello and M Tabor, Painleve expansions for non-integrable evolution equations,
Physica D39, 77-94 (1989)
[5] R S Johnson, A non-linear equation incorporating damping and dispersion, J. Fluid Mech.
42, 49-60 (1970)
[6] L van Wijngaarden, One-dimensional flow of liquids containing small gas bubbles, Ann.
Rev. Fluid Mech. 4, 369-395 (1972)
[7] P Barrera and T Brugarino, Similarity solutions of the generalized Kadomtsev-Petviashvili-
Burgers equation, Nuovo Cimento B92, 142-156 (1986)
[8] W Ma, An exact solution to two-dimensional Korteweg-de Vries-Burgers equation,
J. Phys. A26, L17-L20 (1993)
[9] Li Zhibin and Wang Mingliang, Travelling wave solutions to two-dimensiom
Burgers equation, J. Phys. A26, 6027-6031 (1993)
[10] E J Parkes, Exact solutions to the two dimensional Korteweg-de Vries-Burgers e
J. Phys. A27, L497-L501 (1994)
[11] W D Halford and M Vlieg-Hulstman, Korteweg-de Vries-Burgers equation
Painleve property, J. Phys. A25, 2375-2379 (1992)
Objectification problem, CHSH inequalities for
a system of two spin-1/2 particles
G KAR and S ROY
Physics and Applied Mathematics Unit, Indian Statistical Institute, Calcutta 700035, India
MS received 25 November 1994; revised 13 September 1995
Abstract. The weak objectification and Bell/CHSH inequalities are studied for a particular
type of set of states of two spin-1/2 particles. The restriction on interference term which allows
Bell/CHSH inequalities to be satisfied are found out.
Keywords. Bell-CHSH inequalities; interference term; weak objectification; classicality of
probabilities.
PACSNo. 03-65
1. Introduction
A quantum mechanical observable is generally non-objective unless the system is
prepared in an eigenstate or gemenge of eigenstates. The hypothetical assignment of
eigenvalues of some non-objective observable to a system which is actually in a super-
position of eigenstates known as weak objectification, is incompatible with quantum
mechanics [1,2].
Hypothetical value assignment is also the subject of discussion about hidden
variable theories underlying quantum mechanics. It has been shown that application of
EPR reality criterion and locality on a collection of observables is equivalent to the
existence of their joint distribution. But in certain important special cases the existence
of joint probabilities is equivalent to the validity of a set of probability relations, namely
Bell/CHSH inequalities [3,4], the experimental violation of which can be taken as
evidence against objectification. Recently it has been shown that the same set of
inequalities is necessary and sufficient conditions that these probabilities can be
obtained in the range of classical probability measure [5,6]. One could use such
a criterion of the classicality of probabilities in order to postulate the objectivity of
properties of a physical system.
It has been shown that weak objectification and the classicality of probabilities lead
to different consequences [2]. For example, in the case of two spin-1/2 particles in
a singlet state, it has been shown that joint probability may be satisfied even in cases
where weak objectification fails. The validity domain of Bell's inequalities has been
shown in figure 1 of [2].
The main cause of non-objectification, either in the sense of weak objectification or
classicality condition for probabilities, is the existence of interference term in a pure
G Kar and S Roy
state. Here, considering two parameter family of states of two spin- 1/2 particles which
are intermediate between singlet state and mixture of product states having spin zero
configuration along a particular direction, we shall find out how much interference
term can be allowed so that Bell's inequalities are always satisfied for any set of three
spin- 1/2 observables. This will be done in 3.
In 2 the two parameter family of states will be described and in these states
probabilities for spin- 1/2 observables will be calculated. In 4, the maximum amount of
interference term which allows the CHSH inequalities to be valid for the chosen sets of
four observables which give maximal violation in the extreme pure states will be
investigated. In this case the whole set of states described in 2 will be considered.
2. Two parameter family of states and probabilities
We consider a system of two spin- 1/2 particles which is associated to a 4-dimensional
Hilbert space (<^ 2 (x) ^ 2 ). States are represented by positive, self adjoint and trace class
operator of trace one.
We consider the family of states of the form
(1)
where |0^.(n)>(a = 1,2) are eigenstates of n-d(o being Pauli matrices and n is a unit
vector) for the a-th spin-1/2 particle. P[-] is one-dimensional projection operator
corresponding to the vector state in the third bracket and 1 ^ r ^ 1 and 1^/1^1.
For r = I and A = 0, W corresponds to singlet state and for r = 1 and A = 0,
W corresponds to a pure entangled state given by
)>]. (2)
The spin observables are represented by projection operators. Let P(n ) represent the
projection operator corresponding to the spin observable which measures spin along
the direction n,-.
The probability and joint probability distributions that measurements of n ; -<7 on one
subsystem, n,-a on the other, and n -o and nya jointly, will give + 1 result for the
subsystems as well as jointly for both systems in the state W, are respectively given by
When the two particle system is in the state W, the reduced mixed states of the
subsystems are
(n)>]. (4)
Tr a denotes trace operation on W with respect to the Hilbert space associated with the
oc-th particle.
From the last expressions apparently it seems that the subsystems are in a mixture
of eigenstates where ignorance interpretation can be applied. To check that, let us now
assume that the spin observables n a is weakly objectified with respect to the subsystem
Si in the state W. This implies that the observable n-<r7 is weakly objectified
with respect to the compound system in the state W. Now weak objectification entails
that [2]
Tr[^P( ni )(x)P(n 2 )] =Tr[L(n) WP(n,}P(* 2 }-\ (5)
where P(n t ) and P(n 2 ) are arbitrary test observables and the Luder operation L(n) is
given by
L(n) W= P(n) WP(n) + (I- P(n)) W(I - P(n))
= i^[|0!H(n)>l0 2 -(n)>] + iP[|c/>!.(n)>(x)|0 2 + (n)>]. (6)
Now (5) gives
;-[(nx ni )(nxn 2 )] = 0. (7)
Equation (7) is the condition under which the observable n-a can be weakly
objectified with respect to the state W if the observable n 1 -<rn 2 -<r is used as test
observable. Equation (7) is fulfilled if one of the following conditions are satisfied:
i) one of the factors (n x n t ) or (n x n 2 ) or both are zero.
ii) the planes spanned by (n, n l ) and (n, n 2 ) are orthogonal.
iii) r = i.e. there will be no interference term in W.
From the above observation we conclude that for r ^ 0, in the case of arbitrary choice
of test observables weak objectification fails.
We shall now find the restriction on r for the set of states \V(Q ^ r ^ 1) so that the
probability sequence {pi-,p 2 -,P3>Pi2iP32>Pii} can nave classical representation.
One should take care that pj t has been included in the probability sequence as in the
case of Bell's inequalities one of the three spin observables has to pertain to both the
system. Now in the case of singlet state, p li =Q, and so Bell-Wigner inequality does not
contain such term but for the state W(r 0)
Pi^^Cl-Cn-nJ]. (8)
G Kar and S Roy
Let us now write down one of the CHSH inequalities [6] concerning four observ-
ables PfaJ and P(n 3 ) for one of the particles and P(n 2 ) and P(n 4 ) for the other particle
in the state W.
Now making two observables P(nJ and P(n 4 ) same i.e. n^ = n 4 and putting all the
probabilities from (3) we get
2 - r[l + Hj-n 2 + n 2 -n 3 - n 1 -n 3 ] + (1 - ^[(n^nj^-n)
+ (n 2 -n)(n 3 -n) + (n 1 -n) 2 -(n 1 -n)(n 3 -n)]^0. (10)
To find the maximum value of r for which there will be no violation of the inequality
(10), let us choose the unit vectors for which maximal violation of Bell's inequality
occurs. The choice is given by
2 J '
Let the polar and azimuthal angles of the unit vector n are and ($> respectively. Then
cosdk
where t,j, k are unit vectors along co-ordinate axes.
Then calculating all the scalar product and putting them in the inequality (10) we get
2-|r~i(l-r)sm 2 0[3sin 2 (/)- N /3sin(/)cos^]^0. (11)
The maximum value of sin 2 0[3 sin 2 $ ^/Ssin cos $] will occur at = n/2 and
(f) = 7T/2 + 15. Putting this value of 9 and we get
-2^0 (12)
which gives
r< 5-2,/3 = (M
7-2^/3
For this choice of unit vectors other three inequalities [6] are not violated even for
singlet state.
So as long as r ^ r B , there will be no violation of Bell's inequalities which implies that
the above probability sequence, obtained from three spin- 1/2 observables giving
maximal violation for singlet state, in a. state W, will have classical representation.
4. Non-violation of CHSH inequalities and bound on interference term
In this section we shall study CHSH inequality concerning four different observables
for the whole set of two parameter family of states W given in (1).
. J n/_ \ n/_
which can be written as
r[(nj x n)(n 3 x n) + (n t x n)(n 4 x n) + (n 2 x n)(n 4 x n)
-(n 2 -n)(n 3 -n)K2. (16)
From (16) it is clear that if the observable n-o is weakly objectified in the state V^for all
the pair of test observables n l n 3 . . . etc, the first term in the right hand side of (16)
vanishes and there will be no violation of CHSH inequalities.
Now to the restriction on r, let us first check whether there is any violation for the
state W with r = 1. Putting r - 1 in (15) we get
- [ivn 3 + n x -n 4 + n 2 -n 4 - n 2 -n 3 ] + 2[(n 1 -n)(n 3 -n) +
(n^nXivn) + (n 2 -n)(n 4 -n) (n 2 -n)(n 3 -n)] ^ 2. (17)
Now with the choice,
n 4 = i, n 3 -j,
1 . . -.. 1 . ~ *
(18)
v '2 72'
and
n = sin 6 cos (j)i + sin 6 sin $j + cos Ok (19)
inequality (17) gives
2^2 -2^/2 sin 2 0^2. (20)
So from inequality (20) there will be violation as long as < sin_ l [(1 l/^/z) 1 ' 2 ] and
the maximal violation will occur at = 0. To find out the restriction on r in general, we
shall apply the choice of vectors of (18) for the states W with 1 ^ r < and for those
with < r ^ 1 we shall apply the standard choice giving maximal violation of singlet
state and which is given by
n 2 = i, n 4 =/
n 3 =-/=(-r+/), n 1 =-y=(f+/). (21)
v v
Case 1
For < r < 1, and for the choice of the unit vectors of (21), inequality (15) gives
- 2 + 2^/2r + (1 - r)^/2 sin 2 = /(r, 0) ^ 0. (22)
/(r, 0) is maximum for = 90, and then (22) gives r < y/2 - 1. For = 0, there will be
no violation of CHSH inequality as long as r ^ 1/-J2.
Pramana - J. Phys., Vol. 46, No. 1, January 1996 1 3
whichever choice the vector n takes.
Case 2
For 1 < r < 0, and for the choice of unit vectors given in (18) and putting r = r,
inequality (15) gives
- 2 + 2^/2r - (1 + r)^/2 sin 2 = f(r, 0) ^ 0. (23)
Here /(r, 6) is maximum for 6 = 0. So as long as r ^ 1A/2, there will be no violation
for the choice of observables which give maximal violation in the state W with r= 1.
For both the above choices of observables the remaining CHSH inequalities are not
violated even for singlet state. So in general we can say that for the density operators in
(1), there will be no violation of CHSH inequalities if
V
for the above choices of spin observables.
5. Discussion
We see from the above result that for correlated state of two spin- 1/2 particles with spin
zero configuration along any direction, whether Bell/CHSH inequalities will be
satisfied for spin observables depend on the value of the parameter r, be it a pure or
a non-pure state. But this result in no way forbids from getting four dichotomic
observables violating CHSH inequality for any correlated pure state [7].
One more important thing should be noted. In case where dimension of Hilbert
space is three or more, and if we consider all the observables, no state allows classical
representation of probabilities [8]. So classical representation of probabilities has very
limited scope in quantum mechanics.
Comment
The study of Bell/CHSH inequalities for more general state than (1) can be studied in
the same way and it will be published elsewhere in the near future.
Acknowledgement
The authors are greatly indebted to the referee for his valuable suggestions to rewrite
the paper for further clarification.
References
[1] P Busch and P Mittelstaedt, Found. Phys. 21.-889 (1991)
[2] P Busch, P Lahti and P Mittelstaedt, Found. Phys. 22, 949 (1992)
[3] A Fine, Pkys. Rev. Lett. 45, 291 (1982)
[4] D Muynck, Phys. Lett. A114, 65 (1986)
[5] I Pitowski, Quantum probability-Quantum logic, lecture notes in physics, (Springer- Verlag,
Berlin, 1989) Vol. 321
[6] E G Beltrametti and M J Maczynsky, J. Math. Phys. 34, 4919 (1993)
[7] D Home and F Seller!, Riv. Nuovo Cimento. 14, 1-95 (1991)
[8] D Mermin, Rev. Mod. Phys. 65, 803 (1993)
On the structure and multipole moments of axially symmetric
stationary metrics
S CHAUDHURI and K C DAS
Department of Physics, Gushkara Mahavidyalaya, Gushkara, Burdwan 713 128, India
Department of Physics, Katwa College, Katwa, Burdwan 713 130, India
MS received 5 October 1994
Abstract. The structure of the stationary metrics [1], generated from Laplace's solutions as
seed, is investigated. The expressions for the equatorial and polar circumferences, the surface area
of the event horizon, location of singular points and the Gaussian curvatures of the metrics [1]
are derived and their variations with the field parameter a are studied. The multipole moments
are calculated with the help of coordinate invariant Geroch-Hansen technique. These investiga-
tions expose some interesting properties of the metrics, some of which are known in the literature
and some deserve a new interpretation.
Keywords. General relativity; solutions of Einstein's equations; surface geometry; multipole
moments.
PACSNo. 04-20
1. Introduction
In a paper [1], we constructed the stationary solutions of Einstein's field equations
using two different solutions of Laplace's equation as seed. These solutions describe the
real astrophysical objects and reduce to the Kerr metric [2], when some restrictions are
imposed on the constants appearing in the solutions. The first one is asymptotically flat
and represents the field of a rotating axially symmetric object with mass and higher
multipole moments. The second one is not asymptotically flat and may be interpreted
as the gravitational field of a rotating object embedded in an external gravitational
field. With proper restrictions on the constants, the second solution reduces to the
Kerns and Wild metric [3], which is interpreted as Schwarzschild metric embedded in
a gravitational field. Both the solutions are not only for a better description of the field
of a deformed mass but also for a better description of the gravitational field of
a rotating star which is spherically symmetric in its static limit.
In this paper, the structure of our derived metrics [1] is investigated. The location of
the event horizon and infinite red shift surface of the metrics are studied. It is found that
the event-horizon lies within the infinite red shift surface and both the surfaces, like the
Kerr metric, meet at the poles (6 0, it). The expressions for the surface area and
equatorial and polar circumferences are also worked out which give an overall
knowledge about the surface deformation. The Gaussian curvatures of our solutions
are analyzed. It is found that zones of negative curvature develop around the polar and
variations of the surface area, the circumferences and the curvatures with are thus
discussed. The Geroch-Hansen multipole moments [4, 5,6] of the metric for set-1 are
evaluated using Hoenselaers procedure [7]. As set- 2 metric is not asymptotically flat,
the estimation of its multipole moments is not taken into consideration here.
The mass multipole moments are the measure of deviations from the spherical
symmetry of a gravitating body. In all the gravitation theory the mass multipole moment is
related to the distribution of matter. Mass multipole moments exist in Newtonian
gravitation too. The conventional technique of calculating multipole moments lies in
expanding the metric asymptotically. As the general theory of relativity predicts the
distortion of curvature of space-time surrounding the object, even an asymptotically flat
metric gives rise to a different multipole moment near the body where distortion of
curvature is appreciable, compared to the moments calculated from asymptotic expansion.
According to Geroch [5], it is very hard to see how this information could be faithfully
brought in from infinity over the curved space in order to compare it locally with the
matter distribution. As there are equivalent definitions of multipole moments in
Newtonian theory e.g. as coefficient in a multipole expansion, as moments of source
distribution or objects associated with the conformal group [5], Geroch [5] and
Hansen [6] proposed a relativistic and coordinate invariant definition of the multipole
moments. The procedure for calculating relativistic moments prescribed by Geroch
and Hansen is complicated. Following the prescription of Hoenselaers [8], Quevedo
[7] obtained a useful recurrence relation for calculation of higher multipole moments.
In this paper, we have followed Quevedo's technique and obtained expressions for
coordinate invariant relativistic multipole moments. In 2, the procedure for calculat-
ing Geroch-Hansen (G-H) multipole moments is described in brief. In 3, the surface
geometry of the event-horizon, circumferences and curvatures of our metrics [1] are
analyzed and their properties studied. The singularities on the infinite red shift surface
are also investigated. Finally the mass multipole moments of the asymptotically flat
metric are calculated. In conclusion a discussion of the properties of the metrics is given.
2. Procedure for calculation of G-H multipole moments
Fpr an axially symmetric stationary line element, in prolate spheroidal coordinates (x, y),
* i y
(1)
the Ernst potential [9, 10] is defined as
=/ + i<D, (2)
where, k is a constant, /, y, w and <E> are functions of (x,y) only. $ is known as the twist
potential. The prolate spheroidal coordinates (x, y) are related to Papapetrou coordi-
nates (p, z) by
P 2 = fc 2 (x 2 -l)(l-j; 2 ), (3)
z = kxy. (4)
!+'
Weyl canonical coordinate by
(5)
(6)
and the conformally transformed potential <f on the symmetry axis y = 1, is defined by
z
With the above substitution, Hoenselaers listed the expressions for mass multipole
moments (M,) and current multipole moments (J t ) of the source as
M, = Re(w, + d,), (8)
(9)
where,
J t = Im(m, 4- d,),
m, =
Id'fcl)
''"/! dz 1
(10)
z =
d t (l = 0, 1, 2, 3,. . .) is determined by comparing (8) and (9) with the original Geroch-
Hansen definition. According to them d t can be expressed in terms of m k for k ^ / 1.
The first six values of d { are
m
21" 1 '
Expanding in powers of z, one obtains
k=l
and from (10) and (7):
2=0
*!'
m,=
(/+!)!
2 =
(11)
(12)
(13)
Substituting the value of from (5) in (13), an important recurrence formula for m t is
obtained
m, = -
where
1 d
(14)
(15)
Pramana - J. Phys., Vol. 46, No. 1, January 1996
19
Q/l
/2 ( = ~~ i + 2^ 1 Vi, for 1^2. (16)
Uxj
For static metric, / = e 2 ^, <$ = and k = m, the above calculation becomes a simple
task.
In general, Quevedo [7] summarized the above procedure as follows. (1) Calculate
Ernst potential E and according to (2) and (5), (2) Calculate m l according to (14),
(3) Obtain multipole moments from (8), (9) and (11).
3. On the surface geometry of event-horizon and multipole moments
In this section, the structure of our metrics [1] is investigated. The equatorial and polar
circumferences, the surface area of the event-horizon and the Gaussian curvatures at
the polar and equatorial regions are computed. It is shown that the superposing field
plays an important role on the shape of the infinite red shift surface. Negative curvature
zones are found to exist under certain restrictions on the values of the constants a and
a. The mass multipole moments and the current multipole moments are derived for an
asymptotically flat stationary solution. The multipole moments differ much from the
moments of Kerr.
The axially symmetric line element is written in the form as in eq (1). The metric
functions/, y and w are given in ref. [1].
According to Gutsunaev and Manko [1 1], the Ernst potential E = f + i(f>, can be
expressed as
where a and b are derived from two pairs of first order differential equations given in
[1 1, 1]; 2\fs is the Laplace's solution. Thus different solutions of Laplace's equation will
render different solutions of the axially symmetric field equations.
We considered the following cases in ref. [1].
Set 1: Laplace's solution
2^ = a (x + y)- 1 , (18)
where a is a constant,
a = - aexp[ j;(x + y)~ '], (19)
b = aexp[-a (l+xj;)(x + j;)- 2 ], (20)
f = e* l(x+y) A/B, (21)
(22)
i + k 2 , (23)
20 Pramana-.T PHi/c Vnl Af* Mn ^ T an .,o
A = x 2 -
- a 2 (l - y 2 )
+ a 2 [(l -
(25)
y(e ( ~ ao(jcy + * ))/(JC + y)2
+ a ( 1 y 2 ) [e ( ~ aa(x
x l-a^t-^
+a 2 e ( - ao(1 - y2))/(JC+JF)2 )].
(26)
(a) Singularities, infinite red shift surface and event-horizon
A preliminary analysis of our metric (21)-(26) was published in [1]. Computer analysis
shows that the metric retains its singularity at the poles x = 1, y = 1, and for a
constant value of y, the location of x-coordinate of singular points changes with the
variations of a and a. On the equatorial plane (y = 0), and planes adjacent to the
equator (0-5 ^ y > 0), for a constant value of a, the singular points come closer to x = 1
value when a is increased. When y > 0-5 the x-coordinate of singularity first decreases
and then increases with the increase in a . It is also observed that for a constant value of
a, the range of x-coordinate of singular points decreases when 3; is increased from to 1.
It is interesting that for y ~ 0-9 and a ^ 0-5, singular point shifts away from the origin
when a is gradually increased, while for y = 0-9 and a = 0-9, the x-coordinate of
singularity decreases with increase in a . The location of singular points are shown in
figures l(a)-(e), for pre-assigned values of a and a.
2.0
1.5
1.0
0.5
o.o c
alpha.y
Location of singularity with strength
of superposing field (alphanot)
Location of singularity
* -*- .
" "" l *-. -.i
) 0.2 0-4 0-6 0-8
Alphanot
series B x series c
5,0 forB:.5.-5 for C
Figure 1 (a). Graphs illustrating the locus of singular points due to the variations
of the strength of the superposing field (a ). Figure shows that the singular points
come closer to x = 1, for different set of values of a and 3; (here a = 0-5, v = for series
B and a = 0-5, y = 0-5 for series C) when a is increased from 0-1 to 0-9.
\
-0.0
0.20
0-40 0-60
Alphanot
0.80
H)0
Figure 1 (b). A plot showing the locus of singularities against the variation of a ,
with constant a and y (we have taken a = 0-3, y = 0-7). Singular points are found to
come closer to x = 1 value and then goes away from it.
. When the values of a and a (a ^ 0-5) are kept fixed, it is found that with the increment
in the value of y, the location of singular point increases slightly beyond x = 1 value and
then comes closer to x = 1, However, for a > 0-5, the x-coordinate of singular point is
a decreasing function of y. These are illustrated in figures 2(a) and 2(b). With constant
a and y, as the value of a increases, the singular point shifts away from x = 1 value.
Our metric shows two important surfaces, namely, the event-horizon and the infinite
red shift surface enclosing the event-horizon. The existence of the event-horizon is an
important factor for a black hole which according to Penrose is the boundary of the
asymptotic region from which time-like curves may escape to infinity [12]. An
event-horizon is always a null hypersurface and more than one may be present there. It
is a one way path and there may occur a naked singularity in the absence of an
event-horizon. The event-horizon of the metric (1) is at x = x hor = 1.
The infinite red shift surface can be obtained by equating
/ = (27)
in (21). Static sources and observers can stay only outside the above surface and not on
or inside it. From (21), (24), (25) and (27), it is found that x- lrs = x(y) and x ; r s > x hor for
|y| < 1 . However at the poles y = 1 , the infinite red shift surface and the event-horizon
touches each other. Thus the event-horizon is always covered by the infinite red shift
surface similar to the Kerr metric and the ergosphere (i.e. the region in between the infinite
red shift surface and event-horizon) has analogous properties to that of Kerr metric.
(b) Surface area, polar and equatorial circumferences
At event-horizon i.e. at x = 1 and t constant, our metric (1) can be treated as a two
22
Pramana - .1. Phvs.. Vol. 46. Nn 1 .Tamiaru
strength of superposing field
(alphanot): alpha = 0. 8, Y= 0-9.
0.20
0-AO 0.60
Alphanot
0.80
1-00
Figure l(c). Figure shows the shift in the location of singular points with
variation of a (a = 0-8, y = 0-9 are kept constants).
Location of singularity with the
strength of superposing field
(alphanot): alpha- 0- 5, Y =0.9,
-0-0
0-20
0.40 0.60
Alphanot
0.80
1.00
Figure l(d). The variation of the location of singular points with a Q (for a =
y = 09) are plotted. Singular points are found to shift away from x = 1 value.
(alphanot): alpha =0-9, Y =0.9,
5. o
3 <N
o *7
"o
0.20 0.40 0.60 0.80
Alphanot
1.00
Figure 1 (e). Graph illustrating the locus of singular points due to the variation of
(for a = 0-9, y = 0-9). As a Q increases, singular points come closer to x = 1.
X
2-0
1-5
1.0
0-5
0.0
olph
Location of singularity at different
latitudes for const. alphanot and alpha
coordinate of singularity
' ^~~^^~^^-_
* * i
i i i i
0.2 0.4 0.6 0.8
Latitude
, series A x series B
anot, alpha = 0.6/ 0-3 for A: 0.5. 0.5 B
Figure 2 (a). Graphical illustration showing the influence of external field on the
shape of the infinite red shift surface. For oc = 0-6, a = 0-3, a = 0-5 and a = 0-5, the
infinite red shift surfaces are shown in series A and B respectively.
dimensional line element which under the coordinate transformation
y = cos 0,
fe 1 =(l-a 2 )" 2 and k 2 = -4koc(l -a 2 )' 1 ,
(28)
(29)
latitudes forconst. alphanot and alpha
X coordinate of singularity
0-2 0-4 0-6 0-8
Latitude
series A x series B
alphanot, alpha = 0.5, 0.8 forA; 0.9. 0.9 B
4.
-r
sin 2
- aocos 0/( 1 + cos 0)
H
and
Figure 2(b). Another plot showing the shape of infinite red shift surface for
constant values of a Q and a (a = 0-5, a = 0-8 for series A and a = 0-9, a = 0-9 for
series B).
assumes the form
ds 2 = doodO 2 + ^ d ^ 2 ' (3)
where
k 2 _ ao/(1 + cos <,)
9ao = ~ TT?-"^ > (31)
(32)
(33)
(34)
(1-a 2 )- ' (35)
The surface area thus increases with a . On substituting a = 0, (35) reduces to the
familiar Kerr expression, S = 87tm(m + (m 2 -a 2 ) 1/2 ). For = a = 0, Schwarzschild
expression, S = \6nrn 2 is reproduced.
The latitudinal circumference (i.e. the curcumference at different latitude) can be
obtained from the integral
(36)
H = a 2 [(1 + cos e)e" of2 - (1 - cos 0)<?- ao/2 ] 2 +
The surface area of the event-horizon can be evaluated from the integral [13]
fit r2n
H
Integrating for metric (31)-(33), we obtain
l + aV) _
o L,nauanun ana
and is found to be
(37)
Here, 9 = n/2 represents the equator and = 0, JT are the upper and lower poles
respectively. One thus obtains the expression for equatorial circumference as
(1 + a V)
22
- 2 )[a
} 2 + 4e a ] 1/2 '
(38)
Equation (38) reduces to the corresponding Kerr expression for = and to the
Schwarzschild for a = 0, a = 0.
The polar circumference (A p ) can be computed from the relation
(39)
For metric (30) given by (31)-(33), it is found that
k
(l- 2 )Jo
Evaluation of the integral (40) is not simple. In order to get the exact variations of A lf
A & and A v with a , (37), (38) and (40) are analyzed using computer. It is found that for
L
2.0
1.5
t-0
0.5
0.0
a
Latitudinal Circumference
latitude-y in prolate sph.coordlnates
at.circumference 8pi k units
1
x-~ -^
f ^7 ^"^
7 ^
) 0.2 0.4 0-6 0.8
Latitude
phanot. alpha = 4, .5 for A: 10. -1 forB
Figure 3(a). The plot shows the changes of circumference at different latitudes for
Latitudinal Circumference
latitude;./ in prolate sph. coordinates
Lot- circumference 8pi k units
0.2 0.4 0-6
Latitude
x series C series
0-8
Figure 3(b). Curves represent the variation of latitudinal circumferences for
constant values of a and a. The assumed values of a and a are 7,0-5 and 10,0-5 for
series C, and D respectively.
1
Variation of Polar and Equatorial
circumference with alphanot
'olari. Equatorial circumferences
inn
/
80
an
/
L n
\^ ^^
* * * K x, x H *^"^
(
) 2 4 6 8 10 1
Alphanot
2
a
phas.1 Ap Ae
Figure 4(a). With constant a (a = 0-1), the nature of variations of polar and
equatorial circumferences with the strength of the superposing field (a ) are shown.
The upper curve represents polar circumference (A p ) and the lower one is for
equatorial circumference (A e ). A p always remains larger than A e .
constant values of a and a, the latitudinal circumference first increases with the
latitude and then decreases. The locus of the points on the infinite red shift surface or
perhaps the body itself assumes a dumbbell structure. The deformation is different for
different sets of values of a and a as illustrated in figures 3 (a) and 3(b). It is also noted
that for a constant value of a (but for a < 0-5), when a is gradually increased, the
equatorial and polar circumferences first decrease and then increase with a and
6 8 10
Alphanot
series A x- . series B
alphas 0.9
Figure 4(b). A set of curves, showing the polar (A p ) and equatorial (A e ) circum-
ferences, are given. For a = 0-9 and a Q > 8, A e exceeds A .'
A p always remains larger than A e . However, when a>0-5, both A p and A & are
increasing functions of a . But the rate of increase of A & is larger than that of A p . When
a > 8, the equatorial circumference becomes larger than the polar circumference. The
exact value of a for which A e exceeds A p depends on the value of a. Figures 4(a) and
4(b) show variations of polar and equatorial circumferences with a . With a constant
both A p and A,, increases with the increase in a.
The nature of variation of x with a , a or y as observed in the above analysis points to
the deformation of the infinite red shift surface and perhaps of the body itself to which
the field is due.
Astrophysical objects are not isolated in space. They are embedded in external
gravitational as well as electromagnetic field and thus our above analysis bears some
relevance to those objects. However, the exact correspondence is yet to be discovered.
(c) Gaussian curvature
The Gaussian curvature is a measure of geometry intrinsic to the horizon and it is
independent of embedding space. The Gaussian curvature of the metric (30) can be
computed from the relation [14]
c== __ _
2EGdO\EG
where
The Gaussian curvature in this case becomes
(41)
(42)
2/c 2
(43)
28
Pramana - J. Phys., Vol. 46, No. 1, January 1996
where
B, = a 2 ^ + 2b 2 cos 6 + 6 3 cos 2 " (1 - COS0)/(1 + cose)
_ -(aocos6)/(l +cos9
e:
1-COS0
"(1
-<xocosO/(l + cos0)
(1 + cos 6)
fi 5 =
2a n a (l-cos0)
,ao(l-cosO)/(l+cos0)
1 | o( 1- cos 0)/{1+ cos 0) (44)
e
and
On substituting a = and <x = a = 0, (43) reduces to the corresponding expressions
for Kerr and Schwarzschild metrics respectively. The curvature is found to be a func-
tion of polar angle.
At the pole = 0, the curvature becomes
-^
In the polar region, where a and a satisfy the relation
= 4a 2 (47)
the curvature is zero and the surface in that region becomes a plane.
A zone of negative curvature develops around the pole 6 = 0, if the values of a and
a are such that
provided aV < 1. If aV > 1, the curvature will always be negative. The negative
curvature cannot be visualized because it cannot be embedded in a flat Euclidean
accustomed to in the usual three dimensional Euclidean space. They have positive
curvature everywhere. The second type is very unusual in our familiar three dimen-
sional space. They possess negative Gaussian curvature and global embedding is not
possible for surfaces having C < 0.
Since there exists a singularity at the pole Q = n (i.e. at x = 1, y = 1), it is very
difficult to investigate the nature of curvature at that pole with a computer.
At the equator 9 = n/2, the curvature can be written in the form
2/cV
[(2-2a -aS)P 2 +(R-8a Q)P-8Q 2 ],
(49)
where
It is very difficult to predict the nature of variation of equatorial curvature with a .
However, a computer analysis shows that for a constant value of a, C e=7r/2 is an
increasing function of a . Figure 5 shows the plot of equatorial curvature with the
variations of a . Further it is found that for fixed a , equatorial curvature increases
with the increase in a.
(d) Multipole moments
It has been shown in our earlier paper [1] that ( 19) to (26) determine our new stationary
metric (1) completely. With a = oc = 0, our metric reduces to the Schwarzschild
(
A
3
1
1
Alpha
Equatorial curvature vs alphanot
alphanotzstrenghtof superposing field
Curvature in k unfts
1
,-^*^^
*~~~*^~~ __
^==^^^^
0.2 (U 0.6 OS
Alphanot
. series A + series B x series C
= 0.1 A',0.5 for 6:0.9 forC
Figure 5. A plot of equatorial curvature (CJ with the strength of the external field
is shown. C e is an increasing function of . The values of a taken are 0-1 for series A,
0-5 for series B and 0-9 for series C.
30
Pramana - J. Phvs.. Vol. 46. Nn. 1. .Tannarv IQQfi
kx = r m, y = cosQ, k = mp, a = mq,
k 2 = m 2 -a 2 , p = (l-a 2 )(l+a 2 )- 1 , q = 2a(l + a 2 )~'.
fCerr metric is obtained in its standard form. When no restrictions are imposed on the
constants i.e. a / 0, a * 0, the solution given by (19)-(26), generalizes the Kerr metric
with an arbitrary set of multipole moments determined by the parameter . The first
bur coordinate invariant relativistic Geroch-Hansen [5, 6] multipole moments char-
acterizing the mass and angular momentum distributions are computed as follows
(l-a 2 ) 2
__ fc^ ^ i
^) +4a 2 (l-a
2\-:
(51)
J 1 =2/c 2 a[a (l-a 2 )-(l+a 2 )](l-a 2 )- 2 ,
J 2 = -2/c 3 aa (l-a 2 )" 1 ,
+ 2a 2 {a (l - a 2 )(9 - a 4 ) - 4(1 + a 2 )} (1 - a 2 )'
(52)
Jince J = 0, the metric obtained is asymptotically flat [15]. With a = 0, the multipole
noments so obtained reduce to that of Kerr and the mass monopole term becomes
qual to the total mass of the source.
A computer calculation for the variation of mass multipole moments with the strength of
he superposing field (a ) shows that the monopole moment (A/ ) decreases with increase in
: while the dipole moment (M t ) increases. For a constant value of a (we have taken
: = 0-9), these two moments M and M i becomes equal when a a 9-5. As regards to
[uadrupole (M 2 ) and octupole (M 3 ) moments it is found that the former increases with
: while the latter is a decreasing function of a . The variations of mass multipole
loments with the external field parameter a are shown in figures 6(a) and 6(b).
On the other hand, with constant a , the monopole moment increases with the
ncrease in the rotation parameter a, while the dipole moment is independent of it. The
uadrupole moment first increases with the increase in a and then decreases for values
f lying in the range 1 ^ a < 8. When a ^ 8, M 2 increases with a. The octupole
loment is a decreasing function of a.
10
a
6
4
2
alpha =
Variations of moments wilh alphanot
ser A for m 0; B for ml
^^
""^-^
^X
*""^ \ 1 II 1
024 6 8 10 12
Alphanot
Figure 6 (a). The graph shows the variations of monopole (M ) and dipole (Mj )
moments with , keeping a constant (here we have taken a = 09). Series A repre-
sents monopole moment and series B for dipole moment.
Variations of moments with alphanot
ser.C form 2:0 form3
momentstThou sands)
2. _.
-2
-4
-6
-8
-10
-12
alphas
^'--fr- j,^ ' " " " ^
^v.
x ^
X
N
D 2 1, 6 8 10 12
Alphanot
series C x series D
0-9
Figure 6(b). Variations of quadrupole (series C) and octupole (series D) moments
with are plotted (for a = 0-9).
Set 2: Let us take another Laplace's solution
a and b were found to be [1]
a= -aexp[a z (x-y)],
(53;
(54;
(55;
where z is another constant. The metric functions /, w and y, as in set 1, are given by [1]
(56)
(57)
(x 2 -l)(l-y 2 )l (58)
where k 1 and k 2 are constants and A, B, C are given by
aoo^2
], (59)
_ J) e -oy-j2
aozox ] 2 }, (60)
C = ae~ 2aozoy l(x 2 - l)(e a<!Zoy - a 2 e" ao20) ')
x f g aozox , g-aozox _ y/g-aozox _ g aorox\1
+ (1 y 2 )(e xozox + e ~ aozox )
x {e 01020 ^ - a 2 e~ aozoy + x(e MZoy + a 2 e~ aozoy )}]. (61)
(a) Infinite red shift surface, event-horizon and singularities
The event-horizon of our metric (1) is at x = x hor = 1 and the infinite red shift surface is
obtained by equating / = in (56). At the poles y + 1, x = 1, i.e. the event-horizon
and the infinite red shift surface coincide. However, for values of \y\ < 1, x lrs > x hor ,
and the ergosphere possesses Kerr-like properties.
The reported metric is singular at the poles x = 1, y= 1, and at least one
singular point exists on the equatorial plane y = 0. With proper restrictions on the
constants a and a, our derived metric reduces to the different well-known metrics such
as Schwarzschild, Kerr and Kerns and Wild.
(b) Surface area, polar and equatorial circumferences
At event-horizon i.e. at x = 1, our metric (1) with 2if/ = a xy, assumes the form of a two
dimensional line element, which on substitution
(62)
and
can be written as
0d4> 2 , (63)
where
k 2
a aa = - - =-, H'<r aocosfl , (64)
S Chaudhuri and K C Das
- 2 222 ee^ osd ( ,
{65)
(i~a 2 ) 2
#' = a 2 [(l + cos6>)e ao -(l -cosfl)^* ] 2 + 4e 2aocose , (66)
/ 8 = ^ ao + e- an . (67)
The surface area of the event-horizon is obtained as
,68)
The surface area thus increases with increase in the strength of the superposing
field (a ). On substituting a = 0, (67) reduces to that of Kerr and with = a = 0,
Schwarzschild's expression S = \6nrn 2 is obtained.
The latitudinal circumference (A,) is computed from (36) and is found to be
(69)
With = Ti/2, one obtains the expression for circumference at the equator. A computer
analysis shows that with a = constant and for small value of a (0-l ^ a < 0-5, a < 4), the
latitudinal circumference decreases compared to that at the equator (i.e. equatorial
circumference). Further, for the same value of a but a > 4, the latitudinal circumference
increases gradually as one approaches the pole. With a = 0-5 and = 4, it first increases
with y = cos 6 and then decreases. The variations of A l are plotted in figures 7 (a) and 7(b). It
is observed that for small values of a , the equatorial circumference (A e ) is a decreasing
function of a . However, for large values of a (a > 20), the circumference at the equator
approaches a constant as shown in figure 8.
The polar circumference is given by
p C1 nr 2 \ L
U a ) Jo
(1 COS I
The evaluation of the integral (70) is difficult. However, it is found that for a constant value
of a (but a < 1) the polar circumference is an increasing function of a (see figure 9). It is also
noted that the rate of increase of A p is large for greater value of a. For constant ,
A p increases too with a.
(c) Gaussian curvature
The Gaussian curvature of the metric (63) is now calculated using (41) and is expressed as
~R' 1 9
Lot. circumference in 8pi k uni't
0.4 0.6
Latitude.
. seriesA x-
upper foralphanot=A,alpha=.5; Lower 1,.
series B
Figure 7 (a). The nature of latitudinal circumferences are shown in the figure for
different constant sets of values of a and a. (Series B for a = 4, a = 0-5, and series
Latitudinal circumference
Latitude = y in prolate sph. coordinates
. ,Lat. circumference in 8pi k unit
1 n
/
8
/
x"
6
/
/
^S
o
-~-*^
0.2 0,4 0.6 0*8 1
Latitude
series A
alphanot=10, alpha =0.5
Figure 7(b). Another plot of latitudinal circumference for a = 10, a = 0-5. The
nature of A } differs considerably from that shown in figure 7(a). This is due to the
change in the value of a .
# 3 = (a sin 2 9 - 2cos 0)e aocose ,
B' 4 = [ajsin 2 0-2(1+ 2a cos
B' 5 = 2a 2 (b' 2 + 6' 3 cos 9) + 8a e 2aocos(> 5
F 6 = 2a 2 &' 3 + 16a 2 e 2aocose ,
Pramana - J. Phys., Vol. 46, No. 1, January 1996
(72)
35
E
0-8
0.6
0.4
0.2
0-0
upper f(
tifuuiuuui ciicurmerence vs aipnanot
Aiphanot = strength of superposing field
iqul. circumference dpi k unit
^
V, .
\
\
V~^_
10
20 30 40 50
Aiphanot
>ralpha=0-5, lower for 0.1
Figure 8. The graph represents the variation of equatorial circumference with th<
strength of the superposing field. With the increase in a , A e first decreases and thei
assumes a constant value. The upper curve is for a = 0-5 and the lower one is fo
a = 0-1.
uoo
1000
600
200
Aipha=
Polar circumference vs Aiphanot
Aiphanot * strength of superposing field
3 olar circumference in k unit
/
/
/
/
/
r*^~^
3 2 4 6 s 10
Aiphanot
_. series A
o.i
Figure 9. Variation of polar circumference (A ) with a is shown. A increases wit!
the increase in . The assumed value of a = (H.
and
(72
c
0=0
(l-a 2 )
As the value of is increased, the curvature at the pole decreases. A zone of negative
curvature develops around the pole 6 0, if the values of a and a satisfy the following
relation
(l-2a )^<r (75)
provided a < 1. However, when a > 1, the condition for obtaining a negative curvature is
somewhat different and it is expressed by
(2a -V<. (76)
The curvature at the other pole is obtained by putting = n in (71)-(72) and can be
written in the form
M _/v2\3 rt 3aor On,2 "1
(77)
0=n 4k 2 (l + a 2 ) 2
The curvature thus increases with increasing values of . If the values of and a are such
that they satisfy the following relation
-> 2a 2 ,-.
(l+2a )e < , (78)
a zone of negative curvature develops around the polar region 6 = n.
From the above discussion it follows that zones of negative curvature develops around
the pole whether a < 1 or a > 1. But the restriction on the values of oc is different in each
case. With a < 1, the negative curvature will appear when a < 1/2 and condition (75) is
obeyed. If the value of oc becomes a ^ 1/2, the curvature will then become negative
irrespective of condition (75). On the other hand, when a > 1, zone of negative curvature
will appear if a > 1/2 and restriction (76) is satisfied. However, when a ^ 1/2, the
curvature will be negative whether a and a satisfy (76) or not. At the pole 9 = n, for a < 1, as
long as the inequality (78) is satisfied, there is no other restriction on the value of a in
obtaining negative curvature at that pole. If a > 1, the curvature at the pole = n will
always be negative.
At the equator, 6 = n/2, the curvature becomes
C 9=rc/2 = T- [ L2 ( 2 - o) + 2L ( N + 2 oM) - 8M 2 ], (79)
where
jV = Z/ 3 a 2 + 8a 2 . (80)
, b' 2 , b' 3 are given by (73).
Pramana - J. Phys., Vol. 46, No. 1, January 1996 37
Alphanot= strength of superposing field
Curvature
-0.5
Alphanot
o series A x
Alpha = 0.5 for ser A 10.1'forserB
series B
Figure 10. Plot showing the nature of variation of equatorial curvature (C e )
with a for pre-assigned value of a. Here, a = 0-5 for series A and a = 0-1 for series
B. C e decreases with increase in . The negative curvature region is also shown.
A computer analysis shows that the equatorial curvature decreases with the increase in
, and for a > 0-9, a zone of negative curvature develops around the equatorial region.
The variation of equatorial curvature with the strength of the superposing field a and the
negative curvature region are illustrated in figure 10. Further, it is observed that with the
increase in a, C fl=3t/2 decreases.
4. Conclusion
An analysis of the surface geometry of our derived metrics [1] is presented in this paper.
In set 1, the derived metric is asymptotically flat and on imposing some restrictions on
the constants a and a appearing in the solutions ((21)-(26)) it reduces to the well-known
Schwarzschild and Kerr metrics. The derived solution, thus generalizes the Kerr metric
with an arbitrary set of multipole moments determined by the parameter a . The
singularities of the solution are investigated using computer. The seed function is singular
on the surface x + y = and this singularity is reflected in the derived metric too. Since
1 > y > 1, the singular values of x remain encased within x = 1 surface. Further the
metric is singular at the poles x=+l,)>=l.Itis observed that the location of singular
points depend on the values of the constants a and a. For large y and for a < 0-5, the
x-coordinates of the singular points are found to be located away from the origin when
the strength of the superposing field a gradually increases, while for a ~ 0-9 and for
the same value of v, the values of x are decreasing function of a . When the values of a and
y are kept constant the singular points are found to be located away from x = 1 value with
the increment in a.
It has been stated earlier that with = 0, our derived metric reduces to the Kerr metric.
When a ^ 0, it is found that the infinite red shift surface becomes distorted although the
event-horizon remains the same as that of Kerr.
wuii ut, wiicu oc -# u'u. 111 me msi ua.se me ia.ie 01 increase 01 /i e is greater
than that of A p .
In this connection, it may be pointed out that the results obtained here is contrary
to our previous result [16], where the same seed has been used to obtain two soliton
solution of axially symmetric metric by the inverse scattering method of Belinskii
and Zakharov [17]. It was found that A p increases with a while A e decreases. This is
perhaps due to a greater number of constants appearing in the solution [16] and different
sets of values assigned to the constants. However, when the constants are properly
adjusted, it is found that the solutions obtained by these two methods (viz. the Gutsunaev
and Manko method of the present paper and the inverse scattering method of [16])
coincide with each other.
The Gaussian curvatures of metric (30) are also evaluated. On imposing some restrictions on
the constants a and a, zones of negative curvature are found to develop around the
polar region.
The coordinate invariant Geroch-Hansen relativistic mass and angular momentum
multipole moments are computed and their relative abundances are plotted. On substitu-
ting a = 0, the multipole moments reduce to the moments corresponding to Kerr.
For set 2, it was shown that with proper restrictions on the constants oc and a, our
solutions given in (56)-(61) reduce to the Schwarzschild, Kerr and Kerns and Wild metrics.
The general solution may thus be interpreted as the non-linear super-position of Kerr
metric with a gravitational field. The solutions obtained in this case are not asymptotically
flat. The event-horizon is always covered by the infinite red shift surface. The metric is
singular at the poles x= 1, y= 1 and at least one singular point exists on the
equatorial plane.
The surface area of the event-horizon, the latitudinal and polar circumferences are
evaluated and these are found to vary with a . The Gaussian curvatures in the polar and
equatorial regions are also computed and it is noted that when certain restrictions are
imposed on the constants a and a, zones of negative curvature develop around those
regions.
In the late sixties, Kerr-Newman solution drew much attention of the astrophysicists,
since that was the only solution then, supposed to represent the actual field of a rotating
charged object. Moreover, that solution goes over to Kerr when electrostatic charge is set
to zero.
It is now believed that Kerr metric cannot represent the exact exterior field of an
arbitrary rotating star because of its very special relationship between the multipole
moments and angular momentum [18]. In our previous paper we have given a
Kerr-like metric associated with/without external gravitational field. In this paper,
we studied their structural properties in order to compare them with Kerr metric.
Although our analysis is a positive step towards the goal, it is not proved beyond
doubt whether these solutions prove or disprove Israel and Caster conjecture
[19,20,21] and describe the exterior field of a so called black hole. As regards naked
singularities, Newton-Rapson or equivalent methods of analysis failed to predict
naked singularity outside the infinite red shift surface. Further analysis remains open
for future.
Pramana - J. Phys., Vol. 46, No. 1, January 1996 39
Acknowledgements
Thanks are due to Prof. S Banerji, Department of Physics, Burdwan University, Burdwan
for many useful discussions on the paper. One of the authors (SC) wishes to thank the
UGC for financial support.
References
[1] K C Das and S Chaudhuri, Pramana - J. Phys. 40, 277 (1993)
[2] R P Kerr, Phys. Rev. Lett. 11, 237 (1963)
[3] R M Kerns and W J Wild, Gen. Relativ. Gravit. 14, 1 (1982)
[4] R Geroch, J. Math. Phys. 11, 1955 (1970)
[5] R Geroch, J. Math. Phys. 11, 2580 (1970)
[6] R O Hansen, J. Math. Phys. 15, 46 (1974)
[7] H Quevedo, Phys. Rev. D39, 2904 (1989)
[8] C Hoenselaers, Gravitational collapse and relativity, Proc. 14th Yamada Conf., Kyoto
Japan, 1986, edited by H Sato and T Nakamura (World Scientific, Singapore, 1986)
p. 176-184
[9] F J Ernst, Phys. Rev. D167, 1 175 (1968)
[10] E J Ernst, Phys. Rev. D168, 1415 (1968)
[11] Ts I Gutsunaev and V S Manko, Gen. Relativ. Gravit. 20, 327 (1988)
[12] W Kinnerseley and M Walker, Phys. Rev. D2, 1359 (1970)
[13] W J Wild and R M Kerns, Phys. Rev. D21, 332 (1980)
[14] T Willmore, An introduction to differential geometry (Oxford University Press, Oxford,
England, 1959) p. 79
[15] V S Manko and I D Novikov, Class. Quant. Gravit. 9, 2477 (1992)
[16] S Chaudhuri and K C Das, (Communicated)
[17] V A Belinskii and V E Zakharov, Sov. Phys. JETP, 48, 985 (1978)
[18] J Castejon-Amenedo and V S Manko, Phys. Rev. D41, 2018 (1990)
[19] W Israel, Phys. Rev. 164, 1776 (1967)
[20] B Carter, Phys. Rev. Lett. 26, 331 (1971)
[21] K S Throne, Comm. Astrophys. Space Phys. 2, 191 (1970)
Effect of heavy quark symmetry on the mass difference of
B-system in minimal left right symmetric model
A K GIRI, L MAHARANA and R MOHANTA
Physics Department, Utkal University, Bhubaneswar 751 004, India
MS received 5 July 1995; revised 15 November 1995
Abstract. An estimation of the mass difference of B B system with heajyy quark symmetry
formalism is presented. The effective Hamiltonian describing the transition hd<-*hd (where h = b
for B-system) is considered in a manifest left right symmetric (MLRS) model along with
contribution from neutral Higgs boson. We use the spin and flavor symmetry for heavy quarks to
obtain the transition matrix element <B \3V e[[ (x)\Bj > in terms of Isgur- Wise function. Assuming
that B and 5 states are at rest, we find that Isgur- Wise function turns out to be unity. However
using the experimental values of AM K and AM Brf as input, we find that M R = 835GeV and
Keywords. Left right symmetry; gauge bosons.
PACS Nos 1 1-30; 13-10; 14-80
1. Introduction
In recent years heavy flavor dynamics has proven to be very useful to obtain model
independent information on systems containing heavy quarks [1,2]. When one or more
quarks are heavy compared to hadronic scale, some new symmetries appear in the low
energy effective Lagrangian for QCD. In the limit m Q -> oo(m Q being the mass of heavy
quark), two additional symmetries beyond those of QCD arise [1]. The first one is the
heavy flavor symmetry where mass of the heavy quark is scaled out and the Lagrangian is
same for all flavors. Thus there is SU(N f ) symmetry among the heavy quarks. The second
symmetry is the spin symmetry. In the limit of infinite heavy quark mass, the spin
degrees of freedom of heavy quarks are decoupled and thus SU(2) rotation of the heavy
quark spin becomes a symmetry. These additional symmetries allow many interesting
predictions. In particular they imply model independent relations between form factors
of weak decays. Several relations [3,4] have been recently derived showing that the
excitation spectra and form factors are independent of mass and spin of heavy quark.
Isgur and Wise [5] showed that in the lowest order in QCD all the weak decay
amplitudes are determined in terms of a single function (vv') which is known as
Isgur- Wise function. Falk et al [6] also showed that the transition amplitudes for
inclusive semileptonic JS-meson decay are given in terms of the Isgur- Wise function.
In the present investigation we exploit these ideas to evaluate the transition matrix
element of #J J3j system in a manifestly left right symmetric model (MLRS) [7] and
obtain the mass of the right handed gauge and Higgs bosons. Here we consider effective
41
standard model with gauge group SU(2) L x Sl/(2) R x 17(1). It has the merit of allowing
the gauge group, P and CP to be broken spontaneously at the same time_ and thus
successfully explain the mass mixing [8] and CP violation [9] for M M system.
Besides this, it offers the possibility of a great deal of new physics beyond standard
model at the energy scale of several hundred GeV. It is therefore crucial to find a lower
limit on M R , the mass of right handed partner of W-boson in this model. In evaluating
the mass difference AM = 2 < M | Jf eff (x)|M >, for Bj - Bj system we use heavy quark
symmetry and obtain the matrix elements in terms of the Isgur-Wise function (w').
However Isgur-Wise function turns out to be unity since both B d and Bj states are at
rest. Thus within the limit of above approximation, the expression for the mass
difference contains parameters like M R and M H . -Evaluating K K mass difference in
MLRS model and fixing it with the experimental value of AM K and using
m t = 174 10 GeV [10] we obtain M R which subsequently when used in the expression
for AM B yields the lower limit of M H .
We organize the paper as follows. In 2, we give the outlines of minimal left right
symmetric model and the effective Hamiltonian we use in our consideration. Section
3 is devoted to the evaluation of hadronic matrix elements and the mass differences for
Bj-Bj and X -K system. In 4, we evaluate the Isgur-Wise function. Section
5 contains results and discussion.
2. Minimal left right symmetric model and evaluation of effective Hamiltonian
We review some features of manifest minimal left right symmetric model relevant to our
discussion. The Lagrangian is invariant under S17(2) L x Sl/(2) R x U(\)j_ L . In this
model one can write the effective Lagrangian for K - K and Bj - Bj transition
process as [11]
^eff = -4 D?(x)y"ULn(x) W^ + p(x)yURn(x
V 2
x)S+-} + h.c, (1)
where p(x) and n(x) are p- and w-type quarks defined by
(x)\
c(x)
t(x)j
n(x) =
d(x)
s(x)
b(x)
(2)
with u,c,t,d,s and b being six quark flavors. W L and W R are left and right handed
charged gauge bosons with mass M V and M R whereas S and S are unphysical
charged Higgs bosons [11] (longitudinal components of W+ and W+ respectively).
Left right symmetry of the gauge interactions requires that the two S17(2) gauge
42 Pramana - J. Phys., Vol. 46, No. 1, January 1996
\
(a)
(b)
Figure 1 (a, b). Feynman diagrams for the K K and B B transition ampli-
tudes for (h s, b). S LR are the unphysical gauge bosons corresponding to W L R and
<t> 2 3 are the flavour changing neutral Higgs bosons of the minimal left right
symmetric model.
couplings be equal [12], i.e./ L =/ R =/. The left- and right-handed weak CKM mixing
matrices are taken to be same in the Lagrangian i.e. U L = / R = U. The helicity
projection matrices for the left- and right-handed ones are denoted by L = (1 y 5 )/2
and R = (1 -)- y 5 )/2 respectively. D p and D n are the mass matrices chosen to be diagonal
and are written as
D P =
lm u
m c
O
and
m s
(3)
We observe that the process we consider here can occur through gauge bosons
exchange as depicted in figure 1, in the lowest order Feynman diagrams. Thus we
obtain the effective interaction Hamiltonian as
(4)
e fj-(ljl\.) H- t?Tgj-f (K.R)j
where ^f eff (LL), ^f cff (RR) and ^f eff (LR) represent the Hamiltonian with the exchange
of (W L , W L ), (W K , W R ) and (W L , W R ) gauge bosons respectively in the box diagram
which are written as
and
#> (1 T\- F L y i ; /
&1 & ff\LjLj) - 2 / j "j'^'/V.
47T i,j = u,c,t
x(^"U)(%L4
ffl fRl?^ 3^ /T T ^ T < > P
(5)
(6)
where
(8)
Pramana - J. Phys., Vol. 46, No. 1, January 1996
43
The parameters taken in the above expressions are /I,- = Uf h U id , r\ = (M^/M^) and
x. = (mf/Ml) where m l is the mass of the ith quark flavor. In fact the Hamiltonian given
in the above expressions indicate a transition for K K system with h = s and for
J5J - B d with h = b flavor. The neutral Higgs boson contribution to the effective
Hamiltonian is realized through the exchange of two neutral Higgs bosons 3> 2 and
<J> 3 at the tree level is given by
j^ eff (tf)=-^~^ ( mi ^) 2 (hLd)(hRd). (9)
In the above we have assumed a common Higgs mass M H for both <D 2 and <D 3 [13].
3. Heavy quark symmetry and mass difference for Af M system
To evaluate the hadronic matrix elements for M M system taking the MLRS
Hamiltonian we consider the mass matrix Jt as
M* 2 M 2
where
M u = M 22 = <M|Jf eff (x)|M> = <M K ff (*)|M >, (11)
and
M 12 = <M|^ eff (x)|M> = <M|^ eff (x)|M>. (12)
In (11) and (12) |M> represents the meson state and |M> represents corresponding
anti-meson state. We diagonalize the mass matrix and obtain the mass difference
between M and M mesons as [14]
AM M o = 2M 12 = 2<Mi^ eff (x)|M >, (13)
and we use heavy quark effective theory (HQET) to evaluate the above matrix elements
for B - B system.
In HQET the ground state for pseudoscalar heavy meson containing a heavy quark
Q and a light anti-quark q is given in terms of interpolating fields [2] as
Pt(v)<=q.y s *ijM~ P , (14)
where h[ is a heavy quark of type T with four velocity v and related to the conventional
quark field operator Q f (x) by
Q i (x) = exp(-im Q u-x)^ J (15)
and the light quark g, stands for a column vector in flavor SU(3) space as
u
. (16)
Thus the pseudoscalar heavy meson transforms as a SL7(3) antitriplet. The charge
conjugate state P,(y) can be related to P ; (u) state by charge conjugation convention as
with the charge conjugation matrix for Dirac spinor c = iy 2 y. Hence we obtain
P i (v)=-h i J 5 q v JM~r. (18)
rhus the ground states for J5 and B mesons are
B d (v) = d v y 5 b v ^/M^, (19)
and
M^. (20)
[n order to estimate the mass matrix elements given in (13), we need the evaluation of
the matrix elements of the quark operator contained in the effective Hamiltonian i.e.,
)|/UAYKIM(t/)>, (21)
and
fcUA-R^MV)). (22)
Evaluation of the above matrix elements in (21) and (22) are formally done by
vacuum saturation method [15]. We present here an explicit evaluation of (21) using
the wave functions for B%(v) and B%(v') in HQET as given in (19) and (20)
(23)
Now we consider the first part of eq. (23), with B%(v) state as given in (19)
t(vi>r). (24)
In the above we have used the relations [16]
(25)
and
(26)
where (v-v') is the Isgur-Wise function.
The evaluation of the second part of eq. (23) gives
(vv'), (27)
thus we obtain
(M(v)\h^Ld v ,h v YLd v \M(v') = 2(irt/K>i/)M M o. (28)
The factor 2 occurs because we can choose the current on the side of M in two different
ways. Similarly evaluation of (22) gives
T R
, (30)
where f. K is the X-meson decay constant, related to the pion decay constant by
f K ~ l-22f K where/^ = 93 MeV. Using these relations we obtain [9]
: f/K M K> ( 31 )
and [11]
where p is given by [17]
Ml A
K .2 + 0)- (33)
With the matrix elements as given in (31) and (32) for K - K system and in (28) and
(29) for B-B system we estimate their mass differences in subsection A and
subsection B respectively.
A. Mass difference for K K system
Here we estimate the mass matrix elements M 12 for K - K system. We neglect the
exchange of w-quark in the box diagram (figure 1), since m u = 0. Including the contribu-
tions from c~ and t-quark exchange, and keeping x c only up to first order, we obtain
-/ii. (34)
with
4-
and
with
GJMJ
r *~ 4 /*"!.... \ i A. . i /? *} i^/l\
- ^t) + (4 - 2x t ) lnx t + (1 - x t ) 2 -\nril (37)
46 Pramana - J. Phys., Vol. 46, No. 1, January 19%
Heavy quark symmetry on MLRS model
r here we have kept only terms of order Y\. Similarly the mass matrix element for the
[iggs sector is given as
M ggs =-
(38)
instituting these expressions for M 12 in (13) we obtain
. . 3/ M 2
x 1+- K
, -^ LR
.ILL
27T
(39)
. Mass difference for J9 5 system with heavy quark symmetry
or 5 B d system, we also neglect the exchange of c-quark as its mass is much smaller
lat the &-quark mass of the external line. Hence considering the contribution only
om virtual t-quark along with the QCD correction factors and using (28) and (29) we
3tain the expressions for M 12 as
2
GIM
ith
- 2
(40)
(41)
ith
id
MLR_
12-
fHiggs _ _
ne QCD correction factors are taken [8] to be ^ QCD = 0-83 and r\,
>tain
(42)
(43)
. (44)
= 1-8. Thus we
G 2 M T 2
uiv^ luai. naiuu wii^ic u ^JL, u, u, \J), we uuiaiii ^u f ^ A. vv & nave uouu. 111^ vjxu rr i^uciiiv
model [18] to determine the Isgur-Wise function. In the context of this model the IW
function may be extracted from the overlap integral
f
= d 3 x<&(x)<&,(x)exp(-iAv'-x) (46)
*/
where the labels I and F denote wave function of the initial and final meson respectively.
The "inertia parameter A" corresponds to the mass of light degrees of freedom. We
shall use for A the expression [19],
A = ^K (47)
m b + m d
which accounts for the kinetic effects of heavy quark. The quark masses are taken as
m d = 330MeV and m ft = 5-12GeV. The wavefunctions are chosen to be the eigen
functions_pf orbital angular momentum I, where both the initial and final mesons i.e.
M and M will have I = and thus the wave functions are given by
ixi), (48)
\i x i/
and
^ (x)=7 ofe)^(|x|), (49)
\l x !/
with normalization
[d 3 ^* F (x)0) I/F (x) = [r 2 drc/)* F (r)0 !/F (r) = 1. (50)
J J
Inserting the wavefunctions as given in (48) and (49) into the overlap integral (46) and
choosing the quantization axis of orbital angular momentum in the direction of
velocity, the Isgur-Wise function is given as
(51)
To calculate the above integral we insert the orbital wave function of harmonic
oscillator in the form
02X3/4
(52)
48 Pramana - J. Phys., Vol. 46, No. 1, January 1996
Heavy quark symmetry on MLRS model
ith strength fi B = 0-41 GeV for B meson [18]. With (vv f ) = 1 we obtain the Isgur-
Vise function for J? B system to be
f(u-u') = l. (53)
. Results and discussion
[ere we estimate the masses of W^ and Higgs bosons. To do this we take the
Dnstituent quark masses as m d = 330MeV, m s = 550Me.V, m c =l-8GeV and
! 6 = 5-12 GeV in addition to the experimentally observed masses of K and B mesons as
f jfo = 497-67 MeV and M B o = 5279 MeV. The experimental values for G F and M L are
iken to be G F = 1-16637 x KT 5 GeV 2 and M L = 80-22 GeV [20]. The CKM ma-
ices involved in our calculations are taken as their central values [20]. Next assuming
ic Higgs contribution to be negligible for ^-system and taking the experimentally
leasured value of AM K = 3-51 x 10 ~ 15 GeV eq. (39) yields M R = 835 GeV. Then using
lis value of M R along with the experimental value AM B =3-35 x 10" 13 GeV [20], we
btain from (45) M H ^ 2-9 TeV.
We have attempted here to predict the masses of right handed gauge boson M R and
jggs boson M H basing on heavy quark symmetry formalism. In doing so we have
msidered the effective Hamiltonian for the system describing 5 B transition in
ILRS model along with the contributions from neutral Higgs boson sector and the
adronic matrix elements for B B system which depends only on the Isgur-Wise
mction. However the Isgur-Wise function can be evaluated with GISW quark model
hich is completely determined by considering the kinematics of the system. Thus the
;timated expression of the mass difference for K K system in left right symmetric
odel with vacuum saturation method gives the value of M R which subsequently when
>ed in the expression for AM B , yields a lower limit of M H . However in the earlier
vestigations Beall et al [17] have derived a lower bound on W R mass to be
f R > 1-6 TeV by demanding AM K > and neglecting the contribution from t- quark,
[ohapatra et al [1 1] included the effect of r-quark and considered the effects of Higgs
tnultaneously with those of gauge bosons, obtained M R > 200 GeV for
r H = 100 GeV. But the present experimental limit on M R and M H are beyond their
timations. Considering the K L K s mass difference in MLRS model Ecker et al [9]
:t lower bounds such as M R ^ 2-5 TeV and M H ^ 10 TeV. Donoghue and Holstein
'!] have analyzed the non-leptonic AS = 1 weak decays and concluded that
r R > 300 GeV assuming left right mixing to be the same. Neglecting the ~quark effect
[aharana [22] in a field theoretic quark model obtained M R > 715 GeV whereas for
elusion of the effect of t quark Maharana et al [23] found that M R = 1650 GeV for
, = 162 GeV. However the present investigation has considered m, = 174 + 10 GeV as
i input [10] to obtain M R = 835 GeV and M H ^ 2-5 TeV. Nevertheless, our result with
cent experimental values of m p AM B , AM K and CKM matrix elements may have
itter reliability in its predictions over the earlier investigations.
:knowledgments
a fellowship.
References
[I] N Isgur and M B Wise, Phys. Lett. B232, 113 (1989); Nucl. Phys. B348, 276 (1991)
[2] H Georgi, Phys. Lett. B240, 447 (1990); Nucl. Phys. B348, 293 (1991)
[3] N Isgur and M B Wise, Phys. Rev. Lett. 66, 1130 (1991)
[4] M B Wise, CALT-68-1721, Lectures presented at the Lake Louise Winter Institute, Feb.
17-23 (1991)
[5] N Isgur and M B Wise, Phys. Lett. B237, 527 (1990)
[6] A F Falk, H Georgi, B Grinstein and M B Wise, Nucl. Phys. B343, 1 (1990)
[7] J C Pad and A Salam, Phys. Rev. D10, 275 (1974)
R N Mohapatra and J C Pati, Phys. Rev. Dll, 566 (1975)
R N Mohapatra and G Senjanovic, Phys. Rev. D12, 1502 (1975)
[8] G Ecker and W Grimus, Z. Phys. C30, 293 (1986)
[9] G Ecker and W Grimus, Nucl. Phys. B258, 328 (1985)
[10] F Abe et al, CDF Collaboration, Phys. Rev. Lett. 73, 226 (1994)
[II] R N Mohapatra, G Senjanovic and M Tran, Phys. Rev. D28, 546 (1983)
[12] R N Mohapatra, in Gauge theories of fundamental interactions, edited by R N Mohapatra
and C H Lai (World Scientific Co., Singapore, 1981) p. 1
[13] G Ecker, W Grimus and H Neufeld, Phys. Lett. B127, 356 (1983)
[14] S P Misra and U Sarkar, Phys. Rev. D28, 249 (1983)
[15] M K Gaillard and B W Lee, Phys. Rev. DIG, 897 (1974)
[16] H Y Cheng, C Y Cheung, G Lin Lin, Y C Lin, T M Yan and H L Yu, Phys. Rev. D47, 1030
(1993)
[17] G Beall, M Bander and A Soni, Phys. Rev. Lett. 48, 848 (1982)
J Trampetic, Phys. Rev. D27, 1565 (1983)
[18] B Grinstein, N Isgur, D Scora and M B Wise, Phys. Rev. D39, 799 (1989)
[19] T Altomari, Phys. Rev. D37, 677 (1988)
[20] Particle Data Group; Review of Particle Properties, Phys. Rev. D50, Part 1 (1994)
[21] J F Donoghue and B R Holstein, Phys. Lett. B113, 382 (1982)
[22] L Maharana, Phys. Lett. B149, 399 (1984)
[23] L Maharana, A Nath and A R Panda, Phys. Rev. D47, 4749 (1993)
PR AM ANA (f) Printed in India Vol. 46, No. 1,
journal of January 1996
physics pp. 51-66
Signature inversion in the K = 4 band in doubly-odd
l52 Eu and 156 Tb nuclei: Role of the fc/2 proton orbital
\LPANA GOEL and ASHOK K JAIN
[Department of Physics, University of Roorkee, Roorkee 247667, India
VIS received 13 June 1995; revised 20 September 1995
Abstract. The phenomenon of signature inversion in the doubly-odd nuclei 152 Euand 1S6 Tbis
anderstood within the framework of a two-quasiparticle plus rotor model. It is shown that the
7 9/2 : 1/2 [541] proton orbital plays a crucial role in reproducing this phenomenon.
Keywords. Doubly odd deformed nuclei; 152 Eu and 15t Tb; Coriolis coupling calculations;
signature inversion.
PACS Nos 21-60; 27-70
L. Introduction
During the last several years many investigations have been carried out to study the
anusual features exhibited by the rotational bands of the odd-odd deformed nuclei
"1,2]. One of the most striking and anomalous features has been the signature
aversion phenomenon in the high-K rotational bands of doubly-odd lighter rare-earth
nuclei [3-7]. These bands are usually assigned a high-/ (/ 13/2 neutron h ll/2 proton)
;onfiguration. Unlike most of the K + = (Q p + QJ bands in the odd-odd nuclei which
display a smooth behaviour, these K + bands exhibit a large odd-even effect in their
rotational energy spacings implying a dependence on the signature quantum number.
It is pertinent to give here a brief description of the signature quantum number and
Is origin. The signature quantum number is related to the invariance of the nuclear
wave-function under rotation by n about an axis perpendicular to the symmetry axis.
\t large rotational frequencies, signature and parity are the only two quantum
lumbers which survive.
A rotation by it can be generated either by acting on intrinsic variables and
performing a rotation by the corresponding operator J? or by acting on the collective
variables and performing the rotation by the corresponding operator R e . Invariance of
the system under this rotation implies that [8]
R. = R t (1)
Dr
R, R a = 1 .
r = i ana r = i. \J)
Also,
R.D I UK . Q = R t Y I M = (-lYY z tl . (4)
From (3) and (4), we get
r = (-!)'. (5)
The rotational spectrum for X = band therefore gets divided into two parts:
7 = 0,2,4,6,..., r= + l,
1=13,5,1,..., r=-l. (6)
Whereas only r = + 1 is possible in the even-even nuclei, both the r = + 1 and r = 1
sets are possible in the odd-odd nuclei.
For K ^ 0, the intrinsic states are two-fold degenerate and the corresponding
operator is jR ; = exp( inJ s } which has a value exp( ma), where a is the signature
quantum number. The square of this operator leaves the wavefunction unchanged for
a system having even number of fermions. However an odd numbered system trans-
forms like spinors and consequently changes sign. Thus
where A is the total number of particles in the system. It is therefore clear that the
rotational bands of an odd-odd system having K ^ can also be classified according to
the classification given in (6) for K = bands. The r = + 1 members of the rotational
bands correspond to the signature quantum number a = whereas the members
having r = 1 correspond to the signature quantum number a = 1.
In general, the wavefunction, which incorporates the Ti-invariance and also the axial
symmetry may be written as
1/2
'(/>-}, (8)
where cj) K = |Ka p > = |p p Q p )|p n Q M > for an odd-odd nucleus and </> K =Ri<p K . Since the
rotational Hamiltonian having a Coriolis term breaks the time reversal symmetry,
different contributions are obtained for the a = and a = 1 members of the rotational
bands giving rise to an odd-even shift in energy for K = bands. This odd-even effect is
the prime source of signature dependent features in the odd-odd nuclei. An additional
source of odd-even shift is the Newby term arising due to diagonal n-p interaction for
K = Q bands. However the contribution of this term is very small as compared to the
contribution from the decoupling term in the high-j bands. The Newby term therefore
does not appear to play a significant role in the signature inversion phenomenon.
The signature dependent term in the Hamiltonian dictates that the energetically
favoured signature in these bands is given by a f = l/2(- l)~ 1/2 + l/2(- l)i~ 1/2 .
However, a signature inversion at lower spins is observed in the K + bands having high-;
configuration because the unfavoured spins lie lower in energy up to a critical spin J c . The
signature splitting then reverts to the normal signature beyond the critical spin.
52 Pramana - J. Phys., Vol. 46, No. 1, January 1996
We have recently shown [6] that the Coriolis coupling term is sufficient to explain
the signature inversion in 160 Ho. This is supported by other calculations [5] also. We
could show that a Coriolis mixing of the [(i 1 3/ 2 ) n ,(h 11/2 ) p ] orbitals is sufficient to
explain the weak signature inversion seen in the K n + = 6~ {1/2 ~ [523] p <8) 5/2 + [642],,}
band of * 60 Ho. It essentially represents a transmission of large odd-even shift present in
the K = and K = l bands having the configuration {l/2~[550] p (g)l/2 + [660] n }
through a very high order Coriolis coupling to the K = 6 band. However the same
calculations did not succeed in the other two nuclei namely 1 52 Eu and 1 56 Tb where the
signature inversion is more pronounced. We notice that another high-j orbital belong-
ing to h g/2 namely 1/2 [541] proton orbital lying quite low in energy must also be taken
into account. The systematics of the single particle states also [9] suggest that this
orbital along with the 1/2 [660] neutron orbital is bound to play an important role in
the signature inversion phenomenon. In 2, we give a brief description of the model. In
3, we present the results of our calculations for 152 Euand 156 Tb within the framework
of a two-quasiparticle plus axially symmetric rotor model [TQPRM] and demonstrate
that the signature inversion is a result of Coriolis mixing between the large number of
bands arising from a coupling of the i 13/2 neutron with the h l 1/2 and the h g/2 proton
orbitals. The mechanism of the signature inversion is also brought out. In 4 we
summarise the results.
2. The model and the methodology
The total Hamiltonian of the system in the framework of the TQPRM [1] is divided
into two parts, the intrinsic and the rotational.
The intrinsic part consists of a deformed axially symmetric average field H av , a shorl
range residual interaction H pair , and a short range neutron-proton interaction V np , sc
that
^ H =H +H + V (10
"intr * I av T -"pair ' r up' x
The vibrational part has been neglected in this formulation. For an axially-
symmetric reflection-symmetric rotor,
"rot / <7 \ 3' ' cor" 1 " ppc irrot'
where
H COT =-h 2 /2/(IJ. + I.j + )
;) "T~ I / / ) I \
ps / ^^ n J nz * -J
The particle angular momentum./ is given by the sum of angular momentum of thi
odd proton j p and the odd neutron ;'. The operators I =/ 1 // 2 , j =A. ij 2
i = / + i; and / = / + i; are the usual shifting operators. / is the moment o
Jn> J n\ J 1^2 P + Pi Pz
inertia with respect to the rotation axis.
The set of basis eigenfunctions of H av + h 2 /2/(I 2 + 1\] may be written in the form c
^F t^o. ^rttotinnQl \vavpfnnrf irn D 1 .... anH fhft intrinsi
Alpana Gael and Ashok K Jain
wavefunction | Ka. p > as
27 +1
\_16n 2 (l+5 KO )
1/2
[X>J, x |Ka> +(-
(13)
where the index a p characterizes the configuration (a p = p p p n ) of the odd neutron and
the odd proton. A correct choice of the set of basis functions is very important as all the
states which may couple together and influence each others behaviour should be
included in the calculations.
Diagonalization of the total Hamiltonian matrix for each value of the angular
momentum / gives us the energies th (7, o^a) for all the bands built on the two-
quasiparticle (2qp) configuration, \Ka. p <j} present in the basis set of the eigenfunctions.
s
't
10
152
Eu
", {5/2~C5323 p 3/2*C6513 n }
I / I I 1
18
12
16
18
20
Figure 1. The exnerimental nlots of \E(J -+ 1 - \\ft.l vs J fnr th
The Newby-shift enters as a parameter along with the other parameters such as the
quasiparticle energies E a , the moment of inertia / and the single particle matrix
elements <j + >. The single particle matrix elements are initially taken from the Nilsson
model wavefunctions and some of the important ones are modified during the least
square fitting procedure of the band level energies. A complete Coriolis coupling
calculation thus requires a knowledge of a large number of 2qp states which are often
unknown. We have therefore estimated the excitation energies of the important
unidentified bands by using a semiempirical formulation [9]. In this formulation, the
known properties of the quasiparticle configurations involved are taken from the
neighbouring odd- A nuclei.
3. TQPRM calculation in 152 Eu and 156 Tb
3.1 Signature inversion in 152 Eu and 156 Tb: Empirical data
In figure 1 we plot the experimental data of AE(7 ->/ - l)/27 vs. angular momentum
/ to show the odd-even staggering in energy, and the signature inversion exhibited by
the Kl = 6~ band in 160 Ho [10] and the K* + = 4~ band of 152 Eu [1 1] and 156 Tb [3].
The suggested Nilsson configuration fof the K + = 6 ~ band in 1 60 Ho is (7/2 ~ [523] p <g)
5/2 + [642],,} and for the K n + =4~ band in both the nuclei 152 Eu and 156 Tb is
(5/2~ [532] p 3/2 + [651] n }. The critical spin where the inversion occurs is shown by
an arrow. The critical spin is defined from the higher spin side; it is the point where the
normal behaviour changes into the anomalous behaviour. The pattern of odd-even
staggering in all the three nuclei are similar but considerably enhanced signature effects
are observed in the K 4 band of 152 Eu and 156 Tb. A number of distinguishing
features as compared to 160 Ho can be noted. We observe that the odd-even staggering
in 160 Ho is much less in magnitude as compared to 152 Eu and 156 Tb. Also the
magnitude of the staggering at lower spins is very large in 152 Eu and 156 Tb which
decreases as the point of inversion is approached; after the inversion the magnitude of
the staggering again increases gradually. It appears very natural to explain the large
magnitude as a direct consequence of the shifting of Fermi level of both the proton and
Table 1. The experimental data [9] of the one-quasiparticle bands in the neigh-
bouring odd-A nuclei used to estimate the band energies of the unidentified bands
(given in the first row) included in the TQPRM calculations of 152 Eu. The fitted
values are given in brackets.
p
N
~ 50-0 keV
(systematics)
5/2 [532]
~600keV ~800keV
(systematics) (systematics)
3/2 [541] 7/2 [523]
~900keV
(systematics)
1/2[541]
~1000keV
(systematics)
1/2 [5 50]
345 keV
3/2 [651]
355-7 keV
1/2 [660]
~400keV
systematics
5/2 [642]
400
(175)
405
(405)
450
(450)
500
(250)
505
(505)
550
(550)
945
(945)
955
(955)
1000
(950)
1045 1145 1245
(1045) (1145) (1245)
1055
(1055)
1100
(1050)
1245
(1150)
1255
(1000)
1300
(1300)
1345
(1250)
1355
(1 100)
1400
(1400)
1345
(1245)
1355
(1150)
1400
(1400)
1445
(1345)
1455
(1250)
1500
(1500)
in "tu. 1 ne experimental data 01 yrast Dana is only Known LI i j. Also given are tne
parameter values of , h 2 !!/, E N and those values of <;'+ > which were adjusted
along with the Nilsson model values in the parentheses. The deformation was taken
ase = 0-18ande = -0-03.
Configuration
Proton Neutron
^exp
K K ,I (keV)
cal E a fi 2 /2j? E N
(keV) (keV) (keV) (keV)
5/2 [532] 1/2 [660]
3~, 3
349-3 405-0 9-5
5/2[532]l/2[660]
2~,2
455-0 505-0 9-0
1/2 [550] 1/2 [660]
1",1
1284-3 1150-0 7-2
1/2[550]1/2[660]
Q-,0
1355-8 1250-0 7-2 4-0
3/2[541]l/2[660]
2~,2
852-5 955-0 9-5
3/2 [541] 1/2 [660]
1~,1
843-2 1055-0 9-0
5/2 [532] 3/2 [651]
4-, 5 180-6
146-8 175-0 7-0
5/2 [532] 3/2 [651]
i~,i
253-1 250-0 7-0
1/2 [550] 3/2 [651]
2-, 2
1378-1 1245-0 9-5
1/2[550]3/2[651]
1~,1
1363-2 1345-0 9-0
3/2[541]3/2[651]
3~, 3
874-7 945-0 9-5
3/2[541]3/2[651]
0~,0
1190-1 1045-0 9-0 0-0
l/2[550]5/2[642]
3-, 3
1757-5 1400-0 9-5
1/2 [550] 5/2 [642]
2-,2
1556-2 1500-0 9-0
3/2 [541] 5/2 [642]
4~,4
956-8 950-0 11-9
3/2 [541] 5/2 [642]
1~,1
1055-1 1050-0 11-6
5/2 [532] 5/2 [642]
5~, 5
418-6 450-0 12-0
5/2[532]5/2[642]
0~,0
488-8 550-0 11-6 0-0
7/2[523]3/2[651]
5~,5
1263-3 1145-0 9-5
7/2[523]3/2[651]
2~,2
1268-9 1245-0 9-0
1/2[541]1/2[660]
0~,0
746-2 1000-0 12-9 -40-0
1/2[541]1/2[660]
1M
1135-2 1100-0 12-5
1/2 [541] 3/2 [651]
1",1
1159-9 1150-0 11-9
1/2[541]3/2[651]
2~,2
1564-6 1250-0 11-6
1/2 [541] 5/2 [642]
2~,2
1275-3 1300-0 11-9
1/2 [541] 5/2 [642]
3-, 3
1025-4 1400-0 11-6
<1/2[550]|1/2[550]> P =
5-32(5-79)
<5/2[532]|3/2[541]> p = 4-55(5-55)
<l/2[541]|l/2[541]> p =
-4-32(-3-64)
<7/2[523]|5/2[532]> p = 4-14(5-14)
<3/2[541]|l/2[541]> p =
4-75(0-18)
< 1/2 [660] 1 1/2 [660] > = - 3-74(- 6-69)
<3/2[651]|l/2[660]> n =
3-69(6-66)
<5/2[642]|3/2[651]> n =
3-52(6-52)
the neutron from 7/2[523] p to 5/2[532] p and 5/2[642] n to 3/2[651] n . The dependence
of the magnitude of staggering on the Nilsson orbitals occupied by the odd-neutron
and the odd-proton is also observed in other cases. However, by using a physically
meaningful set of parameters, we have been unable to reproduce the signature inversion
in these two nuclei within the basis space of the [(^i 1/2 )p(ii3/ 2 )J orbitals only; it is
indeed very surprising that our calculations fail in 156 Tb and 1 "Eu. It therefore seems
very natural to explore the role of the 1/2 [541] proton state belonging to the h 9/2
orbital. This orbital lies low in energy and when combined with an z' 13/2 neutron orbital
will give rise to a K = band which has a phase opposite to the normal odd-even
staggering.
Figure 2(a). The AE ' = { A(J -> / - 1) - A(J - 1 -> I - 2) } vs. 7 for the K + = 4 -
band in 152 Eu and 15fi Tb from our calculations; the odd-even staggering as well as
signature inversion is well reproduced.
10
M 2
cs
!"'
LU 6
<1
Expt.
Theo.
K+= 4", (5/2" C5323 p 3/2 + t65H n }
(
'^Eu
152 C
a TO 12 E re 18 20
Figure 2(b). Comparison of the experimental data on odd-even staggering for the
K K + = 4~ band in 152 Eu and 156 Tb with the TQPRM calculations.
3.2 Signature inversion in 152 Eu: Calculations
A total of 26 2qp rotational bands were included in the [TQPRM] calculations of
152 Eu. The positions of all the 25 unknown bands were estimated by using the
experimental data of the single particle states in the neighbouring odd- A nuclei [9]. In
table 1 we have summarized the estimated values of the bandhead energies as E x which
Pramana - J. Phys., Vol. 46, No. 1, January 1996
57
100
*C660: n }
V V V V V V \
K^=1~, {l/2~C54i: p 1/2 + C6603 n }
\AAAAA
50
V V V V V V V
10
Figure 3. The behaviour of the unperturbed K*_ = ~ and K* + = 1 bands used in
the calculation of 152 Eu.
represent a reasonably good first order estimate within 100-200 keV. The splitting between
two GM partners was uniformly assumed to be 100 keV. Besides the [(^ 1 i /2 ) p (Ji3/2)rJ
configurations, 6 bands belonging to the [l/2[541] p (g)z 13/2 neutron] configuration
were also included in these calculations. Positions of 1 2 bands belonging to 6 configura-
tions were adjusted during the fitting procedure; the variation did not exceed 250 keV
in any case. These 12 bands were seen to play the most important role in reproducing
the signature inversion and form the most important chain for transmitting signature
effect. The moment of inertia parameters for the K+ and the K_ bands were chosen to be
9-5 and 9-0 keV respectively; 14 of these were adjusted during the fitting procedure. We also
find that the results are not very sensitive to the values of the Newby-shift E N ; a variation of
this parameter up to lOOkeV did not produce any significant change in the results. The
values of E N given in table 2 are therefore not well determined from these calculations. The
matrix elements were again taken from the Nilsson model. The i 13/2 neutron matrix
elements were attenuated as usual. Of the remaining matrix elements the most
significant adjustment was made in the <3/2[541][; + |l/2[541]) p matrix element; it
was increased from its Nilsson value OT8 to 4-75. It indicates the need for a strong
coupling of the l/2[541] p orbital. In table 2 we summarise the final parameters arrived
at after the fitting of the K = 4 band. The results of our calculation for 1 52 Eu are shown
in figures 2 (a) and 2(b) where a clear signature inversion can be seen at spin / = 14.
The mechanism of the signature inversion in 152 Eu appears to be more complicated
than in 160 Ho [6]. The signature effects for the K = 4 band are seen to follow from the
two X* = 0" (1/2- [550] p l/2 + [660] n } and the Kl = (T (1/2- [541] p l/2 + [660]J
bands. Here the Newby-shifts for the K = bands do not play an important role as the
Coriolis /AK=1 AK=1\Coriolis
AK=0
|l/2"C5A13p(S>3/2' f [651] n } {l/2'C550: p 3/2 + C65i: n } {3/2'C5413p1/2 + [660] n }
Coriolis
AK=1 AK=1\Criolis AK=0 Coriolis AK=
AK=1 Coriolis
AK=0
n } {V2"E550] p V2*E660: n }
{l/2-[550]p1/2*C6603 n }
Figure4. The chain diagram of the K+ andtheM_ bands showing the transmission of
the odd-even effect which is responsible for the signature inversion in the K* + =4~
band of 152 Eu and 156 Tb.
variation of this parameter up to 100 keV has almost no effect on the results. When all
the mixing terms except the decoupling parameters a p and a n for the proton and the
neutron orbitals respectively have been taken to be zero we obtain the results as shown
in figure 3. It may be noted that the two K = bands exhibit a phase opposite to each
other. The corresponding G M partners having K = 1 also exhibit opposite signature
dependence. The opposing phases are easily understood from the sign of the decoupling
parameters involved. However the K + and the K_ bands can couple directly in special
situation where Q p or Q n = 1/2; if Q p = Q n = 1/2 then an extra diagonal term of the form
of (- 1)' + l h 2 /2/ a p -a n contributes to the odd-even shift of the K = band. For the
K*_ = 0~ {l/2~ [550] p 1/2 + [660],, } band both the decoupling parameters are oppo-
site in sign; this favors odd spins in the K = band; the GM partner K = 1 band couples
with K = band by a coupling term [1, 12]
n = 0",
A A A X\ /\ A /
v
Figure 5 (a). The staggering plots of the perturbed K_ and K + bands in 152 Eu
belonging to the most important chain which couples the K n + =4" band to the
K*_ Q~ band are shown in this figure.
200-
Figure 5(b). Same as in figure 5(a)
Since a- p and cf n are opposite in sign, even spins are favoured in the K = I band. On the
other hand, the K n _ = 0, {l/2~ [541] p l/2 + [660] B } band has decoupling parameters
f\ UJ 1 ~
Figure 5(c). Same as in figure 5 (a).
50
I-
A
OO
-100
V v v v \
K>3", (5/2'C5323 p 1/2*6603 n }
10
16
18
20
Figure 5(d). Same as in figure 5 (a).
Table 3. The experimental data [9] on the one-quasiparticle bands in the neigh-
bouring odd-A nuclei used to estimate the band energies of the unidentified bands
(given in the first row) included in the TQPRM calculations of 156 Tb. The fitted
values are given in the parentheses.
p
N
227 keV
5/2 [532]
545-3 keV
7/2 [523]
863 keV
1/2[541]
891-1 keV
3/2 [541]
~1100keV
(systematics)
1/2 [550]
86-5 keV
3/2 [651]
266-6 keV
5/2 [642]
720-6 keV
1/2 [660]
330 430
(250) (350)
500 600
(500) (600)
950 1050
(900) (1000)
650 750
(550) (650)
949 1049
(850) (950)
1129 1229
(1090) (1200)
1583 1683
(1400) (1500)
1000 1100
(1000) (1100)
1100 1200
(1100) (1200)
1600 1700
(1550) (1650)
1200 1300
(1200) (1300)
1350 1450
(1450) (1550)
1800 1900
(1700) (1800)
in '" 1 D. me experimental aata 01 yrast Dana is only Known [JJ. A1SO given are me
parameter values of a , ti 2 /2/, E N and those values of <;'+ > which were adjusted
along with the Nilsson model values in the parentheses. The deformation was taken
as e, = 0-22 and e, = - 0-02.
Configuration
Proton Neutron
^exp
K\I (keV)
ca! E a h 2 /2/ E N
(keV) (keV) (keV) (keV)
5/2 [532] 3/2 [651]
4-, 6 379-0
258-8 250-0 11-85
5/2 [532] 3/2 [651]
1~,1
356-5 350-0 10-0
3/2[541]3/2[651]
3~, 3
802-2 1000-0 10-0
3/2[541]3/2[651]
Q-,0
1036-9 1100-0 10-0 0-0
7/2[523]3/2[651]
5~, 5
696-5 550-0 10-0
7/2 [523] 3/2 [651]
2~, 2
680-2 650-0 10-0
1/2[550]3/2[651]
2~,2
1153-4 1200-0 10-0
1/2[550]3/2[651]
1~,1
1257-4 1300-0 10-0
5/2 [532] 1/2 [660]
3~, 3
1065-6 900-0 10-0
5/2 [532] 1/2 [660]
2", 2
956-5 1000-0 10-0
1/2 [550] 1/2 [660]
1",1
1876-2 1700-0 9-5
1/2 [550] 1/2 [660]
0~,0
1913-3 1800-0 9-5 40-0
3/2 [541] 1/2 [660]
2", 2
1601-5 1550-0 10-0
3/2 [541] 1/2 [660]
1~,1
1514-2 1650-0 10-0
5/2 [532] 5/2 [642]
5~,5
476-6 500-0 12-0
5/2[532]5/2[642]
Or,0
581-0 600-0 12-0 0-0
3/2 [541] 5/2 [642]
4~,4
1054-6 1100-0 12-0
3/2 [541] 5/2 [642]
1",1
1217-7 1200-0 11-0
1/2 [550] 5/2 [642]
3", 3
1475-5 1450-0 10-0
1/2 [550] 5/2 [642]
2~,2
1583-5 1550-0 10-0
1/2 [541] 1/2 [660]
o-,o
1378-0 1400-0 12-0 -40-0
1/2 [541] 1/2 [660]
1~,1
1397-8 1500-0 12-0
1/2[541]3/2[651]
1~,1
855-3 850-0 12-0
1/2[541]3/2[651]
2", 2
1015-3 950-0 12-0
1/2 [541] 5/2 [642]
2", 2
1189-3 1090-0 12-0
l/2[541]5/2[642]
3~, 3
1121-9 1200-0 12-0
<l/2[550]|l/2[550]> p =
3-74(5-74)
<7/2[523]|5/2[532]> p = 3-12(5-12)
<l/2[541]|l/2[541]> p =
-2-74(-3-47)
< 1/2 [660] 1 1/2 [660] > = - 3-60(- 6-60)
<3/2[541] |l/2 [550] >,=
3-69(5-69)
<3/2[651] 1 1/2 [660] > = 3-58(5-58)
<3/2[541]|l/2[541]> p =
3-52(0-22)
<5/2[642]|3/2[651]> n = 2-47(6-47)
<5/2[532] |3/2 [541] > p =
3-51(5-51)
band. These opposing signature effects of the K = and K = 1 bands are transmitted to
the K = 4 band through a coupling of bands in a chain which is shown in figure 4. When
all the mixing terms are turned on, the various bands of the chain are perturbed as
shown in figure 5(a-d). The final band i.e. K K + =4-{5/2~[532] p 3/2 + [651],,} is
observed to be highly mixed in nature. We find that the odd-spin members are favored
in the low spin region mainly because of the 1/2~[541] orbital present in the
calculations. The 1/2" [550] orbital contributes more effectively in the high spin region
and leads to the favoring of even-spins. From this we may conclude that a band-
crossing like phenomenon is taking place in this nucleus. The K" + = 3~ (3/2 ~ [541] p <g)
3/2 + [651] B } band seems to play a crucial role in the transmission of the odd-even shift
100
!
>
- 1
M i
1
.*f ~ 50
< 50
50
'"ID
A A/A
A A A A A A
/ V V
s
V V V V V V N
X "V \/
V V V V V V
/\ ~/\ /\ X /\ /\
' A A /
N/ V V V \/ V \
K+=1", {1/2~C541 D D V2 + C660D n }
^ A A A A A /
x V V
1 1 1
V V V V V V
i I I I. I I
Figure 6. The behaviour of the unperturbed K*_ = and K* = 1 bands used in
the calculation of 156 Tb.
to higher-X bands. This band has location in 160 Ho such that it breaks the smooth
transmission of the odd-even shift of the (l/2"[541] p l/2 + [660] n }* = 1~,0~
bands to the main band. On the other hand, this band is favorably placed in 152 Eu and
156 Tb to allow the transmission of the odd-even effect coming from the h 9/2 orbital.
Also, the matrix element of <3/2[541]|j + |l/2[541]> p needs to be considerably en-
hanced in 1 52 Eu and 1 56 Tb indicating the importance of the h 9/2 proton orbital in these
two nuclei.
3.3 Signature inversion in 156 Tb: Calculations
Calculations for * 56 Tb were carried out by using the same set of the 26 bands as used in
1-5 2 Eu. The positions of 25 unknown bands were estimated by using a similar method.
In table 3, we summarise the estimated values of the band energies; these changed
values reflect the shift in Fermi energy in going from 152 Eu to 156 Tb [9]. The positions
of 16 bands were adjusted within lOOkeV during the fitting procedure. The moment of
inertia parameters ^ 2 /2/ were uniformly chosen to be lOkeV for all the bands; it was
adjusted in 13 cases during the fitting procedure. The matrix elements for the
2 13/2 orbital were reduced as usual. Among the other matrix elements the largest
variation was again made in <3/2 [541] |y + 1 1/2 [541] > p matrix element; it was changed
from 0-22 to 3-52. The 1/2 [541] decoupling parameter was also reduced from the
Nilsson value of - 3-47 to - 2-74. The final set of parameters arrived at after the fitting
are listed in table 4. We must emphasize here the fact that the quality of the fitting in
156 Tb as well in 152 Eu is not as good as in 160 Ho. With our limited resources and time
we have not attempted an extensive fitting of the bands. It is our belief that fitting of the
data can be considerably improved in both the cases. The final results of our
calculations are shown in figures 2(a) and 2(b) where a signature inversion can be
observed at / = 14.
K>3-, (5/2~[5323p 1/2*C6603 n }
^S
18
Figure 7 (a). The staggering plots of the perturbed K_ and K + bands in 156 Tb
belonging to the most important chain which couples the K* + =4" band to the
K*_ = 0~ band are shown in this figure.
The mechanism of signature inversion in 156 Tb is identical to that in 152 Eu. The
signature effects in the main band again follow from the two K = bands shown in
figure 6. We plot both the K = bands and their G M partners for the situation where
all the mixing terms except a p and a n are zero. It is rather interesting to note that the two
K = bands are seen to exhibit the same signature dependence whereas in x 52 Eu they
show opposite phases. Normally we would have expected the K1=0~,
(l/2~ [541] p (g> l/2 + [660] n } band to favor even-spin members. However in this case
the mixing between the odd spin members of the K = and the K = 1 band of
(l/2~ [541] p (g) l/2 + [660] n } configuration is such that the wavefunctions have almost
50-50% admixture of both the bands. In such a situation it is difficult to label one of the
two odd spins as belonging to either K = or the K = 1 band. Even a slight increase in
150
50
-50
50
-50
100
50
-50-
50-
-50-
A AA A
A
\
/\ /\ A A
:*C660] n }
V
10
12
16
18
Figure 7 (b). Same as in figure 7 (a).
one of the two. Thus effectively the odd-spin members of the K = and K = 1 band
have been interchanged in the calculations; this is the reason for the observed behavior.
However, this apparent change does not affect the final results.
The signature inversion in the K = 4 band of 1 56 Tb also occurs through a coupling of
the same chain as for 152 Eu which is already shown in figure 4. The behavior of the
various bands of the chain, when all the mixing terms are introduced, is shown in
figures 7(a-b). The signature inversion in 156 Tb therefore has a similar origin as in
152 Eu.
5. Conclusions
In conclusions, we find that the phenomenon of signature inversion may be understood
within the framework of TQPRM. We could reproduce the general trends of the signature
inversion in 156 Tb and 152 Eu; however we could not. obtain the correct magnitude of
staggering which seems to require a further refinement of our parameters and fitting
procedure. The reproduction of the inversion phenomenon in 152 Eu and 156 Tb
required the inclusion of h 9/2 : 1/2 [541] proton orbital indicating its importance. The
location of the K* + = 3~{3/2~[541] p 3/2 + [651]J band appears to play a crucial
role in the relative importance of the Ji 9/2 :l/2[541] orbital in 152 Eu and 156 Tb nuclei
vis-a-vis 160 Ho nucleus where the very weak signature inversion was reproduced
withrmt inr-lnrlincr tVn=> li /-rJ-\ito1 f^TI Tt urrtnlH KP nf
tr>
explain the whole systematics of signature inversion phenomenon in these nuclei on the
basis of the inputs identified in the present study to be of importance. Work is in
progress in this direction.
Acknowledgement
One of the authors (AG) gratefully acknowledges the financial support received from
CSIR (Government of India).
References
[I] A K Jain, J Kvasil, R K Sheline and R W Hoff, Phys. Lett. B209, 19 (1988); Phys. Rev. C40,
432 (1989)
[2] P Semmes and I Regnarsson, Int. Con/. High Spin Physics and Gamma-soft nuclei
(Pittsburgh, PA, USA) Sept. 17-21, 1990
[3] R Bengtsson, J A Pinston, D Barneoud, E Monnand and F Schussler , Nucl Phys. A389, 158
(1982)
[4] I Hamamoto, Phys. Lett. B235, 221 (1990)
[5] K Hara and Y Sun, Nucl. Phys. A531, 221 (1991)
[6] A K Jain and Alpana Goel, Phys. Lett. B277, 233 (1992)
[7] Alpana Goel, Study of the two-quasipartide band structures in odd-odd and even-even
nuclei PhD Thesis, University of Roorkee (1992) (Unpublished)
[8] A Bohr and B R Mottelson, Nucl. Struct. (Benjamin, New York, 1975) Vol. II
[9] A K Jain, R K Sheline, P C Sood and K Jain, Rev. Mod. Phys. 62, 393 (1990)
[10] J A Pinston, S Andre, D Barneoud, C Foin, J Genevey and H Frisk, Phys. Lett. B137, 47
(1984)
II 1] T Von Egidy et al, Z. Phys. A286, 341 (1978)
[12] A J Kreiner, D E Gregorio, A J Fendrik, J Davidson and M Davidson, Phys. Rev. C29, 1572
(1984)
Conservation of channel spin in transfer reactions
V S MATHUR and ANJANA ACHARYA
Department of Physics, Banaras Hindu University, Varanasi 221 005, India
MS received 26 July 1995; revised 15 November 1995
Abstract. The conservation of channel spin implying that the spin of the initial bound
pair coupled to that of the initial free particle should result in the same channel spin as the spin
of the final bound pair coupled to the spin of the final free particle, follows as a consequence
of three-body theory of transfer reactions with the assumption of separability of two-body
r-matrix. To test the validity of this principle we look at the experimental data on stripping
reactions on even-even nuclei. We find that although reactions to channels not conforming
to channel spin conservation are not altogether ruled out, the cross-sections of reactions
violating channel spin conservation are much smaller than those conforming to channel spin
conservation.
Keywords. Transfer reactions; three-body theory; channel spin.
PACS Nos 24-40; 25-70
1. Introduction
The three-body formulation of transfer reactions of the type
A + a(b + x) = B(A + x) + b,
wherein we treat the particles a, b and x as cores (ignoring their structures) and use
three-body equations (with separable two-body interaction) to describe the process,
a consequence that follows is the conservation of channel spin. Accordingly the spin of
the initial bound pair coupled to that of the initial free particle results in the same
channel spin as the spin of the final bound pair coupled to the spin of the final free
particle. The deduction of this result is presented and to see the validity of this principle
experimental data on (d, p) and (d, ri) reactions on various closed shell nuclei is observed.
It is found that although reactions to channels not conforming to the conservation of
channel spin are not altogether ruled out, their cross sections are far smaller than those
conforming to this principle.
In 2 we give the deduction of this principle on the basis of three-body theory and the
assumption of separability of two-body ^-matrix. In 3 the experimental data is
analyzed in this context and inferences are drawn.
2. Reaction amplitude in terms of the solutions of coupled integral equations
In three-body calculations for transfer reactions, the use of Alt-Grassberger-Sandhas
(AGS) [1] version of three-body equations viz.
U tJ (z) = (1 - Sy)(* - H ) + (1 - c5 fc ) T fe (z)G (z) t/ kj (z), (2.1)
where G (z) = (z H^' 1 is the free resolvant operator and T k (x) is the two-body
transition operator in three-body space, is preferred over other versions of three-body
equations because the matrix element of the AGS operator U (j (z) between the initial
and final asymptotic states, i.e.
hereafter referred to as the reaction amplitude, is very simply related to the cross-
section of the processy - i. Here Qj and d j are, respectively, the on-shell momentum and
the spin projection of the jth particle and I ILjSj ^ n ( 1 LjSj)Jj is the bound state of theyth pair,
assumed to be mixed L j S j state. (In case the bound state is a pure LjSj state, the
summation over LjSj may be taken to be non-existent). To evaluate this matrix element
one has to solve the AGS equations (2-1) choosing a suitable basis of representation.
Choosing (i) the angular momentum basis viz.
<p l q t ((L t S t )J l s l )k i l t : JM\ == <p t q i (L t S l )ft t : JM\ = < Mj a,: JM\,
where p t is the magnitude of relative momentum of ith pair and q t is the momentum of
the ith particle in the centre of mass frame and ((L^J^k^: JM is the angular
momentum coupling scheme and (ii) invoking a separable approximation for the
two-body t-matrix in three-body space, i.e.
<W(LkSJfc JM\T k (z)\r' k u' k (L' k S' k )F k : JM}
we can reduce the AGS equations to one-dimensional coupled integral equations
(Mathur and Prasad [2,3], Mathur and Padhy [4]), i.e.
du.JK^u,^^: J)
J}. (2.3)
Here the Born term K ik (or K tj ) is defined by
K^uM.: J) = (1 - a tt ) X X
L!s t L k s k j j z ~Pi qt
<P i q i (L i S i )^. JM\r k u k (L k S k )p k : JM> (2.4)
and the quantities T^q^jft^ji J) occurring in (2.3), are related to the matrix element of
AGS operator between the angular momentum basis states as follows:
T.(qQ'BB-J}=Y
lj(qWi ' } & -- z-p-q
ji J>. (2.5)
bound state <t>* ( i iSdJl (p'j) by
(2-6)
Using (2.1) to (2.5) it is possible to express the reaction amplitude in terms of the
on-shell solutions of the one-dimensional coupled integral equations (2.3) as follows:
Introducing, in the above mentioned matrix element (reaction amplitude), two unit
operators (one before and one after (/ y ) we have
i4 z *rw,"W*)i<w z ^
Now
= ZZ Z
a- JM
\ i/2
Z G,
L,Si
<M
JL,S,
^ dpjdtf
(2.7)
Sd
where the normalization condition <Q l |q > = (l/2Q,) 1/2 (q; - Q f ) has been used. The
diagram represents the expression
(KjMjJ J,Af
M, r m^'
(Elbaz and Castle [5]).
Also
L,S,
JM , ,*$<<;.
J-.Mi
We can use (2.8) and (2.9) in (2.7) to get
Z fev,-M,.
A Z
i /2
z z z a-e;
JM
z z
JM
JiM;
JjMj
(2.10)
One can do summation over M by joining the two JM lines (shown by dotted lines)
(Elbaz and Castle [5]). Since K[, /;, KJ and I] are dummy suffixes, one can replace them
by K { , / Xj and /_,, respectively. Using the relation between the bound state wave
function in momentum space ^"i jS;)Jj (pJ ) and the form factors g1' LiSi)Jj (p'j ) (2.6) and using
(2.5) we get
-L/Sf LjS)
l/2
Z Z
/J,- KA J
Q:
Now cutting the diagram across dotted lines we get
iQj^NtNj Z Z Z ^(Q.fi^lC.^^Oi
Kifi Kjlj J
Sid
'J U J
J J M J
Sid;
Kj
(2.1.1)
(2.12)
while the second and third, respectively, stand for ([/ ( .] 2 /\(cos0))/47r and {IjKjJj}.
Finally, the square of the modulus of the reaction amplitude, summed over final
magnetic quantum numbers and averaged over initial magnetic quantum numbers,
which in turn is proportional to the differential cross-section of they-* i process, is
given by
Li S,-
I
avg
P,(cos0)P,(cos0)
iKJlJjKfrJ') ' A ^f - '-.
(2.13)
From (2.12) we observe that the reaction amplitude contains a factor
Wfc (JiM^fdf | KjM kj )(JjMjSjdj\KjM ki ) and 5 KiKj . It implies that, for the amplitude to
be nonzero, the spin Jj of the initial bound pair coupled to the spin Sj of the initial free
particle must give rise to the same channel spin Kj as the spin J ( of the final bound pair
coupled to the spin s i of the final free particle. This inference of conservation of channel
spin (K t = K^ rests, of course, on our assumption that the two-body t-matrix is
essentially separable, its non-separable part being negligible and the bound state in the
particular channel delineates the interaction in that channel. It is of interest, however,
to test the truth of this conjecture on the basis of experimental data.
3. Discussion
Let us consider (d, p) and (d, n) reactions on even-even target nuclei (spin Sj = 0). Since
the deuteron spin Jj equals one, the initial channel spin Kj = 1. According to this
principle the final channel spin K i should also be equal to 1. Since s i (the spin of the
outgoing proton or neutron) is | the spin of the residual nucleus J z - should be \ or
f . Thus, according to this hypothesis, stripping reactions leading to residual nuclei with
spins other than 5 and f would not be permitted.
On examining the data on (d, p) and (d, ri) reactions on various even-even nuclei, we
find that although reactions leading to residual nuclei with spins other than f and \ are
not ruled out, the differential cross-sections of reactions leading to spin states \ and \
dominate over all others. From this, one can infer that the two-body interaction is
essentially separable and has only a small non-separable component. The data are
reproduced in table 1, which shows peak values of <r(0) for (d,p) or (d, n) reactions on
various even-even nuclei, leading to various excited states of the residual nuclei.
cneigy
in iviev;
(,mD/srj
Keterences
6 Li(d,p) 7 Li
v = -0(p 3/2 )
9-3
Schiffer et al [6]
d = 12MeV
^ = 0-98(p 1/2 )
6-8
12 C(d,p) 13 C
x = 0-0(p 1/2 )
21-3
Schiffer et al [6]
d =12MeV
x =3-68(p 3/2 )
21-2
16 O(d,p) 17 O
E x = Q-Q(d sp )
31-1
Exfor [7]
d = 5-03 MeV
x = 0-87(s 1/2 )
137-5
(Courtesy IAEA)
16 O(d,p) 17 O
x = 0-0(cf 5/2 )
15
Alty et al [8]
E d 12 MeV
x = 0-87(s 1/2 )
30
, = 5-80(rf 3/2 )
28
16 O(d,n) 17 F
E x = 0-0(d 5/2 )
60-2
Oliver et al [9]
d = 7-73 MeV
, = 0-5(s 1/2 )
249-0
16 O(d,n) 17 F
x = 0-0(^ 5/2 )
50-0
Oliver et al [9]
d =12MeV
x = 0-5(5 1/2 )
150-0
35 Ar(d,p) 37 Ar
je = g.s(/ 7/2 )
5-21
Fitzetal [10]
d = 10-06 MeV
E x = l-27(p 3/2 )
25-51
E x = l-52(d 3/2 )
0-53
36 Ar(rf,p) 37 Ar
x = g.s(/ 7/2 )
5-21
Sen etal [11]
= 10-86 MeV
x =l-259(p 3/2 )
25-51
x =l-509(rf 3/2 )
0-53
38 Ar(d,p) 39 Ar
s = g.s(/ 7/2 )
0-8
Ipson et al [12]
d =HMeV
E x l-262(p 3/2 )
1-5
x -2-650(p 3/2 )
0-9
40 Ca(d,p) 41 Ca
jc = 0-0(/ 7/2 )
2-30
Leighton et al [13]
E d = 5 MeV
E x = 2-0(p 3/2 )
10-3
40 Ca(d,p) 41 Ca
x = 0-0(/ 7/2 )
4-0
Schmidt-Rohr et al [14]
d = 11-8 MeV
E x = l-97(p 3/2 )
17-0
* = 2-47(rf 3/2 )
10-0
48 Ca(d,p) 49 Ca
X == O'Olp^in)
25-0
Roy and Bogaarde [15]
d = 5-5MeV
jc == 2'03(pj/ 2 )
23-0
48 Ca(d,p) 49 Ca
JC = 0-0(p 3/2 )
both cross-
Belole et al [16]
d = 7-2 MeV
x = 2-03(p 1/2 )
section are
comparable 20
48 Ca(d,p) 49 Ca
x = 0-0(p 3/2 )
50-0
Metzetfl/[17]
d = 13MeV
x = 2-03(p 1/2 )
40-0
x = 4-01(/ 5/2 )
11-0
52 Cr(d,p) 53 Cr
O Q I n \
x o* Vi^3/2/
10-5
Rao eta/ [18]
d = 7-5MeV
x = 0-565(p t/2 )
3-71
x =l-008(/ 5/2 )
0-73
58 Fe(d,p) 59 Fe
x = 0-287 3 1/2 )
7-29
1-99
Klema [19]
x = 0-470 (/ 7/2 )
1-16
x = 0-728(p 3/2 )
3-79
x =l-026(/ 7/2 )
0-29
(Continued)
12
Pramana - J. Phys., Vol. 46, No. 1, January 1996
Reaction and
deuteron lab.
energy
State of residual
nucleus
(E x in MeV)
(mb/sr)
References
58 Ni(d,p) 59 Ni
d = 12MeV
x = 0465 3 ( / i 5/2 )
x = 0-881(s 1/2 )
11-37
1-176
4-437
Chowdhury and
Sen Gupta [20]
d = 12MeV
t^O^l)
0-2
0-4
Detorie<?r<3/[21]
E*=l-430(5," 2 )
5-0
120 Sn(d,p) 120 Sn
d =15MeV
l-o-o^l)
3-17
1-93
Schneid [22]
x = 0-93(^ 7/2 )
0-276
,= l-91(p 3/2 )
x = 2-06(/ 7/2 )
0-125
0-047
d =17MeV
E x = g.s(d 3/2 )
x = 0-058(s 1/2 )
^=l-700(^ 5/2 )
x =l-857(p 3/2 )
3-696
1-303
0-110
0-157
Bechara and Dletzseh [23]
d = 7-5MeV
x = 0-106 3 $ 11/2 )
x = 0-179( Sl/2 )
0-35
0-065
6-30
Haidenbauer [24]
d =12MeV
; = ll05(t 2 )
2-40
0-56
Ehrenstein et al [25]
x = 0-364(5 I 3 / / 2 2 )
x =l-100(p 3/2 )
0-08
2-1
0-80
References
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[16] T A Belole, W E Dorenbuch and J Rapaport, Nud. Phys. A120, 401 (1968)
[17] W D Metz, W D Callender and C K Bockelman, Phys. Rev. C12, 827 (1975)
[18] M N Rao, J Rapaport, A Sperduto and D L Smith, Nucl. Phys. A121, 1 (1968)
[19] E D Klema, Phys. Rev. 161, 1136 (1967)
[20] M S Chowdhury and H M Sen Gupta, Nucl. Phys. A205, 458 (1973)
[21] N A Detorie, P L Jolivette, C P Browne and A A Rollefson, Phys. Rev. CIS, 991 (1
[22] E J Schneid, Phys. Rev. 156, 1316 (1967)
[23] M J Bechara and O Dletzseh, Phys. Rev. C12, 90 (1975)
[24] Haidenbauer, Nucl. Phys. A104, 327 (1967)
[25] D Von Ehrenstein, G C Morrison, J A Nolen and N William, Phys. Rev. Cl, 2066
Perturbation theory of polar hard Gaussian overlap fluid mixture*
SUDHIR K GOKHUL 1 and SURESH K SINHA
Department of Physics, L. S. College, B. B. A. Bihar University, Muzaffarpur 842001, India
1 Permanent address: Department of Physics, S N S R K S College, B N Mandal University
Saharsa 852 201, India
MS received 16 August 1995; revised 6 January 1996
Abstract. A perturbation theory of polar hard Gaussian overlap fluid mixture is discussec
Explicit analytic expressions for the second and third varial coefficients are given. Numerics
results are estimated for the thermodynamic properties of quadrupolar hard Gaussian overla]
fluid and fluid mixture. It is found that the excess free energy and internal energy depend 01
concentrations c l5 c 2 , molecular diameter ratio R, shape parameter K and the quadrupol
moments QT, <2*.
Keywords. Polar hard Gaussian overlap fluid; quadrupole moment; residual Helmholtz fre
energy; internal energy; equation of state.
PACSNos 05-70; 61-25; 65-50
1. Introduction
In recent years theoretical and experimental efforts have been put in to understand th
structural and thermodynamic properties of polar non-spherical molecule fluids [l-6~
Several potential models have been proposed for molecular fluids of non-sphericz
molecules [1]. Recently the Gaussian overlap (GO) model of Berne and Pechukas [7
has been used by many authors. The GO model is of special interest, because it provei
to be a solvable one. The hard Gaussian overlap (HGO) model has a close connectio
with the hard ellipsoid of bodies and is a useful reference system for molecular fluids c
non-spherical molecules.
Considerable progress have been made in the study of fluids of HGO molecules wit
additional electrostatic interactions [3, 5]. This is confined to pure fluid only and n
attempt has been made to investigate the thermodynamic functions of non-specn
molecule fluid mixture with electrostatic interactions.
In the present paper, we calculate the thermodynamic properties of a polar HG<
fluid mixture, using a perturbation theory with a HGO model as a reference an
electrostatic interaction as a perturbation. :
In 2, we describe the perturbation theory for evaluating the Helmholtz free energ
of a polar HGO fluid mixture. The virial equation of state for pure HGO fluid i
discussed in 3. Analytic expressions of the second and third virial coefficients for pur
HGO fluid are given there. Section 4 is concerned with the calculation of the thei
modynamic properties of dense pure polar HGO fluid. Section 5 is devoted to dens
polar HGO fluid mixture and the summary is given in 6.
We consider a multi-component system of molecule interacting via the pair potential
written as a sum of two terms
afc(^i > a> 2 ) = uJJ, 00 ^ , co 2 ) + t& s (ri , co 2 ) (1)
where wJJ, GO is the hard Gaussian overlap (HGO) potential acting between molecule 1 of
species a and molecule 2 of species b separated by a distance r = \r l r 2 \ and given by
= />CT ab (co 1 co 2 ). (2)
O' a6 (co 1 co 2 ) is the distance of closest approach between two molecules of species
a and b. We can take the expression for a ab given by Berne and Pechukas [7] in terms of
the Euler angles
a + cos 2 9 b
-2x ab co S d aC ose b cos6 ab )/(l-x 2 ab cos 2 6 ab )T 112 - (3)
<r b is a width and
is an anisotropy parameter. The parameter K aa is the length (2a fl )-to- width (25 J ratio of
a molecule of species a. The effective value of K ab between hard molecules of unlike
species may be given by [8]
^12=(U+22)/^U + & 22 )=(^ 11 ^l + ^22^2)/Kl + ^2)- ( 5 )
The effective value of a ab is given by
*! 2 = l0"?i+^ 2 )/2. (6)
The second term in (1) is the electrostatic interactions due to the permanent
multipole moments and can be written in the form [1,9],
2 = foW^aJXwz) + (3/4r*)|> a e 6 4g( Ml a) 2 )
+ V h Q a q a > l a> 2 )-} +(3/4)(<2 fl Q & /r 5 )^(o) 1 co 2 ) (7)
where
flb ] (8a)
os(/) flb (8b)
15cos 2 fl cos 2 b
4cos6' a cos0 b ) 2 . (8c)
Here 8 as B b and ab = a - ^ are the angles which determine the orientation of the
molecules with respect to the line joining the centres of the molecules. /i fl and Q a are,
respectively, the dipole moment and quadrupole moment of a molecule of species a.
Using this division of the pair potential, where the HGO model represents a reference
system, the perturbation expansion of the residual Helmholtz free energy of the polar
Polar hard Gaussian overlap fluid
hard Gaussian overlap fluid mixture can be written as
(A-A*)/NkT = ((A HGO -A*)/NkT) + (Af/NkT) + (Af/NkT)+ (9)
where A* represents the Helmholtz free energy of an ideal gas and A HGO A* the
residual free energy of the HGO fluid mixture. A^ is the nth order perturbation term
due to the electrostatic interactions. Like one-component system [3], the first order
perturbation term vanishes in the present case also.
The second order perturbation term A^/NkTfor molecular fluid mixture is given by
a,b
(10)
where ft = (kT)~ 1 , g^ b 30 (ro) 1 a> 2 ) is the molecular pair distribution function (PDF) of
the reference HGO fluid mixture, p - N/V is the number density and c a = N a /N is the
concentration of species a. Here ((. . .)>< 0j w 2 represents an unweighted average over the
molecular orientations co 1 and co 2 .
The third order perturbation term is expressed as
Af/NkT = (AfJNkT) + (Af 2 /NkT) (1 1)
where
a,b
a,b,c
?(U)rfni.2,3)> mtWi dr 2 dr 3 ^ (13)
Here g (1, 2, 3) is the triple distribution function of the HGO fluid mixture.
In order to evaluate the perturbation terms, we introduce the new variable r* defined
by r* = r/tT fl6 (ca 1 a) 2 ), then the potential (2) transfers to the central form i.e. the
hard-sphere potential. In the same approximation, the molecular PDF of the HGO
fluid mixture becomes that of the hard sphere (HS) fluid mixture i.e.
When (7) is substituted in (10), the integrals appearing there can be written as
_/_0\-m + 3 rm(: ;\jmn
\ a ab) J ab( l 'J) 1 ab
where
J%(iJ) = <S(fi) 1 ( 2 )Cff 8fc (a) 1 a) 2 )/((ri)]-' +3 > aiWi (16)
and
" d(r*)(r*T mn * 2 dr*. (17)
using \. 1J J? U1C scuuuu uruer peruiruiuujn term /i 2 can uc wuucii
]. (18)
In (18), we have used the following reduced quantities
In a similar way, the leading contribution to the third term, A*\ , can be written as
a,b
] ; (20)
where Jj ft 1(1) and Jj b 1(2) are the coefficients corresponding to (ii 2 ab ) 2 (Ql b ) and
(^nbXMabSab) 2 * respectively, while J a l b 3 is the coefficient corresponding to (^ ab Q ab ) 2 (Qlb}-
Using the superposition approximation for 0JJJ (1, 2, 3), the term ^32 can be written
as
a,b,c
__ (21)
where
(22)
ky 2 do> 3 (23)
j j j
and
^"j^COjCtJj) = ^ ab ( l ^i ( ^-)[ff ab ((Ji) i O)-}/ff^ b ]~ n + 2 . (24)
Here A denotes integration over r* 2 , r* 3 and c? 3 which form a triangle.
The ./-integrals appearing in (18) and (20) can be determined by the Conroy
integration methods [10,4]. The numerical values for J are available for different
values of K [3]. Numerical integration of L abc in general is time consuming except for
the \n \JL \L and Q Q Q interactions.
The total electrostatic contribution to the Helmholtz free energy A ES is evaluated
from the Fade approximation [11]
(25)
Polar hard Gaussian overlap fluid
The total residual Helmhotz-free energy of the polar HGO fluid mixture is deter-
mined as
(A - A*)/NkT = ((^ HGO - A*)/NkT) + A ES /NkT. (26)
The other thermodynamic functions follow from the respective derivatives of the free
energy.
3. Virial equation of state of dilute pure polar hard Gaussian overlap fluid
The equation of state for pure polar HGO fluid, obtained from (9), can be expressed in
the virial form i.e. in the power of density p
Cp* + --- (27)
where A 1, B and C are the second and third virial coefficients, respectively.
The second virial coefficient can be written as
+ ... (28)
where
B HGO = (27r/3)a 3 <((7(o; lC o 2 )/a ) 3 ^ (29)
is the second virial coefficient of the HGO fluid. Here <(o"(co 1 aj 2 )/' ) 3 >w 1 w 2 can ^ e
expressed as [12]
+3a)/4 (30)
where a is a parameter of non-sphericity
(31)
Here & is the (l/47r)-multiple of the mean curvature integral, < the surface integral and
K HER = na 3 K/6 is the volume of the hard ellipsoid of revolution. It has been assumed
here that K HGO = K HER . <(o"(co 1 to 2 )/cr) 3 > WiW2 can also be expressed as [8]
<(cr(co 1 a) 2 )/(T ) 3 > 0it02 = KF (x) (32)
where
---]. (33)
.Bf s and B^ s are, respectively, the second and third order perturbation terms due to the
electrostatic interaction. In the present case, they are given by
J5f = -7r(T 03 [(^* 2 )
Bf =
(32* 2 /4) 2 J 10 (22)7 10 ] (34)
2) + 3J rll(2) (12))7 11
+ (3(2* 2 /4) 3 J 15 (22) 7 15 ] (35)
+ (27/224)(Q* 2 ) 2 J 10 (22)] (37)
Bf = (2/3)7T < r 03 [(9/64)( J u* 2 ) 2 (Q* 2 )(J 11(1) (12) + 3J 11(2) (12))
+ (81/320)(/r* 2 )(Q* 2 ) 2 J 13 (12) + (9/512)(Q* 2 ) 3 J 15 (23)]. (38)
The third virial coefficient can be expressed in a similar way as
C = C HGO + Cf + Cf + (39)
where C HGO is the third virial coefficient of the HGO fluid and Cf and Cf are,
respectively, the second and third order perturbation terms due to the electrostatic
interactions. The third virial coefficient C HGO can be expressed as [12]
C HGO = [1 + 6a + (3/2)a 2 (l/t + 1)] K 2 ER (40)
where a is defined by (31) and T is 'needleness' parameter defined as
&. (41)
The second order perturbation term Cf can be expressed as
+ (3Q* 2 /4) 2 J 10 (22)X 10 ] (42)
where
X m =
Here HS (r*) is the cluster integral for the hard sphere fluid of the effective hard sphere
diameter er ff = <rK 1/3 . An analytic expression for a HS (r*) is available [9]. Then (43)
can be evaluated as
X 6 = (47E/3)ff 3 K[(l/16)((l/6) + ln2)]. (44)
and
X m = (47r/3)o- 03 K[{(l/(m - 3)) - (3/4(m - 4)) + (l/16(m - 6))}
for m > 6. (45)
The third order perturbation term C|p can be written as
Cf = Cf l + Cf 2 (46)
where
Cf 1 =(2/3)^ 03 [(9/4)(^* 2 ) 2 (Q* 2 )(J 11(1) (12) + 3J 11(2) (12))X 11
)(Q* 2 ) 2 J 13 (12)X 13 -f-(27/64)(Q* 2 ) 3 J 15 (22)A r15 ] (47)
* 2 ) 3 T^ + (9/4)(^* 2 ) 2 (e* 2 ) T"" Q
)(Q* 2 ) 2 T^ + (27/64)(Q* 2 ) 3 T^Q] (48)
r rnmp
-/?[w HS (r* 2 ) + u
HS
(r* 3 )] }
M" mp is the function corresponding to (23) for one-component system. The triple
integrals T have been evaluated for the hard sphere fluid [13]. We may evaluate these
integrals for general values of K approximately. For this, we may write (24) as
Under this approximation, (49) can be written as
T-vimp V- (('i + m + p)/3 + 2) nrnmp
1 JV ^ HS
(50)
(51)
where T HS is the value of T for the hard sphere fluid, where the value of T for [i - p - /^
Lifi Q, fi Q Q and Q > 2 interactions may be obtained by the method of
Larsen et al [13]. Thus the results are
= 0-023 5/X, (52a)
(52b)
(52c)
(52d)
Values of the integrals are listed in table 1. The Monte Carlo (MC) numerical results for
ywu an( j J-QQQ obtained by Boublik [4] are also reported there for comparison. The
agreement is good for K < 1-5. For K > 1-5, the deviation appears in the fourth place of
decimal and increases with K.
Knowledge of B* s and Bf* as well as C^ s and Cf s allows us to write the Fade'
approximant [1 1] which may be employed to determine the whole polar contribution
to the virial coefficients. The whole quadrupolar contributions to the virial coefficients
are negative, the magnitude of which increases with increase of Q* 2 . The virial
Table 1. Values of the integral T for different values of K.
K
fQQQ
Present Boublik Present Boublik
1-00
0-0235
0-0235
0-0155
0-0155
0-0118
0-0118
1-20
0-0196
0-0196
0-0090
0-0090
0-0087
0-0077
1-35
0-0174
0-0173
0-0063
0-0063
0-0072
0-0059
1-50
0-0157
0-0154
0-0046
0-0046
0-0060
0-0046
1-65
0-0142
0-0139
0-0035
0-0036
0-0051
0-0037
1-80
0-0131
0-0126
0-0027
0-0029
0-0044
0-0030
2-00
0-0117
0-0112
0-0019
0-0028
0-0037
0-0027
2-20
0-0107
0-0101
0-0015
0-0018
0-0032
0-0019
2-50
0-0094
0-0088
0-0010
0-0013
0-0026
0-0014
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-12.0
Figure 1. The second virial coefficient B/V of the HGOQ fluid as a function of Q* 2
for K = 1-0, 1-5 and 2-0.
coefficients B HGO and C HGO of the HGO fluid, which are positive, are minimum at
K = 1-0 and increases as K moves away from 1-0. On the other hand, the magnitude of
the quadrupolar contributions to the virial coefficients are maximum at K = 1-0 and
decreases as K goes away from 1-0. The resultant of these two contributions gives B and
C of the quadrupolar hard Gaussian overlap (HGOQ) fluid. Thus the quadrupole-
quadrupole interaction lowers the virial coefficients. The values of J?/Kand C/V 2 of the
HGOQ fluid as a function of Q* 2 are reported in figures 1 and 2, respectively, for
K = 1-0, 1-5 and 2-0. They decrease with increase of Q* 2 at a given value of K. Further
they are minimum at K - 1-0 and increase as K moves away from 1-0.
4. Pure polar hard Gaussian overlap fluid
In order to test theory we consider a pure HGOQ fluid, which was studied earlier by
Boublik [3] neglecting the A 32 term and Boublik etal [3] including the term
approximately. A 32 for the pure HGOQ fluid can be obtained from (21) as
A 32 /NkT =
(53)
where L is defined by (22). Boublik et al [3] have roughly estimated it. But a good
knowledge of A 22 terms is important for polar fluid. Here we estimate the A 32 term in
-22.0
0.0 0.5 f.O 1.5 2.0 2.5 3.0
Figure 2. The third virial coefficient C/V 2 of the HGOQ fluid as a function of Q* 2
for K = 1-0, 1-5 and 2-0.
a better way. For pure Q Q Q interaction, L is given by Monte Carlo numerical
integration, which is fitted to the formula [3]
L QQQ = O0145exp(4-3158j7)/K 2 ' 65265 . (54)
We employ this formula to calculate the A 32 term.
The thermodynamic functions of the reference HGO system can be determined from
the hard convex body equation of state [12].
-tf (55)
where Y\ = na 3 K/6 is the packing fraction.
The parameter of non-sphericity a of corresponding HER is determined for the given
value of 1C and the volume F HER given by [3].
03 <^0))/cr ) 3 > WW (56)
(57)
where <((j(a} 1 a) 2 )/cr ) 3 > a , ia , 2 can be expressed as [8]
<(a(co 1 co 2 )/cr ) 3 > a)iWi = KF (x)
where F is given by (33). Thus a can be determined.
HGO fluid (K = 1-792).
Present SSS Boublik MC
0-30
4-51
4-33
4-50
4-56
0-35
6-00
5-70
5-98
6-08
0-40
8-11
7-63
8-07
8-20
0-425
9.49
8-88
9-43
9-57
Table 3. Values of A 2 /NkT and A 3 /NkT for the HGOQ fluid for K = 1-792.
X*
1
-A 2 /NkT
A 3 i/NkT
A 32 /W/cT (
'.A 3 i + A 32 )/NkT
0-9914
0-30
0-264
0-026
0-004
0-030
0-35
0-341
0-034
0-007
0-041
0-40
0-432
0-045
0-011
0-056
0-425
0-435
0-051
0-014
0-065
1-9828
0-30
1-056
0-207
0-032
0-239
0-35
1-365
0-274
0-055
0-329
0-40
1-729
0-358
0-089
0-447
0-425
1-935
0-407
0-111
0-58
2-9743
0-30
2-377
0-698
0-109
0-807
0-35
3-072
0-926
0-184
1-110
0-40
3-891
1-201
0-299
1-507
0-425
4-353
1-374
0-376
1-750
Using the relation
(A-A*)/NkT- I E(j8P/p)-l](d/7w') (58)
o
the expression for the residual Helmholtz-free energy is given by
(59)
To test theory, we calculate the compressibility factor /5P HGO /p of the HGO fluid
(with K = 1-792) at r\ = 0-30, 0-35, 040 and 0-425 for which Monte-Carlo (MC) data are
available [3]. Results are compared with the MC data in table 2. The results obtained
by Singh et al [8] denoted by SSS and Boublik [3] are also shown there. The agreement
is good and is better than SSS and Boublik data.
We calculate the contributions due to the quadrupolar interactions for the reduced
quantity X* = 3Q 2 /4/eT a QS at K = 1-792 for which J 10 = 1-0405 and J 15 = 0-4720 [3].
The integral /" can be evaluated using the equation of Larsen et al [13].
In table 3, we report the values of A 2 and A 3 of the HGOQ fluid for K = 1-792 at
X* = 0-9914, 1-9828 and 2-9743. From these data, we find that A 32 is less than A 31 .
However the relative contribution of the A 32 term increases with increase of density. So a
good knowledge of the A 32 is important especially at high density and at high value of X*.
- ~.x. -,. >_-win.i.iwi.xjii nuv/ iv uiv/ pL-iiiicuiuiu yjuauiupuic UJUUJCiU IU LilClIllU-
dynamic properties of the HGOQ fluid (K = 1-792).
A"* = 0-9914 **
= 1-9828
X* = 2-9743
/
1
2
1
2
1
2
-(A
~A HCO
)/NkT
0-40
Sum
0-39
0-38
1-37
1
28
2
68
2-38
Fade
0-39
0-38
1-43
1
37
2
97
2
80
MC
0-39 + 0-01
1-36 + 0-01
2
70 + 0-01
0-425
Sum
0-43
0-42
1-53
1
42
2-98
2-60
Fade
0-44
0-43
1-60
1
53
3'
31
3'
10
MC
0-44 0-03
1-49 0-05
2-
96 0-08
-(V
_ fjHGO
)/NkT
0-40
Sum
0-73
0-70
2-38
2'
12
4-
16
3-
26
Fade
0-75
0-72
2-62
2-46
5-
23
4-82
MC
0-73 0-01
2-36 0-01
4-47 0-01
0-425
Sum
0-82
0-77
2-65
2-31
4-
59
3-
46
Fade
0-83
0-80
2-92
2-
73
5-
82
5-
32
MC
0-79 + 0-01
2-57 0-01
4-88 + 0-05
.esults obtained for (A - A" GO )/NkT and (U-U HGO )/NkT of the HGOQ with
= 1-792 with and without the A 32 term are reported in table 4 for r\ = 0-40 and 0425,
;re 1 and 2 represent the values without and with the A 32 term, respectively. The sum
>ry like previous study [3, 5] gives better results when the A Z2 term is neglected,
en the Fade' approximation is employed, the inclusion of the good form of A 32 term
Is slight improvement over the previous one. However it is clear from table 4 that
sum theory without the A 32 term predicts good results in general.
'olar hard Gaussian overlap fluid mixture
adopt the extended Van der Waal one (EvdWl) fluid theory of mixture [14, 8] to
ulate the properties of the HGO fluid mixture. This theory approximates the
Derties of a mixture by those of a fictitious hard-body fluid with the parameters
a,b
a,b
(60a)
(60b)
i the EvdWl fluid theory of mixture the pressure and residual Helmholtz-free
gy of the HGO mixture are given by Singh et al [12] and Kumari and Sinha [15].
smploy these expressions in the present calculation.
order to evaluate the integrals I" ab , which appear in the polar-dependent terms, we
in approximation in which the mixture PDF g is equal to the zeroth order term in
nformal solution of Mo et al [16].
Here
-03
'ab
and I" is the integral of the pure fluid at the packing fraction Y\ O where
i\o = 1\\ + ^c 2 ((2V i2 - F n - V 22 )/(c, F n + c 2 F 22 ))]
with
and evaluated using the equation of Larsen et al [13].
The three-body integral L p can be expressed as
(61)
(62)
(63)
(64)
.'to ,K ) (65)
where K p is the pure fluid integral at the packing fraction 77 . The values of the
integral for the Q Q Q interaction can be obtained by (54) using rj in place of rj.
2.0
1.0
Ul
0.0
-1.0
1.0
2.0
3.0
Figure 3. The excess Helmholtz-free energy A^/NkT of the HGOQ fluid mixture
with Q* = Q* = V2 as a function of R at r\ = 0-3 for c t = c 2 = 0-5 and for K = 1-0 and
2-0. The HGOQ value is denoted by solid line while the HGO value by dashed line.
I 1 I I
0.0 0.5 t.O 1.5 2.0 2.5 3.0
-1.0
Figure 4. The excess Helmholtz-free energy A E /NkT of the HGOQ fluid mixture
withet = 8* = V 2asafunctionofJ ^ at '? = ' 3forc i =c 2 ==0 ' 5anciforK= I'l and
5/3. The key is same as figure 3.
We calculate the numerical results for a binary HGOQ mixture. In the present case,
we assume that all the constituent molecules have the same shape but different sizes.
In figure 3, the values of the excess Helmholtz-free energy A E /NkTfor HGO mixture
(61=62=) and HGOQ mixture (with <2t = 6* = V 2 ) with c 1 = c 2 = 0-5 are
. demonstrated as a function of diameter ratio R for r\ = 0-3 and K 1-0 and 2-0. The
HGO value is maximum at jR = 1 -0 and decreases as R moves away from 1 -0 for a given
value of K. Further the HGO value increases with increase of K. The contribution of
the quadrupole interaction is negative, which decreases the value ofA E /NkT. The effect
of the quadrupole interaction is maximum at R = 1-0 and K = 1-0 and decreases as
R moves from 1-0 as well as K goes to the higher value. The influence of the quadrupole
interaction at a given K vanishes at R = and oo. This behaviour can be explained from
(20), which gives Q* 2 at JR = and oo and the system behaves as the HGO mixture.
The values of A E /NkT for the HGO mixture and HGOQ mixture (with
<2* = Q* = ^2) for c i = c 2 = O5 are reported in figure 4 as a function of K for 77 = 0-5
and R = 1-1 and 5/3. The values of both HGO and HGOQ are minimal at K = 1-0 and
increases steadily as K moves away from 1-0 at a given R. This behaviour depends on
r\ and/or R. The HGO value of A E /NkT at a given K decreases with increases of R,
whereas the quadrupole contribution AXf/JV/cT, which is negative, increases. As
a result, the HGOQ value at R = 5/3 varies more rapidly than that at R = 1-1, as
-5.0
Figure 5. The excess internal energy U/NkT of the HGO fluid mixture with
Q* = Q* = J2 as a function of q at v\ = 0-4 for R = 5/3 and K = 1-0 and 1-792.
and 5/3 intersect at two values of K i.e. at K = 0-8 and 1-2. From the figure it is clear that
the influence of the quadrupole interaction is maximum at K = 1-0 and decreases as
K goes away from 1-0.
Figure 5 shows the value of excess internal energy U/NkT of the HGOQ mixture
(with Q* = Q*~ ^2) as a function of ^ at r\ = 04 for X = 1-0 and 1-792 and for R = 5/3.
It is found that the excess values are zero at q = 0-0 and 1-0, and non-zero in the
intermediate range of c t .
Thus we come to the conclusion that the excess free energy and internal energy
of the HGOQ mixture depend on the concentration c ls c 2 , particle diameter ratio
-v. *.
R = <r5 2 /<r5i, shape parameter K and the quadrupole moments Q*[, Q
6. Summary
The purpose of the present paper has been to develop a theory for evaluating the
thermodynamic properties of a polar HGO fluid mixture. We have given explicit
analytic expressions for the second and third virial coefficient of the pure polar HGO fluid.
We have also derived explicit expression for the Helmholtz-free energy for the
HGOQ fluid and fluid mixture. It is found that the contribution of the quadrupolar
interaction depends on the quadrupole moments Q*,Q*, the concentration c x , c 2 , the
molecular diameter ratio R and shape parameter K in general and on the packing
on c l , c 2 and R in the same way as those of the quadrupolar hard sphere mixture [17].
The effect of the quadrupole interaction is maximal for K 1-0 (hard sphere) and
decreases as K deviates from 1-0. Since no simulation results are available for the polar
HGO fluid mixture, no comparison has been made in this case. Extension of the theory
to some real polar Gaussian overlap fluid mixtures will be discussed in future
publications.
Acknowledgements
One of the authors (SKS) acknowledges the financial support of the University Grants
Commission, New Delhi.
References
[1] C G Gray and K E Gubbins, Theory of molecular fluid (Oxford, Clarendon, 1984) Vol. 1
[2] M C Wojcik and K E Gubbins, Mol. Phys. 51, 951 (1984)
M C Wojcik and K E Gubbins, J. Phys. Chem. 88, 4559 (1984)
[3] T Boublik, Mol. Phys. 69, 497 (1990)
[4] T Boublik, Mol. Phys. 76, 327 (1992)
T Boublik, Mol. Phys. 77, 983 (1992)
[5] T Boublik, C Vega, S Laga and M Siazpena, Mol. Phys. 71, 1193 (1990)
[6] C Vega and S Laga, Mol. Phys. 72, 215 (1991)
[7] B J Berne and P Pechukas, J. Chem. Phys. 56, 4213 (1972)
[8] T P Singh, J P Sinha and S K Sinha, Pramana - J. Phys. 31, 289 (1988)
[9] J O Hirschfelder, C F Curtiss and R B Bird, Molecular theory of gases and liquids (Wiley,
New York, 1954)
[10] H Conroy, J. Chem. Phys. 47, 5307 (1967)
[11] G Stell, J C Rasaiah and H Narang, Mol. Phys. 27, 1393 (1974)
[12] T Boublik and I Nezbeda, Coll. Czech. Chem. Commun. 51, 2301 (1986)
[13] B Larsen, J C Rasaiah and G Stell, Mol. Phys. 33, 987 (1977)
[14] T W Leland Jr, J S Rowlinson and G A Sether, Trans. Faraday Soc. 64, 1447 (1968)
[15] R Kumari and S K Sinha, Physica A211, 43 (1994)
[16] KG Mo, K E Gubbins, G Jacucci and I R McDonald, Mol. Phys. 27, 1 173 (1974)
[17] B Rai, N Prasad and S K Sinha, Pramana - J. Phys. 35, 533 (1990)
Structural study of aqueous solutions of tetrahydrofuran and
acetone mixtures using dielectric relaxation technique
A C KUMBHARKHANE, S N HELAMBE, M P LOKHANDE,
S DORAISWAMY + and S C MEHROTRA
Department of Physics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad
43 1004, India
+ Chemical Physics Group, Tata Institute of Fundamental Research, Bombay 400 005, India
MS received 4 September 1995; revised 8 December 1995
Abstract. The complex permittivity, static dielectric constant and relaxation time for tetra-
hydrofuran-water and acetone-water mixtures have been determined at 0, 10, 25 and 35C
using time domain reflectometry technique (TDR). The behaviour of relaxation time of the
mixture shows a maxima for the mixture with 30% of water by volume. This suggests that the
tendency to form cluster between water and solute molecule is maximum for this mixture. The
excess permittivity for both tetrahydrofuran-water mixture and acetone-water mixtures, are
found to be negative. The Kirkwood correlation factor has been determined at various
concentrations of water. Static dielectric constant for the mixtures have been fitted well with the
modified Bruggerhan model. The values of the Bruggeman parameter a for tetrahydrofuran is
found to be more than the corresponding value for acetone.
Keywords. Dielectrics; microwaves; tetrahydrofuran; acetone; excess permittivity; Kirkwood
factor.
PACS No. 77-22
1. Introduction
Considerable dielectric relaxation study have been done in aqueous solutions such as
dimethylsulfoxide-water [1], JV.JV-dimethylformamide-water [2], N, JV-dimethyl-
acetamide-water [3], hexamethylphosphoramide-water [4] and acetonitrile-water [5]
mixtures. For the systems studied it has been observed that the value of the relaxation
time shows a maxima by addition of water in solutes, This has given the information
about solute-solvent interaction, molecule packing of hydration shell to the solute
molecules and molecular volume size. Tetrahydrofuran and acetone [6] are interesting
systems because they are completely miscible in water and are of different structural
types.
Recently, nuclear magnetic resonance (NMR) relaxation in tetrahydrofuran-water
and acetone-water mixtures were studied [7], by using NMR technique at various
composition of tetrahydrofuran and acetone in water solutions. The proton relaxation
time rates as a function of water composition in tetrahydrofuran and acetone were
observed at maximum concentration 0-8 mole fraction of water. It is interesting to
compare the results with the results of dielectrics.
mixtures with varying concentrations (0-100%) is reported over the frequency range
lOMHz-lOGHz by using the TDR technique. The dielectric parameters, excess
dielectric permittivity, the Kirkwood correlation parameters and the Bruggeman
factor have also been determined for these systems.
2. Experimental
Materials
Tetrahydrofuran (THF) and acetone (ACT) of spectroscopic grade were obtained
commercially and used without further purification. The water used in the preparation
of the mixtures was obtained by double distillation procedure and deionized before use.
The mixtures of various compositions (0 to 100%) were prepared by volume before
mixing.
Apparatus
The complex permittivity spectra was studied using (TDR) method [8, 9]. The detail of
the experimental method are described earlier [8,9]. The Tektronix 7854 sampling
oscilloscope with 7S12 TDR unit has been used. A fast rising step voltage pulse of
25 psec rise time generated by a tunnel diode was propagated through a coaxial line
system. The sample was placed at the end of coaxial line in standard military
application (SMA) coaxial cell of 3-5 mm outer diameter and 0-486 mm effective pin
length. All measurements were done under open load condition. The change in the
pulse after reflection from the sample placed in the cell was monitored by the sampling
oscilloscope. In this experiment, time window of 5 ns was used. The reflected pulse
without sample and with the sample were digitized in 1024 points and transferred to
computer through GPIB (general purpose interface bus).
The Fourier transformation was done to yield the complex reflection coefficient
spectra p*(co) over the frequency range 10 MHz to 10 GHz. The complex permittivity
spectra e* (CD) was determined from the p* (co) by applying the least squares fit method as
described in our earlier publication [4].
3. Results and discussion
The static dielectric constant (s ) and relaxation time (T) have been determined by
fitting the complex permittivity spectra e*(co) with the Debye equation,
with e , T as fitting parameters in (1). Since the permittivity spectra in the present study
is in the frequency range of 10 MHz to 10 GHz, the e OT in (1) as determined from this
study is just a fitting parameter, which does not correspond to the permittivity at high
frequency, related to vibrational and electronic motions. It was found to be a reason-
able satisfactory procedure to keep the value of e m as a fixed parameter (3-5), for the
determination of and T [4].
Table 1. Dielectric parameters for acetone-water
mixture.
Vol % of
acetone
CD
r(ps)
CD
t(ps)
C
10
C
100
29-9(5)
6-2(4)
21-6(1)
4-2(3)
90
31-5(1)
11-4(1)
26-8(1)
7-4(3)
80
37-6{2)
15-5(5)
33-9(1)
10-1(4)
70
42-1(4)
18-3(10)
41-7(7)
14-0(16)
60
42-0(1)
16-9(2)
45-0(2)
13-2(2)
50
52-3(11)
17-7(16)
51-3(9)
12-0(14)
40
67-2(11)
27-9(19)
58-0(15)
17-3(3)
30
70-0(2)
21-2(4)
69-6(5)
14-8(7)
20
77-0(6)
21-1(3)
75-4(1)
15-4(6)
10
80-0(5)
17-3(6)
77-2(6)
11-8(7)
87-9
17-7
83-9
1.2-9
25'
J C
35
c
100
20-8(1)
3-4(5)
18-9(5)
2-9(5)
90
25-8(1)
7-3(4)
25-5(1)
4-8(3)
80
31-0(2)
8-6(3)
31-0(1)
7-5(4)
70
40-0(4)
10-1(6)
35-8(3)
9-1(9)
60
44-5(3)
11-9(6)
44-0(5)
10-8(9)
50
51-6(7)
10-8(10)
49-1(7)
9-3(11)
40
57-3(3)
11-8(5)
54-7(2)
10-7(4)
30
62-5(9)
10-7(12)
61-4(1)
9-4(10)
20
69-1(3)
12-2(4)
66-1(2)
9-5(8)
10
75-4(8)
10-2(6)
66-1(6)
9-5(9)
78-3
8-2
75-0
6-5
Numbers in brackets denote uncertainties in the last
significant digit as obtained by the least squares fit
method, e.g. 29-5(5) means 29-5 0-5. The realistic error in
e,, and T is estimated to be about 2% of their values.
Table 2. Dielectric parameters for tetrahydrofuran-water mixture.
Vol % of
THF
10 C
C
25
C
35
C
eo
-c (ps)
%
T(pS)
*o
r(ps)
100
10-4(2)
5-3(2)
6-7(1)
4-0(1)
6-0(3)
3-2(4)
90
15-0(1)
14-3(2)
17-9(2)
13-2(1)
17-4(1)
11-8(3)
80
23-6(1)
18-6(1)
20-5(4)
15-5(2)
23-7(1)
14-8(4)
70
34-1(1)
24-3(3)
35-4(1)
17-4(2)
30-7(1)
14-9(2)
60
45-7(1)
26-4(2)
40-9(1)
17-0(4)
40-2(1)
14-5(3)
50
53-8(1)
25-9(2)
54-6(2)
15-6(4)
50-8(3)
14-6(8)
40.
59-1(1)
24-1(2)
54-9(2)
14-8(5)
50-9(3)
13-0(7)
30
65-0(2)
21-5(3)
64-9(1)
14-0(2)
63-8(1)
9-8(2)
20
70-6(4)
17-9(6)
73-1(1)
12-6(1)
68-7(1)
9-4(2)
10
80-6(6)
15-2(7)
76-4(2)
10-6(2)
72-5(1)
7-3(1)
83-9
12-9
78-3
8-2
75-0
6-5
THF-water mixtures. It can be seen from tables 1 and 2 that the relaxation time
increases with water concentration in the mixtures. The maximum relaxation time were
observed at 30% of water in both THF and ACT mixtures. Similar behaviour were
observed for proton relaxation rates by NMR relaxation technique for both THF and
ACT systems in water mixtures [7]. This maxima behaviour may be because the water
molecule with solute molecules in the mixture creates a cluster structure such that both
THF and ACT molecules rotate slower in the mixture.
The excess permittivity Q and the excess inverse of the relaxation time I/T E for THF
and ACT in water mixtures are determined using the following equations as follows [2]
*o = (o - On ~ ((*o - oo)w* m + (e - JsU ~ *)) ( 2a )
l/t E = (l/r) M - [(l/i) w X m + (1/ T ) S (1 - XJ] (2b)
where subscript M, W, and S correspond to the mixture, water and solute, respectively,
and X m is mole fractions of water in solute.
The viscosity data is taken from the references [10, 11]. The excess viscosity r\ E are
determined using the equation
* B ^ M Mfow*Xm) + fos*(l-*J)). (3)
The variation of Q , 1 /T E (in GHz), rf (in cP) with mole fraction of water (X m ) in ACT
and THF are shown in figures 1 (a, b, c) and 2 (a, b, c) respectively. The excess dielectric
constant and inverse of relaxation time for both THF-water and ACT-water mixtures
show negative behaviour. This shows that the oxygen in both THF and acetone helps in
formation of hydrogen bonding in the mixtures. The corresponding plots of excess
viscosity show positive behaviour. In both mixtures, the maxima is found to be in
water-rich region. The molecular interaction responsible for viscous motions seems to
be different from the interactions responsible for dielectric behaviour.
The Kirkwood correlation factor g provides information regarding the structural
information of molecules in the polar liquids. The value of g in a pure liquid can be
obtained by the following equation [12, 13]
(p - e co )(2s + e J
9/cTM
where fi, p and M correspond to the dipole moment in gas phase, density and molecular
weight respectively, k is the Boltzmann constant and N is the Avogadro number.
The modified form of (4) is used to study the orientation of the electric dipoles in
the binary mixtures as follows [2-4]
9kTl M w w ' M s v w
where g ef[ is Kirkwood correlation factor for binary mixture, F w represent the
volume fraction of water in solute. e 0m and oom are the static dielectric constant and
dielectric constant at high frequency, respectively. To calculate the g e{[ we have taken
jt= 1-83, 1-63 and 2-8 D, for water [14], THF [15] and ACT [16], respectively. The
values of densities are 0-8892 and 0-7899 for THF [16] and ACT [16], respectively
-10
-16
-20
0.2
0.4 0.6
-10
-15
-20
-J L.
0.8
0.2 0.4
0.6 0.8
Figure 1. The (a) excess dielectric permittivity (e|j), (b) excess relaxation time (!/T E )
in GHz and (c) excess viscosity (if) in (cP) versus mole fraction of water X m in
acetone at 25C: (d) Experimental values. The solid line describes the best possible
curve as obtained by the commercial (Harvard) graphic software.
at 25C. The e^ values are taken as the square of the refractive index data [16] for
THF and ACT. The values of # eff for THF and ACT in water mixtures are given
in table 3. Errors are also estimated in these values by assuming 2% error in values
of permittivity. The values of the excess dielectric parameter E decreases rapidly
near the dilute region of solute molecules. It has also been observed that in this region,
the relaxation time increases rapidly with the solute concentration. This suggests
the formation of cage-type structure around solute molecules. This causes the slower
Pramana - J. Phys., Vol. 46, No. 2, February 1996
95
-6
-20
-26
0.4 0.6 0.8 1 0.2 0.4 0.6
Figure 2. The (a) excess dielectric permittivity (e^), (b) excess relaxation time (!/T E )
in (GHz) and (c) excess viscosity (r] E ) (in cP) versus mole fraction of water X m in
tetrahydrofuran at 25C: (Q) Experimental values. The solid line describes the best
possible curve as obtained by the commercial (Harvard) graphic software.
relaxation and lower values of orientational polarization. As the number of solute
molecules increases, there are not enough water molecules available to form the
cage-type structure.
The Kirkwood correlation factor for THF and ACT are smaller than the cor-
responding values in water. The static dielectric constant of e 0m , e 0w and e 0s of the
mixture, water and solute can be related with the Bruggeman mixture formula [17, 18]
with volume fraction of X of water. The Bruggeman's mixture formula is given by the
factor for THF and ACT in water
mixtures at 25 C.
Vol % of
solute
in water
g,
rfr
THF
ACT
100
1-17(7)
1-02(6)
90
1-37(8)
2-17(13)
80
1-55(9)
2-00(11)
70
1-90(11)
2-72(16)
60
2-00(12)
2-65(16)
50
2-20(13)
3-07(18)
40
2-32(14)
2-71(16)
30
2-41(14)
2-87(17)
20
2-54(15)
2-92(17)
10
2-65(15)
2-79(16)
2-67(16)
2-67(16)
ression[17, 18]
~
0w
Sow)
1/3
(6)
: experimental data does not show the linear relation between / BM and X. The
srimental. data is well fitted to the modified Bruggeman equation as follows [2]
f BM =l-(aX~(l-a}X 2 )
(7)
:re a is an arbitrary parameter, a = 1 correspond to Bruggeman equation (6).
values of a have been determined by the least squares fit method. The values
; for THF-water and ACT- water mixtures are determined to be 2-30 and 1-57,
tectively.
Conclusion
dielectric relaxation parameters, excess dielectric constant, the Kirkwood correla-
. factor and Bruggeman factor have been reported for THF-water and ACT-water
tures for various temperatures. The experimental dielectric relaxation data con-
s valuable information regarding solute-solvent interactions in the mixures. By
ig recently developed theories [19,20], one may get the quantitative information
ut solute-solvent interactions.
nowledgements
authors thank Dr P B Patil and Dr G S Raju for discussion and helpful
>estions. The financial support from UGC, New Delhi is thankfully acknowledged.
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Composite Anderson-Newns model and density of states
ue to chemisorption: Quasi-chemical approximation
GULERIA, P K AHLUWALIA and K C SHARMA
lysics Department, Himachal Pradesh University, Shimla 171 005, India
S received 6 July 1995
jstract In this paper a variation in density of states (DOS) of the substrate due to chemisorption
hydrogen on transition metals using composite Anderson-Newns model has been inves-
;ated for different coverages in quasi-chemical approximation of Fowler and Guggenhiem,
rich in the limit z - oo gives the Bragg- Williams approximation as a special case. Variation in
nsity of states has been studied for one-dimensional periodic substrate with change in adatom
:eraction energy and coverage. With increase in coverage, the bonding and antibonding
-AB) peaks are found to shift towards higher energies and at the same time relative height of
3 peaks also increases. The interesting feature to observe is that both approximations for
particular coverage, give split-off states with different height for both (B-AB) peaks. It
rticularly indicates change in B-AB states, representing amount of chemisorption, with the
ange in interaction energy between adatoms. At the same time bond strength is also found to
crease with interaction between adatoms.
sywords. Chemisorption; density of states; quasi-chemical approximation and coverage.
ICSNos 68-45; 71-20
Introduction
ne of the most interesting problems in solid state physics is chemisorption of gases on
rfaces of metals. This has deepened our understanding of various processes taking
ace on the surface like corrosion, hydrolysis and catalysis. At present, chemisorption
eories can be divided into two groups: semi-emperical model Hamiltonian theories
id first principal methods. A good progress has been made in ab initio calculations
ing density functional theory and its local approximation, cluster calculations or
>herent potential approximation. These methods allow us to carry high precision
Iculations of space distribution of electric charge, binding energy, work function
.anges and other experimental characteristics of adsorbate-adsorbant system,
it, there are numerous difficulties with these methods when many body effects
id correlations in electronic and ionic component of adsorbate are considered. On
e other hand, the model Hamiltonian method appears to be well suited for under-
inding the experimentally observed features due to relatively simple microscopic
cture of the system and possibility of including correlation effects and additional
teractions in the adsorbed system by using different approximations [1,2]. Further-
ore equation of motion method can be useful, especially when one is trying to include
e correlation effects [1]. In the model Hamiltonian method, the major step was taken
Anderson model-Hamiltonian for magnetic alloys, to treat the chemisorbed atom as
an impurity on the surface and then solved the problem within the self-consistent
Hartree-Fock scheme. A more generalized version of Anderson- Newns model called
composite Anderson-Ising model, was proposed by Gavrilenko [5] in which an
attempt has been made to incorporate the Coulomb correlations and interactions in
the adsorbed system. This model was applied to the case of stochastic arrangement of
adatoms on metal surface by Gavrilenko [6] and it was shown that the Anderson
criterion for impurity magnetism in metals holds for this type of phase-transition as
well. Later, using this model, characteristics of chemisorption were further investigated
by Cardena [2]. Calculations were carried out within the framework of Hartree-Fock
approximation for electronic component of adsorbate and using Bragg-Williams
approximation for ionic component. Anderson-Newns model requires proper par-
ameterization to take into account all the correlation effects [7]. The composite
Anderson-Newns model is a step in that direction but restricting to the most simple
Bragg-Williams approximation (BWA) to include correlation among adatoms on
the surface.
In this paper we improve upon the situation by introducing correlators in the
quasi-chemical approximation (QCA). This approximation was first proposed by
Guggenheim and Fowler [8] in the context of chemisorption problem, by introducing
an interaction energy term c = [1 -4(9(1 0)(1 e\p(-2W/ZKT))'], where 6 is
coverage and XT is the thermal energy, subject to the restriction that the interaction
term W/KT^2, a critical value below which desorption starts. They pointed out
that the BWA is too crude an approximation as the value of critical temperature (T c ) for
the phase-transition during chemisorption is found to be very much off its correct
value. On the other hand, QCA takes care of long- as well as short-range interactions,
which improves the situation and brings T c closer to the correct value. It is interesting
to note that in the limit Z- oo, QCA gives the same results as the BWA. But, since
Z cannot be equal to infinity, the BWA is not a realistic approximation and thus QCA
is supposed to give better results [8]. Therefore, it will be more appropriate to apply
QCA to study the chemisorption characteristics within the composite Anderson model.
However, no systematic attempt has been made so far to analyze the chemisorption
behaviour as a function of the interaction between adatoms within the framework of
the above mentioned QCA.
In this paper we have studied the change in density of states (DOS) on chemisorption
of hydrogen on transition metals using this composite Anderson-Newns model [5]
using QCA. We solve this model self-consistently within the Hartree-Fock scheme.
Later, we take the specific form of weighted density of states (WDOS) for atop case, to
be a semi-elliptical one which has been used by most of the workers [2, 3, 6] to make the
problem analytically tractable. The possibility of other (triangular) weighted DOS has
been discussed by authors in a recent paper [9]. Our aim is to study chemisorption
behaviour as a function of the coverage using the semi-elliptical WDOS in QCA.
In 2 we discuss the general formalism and its assumptions. In 3 we define the
quasi-chemical correlators to derive a basic expression for the DOS. Later, we choose
specific form of WDOS to perform numerical calculations. Finally, in 4, we discuss the
numerical results obtained followed by the conclusions.
Quasi-chemical approximation
Basic equations and formalism
e consider a system of N A adsorbed hydrogen-like atoms (adatoms) distributed over
active centres (adsorption centres) on the metal surface (substrate), N A ^ N. Each
>m has a rigid bond with substrate. The configuration of the adatom arrangement
zr the adsorption centres is not fixed. In the simplest case, Hamiltonian of this system
i be written [5, 9] in the form
k,a
+ Un^ + I (V ak bl* ka + h.c)
fc.tr
(1)
re, the sum over a in (1) is over all the adsorption centres, the operator N a = C*C a
5 eigen values or 1. Cj, C a are the Fermi amplitude of the creation and annihila-
n operators of adsorption centre a,s fc is the energy spectrum of clean substrare, E
:he ionization potential of adatom. n ka = al a a ka , n aa =-b\ g b aa , where a ka , b aa are
electron variables for substrate electrons and adatoms electrons, respectively.
describes the Coulomb interaction of atomic electrons and h.c. means Hermitian
ijugate.
iere, the difference from usual form is because of the hopping interactions term
iportional to V ak (V kac ) which takes into account the indirect interactions between
adatoms. Although this model is linear with respect to ion operator JV a , it is
: equivalent to ideal lattice model due to these interactions. To solve this model,
take only one localized impurity atom, consequently N a = 1 if a = A and N a =
K 7^ A. Using unrestricted Hartree-Fock approximation which considers the
raged interactions of all the adatoms and the substrate, the Hamiltonian reduces
a form which can be easily solved [9]. The electronic properties of the sys-
t are described by correlators of type <n a(T >, <N a aff >, <N a fyLr a fcc- >> whereas
ic properties are described by correlators of Ising type i.e., <AT a >, (N^Np >,...,
z N f ...N,y,....
\> derive the properties of the system, we shall use double-time Green's functions
ch are derived by equation of motion method. All the related electronic properties
ystem, can be calculated from these Green's function equations [9]. We shall discuss
state of the system at a temperature restricted to where W/KT ^ 2, below which,
Drption occurs [8].
^uasi-chemical approximation and correlators
obtain the electronic properties of the system, in final form, Gavrilenko used the
A which assumes the distribution of spins as random and consider interactions only
veen nearest neighbours. In BWA
- e) + e 2 = 0[A a/? (i - 0) + ff]. (2)
approximated as
where
2
. (4)
Here Z is the number of nearest neighbors of an atom (coordination number) and
exponential factor (e~ 2W i ZKT ) takes care of both (short as well as long range) interac-
tions between adatoms. It may be noted that in the limit Z-> oo, QCA (eq. (3)) reduces
to (2) of BWA. Of the two approximations, the QCA is certainly better [10] as it gives
the exact solution in the (artificial) one-dimensional case which BWA does not,
moreover it also gives the temperature dependence of chemisorption characteristics.
The QCA is exact in one-dimension not only for nearest neighbor systems but also for
arbitrarily higher neighbor systems [10]. We shall study the situation at and below
a critical temperature above which the desorption starts. In the QCA, the function
<2 q (co) has the form
l ~
where L(a>) is the Newns function and P q (co) is the ^-representation of Grimley's
function [9]. For single site, Green's function (a = j8) describing the charge distribution
on impurity level becomes
20
v ' C + 1 9V
'(6)
To derive (6), we have used approximations described by (3) and (5). Now, as 6 -> and
z->oo (6) gives us Newns theory [3], and as B-+ 1, z-oo it gives solvable limit of
Grimley's model. Equation (6) is also solvable for all intermediate values of 6. The
interesting feature of the composite Anderson model is that it takes care of both the
cases as well as the intermediate values of the coverage. The density of states (DOS) of
this system using (6) and the assumptions [6] involved, can be written in the form
dco
(7)
In our calculations we take one-dimensional model of periodically arranged atoms
with lattice constant a = 1 interacting only with its nearest neighbours and using
periodic boundary conditions. Here, we have taken the energy spectrum S K = F cos(K)
;. For this case after some simplifications (7) can be written as
c+i
A(co) 2
;
(8)
ere y(co) + z'A(co) = L(cu id); A(co) = x(] 2 p(co + B F ) is the weighted DOS states and
)) is the Hilbert transform of A(co)-function and is given by
/* A / \
y(o}) = dy, where rc < arctg ^ n. (9)
j \ y>
fhe functions A(co) and y(co) are called chemisorption functions, latter being the Hilbert
nsform of the former. These are related to each other through G aa (a)) = y(co) + z'A(co).
re we have chosen the semi-elliptical form of WDOS given by
[j9 2 (l-co 2 ) 1/2 for|o)|<land
10 for|o)|>l. (10)
s Hilbert transform of A(co) is
(11)
Numerical results and conclusions
Drder to have numerical values of DOS of substrate on chemisorption of hydrogen
the transition metals using composite Anderson-Newns model, the basic input
ameters are: ionization level E, the electron affinity level E+U, where U is the
ulomb interaction between electrons on impurity level; s = Fermi level, /? = hybrid-
tion parameter and 2F = fo - 8 ) is the band width of the unperturbed system. ^ , e
the upper and lower edges of the band respectively. There are three situations
responding to different positions of ionization and affinity level with respect to
mi level parameterized by n = E % 4- U/2. We shall concentrate on the case
= corresponding to symmetric Anderson's model. Here all the parameters are
sen within the Anderson's model [1 1], and all energies are measured in terms of half
id-width (T), and the position of energy levels is with respect to substrate band
tre, also e = 0. In the present case we take Z = 6, which is also the most suitable
dee for Z in two-dimensional case [8].
4ow, if U/p is small, we have non-magnetic case. In this paper, we shall restrict the
sraction term ( W/KT) to values beyond the critical one i.e. 2, and thus avoid the
sibility of desorption [8].
"he interesting features of the results (associated with QCA) obtained are as follows:
1-00
Q
0-50 -
o-oo
-i.oo
o.oo
. CJ -
1.00
Figure 1. Comparison of DOS on chemisorption using curve (a) Bragg- Williams
approximation and curves (b), (c) quasi-chemical approximation, with two different
values of interaction energy term W/KT 2 and 8, respectively for high coverage,
= 0-8.
4.1 Comparison of the values of DOS using Bragg-Williams and quasi-chemical
approximation for fixed coverages
The DOS corresponding, respectively, to high (9 - 0-8) and low (6 = 0-2) coverages are
presented in figures 1 and 2 in both the approximations, curve (a) represents the BWA
and curves (b) and (c) correspond to QCA. The latter shows a decrease in depth between
B-AB peaks as compared to that for curve (a), implying, the effect of interaction energy
term in both the cases is to cause decrease in bond strength. Furthermore, the BWA is
an approximation at OK, whereas the QCA corresponds to some finite temperature
within the restriction as mentioned above. This implies that with rise in temperature
there is decrease in adatom-substrate bond-strength. As expected, it provides activa-
tion energy fordesorption in form of vibrational energy [12].
4.2 Effect of coverage on bond strength and number of B-AB states
The curves (a), (b) and (c) in figure 3 show variation of DOS for 6 = 0-001, 0;2, and 0-8
respectively, using the QCA by keeping the interaction term constant i.e. W/KT = 2.
Once again we get the split-off states (B-AB peaks) similar to that obtained using the
BWA. There appears to be a critical value of the coverage beyond which the split-off
states between B-AB appear. The splitting is found to increase with the increase in 9. It
indicates an increase in bond strength with the coverage. At the same time, the depth
between the B-AB peaks also gets enhanced, thereby implying an increase in the bond
104
Pramana - J. Phys., Vol. 46, No. 2, February 1996
1.00 -
o
o
0-50 -
0-00
-1-00
0-00
CO -
1-00
Figure 2. Comparison of DOS on chemisorption using curve (a) Bragg- Williams
approximation and curves (b) and (c) quasi-chemical approximation, with two
different values of interaction energy term W/KT = 2 and 8, respectively for low
coverage, 9 = 0-2.
1-50
1-00
o
Q
0-50
0-00
-1-00
0-00
- CO -
1.00
Figure 3. Coverage stimulated non-magnetic case ([///? small) DOS vs u> for
different coverages, represented respectively, by curves (a) 8 0-001, (b) 9 = 0-2 and
(c) 9 = 0-8, keeping the interaction energy term fixed (- W/KT= 2) in QCA.
_l Af 1VT_
1.50
1.00 -
o
o
-1-00
o.oo
1-00
Figure 4. Coverage stimulated non-magnetic case (U/{$ small) DOS vs a) for
different values of interaction energy term (W/KT} = 2, 6, and 10 plotted respective-
ly by curves (a), (b) and (c), keeping coverage fixed, 9 = 0-8 in QCA.
strength between the adatom and the substrate, consistent with the results obtained
within other approaches [13, 14] and that of Gavrilenko [6]. Furthermore., an increase
in the heights of B-AB peaks with 6 depicts an increase in the number of B-AB states
and is found to be consistent with the findings of Brenig and Schonhammer [15].
4.3 Effect of interaction energy on bond strength and number of B-AB states
Both the bond strength and the number of B-AB states are found to decrease with
increase in the interaction between adatoms as shown in figure 4. The typical curves
DOS vs co corresponding to different values of interaction energy term ( W/KT] = 2,
6 and 10 are plotted as (a), (b) and (c) respectively, at a fixed converage 9 = 0-8. These
curves clearly show that with the increase in the interaction between adatoms, the
distance between peaks is decreasing, indicating the decrease in the bond strength
between adatom and substrate. The depth between B-AB peaks is also found to
decrease. This implies the weakening of binding of adatoms with surface.
The magnetic case (U/fi large) can also be treated similarly.
References
[1] E Taranko, R Taranko, R Cardena and V K Fedyanin, Vacuum 45, 307 (1994)
[2] R Cardena, preprint ICTP Dec. (1989)
[3] D M Newns, Phys. Rev. 178, 1123 (1969)
[4] J P Muscat and D M Newns, Prog. Surf. Sci. 9, 1 (1978)
[7] T L Einstein, J A Hertz and J R Schrieffer, in Theory of chemisorption edited by J R Smith
(Springer, Berlin, 1980)
[8] R Fowler and E A Guggenheim, Statistical thermodynamics (Cambridge University Press,
Cambridge, 1986)
[9] R Guleria, P K Ahluwalia and K C Sharma, Phys. Status Solidi. B181, 397 (1994)
[10] G S Rushbrooke, Introduction to statistical mechanics (Oxford Univ. Press, London, 1949)
[11] P W Anderson, Phys. Rev. 124, 41 (1961)
[12] F C Tompkins, Chemisorption of gases on metals (Academic Press Inc., London, 1978)
[13] H Ishida and K Terakura, Phys. Rev. B36, 4510 (1987)
[14] K Masuda, Phys. Status Solidi B87, 739 (1978)
[15] W Brenig and K Schonhammer, Z. Phys. 267, 201 (1974)
PR AM AN A (0 Printed in India Vol. 46, No. 2,
journal of February 1996
physics pp. 109-126
Transient and thermally stimulated depolarization currents in
pure and iodine doped polyvinyl formal (PVF) films
P K KHARE
Department of Postgraduate Studies and Research in Physics, Rani Durgavati Vish-
wavidyalaya, Jabalpur 482001, India
MS received 18 May 1995; revised 26 October 1995
Abstract. Transient currents, measured with pure and iodine doped polyvinyl formal (PVF)
films as a function of poling field (1 5-100kV/cm) and temperature (30-95C), have been found to
follow Curie-von Schweidler law characterized with two slopes in short and long time regions.
The isochronals (i.e. current/temperature plots at constant times) have been found to give rise to
a peak located at 75 r C. The order of current has been found to increase with increase in poling
field, temperature and iodine mixing. The comparative studies of the isochronals with the
thermally stimulated discharge current. (TSDC) indicated the strong resemblance between the
two studies. It is suggested that both the dipolar orientation due to molecular mechanism of
motions with the side chains and space charge due to trapping of injected charge carriers in
energetically distributed traps may be responsible for the observed currents. The dependence of
current and activation energy on iodine mixing is explained on the basis of a charge transfer type
of interaction.
Keywords. Transient and thermally stimulated depolarization currents; dipolar orientation;
space charge relaxation mechanism; charge transfer complexes.
PACS Nos 77-30; 81 -20; 81 -60
1. Introduction
In recent years the thermally stimulated discharge current (TSDC) technique has been
evolved as a powerful tool to advance our understanding about the molecular
relaxation mechanism, trapping parameters and charge storage behaviour of insula-
tors including polymers which find very wide industrial application [1]. Polymers
contain a large- number of structural disorders and, therefore, discrete trap levels in
their bulk. Various reports on TSDC behaviour of polymers and different relaxation
processes contributing to the observed peaks in the corresponding thermograms are
available [1-5]. However, the role of various polarization processes and their relative
contribution to the electret state of the polymer is not yet fully understood. Particular-
ly, the space charge structure (including the trap distribution of energy and also over
the volume of the polymer) are still to be well understood. Such informations are also
being obtained by carrying out the measurements of absorption and short circuit
P K Khare
of TSDC measurement, as it determine the discharging current at constant tempera-
tures instead of varying temperatures. The d.c. step response technique has been
considered to be an attractive alternative for analysing the origin of various peaks
appearing in TSDC thermograms.
Transient currents observed upon the application or removal of a step voltage have
been studied extensively [6-11] to give an insight into the polarization processes in
these materials. A systematic analysis of transient currents has indicated how a combi-
nation of time, temperature and field dependence can lead to a fairly unambiguous
conclusion as to injection mechanism and the amount of trapping taking place. It is
generally accepted that the transient currents in an insulating material, on the
application or removal of a step voltage, may be attributed to one or more of the
following mechanisms [12-19]; (i) electrode polarization, (ii) dipole orientation; (iii)
charge storage leading to trapped space-charge effects; (iv) tunnelling of charge from
the electrodes to empty traps; (v) hopping of charge carriers through localized states. It
has been established that the observed time dependence in isolation does not permit
any discrimination to be made between various mechanisms. The argument for and
against a particular mechanism is to be found by considering the variation of transient
currents with various experimental parameters also, such as temperatures, field and
frequency, etc. Recently [20-23] we have attempted to identify the nature of the
transient currents in pure and polyblend films by comparing the observed dependence
on parameters such as electric field, electrode materials, sample thickness, temperature,
time and establishes relation between the charge and discharge currents, with the
respective characteristic features of the above mentioned mechanisms.
Generally, one does not find all requisite properties in a particular insulating
material. This has motivated various researchers to develop mixed systems for obtain-
ing desired properties. It has been shown [24-27] that carrier mobility can be greatly
affected by impregnating the polymer with suitable dopants. Considerable attention
has been devoted to the problems of the change in the electrical conduction in polymers
due to intentional doping with low molecular weight compounds [28-30].
Poly vinyl formal (PVF) is a thermally stable and weakly polar polymer that exhibits
excellent chemical resistance and good mechanical properties [31]. Inspite of its
activeness for many applications, the conduction mechanism is presently not well
understood [32,33]. It has a non-polar chain with weakly polar group CO rigidly
attached to the main chain [34]. Iodine is a linear molecule having an atomic radius of
1-35 A and therefore, being of very small size, it can diffuse vigorously in the polymer
structure.
The present paper reports the results of simultaneous studies of absorption and
discharging currents and thermally stimulated discharge of pure and iodine doped
PVF thermoelectrets, under field and temperature conditions. Explanations of the
observations made are given on the basis of available theories.
2. Experimental details
5%, by weight) was mixed with it. The solution was continuously stirred for 30min
means of a teflon-coated magnetic stirrer. Thereafter, it was stirred and heated to
'C to yield a homogeneous solution. The glass beaker containing the polymer
ution was then immersed in a constant temperature oil bath. Ultrasonically cleaned,
:uum metallized microscopic glass slides were immersed vertically into the solution
a period of about 20min. After deposition of the film, glass slide was taken out and
ed in an oven at 60C for 24 h. This was followed by room temperature outgassing at
5-33 x 1 " 5 N/m 2 for a further period of 24 h. The upper electrode was also vacuum
Dosited. The thickness of the samples was of the order of % 25 /mi, which was
imated by measuring the capacitance of the fabricated sandwiches taking the value
iielectric constant e of PVF [34] as 3-7. Samples of different thicknesses (5, 10, 25 and
^m) used to study thickness variation were obtained by changing the concentration
polymer solution.
^n the present study the samples were thermally poled with polarization fields of 15
lOOkV/cm, at different temperatures (30-95C) for ISOmin during which the
nsient current is charging mode has been observed 2 min after the application of
d. The current was also observed in discharging mode 2 min after the removal of the
d for the same period of time. Different steps for the preparation of a thermoelectret
as follows: (i) the sample is heated to the desired polarizing temperatures (T p ); it is
)t at T p for sometime (in the present case, 1-5 h) to reach thermal equilibrium; (iii)
n, an electric field (E p ) is applied at T p and kept on for a period of polarizing time (t p );
it is cooled slowly under the field application, to room temperature. The field is then
loved. For TSDC measurements the samples after being polarized for 90 min, with
ing field 20 kV/cm at temperature 65C, was cooled to room temperature under the
3lication of field. The samples were polarized in different fields (ranging from 10 to
)kV/cm) and temperatures (50 to 100C). The representative results for samples
ed at 65C with 20kV/cm are reported here. The polarization was carried out by
meeting a d.c. power supply (EC-H V 4800 D) in series with the sample and with
^eithley 600 B electrometer (for measuring the current), which was carefully shielded
i grounded to avoid ground loops and extraneous electrical noise. The sample
paration, vacuum deposition of electrodes, effective electrode area and geometry,
preconditioning of the samples and the measurement procedure for transient and
DC in this work were exactly the same as reported in the earlier work [20-23, 35, 36].
Results and discussion
^ various results summarized in the paper may help to distinguish between various
icesses. The polarization of polymeric materials may be due to dipolar orientation,
.ce charge formation, trapping in the bulk, tunneling of charges from the electrodes
;mpty traps, or hopping of charge carriers from one localized state to another. The
st probable mechanism responsible for the discharging current in the dielectric can
)rinciple be determined by considering the variation of such currents with tempera-
e, time, field, etc.
n fact, the charging and discharging currents observed in most of the polymers
estigated show the expected behaviour of a dipolar relaxation mechanism in all
compatiDie wim sucn a aipoiar process since me relaxation pnenomena in poiymers are
generally characterized by a distribution in relaxation times leading in the usual
formation of Cole and Cole [37] to a t~" transient current.
So far as the nature of the trapping sites are concerned, there is, and has been much
speculation on this topic, but, at present, it is generally conceded that charge trapping is
primarily due to the basic polymer structure which can be of various traps. This, one
can envisage physical traps in cavities due to defects and free volume inherent in the
bulk polymer structure, the binding energy resulting from a polarization of the
surrounding molecules. Another type of trap or hopping site may be due to chemical
heterogeneities in the polymer structure, such as C=C double bonds, oxidation faults
etc. It seems reasonable to suggest that the C^O groups may represent at least a major
fraction of the localized centres by which the transport of the injected electrons takes
place. The above consideration indicate that the observed discharging current in the
present case may partially be due to dipole orientation and partially due to space
charge mechanism.
The current in the time domain for the short time region was characterized by the
relation
/(t)ocr"; 0<n<l; tl/w p .
i.e. the frequencies which are larger than the loss peak frequency w p , and for the long
time region
/(fleer 1 -"; 0<n<l; tl>l/w p .
with logarithmic slope steeper than unity. The two power laws determine the time
domain response of dipolar system in which a loss peak is seen in the frequency domain.
Similar behaviour is observed in carrier dominated systems, however, low frequency
dispersion below a frequency vv c , which corresponds to long time region is described by
the above power laws with small value of n.
Let n = 1 p, with p close to unity for low frequency dispersion regime. The long
time response of charge carrier system will then be denoted by
/(flocr 1 -"; p = l; tl/w p .
which corresponds to a very slow time varying current.
The complete representation of the universal dielectric response in the time domain,
covering both dipolar loss peaks and strong low frequency dispersion associated with
the charge carrier dominated system may be represented by
/(flocr"; 0<n<2
with the exponent n taking values in different ranges at long and short time respec-
tively [38].
The time dependence of the transient charging and discharging current in pure and
iodine doped poly vinyl formal (PVF) films was investigated over a period of l-180min
in the temperature range 30-95C. Figure 1 shows the discharge current transients for
1 1 2 Pramana - J. Phys., Vol. 46, No. 2, February 1996
Transient and thermally stimulated currents
6 8 W 2 2xlO :
TIME (min)
Figure 2a.
PURE PVF
IODINE DOPED PVF
6 8 10' 2
TIME (mm.)
8 1Q2
2x102
pure and iodine doped PVF within this temperature range with a constant field of
20kV/cm, while plots of charging and discharging current for various poling fields (15
to 100 kV/cm) at 40C has been shown in figure 2. It may be observed from these data
114
Pramana - J. Phvs.. Vol. 4* NA r i7*Kran, 100*
-11
10
PURE PVF
IODINE DOPED PVF
1* 6 8 101 2 **
TIME (mm.)
6 B
2X10'
Figure 2(a and b). Time dependence of charging (o) and discharging () currents in
pure and iodine-doped polyvinyl formal at 40C for various charging fields.
with temperature. There appears to be a process of thermal activation over the whole
range of temperature. It is evident from figures 1 and 2 that both charging and
discharging current obey the well-known expression [7].
(i)
where 7 a is the absorption current, t the time after application or removal of the external
field and A(T) a temperature-dependent factor. It is found that discharging current has
been characterized with logarithmic slope smaller in magnitude than l(n < 1) during
the range of short times, and then goes to the longer time region (where the slope is
steeper, with n lying between 1 and 2). Similar to discharging, the charging current has
also been found to be characterized with n< 1, in short time region, however at longer
times this current tends to approach the steady state conduction current.
In the present case n values for shorter time region were observed to vary from
0-5-0-8 and for long time region these values are observed to vary from 1-03-1-78.
Also, discharging current vary linearly with the field strength which is a characteristic
of dipolar mechanism. These findings indicate that the dipolar polarization is operative
in the present case. The dipolar polarization is further supported if the polar nature
of PVF is considered. The structure of PVF is such that it is essentially a weakly
polar polymer having a small dipole moment. In PVF, CO group is rigidly attached
to the main chain. The nature of current in the observed temperature range may
thus be attributed to a dipolar process involving structural units with a small dipole
moment and a broad distribution of relaxation times, this predominates over any
hopping mechanism. The partial dipolar nature of sample is expected to manifest
itself in the form of a peak in the isochronals. The isochronals constructed from
current-time characteristics are found to be characterized with a peak located at 75C
(figure 4). It is expected that the current peak observed in the T g range of PVF,
corresponds to dipolar orientations due to molecular motions associated with the
side chains.
More direct confirmation of the dipolar hypothesis may be found, correlating the
temperature dependence of the transient currents with thermally stimulated depolariz-
ation currents for which and essentially dipolar origin is generally accepted.
Although the dielectric response is commonly associated with orientation of perma-
nent dipoles, it is undesirable that hopping charges of either electronic or ionic nature
may give rise to a very similar dielectric behaviour. The important distinction lies in the
degree of localization of these carriers [11]. An electron or an ion confined to hopping
between two preferred positions is indistinguishable from a dipole, while a distinctly
different situation arises where the carrier is free to execute hops over finite paths, some
of which may eventually extend all the way from one electrode to the other [11]. We
have to consider four components of the current in such a system; (a) the current
controlled by various polarization mechanisms; (b) the current controlled by the
charging of the capacitor through a resistor R; (c) the conduction current, which is time
dependent. The former components gradually fall off to zero within a hundredth of
a second. The third component is due to formation of space charge. The residual former
current is referred to as bulk current, which may be ionic, electronic or both [1,20].
Struik [39] showed that solid like polymers are not in thermodynamic equilibrium at
tropy are greater than they would be in equilibrium state. The gradual approach to
uilibrium affects many properties, for example the free volume of the polymer may be
creased. The decrease in free volume lowers the mobility of chain segments and also
arge carriers. The decrease in mobility may be expected to reduce DC conductivity,
higher electric fields the change in mobility may take place faster than at lower fields
d recombination of charge carrier may be more.
The Curie-Von Schweidler type of time dependence has been observed for many
lymers with the index n close to unity. A number of mechanisms may be used to
plain such time dependence. It is, therefore, not possible to specify the origin of
insient currents from the analysis of time dependence alone. At temperatures much
ver than the glass transition temperature and for the low to moderate fields used,
/eral of the concepts previously postulated to account for the transient conduction
enomenon can be ruled out on the basis of the experimental facts.
The electrode polarization predicts the strong dependence of the electrode material
the decay of the transient currents [42]. Moreover, uniform and electrode polariz-
ons require the charging and discharging currents at a particular instant to vary
early with charging field [43-45]. Furthermore, the superposition principle, accord-
; to which the charging and discharging currents should be equal but opposite at
uivalent instants is supposed to be valid in such type of polarization [43,44].
>wever, in the case of space charge polarization the superposition principle is not
eyed and the charging and discharging currents depend more strongly than linearly
the applied field [22].
The convincing criterion of the validity of the superposition principle can be
:>vided by the discharging and charging currents ratio of various times, where the
arging current value obtained by subtracting the steady state component [22].
value of unity throughout the transients would indicate the origin of the transients
e to uniform or electrode polarization. In the present case the charging current
ntinued to decay although slowly, even at the end of charging process. Under such
cumstances, accurate estimation of steady state current was not possible and hence
: reliable evaluation of discharging/charging ratio could not be made. However, the
irging and discharging at various times after the application or termination of
irging field are found to follow the power law dependence on field. Log-log plots of
irging and discharging currents and fields at different times (figure 2) show that both
irging and discharging currents reasonably follow a relationship V m , voltage
Dendence with the power m lying between 1 and 2. The log I d versus log fields plots for
re and iodine doped PVF at five different times (ISOmin, SOmin, 20min, 8 min and
lin) show that the I d versus V m relationship, with 1 > m > 2, is reasonably obeyed
;ure 3). The divergence from Ohm's law and failure of the superposition principle
licates the space charge formation. Thus, the observed currents can be best described
terms of space charge mechanism [45].
in fact, the d.c. step response measurements, in which the current response is
asured as a function of time after a d.c. voltage is applied to or removed from the
nple, are quite similar to TSDC measurements except for the temperature being
istant. However, when the measurements are made at various temperatures, it is
ssible to collect the d.c. step data at a specific time and plot them as a function of
10
,-7
8
6
16"
8
6
2
01
DC.
ac
o
ce
<c
in
o
A =180 min
B = 80 min.
C = 20 mm.
0=8 min
E= 2 mm
10 20 30 50 70 10 2
FIELD (kV/cm)
Transient and thermally stimulated currents
mperature. In various cases, such collection of d.c. step data has been found to resolve
e different relaxation peaks [46-48].
The observed divergence from Ohm's law at moderate high electric fields (figure 2)
id the thermal activation of discharge current at various prescribed times indicates
e space charge formation. Polyvinyl formal is known to be a weakly polar. The
arging and discharging currents observed in PVF are also expected to show the
haviour of dipolar relaxation mechanism. The linear region in charging current
rsus poling field curves (results not shown) at lower fields seems to be in favour of the
ternal dipolar polarization. However, the possibility of weak carrier injection at such
Id values cannot be ruled out.
In the case of transients governed by space charge, the peak in the current time curve
ould occur at a time
0-786 a
icre F is the applied field, a is the sample thickness and /j. the carrier mobility. To have
rough estimate of the time at which this peak should occur we used the values of a,
and fj. to be 25jum, lOOkV/cm and 10~ u cm 2 /V. It was found that t m , should
iproximately be equal to 3-782 x 10 3 s. Thus, there is possibility of space charge
laxation occurring at sufficiently longer times. The above considerations indicate
at the observed currents in the present case may partially be due to dipolar
ientation and partially due to space charge mechanism.
The thermal dependence of the discharging current for pure and iodine doped PVF
mples is seen clearly when the current (measured at constant time, i.e. isochronal) is
Dtted against temperature.
Discharging current measured at various prescribed times (i.e. 180, 80, 20, 8 and
nin) versus temperature plots are shown in figure 4. It is clearly seen that the various
>chronals are characterized by a single peak located at 75C. However, no shift is
iserved in the peak temperature with time of observation. It is observed that the peak
nperature decreases with increasing time and is a characteristic of relaxation process.
nilar qualitative behaviour is observed in the other sample. The isochronal peak is
oad and probably it contains several minor processes, one of which may be
sociated with the glass transition of the polymer and the other may be due to thermal
ease of trapped carriers. The broadness of the peak may be explained by assuming
distributed or multiple dielectric relaxation which may be due to distribution in
tivation energy when the rotation of the dipoles does not proceed in the same
vironment. Alternatively, it may be due to distribution in relaxation time, when the
tational masses of the dipoles are not equal. The broadness of the peak in the present
se is, however, most likely to be distribution in relaxation time, because the peak is
curring near T g of the polymer [34], where the side groups move in unison with the
dn chain differing in masses [40,41]. Because the distribution of dipoles in the
lorphous phase is most likely to be random, it is expected therefore, this is a complex
~x-.on ;,-.,,^U, I.-,,-, U/->tli +V./ /lip+fiKiii-i/Mi nf tVo artii7ati/Yn f*nt*rn\r or\A r*1 <a Y Q ti rm time* It
P K Khare
10
A = 180 min.
8 = 80 min.
C = 20 min.
0=8 min.
E = 2 min.
50
60
70
80
90
100
TEMPERATURE ( C)
Figure 4. Temperature dependence of discharging currents (isochronals) at vari-
ous discharge times for pure and iodine doped polyvinyl formal samples: Curves A,
itv ROmiri' 90rrnrv 8 min and
Transient and thermally stimulated currents
lues of n, and thermal activation of current over a certain temperature range as
served in the present case indicate that the space charge due to accumulation of
arge carriers near the electrodes and trapping in the bulk may be supposed to
;ount for the observed current. The decay of current in the long time region for
Ferent samples indicates the existence of energetically distributed localized trap levels
the sample [20,21]. It seems that at shorter times only shallow traps get emptied,
ntributing to stronger current. However, at longer times deeper traps with long
trapping times release their charges and the current decays at longer times. As
nperature increases mobility of carriers also increases, hence all the deeper traps are
ed. Release of a large number of charge carriers from the traps during the process
ty then result in high return rate of carriers leading to blocking of electrodes causing
lecrease in current. The charge injection from electrodes with subsequent trapping of
ected charges in near surface region gives rise to homospace charge and the thermal
ease of charge carriers from the traps. Before the trapped space charge injected at
'her fields is thermally released, a space charge barrier is presented to the electrode
tich suppresses the entrance of charge carriers into the sample. Thus, the observed
rrent remain smaller than its corresponding value.
The bulk of the measurements of the transient currents in pure and iodine doped
/F samples were made with 25 fim. A limited number of measurements, however,
;re made with samples of 5, 10 and 40 /an thickness over a temperature range with
iminium, silver, copper, lead and tin electrodes (results not shown) and it is found
it there is no evidence of thickness and electrode dependence. The observed
nperature dependence (i.e. thermal activation) and the absence of any significant
sets of electrode materials and sample thickness, makes the tunnelling unlikely as
>ossible mechanism to explain the nature of transient currents. Also, the observed
tivation energy values do not show any regular variation with time of observation so
s difficult to understand the observed currents in terms of hopping mechanism as
sorted by Lewis [16].
The isochronals discharge currents I d , and TSD currents /,, can be shown to be equal
functions of temperature and a TSDC thermogram can be compared with an
ichronal I d versus T plot, if l d is considered at a constant equivalent time, t e , value
itered at the temperature written, as [45, 47].
UT)*I d {t e (T m \T}. (2)
The iodine doped PVF samples were charged in different fields (10 to 100 kV/cm) at
Ferent temperatures (50-100C). The representative results for samples charged at
3 C with 20 kV/cm has been reported here (figure 5). The TSDC thermogram shows
ingle peak centered at 80C. The activation energy, E, obtained from initial rise of
IDC with temperature (figure 6) has been found to be 0-43 eV. This peak may be
her due to dipolar origin or migration of charge carriers through microscopic
itance with trapping. Since, in polyvinyl formal, the dipolar group is rigidly attached
the main chain, hence, the observed peak may not be completely due to alignment of
inlfc Tn flHHitinn tn thp rlinnlar r.nntrihntion there must he some contribution from
TEMPERATURE ( C)
Figure 5. Short-circuit TSDC thermogram of (0-5%) iodine-doped poly vinyl
formal thermoelectrets polarized at 65C with 20kV/cm.
of the same order as that of the energy required for molecular motion from one
equilibrium, to another in high molecular weight compounds [49]. This value of E is
also small compared to 1 eV typically required for the movements of ions [50], hence
the simultaneous action of the dipolar orientation process due to the alignment of
dipolar molecules attached to the main polymer chain along with the migration of
electrons/holes released from the valence band through microscopic distances with
subsequent trapping [33].
Activation energy () values obtained from plots of the current versus 10 3 /T at
various prescribed times (180, 80, 20, 8 and 2min) for pure and iodine doped PVF
samples are shown in figure 6. The E values for such a peak are found to vary from 0-37
to 049 eV for PVF samples, while for doped samples it varies from 0-28 to 0-35 eV. The
activation energy are found to decrease with time of observation and for doped
-8
B = 80 mm_
C = 20 mm.
D - 8
E = 2 min.
o
z
o
C
XI
3O
m
z
-I
>
3-3
Figure 6. Initial rise plots (i.e. current versus 10 3 /T) for TSDC and isochronal
peaks.
P X Khare
samples. The value of E agrees well with the activation energy of 0-43 eV obtained for
TSDC peak at 80C in the present case and is also reasonably compared with the
activation energy value reported in the literature [33].
The polymer films are known to be a mixture of amorphous and crystalline regions.
The presence of localized states may lead to the localization of injected charge carriers
giving rise to the accumulation of trapped space charge [51]. The hopping mechanism,
is considered to lead to the increase in activation energy. However, in the present case,
the activation energy is observed to decrease with increase in time. Such behaviour
suggests that hopping of charge carriers is not expected in the present case.
As PVF is a weakly polar polymer, the probability that charge carriers are present in
it, the only charges present under a field are those injected through the electrodes. The
injected charges are trapped at different trapping sites leading to a space charge which
fundamentally influences all the transport phenomena and the effects at the electrode
[31]. It has been found that doping of iodine enhances the current and lowers the
activation energy of the charge carriers.
Iodine when doped in polymers, may reside at various sites [52-56]. It may go
substitutionally into the polymer chain or reside at the amorphous/crystalline bound-
aries and diffuse preferentially through the amorphous regions and form charge-
transfer complexes (CTC) or it may exist in the form of molecular aggregates [57]. In
the present case, the iodine may be held between the polymer chains by weak
electrostatic forces between iodine and hydrogen atoms and hence the decrease in
activation energy may be attributed to the increase in crystallinity of the polymer
matrix due to the alignment of entangled chains in the amorphous regions as a result of
electrostatic interaction between iodine atoms and molecular chains. The CTC, if
formed, will provide conducting pathways through the amorphous regions of the
polymer and would result in the enhancement of its conductivity [58]. The observation
that the current in PVF films does in fact increase with the iodine doping supports the
formation of CTC. Furthermore, the presence of CTC causes a reduction in the barrier
at the amorphous/ crystalline interfaces of the polymer and the observation that the
activation energy of the charge carriers responsible for conduction decreases with the
iodine doping confirms the formation of CTC.
The formation of charge transfer complexes can be inferred from the appearance of
broad and intense absorption bands in the UV visible region of the spectrum. The
maximum of absorption peak for pure PVP film occurred at the wavelength of 220 nm.
The absorption peak for iodine doped PVF was broadened when compared to the
absorption peak of the pure PVF. Iodine doping created extra absorption peaks at 276
and 342 nm (results not shown). This agrees with the earlier findings [58].
4. Conclusion
Considering the effects of various parameters on the transient discharging currents in
pure and iodine doped PVF samples, we conclude that the time dependent polarization
is due to, simultaneouslv. dinolar reorientation due to
e significant help provided by Mr A K Khare, Jabalpur is gratefully acknowledged.
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lobile interstitial mode! and mobile electron model of
lechano-induced luminescence in coloured alkali halide crystals
P CHANDRA, SEEM A SINGH, BHARTI OJHA and R G SHRIVASTAVA+
epartment of Postgraduate Studies and Research in Physics, Rani Durgavati University,
.balpur 482 001, India
Department of Physics, Government Engineering College, Jabalpur 482001, India
;S received 7 April 1993; revised 4 September 1995
bstract. A theoretical study is made on the mobile interstitial and mobile electron models of
echano-induced luminescence in coloured alkali halide crystals. Equations derived indicate
at the mechanoluminescence intensity should depend on several factors like strain rate, applied
ress, temperature, density of F-centres and volume of crystal. The equations also involve the
ficiency and decay time of mechanoluminescence. Results of mobile interstitial and mobile
sctron models are compared with the experimental observations, which indicated that the
tter is more suitable as compared to the former. From the temperature dependence of ML, the
lergy gaps between the dislocation band and ground state of F-centre is calculated which are
08, 0-072 and 0-09 eV for KC1, KBr and NaCl crystals, respectively. The theory predicts that the
;cay of ML intensity is related to the process of stress relaxation in crystals.
eywords. Mechanoluminescence; triboluminescence; colour centres; dislocations; alkali halides.
ACS No. 78-60
Introduction
or y-irradiated alkali halide crystals exhibit intense mechanoluminescence (ML), i.e.
ght is emitted during their mechanical deformation. Involvements of mobile disloca-
ons and F-centres in the ML emission are indicated by several experimental facts like
spendence of ML intensity on the density of F-centres, mechanical bleaching of
-centres, dependence of ML intensity on the number of newly created dislocations,
isappearance of ML immediately after interruption of deformation, etc [1-13].
Several dislocation models proposed for the ML excitation in coloured alkali halide
rystals are dislocation unpinning model, dislocation annihilation model, dislocation
efect stripping model and dislocation interaction model [4]. According to dislocation
npinning model, when a dislocation in X or y-irradiated alkali halide crystal is
npinned by applying an external stress, then the pinning point like F k centre [14] may
e released whose subsequent recombination with F-centre may give rise to the light
nission. According to the dislocation annihilation model, a high local temperature
lay be produced during the annihilation of dislocations of opposite sign, which may
luse the diffusion of the trapped interstitial atoms to the colour centres or it may
irectly ionize the colour centres. According to dislocation defect stripping model,
le moving dislocations may strip interstitial halide atoms which may recombine
10*7
radiatively with the F-centres with the creation of normal ions at the normal sites.
According to dislocation interaction model, the moving dislocations may interact
electrostatically or mechanically with the colour centres and the electrons released
from F-centre during the interaction may subsequently recombine with the holes and
give rise to luminescence.
The simultaneous measurements of stress-strain and ML-strain curves of X-irra-
diated alkali halide crystals show that the ML peaks lag considerably with the onset of
more rapid plastic flow [1]. These results indicate that the ML does not occur during
the unpinning of dislocations, but occurs during the movement of dislocations in the
crystal. Moreover, the ML in coloured alkali halide crystals is observed during elastic
deformation of the crystals and during release of pressure where unpinning of the
dislocations is not possible. These facts indicate that the dislocation unpinning may not
be the dominating source for the ML excitation.
The dislocation annihilation model does not seem to be applicable because of the
following reason. The extent of heating during the annihilation of dislocations is
usually very small. The upper limit for the increase in the temperature during
annihilation may be given by T a = QJ(ji^pC\ where Q d is the energy which is released
during the annihilation of a unit length of a dislocation, A ph is the free path length of
a phonon, p is the density of the crystal and C is the specific heat capacity. The elastic
strain energy per atom length of an edge dislocation is given by the equation [15-17].
E Gb 3 /[4n(\ v)] In (R/r ), where b is the Burger's vector, G is the shear modulus, v is
the Poisson's ratio, R and r are the upper and lower limits of the separation of
two edge dislocations, respectively. Thus, the annihilation energy per atomic length,
for two edge dislocations is Q d = Gb 3 /[2n(l - v)ln(/?/r ). For LiF, G = 2-89 x 10 1 1 dyn
cm" 2 , b = 2-01 x 10~ 8 cm, v = 032, r = 5 x 10~ 8 cm and R = 10 ~ 3 cm, therefore, Q d
comes out to be 3-39 eV. For LiF, p = 2-64gcm~ 3 , C = 0-39 calgm~ Meg" 1 and
A ph ^ 10" 6 cm at room temperature [18], therefore, T a comes out to be 2-29C, which is
very small. Hence, the annihilation of dislocations is not capable of bringing about
a thermal flash in the luminescence of coloured alkali halide crystals. However, the
dislocation annihilation model may be realized at very low temperature. It has been
shown [19, 20] that the liquid helium temperatures are conductive to an enhancement
in the effect of an increase in the temperature on the slipping bands of alkali halide
crystals. As the temperature is lowered, the work of plastic deformation increases on
account of the increase in the yield point, the thermal conductivity of the crystal
becomes worse and its heat capacity decreases in proportion to the third power of the
absolute temperature in accordance with Debye's law. If the temperature of the crystal
prior to deformation was equal to 4-2 K then, according to the calculation [20], the
temperature in the stripping bands is increased by 5 to 50 K. The liberation of hole
centres from traps [21] and the excitation of luminescence during the recombination of
mobile holes with F-centres are possible at the upper limit of these temperatures.
To date there is no satisfactory analysis related to the suitability of dislocation defect
stripping model and the dislocation interaction model. The dislocation defect stripping
model shows that the mobile interstitial atoms produced during deformation of the
crystal, are responsible for the light emission. However, the dislocation interaction
model shows that the mobile electrons produced during deformation of crvstals are
_
model) and the mobile electron model (dislocation interaction model) and their
suitability is analysed by comparing the theoretical results with the experimental
observations.
2. Mobile interstitial model for the ML in coloured alkali halide crystals
Suppose a crystal contains N d dislocations of unit length per unit volume. When
a dislocation of unit length moves through a distance dx, then the number of interstitial
atoms interacting with the dislocation is r { dx JV i5 where r { is the radius of interaction of
the dislocations with interstitials and N, is the density of the interstitial atoms (hole
centres) in the crystals.
If p { is the probability of the sweeping of activated interstitial atoms with the
dislocations, then the number of interstitial atoms swept out by AT d dislocations is given
by
dN is = p,JV d r,JV,dx. (1)
As the diffusion of atoms takes place only from the compression region above the
dislocation line and not from the expansion region below the dislocation line [22], the
factor 2 has not been included in the above equation.
If a dislocation moves the distance dx in time dt, then the rate of generation g of the
interstitial atoms being swept out by moving dislocations may be given by
dx
^ = p i N d r i N i
or
^ = p i N d r i N i w d (2)
where u d is the average velocity of the dislocations.
Equation (2) may be written as
where, e = N d b u d , is the strain rate of the crystal and b is the Burgers vector [23, 15].
After room temperature irradiation, the defects incorporated in alkali halide crystals
are F-centres (their aggregates) and the clusters of interstitial halogen atoms. Thus, in
an irradiated alkali halide crystal, the interstitial is a hole centre i.e. electron deficient
centre like X or X 2 , where X is a halogen atom. According to the dislocation defect
stripping model, the moving dislocation may release interstitial halogen atoms from
clusters of various sizes and their subsequent recombination with F-centres may cause
light emission with normal ions created at normal sites.
Now, the rate equation for the change in the number of interstitial halogen atoms
being swept out by moving dislocations may be written as
d is _
g ff r n F v d n- ts
"-
where n is is the number of interstitial halide atoms being swept out by moving
dislocations at any time t, n F is the density of recombination centres, i.e. F-centres, o r is
the capture cross-section of these centres and r = l/o- r n F u d , is the lifetime of activated
interstitial atoms. Here, the velocity of activated interstitial atoms has been taken to be
equal to the velocity of dislocation because, in the interacting region they will be swept
out with the velocity of dislocation [22, 24]. In this case, the recombination of activated
interstitials with deep hole traps and the retrapping of interstitials have been neglected
because the density of deep hole traps may be very small and the retrapping involves
considerably higher activation energy.
Integrating (4), we have
Taking n [s = at t 0, C 1 comes out to be log(g) and we get
or
Thus, the ML intensity due to the recombination of interstitial halide atoms with the
F-centres may be given by
/ = r] x rate of recombination
or
where r\ is the probability of radiative recombination of halide atoms with F-centres.
Substituting the value of n is from (5), we get
ff r n F d t)]. (6)
In a crystal of volume V, there will be N A V dislocations. Thus the ML intensity may
be given by
p-r-N-eV
1 = n ' V [1 ~ exp( ~ ** n * v M' ( ? )
The above equation shows that the ML intensity / will initially increase with time
and then it will attain a saturation value I s for the longer duration of straining time. The
value of / s may be expressed by the equation
mobile dislocations in the crystal is of a suitable density and their velocity satisfy the
equation, e N d bv d . If the cross-head of the deforming machine is stopped, then the
stress in the crystal does not remain constant but decreases up to a certain extent with
time. The mobile dislocations do not stop immediately but the cross-head does and
continue to move, assisted by the thermal fluctuations. Thus, although the cross-head is
stationary, the plastic deformation increases. This is stress relaxation and it is allowed
to continue for a significant time [23]. The thermal fluctuations are able to assist the
mobile dislocations over all the short range obstacles and the stress in the crystal which
is equal to the applied stress, falls to the value of the long-range internal stress and
thereafter the barrier cannot be surmounted with the aid of thermal fluctuations.
Etching experiment suggests that there is no dislocation multiplication during the
process of relaxation [25].
Suppose a crystal is being deformed at a constant strain rate s, and then the cross-
head is stopped at a time t = r c , at which the ML intensity had attained a saturation
value / s . The experimental observations of Hagihara et al [10] show that the stress
in y-irradiated KC1 crystals decreases slowly from its value at t = t c to some lower
value. On the basis of this result, let us assume that the dislocation velocity decreases
exponentially from its value t; do at t = t c , and follows the relation v d = u do exp[ a(t t c )],
where v d is the dislocation velocity at any time (t tj and a is the rate constant.
Substituting the value of g from (2) and expressing u d = u do exp[ ct(t rj], eq. (4)
may be written as
dn.
a(t-t c )]. (9)
Integrating equation (9) and taking n is = n iso at t = t c , we get
^p2{exp[-a(t--f c )]-
(10)
As n iso = p^N^^N-Jff^ip, equation (10) indicates that dn is /dt = 0, i.e. the equilibrium
is still maintained where the rate of generation will be equal to the rate of recombina-
tion. Thus, the decay of ML intensity in a crystal of volume V may be given by
I = Wi^Vi-Nii'do 7exp[- <x(t - t c )]
or
T Wi^Nj V& ,,_
/= ' exp[-a(r-t c )]. (11)
o
After the completion of stress relaxation, if the accumulated interstitials are left, they
may diffuse slowly towards the F-centres. Thus, the ML emission having comparative-
y longer decay time may be observed after the completion of stress relaxation process.
Equation (7) shows that when a crystal is deformed at a fixed strain rate, initially the
ML intensity will increase with time and then it will attain a saturation value. When the
deformation is stopped, the ML intensity will decrease with the rate constant controlled
intensity should increase linearly with strain rate and volume of the crystal. Equation
(8) also shows that the ML intensity I s should increase with the density of interstitial
atoms or V-centres in the crystal. However, for higher values of the strain, the density of
V-centres (N { ) will decrease due to the deformation bleaching i.e. due to the electron-
hole recombinations, and therefore / s should decrease with the deformation of the
.crystal. As the strain rate increases with the applied stress [10], an increase in the ML
intensity with the applied stress is expected.
When the temperature of a crystal is increased, N { will decrease because of the
thermal bleaching and p } will increase because of the increased mobility of inters titials
[22], Thus, initially the ML intensity should increase with increasing temperature,
attain an optimum value and then it should decrease and disappear at higher
temperatures. As v\ and N { are different for different crystals, the ML intensity may be
different for different crystals.
3. Mobile electron model for the ML in coloured alkali halide crystals
During the plastic deformation, the dislocations only bend between pinning points.
When the stress exceeds the yield point, the dislocations are detached from the pinning
points, and move throughout the crystal. The dislocation D moving under the action of
external stresses, interact with F-centres and capture electrons. In the dislocation
energy band, an electron participates in two motions. It may travel along the
dislocation (because the dislocation band is one dimensional) and it can travel with the
dislocations [26]. If a dislocation containing electrons encounters a defect centre
containing holes, the electron may be captured by this centre and luminescence may
arise, in which the position of the peaks will be identical with the position of the
luminescence emission of the defect centre. From the comparison of ML spectra with
the spectra of other types of luminescence in coloured alkali halide crystals, it has been
proved that the ML arises due to the recombination of electrons from F-centres with
the V 2 centres [27]. Schematically, the ML process can be described by the following
equations
F + D->e d + [-] (A)
V 2 ( = X~ + X~ + X) + <?- 3X~ + D + hv (B)
where F and D represents F-centre and dislocation, respectively, e d is the dislocation
electron i.e. the electron captured by dislocation, [ ] is negative ion vacancy, X~ is
halogen ion and X is self-trapped hole.
Suppose a crystal contains N d dislocations of unit length per unit volume. When N d
dislocations move through a distance dx, then the area swept out by the dislocations is
N d dx. According to dislocation interaction model, the ML excitation in coloured alkali
halide crystals takes place due to the transfer of electrons from F-centre to dislocation
band where the recombination of dislocation electrons with hole containing centres
gives rise to luminescence. Near the edge dislocation, some of the F-centres lie in the
expansion region and some of them lie in the compression region. In the expansion
region, the energy gap between ground state of F-centre and dislocation band (lying
just above the F-centre level) decreases due to the decrease in local density of the
increases in the compression region of the dislocation due to the increase in the local
density of the crystal [28], As a matter of fact, there is a greater probability of the
transfer of electrons from the F-centres lying in the expansion region rather than from
the compression region of the edge dislocations. Therefore, the interaction volume may
be taken only along the expansion region of the crystals and consequently the volume
in which N d dislocations interact while moving through a distance dx may be given by
N d dxr F , where r F is the distance up to which a dislocation can interact with the
F-centres.
If F is the number of F-centres in unit volume, then the number of colour centres
interacting with the dislocations will be N d n F r F dx. If a dislocation moves the distance
dx in time dt, then the number of F-centres interacting per second with the dislocations
is given by
JV d n F r F (dx/df) = N d n F r F v d
where v d is the average velocity of dislocations.
During the interaction of moving dislocations with the F-centres, electrons are
excited from F-centre to the dislocation band. If p F is the probability of transfer of
electrons from F-centres to the dislocation band during the interaction, then the rate of
generation g l of the electrons in the dislocation band is given by
r F v d . (12)
As = N d bv A , g l may be expressed as
.
g- b (13)
When the dislocations containing electrons are moving in a crystal, then the
electrons may recombine with the defect centres containing holes, and also with the
deep traps present in the crystals. The retrap'ping of dislocation electrons in the
negative ion vacancies may also take place. Thus, the rate equation may be written as
-^ = g l - ff t N { v d d - ff l N l v d d - ff 2 N 2 v d d (14)
where n d is the number of electrons in the dislocation band at any time t. N-^N^ and N 2
are the densities of recombination centres, deep traps and negative ion vacancies
(without trapped electrons), respectively, and a r , cr i and a 2 are the capture cross-
sections of the recombination centres, deep traps and negative ion vacancies, respect-
ively. Here, the velocity of electrons has been taken as the velocity of dislocations
because the electrons are moving with dislocations.
Integrating equation (14), we get
where,
__ 1 _
^"KIVi + c-^ + ^N^ 9 (
is the lifetime of the electrons in the dislocation band and C 2 is a constant.
nearby F-centres. Subsequently the dislocation captured electrons may disaj
during their recombination with the holes being diffused towards the dislocation
The dislocation captured electrons may also disappear due to the electron
recombination during the movement of dislocation electrons along the disloc
lines [26]. Once the electrons from the F-centres lying within the interacting dis
are captured by a dislocation and subsequently annihilated, the stationary disloca
cannot capture electrons from other F-centres without change in temperatu
without their movement. Thus, for the crystal which is not under deformation, th<
of thermal generation of dislocation electrons may be negligible and we may as
n d ~ at t = 0. This gives C 2 = log(g'), and therefore, we get
If rj l is the probability of radiative recombination of electrons with hole conta
centres, then the ML intensity may be written as
or
/ = ^'ff r Nii> d fir l T d [l - exp(-
or
As the recombination entities are different in both the cases, n may not be equal
Since a crystal ~of volume V will contain N d V dislocations of unit length, the
intensity may be given by
Equation (18) shows that for a given strain rate, the ML intensity will ini
increase with time and then it will attain a saturation value / s for longer duration <
deformation time. The value of J s may be written as
As the dislocations move in a limited region of the crystal, the interacting volui
low strain rate is much less as compared to the total volume of the crystal [12]. r
for the limited deformation, n F may be considered effectively to be a constant. How
for higher values of the strain, the mechanical bleaching may be significant and n p
decrease considerably. According to (19), / s may decrease with the strain of the cr
Butler [29] and Chandra [30] have reported the decrease of / s fpr jhigher deform,
of the crystals.
The dependence of ML intensity / s on the density of F-centres may be unders
from (19) in the following way. Since in irradiated alkali halide crystals the dens
deep traps N l is much less as compared to density of holes N { [11], the factor a^
the denominator of (19) may be neglected as compared to <r r A".. Furthermore, ne
h -centres is greater than the probability of capture of dislocation electrons by th
nearby negative ion vacancy. Hence, the probability of retrapping of dislocation
captured electrons may be negligible and consequently the effective value of <r 2 may b
negligible. As a matter of fact, the factor a 2 N 2 in the denominator of (19) may also b
neglected. Thus, the value of / s from (19) may be expressed as
- (20
The above equation shows that 7 S should increase linearly with the density o
F-centres.
As p F , r/ 1 , n F and r P are different for different alkali halide crystals, equation (20
shows that some alkali halide crystals may show higher ML, however, some alkal
halide crystals may show weak ML.
With increasing temperature, the probability p F of the transfer of electrons fron
F-centres to the dislocation band will increase, following the relation
p F = p FO exp(-E a //cT) (21
where a is the energy gap between the dislocation band and the ground state o
F-centres [26, 18, 11].
From equations (20) and (21), I s may be written as
(22
Since the movement of dislocations does not stop just after stopping the cross-hea(
used to deform the crystal, for some time the generation and recombination o
dislocation electrons may take place and the ML may appear even after stopping th<
cross-head. Following the derivation of (11), in the present case the decay of MI
intensity may be given by
t _ a(t _ f>)] . (23
After the completion of stress relaxation process, the dislocation velocity
u d = y exp[ a(f ? c )] becomes negligible. If the dislocations still possess capturec
electrons, then the captured electrons may recombine with the holes, firstly, due to thi
movement of electrons along the dislocations and, secondly, due to the diffusion o
nearby interstitial atoms from the compressed region of dislocations towards thi
dislocation lines. Thus, the ML emission having comparatively longer decay time ma;
be observed after the completion of stress relaxation process.
4. Experimental support to the proposed models
To analyse the suitability of the proposed models, the ML measurements wen
performed on KC1, KBr and NaCl crystals grown by Czochralski technique. Thi
crystals were coloured by exposing them to 60 Co source. The absorption spectra wer
recorded using Shemadzu UV spectrophotometer and the density of F-centres wer
TIME (SECOND)
Figure 1. ML versus strain and stress versus strain curves of a "/-irradiated KC1
crystals (dimension = 5 x 5 x 5mm 3 , F 10 17 cm~ 3 , s= lO'^sec" 1 ).
calculated using Smakula formula. The ML versus strain and stress versus strain curves
were determined at different strain rates using a table model Instron testing machine
where the ML intensity was measured with the help of an RCA IP28 photomultiplier
tube. The stress was measured by 907-2 kg capacity Lebow Load Cell (Model No.
3354-2 K), and the strain was measured using a linear variable differential transducer
(LVDT) (Model No. 025 MMR, Schaevitz Engineering Company). The ML, ther-
moluminescence (TL) and after-glow spectra were recorded by using a Baush and
Lomb 1/2 m grating monochromator and EMI 9558, photomultiplier tubes, following
the technique described previously [31]. For the measurement of ML below room-
temperature, one end of a spiral copper tubing immersed into liquid nitrogen was
connected to a cylinder of dry nitrogen and the cooled gas coming out of the other end
of the copper tubing cooled the crystal. By changing the rate of flow of nitrogen gas, the
crystal could be cooled to different temperatures. The temperatures of the crystal was
measured by a copper constanton thermocouple. The TL appears during heating of
y-irradiated crystals and AG appears when the crystals are removed from 60 Co source.
Figure 1 shows that during the deformation of a y-irradiated KC1 crystal at a strain
rate of 10" 4 s~ l , initially the ML intensity increases and then it attains a saturation value
after a particular strain. When the deformation is stopped, it is seen that the ML intensity
decays, and disappears beyond a particular time. The initial rise and attainment of
saturation in the ML intensity are predicted by the mobile interstitial model as well as
by the mobile electron model. Both the models show that initially the ML intensity
should decay exponentially and later on it should decay slowly with a comparatively
Mechanoluminescence
io
10'
8
6
KBr
J L
J L_L
4 6 8 ID'
6 8 10'
fS' 1 )
Figure 2. Plot of log(J s ) versus log(e) (dimension = 5 x 5 x 5 mm 3 ,
longer value of the decay time. Figure 1 shows that after stopping of the cross-head,
initially the ML intensity decays with a fast rate and later on it decays with a slow rate.
Figure 2 shows the plot of log(/ s ) versus log(e) is a straight line where the slope is
n/aorK; prmal tr rmf TViic tvcnlt chnwc tVint thp A/f T intpncitv ic linear \x7itl^ thf ctrsiin rdtA
27.0
60 90 120
ABSORPTION COEFFICIENT (cm' 1 )
150
180
Figure 3. Dependence of ML intensity, I s , on the absorption coefficients and
density of F-centres in KC1 crystals (e = lO'^sec" 1 ).
be noted that both N l and n F increase in a similar manner with the radiation doses given
to the crystals.
Both the models predict that for a given density of F-centres, the ML intensity should
increase linearly with volume of the crystals. As the applied stress increases the strain
rate, both the models suggest that the ML intensity should increase with the applied
stress.
Figure 4 shows the ML, after-glow (AG) and thermoluminescence (TL) spectra of
KC1 and KBr crystals. For KBr crystals, the peaks of both the AG and TL spectra lie
nearly at 470 nm, however, the peak of their ML spectra is slightly shifted towards the
shorter wavelength side and lie at 463 nm. For KC1 crystals, the peaks of both the AG
400 500
WAVELENGTH (nm)
600
Figure 4. Mechanoluminescence, after-glow, and thermoluminescence spectra c
y-irradiated KBr and KC1 crystals.
and TL spectra lie nearly at 460 nm, however, the peak of their ML spectra is slightl;
shifted towards the shorter wavelength and lie at 455 nm. As the spectra of KBr an<
KC1 crystals shift with pressure at the rate of 0-07 and 0-025 nm/bar, respectively [3], i
seems that the spectral shift in the ML spectra as compared to after-glow an<
thermoluminescence spectra may be due to the local pressure during deformation. It i
well established that the light emission in after-glow and thermoluminescence pheno
mena of alkali halide crystals is mainly due to recombination processes involvin;
liberated electrons from F-centres and holes in V 2 -centres. Thus, the similarity of Ml
spectra with the after-glow and TL spectra suggests that the ML is essentiall;
a recombination process between electrons and holes.
The nature of the ML, TL and AG emission spectra in y-irradiated KC1 and KB
crystals can be understood as follows. It has been proposed that the emission is due t<
the recombination of F-centre electrons with the V 2 hole centres; Thus, the energ;
corresponding to the peak of the ML spectra should correspond to the energ
Pramana - J. Phys., Vol. 46, No. 2, February 1996
13'
Crystal
c (eV)
[27,32]
[33]
Calculated
value of A m
(nm)
Experimental
value of A m
(nm)
(figure 4)
KC1
KBr
8-1
7-3
5-40
4-69
458
476
455
463
difference between the bottom of the conduction band ( c ) and the energy level of
V 2 -centre (E V2 ). The wavelength A m corresponding to the peak of ML spectra is
calculated from the relation A m = [ch/(E e - E V2 )'], where c is the velocity of light and
h is the Planck constant. Table 1 shows that the calculated value of the emission peak is
approximately the same as that found from the experimental observations.
Some optical measurements may directly decide the reliability of the two models, for
example, if the electron from F-centre recombines with the hole centre (mobile
interstitial model) or the dislocation-captured electrons recombine with the hole
centres (mobile electron model). So far as known to us, such optical measurements have
not been made in the past. We were not able to perform such measurements because of
certain limitations.
Since the probability, p F increases with the temperature and the density n F of the
F-centres decreases with the temperature, the mobile electron model shows that
initially the ML intensity should increase with temperature, attain an optimum value,
then it should decrease and disappear beyond a particular temperature. Since the
probability p { (probability of sweeping of interstitials with dislocations) increases with
temperature and the density of interstitials decreases with increasing temperature, the
mobile interstitial model also indicates the occurrence of an optimum ML intensity at
a particular temperature. When the value of activation energy is determined by plotting
a curve between log(/J and 1000/7; it is found to be 0-08, 0-072 and 0-09 eV for KC1,
KBr and NaCl crystals, respectively (figure 5). This result shows that the activation
process is involved in the occurrence of ML.
Although both the mobile interstitial and the mobile electron model are able to
explain the dependence of ML intensity of coloured alkali halide crystals on several
parameters, the mobile interstitial model fails to explain the following facts:
(i) Photons as well as electrons are emitted during the deformation of coloured alkali
halide crystals, where they depend similarly on the deformation, strain rate and
F-centre density of the crystals [11, 18]. On the basis of the fact involving
dislocation electrons, the dislocation exo-electron emission can be understood in
the following way. The recombination of the electrons carried out by dislocation
with the deep traps in the crystal, may cause Auger ionization of other dislocation
electrons to the conduction band bottom. The subsequent thermal ionization of
electrons from the conduction band bottom into vacuum may give rise to the
dislocation exo-electron emission. Thus, the mobile electron model is able to
explain the simultaneous emission of photons and electrons during the plastic
140 Pramana - J. Phys., Vol. 46, No. 2, February 1996
10'
1/5
z
?10
KBr
KCl
Nad
1000/T (K"')
Figure 5. Plot of log(/ s ) versus 1000/T for y-irradiated KCl, KBr and NaCl
crystals. (n F 10 17 cm~ 3 , e= 10~ 4 sec~ 1 ).
deformation of coloured alkali halide crystals. However, the mobile interstitial
model is not able to explain the simultaneous emission of photons and electrons
during the plastic deformation of coloured alkali halide crystals,
(ii) Molotskii and Shmurak [1 1] have reported that additively coloured KCl crystals
exhibit weak ML, where the peak of the spectra lies around 2 eV, The ML emission
can be schematically represented by the following equations
(Q
D (D)
where L is the deep trap and L~ is the deep trap possessing captured electrons.
Since the additively coloured crystals do not possess holes, the mobile hole
model is not able to explain the appearance of ML during the plastic deformation of
deformation of additively coloured crystals supports the movement of electrons
with dislocations and their subsequent recombination.
(iii) In X or y-irradiated monovalent impurity doped alkali halide crystals, the holes are
captured by the monovalent impurities which change from M + to M 2+ [26]. If the
mobile hole model of ML is applicable, then for a given density of F-centres, the ML
intensity should decrease with increasing dopant concentration. In contrast, for
a given density of F-centres, the intensity of ML in X or y-irradiated alkali halide
crystals increases with the increasing monovalent impurity concentration [26].
This can be understood with the help of mobile electron model in the following way.
Me 2+ +e d -(Me + )*->Me + + /i + D
Conclusively, it may be said that the mobile electron model provides a dominating
process for the ML excitation in coloured alkali halide crystals.
Acknowledgements
The authors are thankful to the Council of Scientific and Industrial Research,
New Delhi, for financial assistance. One of the authors (BO) gratefully acknowledges
the University Grants Commission, New Delhi for the award of a Teacher Fellowship.
References
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Spatial and time resolved analysis of CN bands in the
laser induced plasma from graphite
S S HARILAL, RIJU C ISSAC, C V BINDHU, GEETHA K VARIER,
VPN NAMPOORI and C P G VALLABHAN
Laser Division, International School of Photonics, Cochin University of Science and Techno-
logy, Cochin 682 022, India
MS received 28 June 1995; revised 16 December 1995
Abstract. Analysis of the emission bands of the CN molecules in the plasma generated from
a graphite target irradiated with 1-06 ^m radiation pulses from a Q-switched Nd: YAG laser has
been done. Depending on the position of the sampled volume of the plasma plume, the intensity
distribution in the emission spectra is found to change drastically. The vibrational temperature
and population distribution in the different vibrational levels have been studied as function of
distance from the target for different time delays with respect to the incidence of the laser pulse.
The translational temperature calculated from time of flight is found to be higher than the
observed vibrational temperature for CN molecules and the reason for this is explained.
Keywords. Laser induced plasma; emission spectroscopy.
PACSNos 52-50; 52-70
1. Introduction
Irradiation of a target material with a high power laser pulse generates intense plasma
emission from the target surface. Such laser generated plasma is a rich source for
atomic, ionic and molecular species in various" states of excitations [1,2]. The abun-
dance of molecular, atomic and ionic species in the plasma will depend on various
parameters like nature of the target, laser power and pressure of the residual gas in the
plasma chamber [3-6]. Different types of studies of laser induced plasma such as
charge and velocity distribution of ablated species, second harmonic generation, X-ray
emission, etc. using high power laser pulses have been carried out in detail by many
workers [7-9]. The composition of the plasma will also depend on the spatial distance
of the point of observation from the target. The detailed studies of composition and
temperature in terms of distance from target have great importance with regard to
certain practical applications of laser ablation process like deposition of diamond like
carbon films [10, Ill-
Laser induced plasma from graphite target will contain, in addition to different
clusters, atomic and ionic species of carbon and transient species like CN in a partially
evacuated plasma chamber. Even though a few studies are available in the literature
related to CN species in the plasma, a systematic investigation of the spatial and
temporal variations of the characteristics of the plasma plume have not been reported
Plasma
Chamber
Figure 1. Schematic diagram of the experimental set up. BS, beam splitter;
EM, energy meter; L, lens; S, sample; M, monochromator; P, PMT; BI, boxcar
averager/gated integrator; CR, chart recorder; DSO, digital storage oscilloscope.
yet. In the present paper the spatial variation of the vibrational temperature of CN
molecules at different points of time during the evolution of the plasma is studied using
a Q-s witched Nd : Y AG laser as the pump source by analyzing the emission spectrum of
the violet system of CN molecule corresponding to the B 2 E + - X 2 1 + transition. This
will provide information regarding the vibrational distribution of CN molecules in the
plasma.
2. Experimental technique
The schematic diagram for the experimental set up is shown in figure 1. Plasma was
produced by the irradiation of a high purity graphite target with 1 -06 /mi laser radiation
(pulse width 9 ns and 1-1 x 10 1 * W cm" 2 maximum power density) from a Q-switched
Nd:YAG laser (Quanta Ray DCR II) at a pulse repetition frequency of 10 Hz. The
target was placed in a partially evacuated chamber (20mTorr) with quartz windows.
The target was mechanically rotated so as to minimize the surface etching and after
every five minutes' scan the focal spot was laterally shifted to different positions on the
target surface in order to provide fresh surface for ablation. In the absence of this
arrangement, emission line intensities tend to fade due to etching of the target surface.
The emission spectrum from the plasma was viewed normal to its expansion
direction by imaging the plasma plume using appropriate collimating and focussing
lenses onto the slit of a one meter Spex monochromator (1200 grooves/mm, 100 mm by
100mm grating blazed at 500 nm). The scan rate of the monochromator was adjusted
by using Spex CD2A compudrive arrangement. The recording was done using
a thermoelectrically cooled Thorn EMI Photo Multiplier Tube (PMT, model KQB
9863) which was coupled to a boxcar averager/gated integrator (Stanford Research
Systems, SR 250). The total extension of the plasma in the present set up was about
spatially resolved observations, different regions of the plume was focussed onto the mono-
chromator slit. In these studies, accuracy in spatial dimensions was better than 0-2 mm. The
output from the gated integrator (gate width 100 ns), which averaged out emission intensi-
ties from ten consecutive pulses, was fed to a chart recorder. The spectrum in the region
355-475 nm was normalized using the optical response curves of the monochromator-
PMT assembly. For temporal studies the PMT output was fed to a 200 MHz digital
storage oscilloscope (Iwatsu, DS 8621) with 50 Q. input impedance. This set up essentially
provides velocity as well as decay times of the constituent species [12] at a specific point
within the plasma and these are extremely important parameters related to the
evolution of laser ablated materials in a direction normal to the target surface.
3. Results and discussion
The spectrum of the graphite plasma contains different vibrational bands of CN
molecules along with emission lines from atomic and ionic species of carbon. Atoms
and ions of carbon ejected from the target due to laser ablation combine with the
ambient nitrogen inside the plasma chamber producing CN molecules through
recombination process. Characteristic spectral emission of CN molecule was obtained
~ CM
3800 3820 3840 3860 3880 3900
Wavelength (A)
3920 3940
Figure 2. CN violet band for Ay = sequence at different spatial distances from
the target"(laser irradiance 7-3 x 10 9 Wcm~ 2 , time delay 5jus) (a) 2mm, (b) 6mm,
(c) 10mm, (d) 14mm.
13
12
10
Figure 3. The vibrational distribution of CN violet band (distance 10mm, laser
irradiance 7-3 x 10 9 Wcm~ 2 ).
in the violet region due to the B 2 Z + --X 2 L + transition [13]. Depending on the laser
fluence, time of observation and position of the sampled volume of plasma, the intensity
distribution of the emission spectra change drastically as the plume expands. Spectra
for sequences Au=l,0, 1, 2 are recorded where, Au = u' v" is the difference
between the vibrational quantum numbers of the upper (B 2 L + ) and lower (X 2 Z + )
electronic states. Figure 2, gives the typical CN-violet band (Au = 0) for different
distances from the target at a laser irradiance 7-3 x 10 9 Wcm~ 2 (estimated laser spot
size being ^200 j wm in radius). Spectrum show a gradual increase in the emission
intensity up to a distance 10mm away from the target and beyond this distance the
intensity decreases rapidly. Contrary to this, the singly ionized carbon (CII) line
intensity decreases continuously as we move away from the target. It has also been
observed that the intensity of CN bands increases up to a laser irradiance of
7-3 x 10 9 Wcm~ 2 and it levels off above this power density.
The band emission intensities were used to calculate molecular vibrational tempera-
ture T vib , details of which are available in the literature [14]. The vibrational distribu-
tion in the excited states of CN molecules at distance 10mm away from the target is
shown in figure 3 at a laser irradiance 7-3 x 10 9 Wcm~ 2 . The inverse distribution
observed for u < 2 is in accordance with the Frank-Condon principle. Similar inverse
distributions were also observed in certain other molecules [15, 16].
The spatial variation of the vibrational temperature for 2 fis and 5 fts delay times after
the onset of the plasma is given in figure 4. It was found that at a particular laser fluence,
depending on the time of observation and the position of the sampled volume, the
vibrational temperature of CN molecules varies. Spatial variation of vibrational
temperature after 2 /is from the onset of the plasma peaks (2-14 x 10 4 K) at a distance
3 mm away from the target. For 5 ^s delay time, the vibrational temperature was
maximum (1-96 x 10 4 K) at 8 mm from the target surface. This is because of the fact that
near the target surface the temoerature is so hieh that collision induced nrocesses
2.05
1.80
1.55
1.30
1.05
0.8
10
distance (mm)
15
20
Figure 4. The variation of vibrational temperature of the CN violet band with
distance from the target for 2 ^s (o) and 5 ^s (n) delay time.
f
9.0
7.4
5.6
o 4.2
2.6
1.0
6 8 10
Distance (mm)
12
14
16
18
Figure 5. The change in the expansion velocity of CN molecules (388-3 nm) as
a function of distance from the target (laser irradiance 7-3 x 10 9 W cm" 2 ).
predominate and cause a decrease in vibrational temperature due to de-excitation of
the higher vibrational levels. As we move away from the target, the collisional effects are
reduced so that effectively vibrational temperature was found to be high. At distances
farther than this optimal distance, the decrease in plasma temperature will cause
vibrational temperature was different for 2 j/s and 5 /us delays (for 2 ^us maximum is at
3mm and for 5^s at 8mm). Such an effect takes place because different physical
processes like collision between neutrals, ions or electron capture by CN~ etc.
predominate at different times within the plasma and the evolutionary history of CN is
fairly complex. This causes the CN number densities to vary with respect to time as well
as space in the laser generated plasma from graphite.
From the observed time delays, one can evaluate expansion velocities of these
transient species. Figure 5 shows the change in the expansion velocity of CN molecules
as a function of distance from the target at laser irradiance 7-3 x 10 9 Wcm~ 2 . It is
found that the expansion velocity of CN molecules was increasing up to a certain
distance from the target (8 mm) and thereafter they slow down rapidly attaining a much
smaller expansion velocity, which corresponds to plasma cooling.
The maximum molecular vibrational temperature for CN molecules was found to be
around 2-14 x 10 4 K, which is much higher than the melting point of graphite
(4 x 10 3 K). This large vibrational temperature may arise due to the direct heating of
the plasma plume. This is supported by the measurement of the temperature equivalent
of translational energy which varies from 2 x 10 4 K to 7 x 10 4 K at a laser irradiance of
7-3 x 10 9 Wcm~ 2 . The large variation in the translational temperature implies that,
the observed time delays are not only due to time of flight (TOF) phenomenon alone
but also due to those arising from other processes like recombination/dissociation of
the species, collisional excitation process etc. Further experiments like mass spectral
measurements may shed some light on these aspects.
In conclusion, laser irradiation of graphite in a low pressure air chamber generates
plasma containing CN molecules. From the spectroscopic studies of the emission
bands of the CN molecules, the population distribution and vibrational temperature at
different regions of the plasma plume have been obtained. It is found that the
vibrational temperature of the CN molecules varies with the position of the sampled
volume within the plasma plume.
Acknowledgements
The present work is supported by Department of Science and Technology, Govern-
ment of India. One of the authors (SSH) is grateful to the Council of Scientific and
Industrial Research, New Delhi for a research fellowship. CVB and RCI are thankful to
the University Grant Commission, New Delhi for their research fellowships.
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Self-similar solutions of laser produced blast waves
K P J REDDY
Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India
MS received 3 May 1995; revised 27 July 1995
Abstract. The aerodynamics of the blast wave produced by laser ablation is studied using the
piston analogy. The unsteady one-dimensional gasdynamic equations governing the flow are
solved under assumption of self-similarity. The solutions are utilized to obtain analytical
expressions for the velocity, density, pressure and temperature distributions. The results predict
all the experimentally observed features of the laser produced blast waves.
Keywords. Ablation; laser; blast wave; shock wave; gasdynamics.
PACS Nos 42-60; 52-50; 36-20; 61-80
1. Introduction
The laser ablation is a dry photoetching technique with submicron feature resolution
[1] and ability to remove material in submicron layers [2]. This technique is useful in
photoetching of polymers for lithographic applications [3] and biological tissues as in
laser surgery [4]. Recent studies on the laser ablation of biological tissues and synthetic
polymers like polymethyl methacrylate (PMMA), polyimide, and polyethylene- tereph-
thalate (PET) has led to the new phenomenon called ablative photodecomposition
which results in the ejection of the ablated material at supersonic velocities [5]. These
studies have addressed various aspects of laser surface ablation processes such as
identifying the emission products in the plume through spectroscopic means [6], the
measurement of their expansion velocities through ultrafast microscopy [7] and
self-focusing of the laser pulse in the medium and delay in the appearance of the gas
phase products above the surface [8].
In vacuum, due to the absence of any resistance from the surrounding gas, the ejected
target material particles disperse ballistically at Mach numbers ~ 150. While in
presence of the background gas, such as air, nitrogen, oxygen and argon, the ejected
material travels at Mach numbers ~ 20 and produces a strong spherical shock wave
which expands into the surrounding ambient gas engulfing more and more back-
ground gas. When the mass m of this shocked gas exceeds that of the ablated, gas m a the
shock wave develops into a blast wave. Since the energy contained in the laser pulse is
finite, the expansion velocity of the shock front decreases with increasing time. Most of
the studies reported are the experimental investigations of the laser ablation process
and the spectroscopic analysis of the plumes produced due to ablation. However, very
few theoretical studies have been reported on the propagation dynamics of the ejected
material from the sample surface. It is important to analyze the processes of expansion
assumption m m a is considered in the present analysis. J ms analysis uses me piston
analogy where the flow is governed by a set of unsteady one-dimensional gasdynamic
equations which are solved using the self-similar solutions method [9, 10]. The
solutions are used to get analytical expressions for the pressure, density and the
temperature distributions along the radial direction of the spherical blast wave
produced by laser ablation starting from the centre of explosion. The analysis of
cylindrical blast waves where m ^ m a has been presented recently [11].
2. Governing equations
In an idealized situation the problem of the blast wave generated by the irradiation of
a polymer surface or a biological tissue with ultrashot laser pulse of high energy in
presence of ambient gas is equivalent to an explosion at time t = in a gas at rest at the
centre of symmetry with an instant liberation of finite amount of energy (which is
a sum of kinetic energy and the heat energy), given by
i i
(1)
where u, p, and p are the velocity, density and the pressure of the gas in the blast wave,
r is the radial coordinate and y is the specific heat ratio of the ambient gas.
The ejected material due to ablation acts as a piston which drives a spherical blast
wave into the ambient gas whose radius R increases with the increasing time as
Po
More and more background gas gets swept by the shock front at the leading edge of
the blast wave which travels at a velocity U defined by
dR_2 E i
-
where, is a constant related to the energy through the relation = (?) in which
a(y) = 0-851 for air or nitrogen [9].
The energy of the blast wave defined in (1) can be determined from the
experimentally measured radius of the expanding spherical blast wave for the known
values of the incident laser pulse by rewriting (2) in the following form
~\ogR-\ogt = l-\og(^-\ (4)
L ^ \Po/
For the narration of the usefulness of the above equation we replot the data given in
[12] in the form shown in figure 1. This data has been obtained using a laser pulse of
energy 370 /rj (incident power of 54 GW/cm 2 ) at 532 nm to generate the blast wave from
1 54 Pramana - J. Phys., Vol. 46, No. 2, February 1996
- 55 -
-6.0 -
-6.5
-9.5
-8-0
Figure 1. Experimentally measured radius R (cm) of the laser produced blast wave
as a function of time (sec) (data replotted from [12]).
the surface of a PMMA. The laser pulse is generated by frequency doubling the output
from a regenerative amplifier which amplifies 80 ps duration pulses from a mode locked
Nd:YAG laser at 1-06 /mi. This pulse is divided into damage and probe pulses and
focused on to the sample surface in a vacuum chamber. The probe pulse with flux < 1 %
of the damage pulse is imaged by a camera with diffraction-limited imaging ability. The
experimental data agrees with the theoretical formula given by (4) if we assume
l/21og(//3 ) = 2-993, which yields E = 163-86 nj. Assuming an ambient air pressure of
1 atm. and temperature of 298 K this yields the blast wave energy E = 139-446 juj. This
energy agrees with the blast wave energy estimated earlier [12].
The motion of the gas in the blast wave is governed by the following set of unsteady
gasdynamic equations [13],
du du I dp
L y I _ _ Q
dt dr pdr
dp dp du 2u
(5)
(6)
(7)
along with the equation of state p = pRT. Equations (5) to (7) are the momentum,
continuity and energy equations, respectively.
i lie cum ui uit picaout aiiaijais i tw suivc me auuvc sci ui C4uamjiia aiiu uuuaui
analytic expressions for the distributions of the flow variables along the radial
coordinate r starting from r = at the centre of the explosion.
3. Self-similar solutions
It is well-known that an abrupt jump in the flow parameters takes place on the boundaries
where the perturbed gas due to the blast wave is separated from the unperturbed gas by
a shock front of radius R. These jump conditions given by the Rankine-Hugoniot
conditions, define the velocity, density and the pressure behind the shock front as
~ If 2 /. W
/.
where, p is the density of the undisturbed gas, f l = l a 2 /U 2 , f 2 = (1 + 2a 2 /
U 2 (y 1))~ l , / 3 = 1 a 2 (y l)/(2yU 2 ), and a is the speed of sound in the medium.
In the laser produced blast wave it is shown that the pressure behind the shock front
is very high compared to the pressure ahead of the shock wave [7]. The pressure ahead
of the shock wave can be neglected in comparison with the pressure behind the shock
wave. However, it is important to estimate with what accuracy and for which shock
waves this assumption is valid. The values of /\ , f 2 , and / 3 in the above equations differ
from unity by < 5% when the ratio a/U < 01 (that is for the stronger shocks). Thus if
we use a/U = and f i =f 2 f 3 = 1, which is same as assuming counter pressure
Po = 0, then an error of < 5% is introduced into the values of u, p, and p. Then the
Rankine-Hugoniot conditions reduce to the following form after substituting U from (3),
4 /
(12)
The corresponding temperature is determined using the equation of state.
The system of characteristic parameters influencing the motion of the perturbed gas
after the explosion, governed by (5)-(7), under adiabatic conditions is represented by
the quantities p Q ,p ,E ,r,t and y. By non-dimensionalizing the basic equations we can
show that the non-dimensional variables depend only on the dimensionless parameters
y, A = pj /5 r/j /5 t 2/5 and T = pJ /6 t/j /3 /)o /2 of which A and T are variables. By neglecting
the counterpressure the variable i disappears. In this case the flow variables change
with time in a manner that their distributions with respect to the coordinate variable
Laser produced blast waves
always remain similar in time. Then such a flow can be considered as self-similar,
lowever, as the shock wave travels away from the centre of explosion it becomes
weaker and the pressure on either side of the shock front become comparable,
"herefore in such a situation the counterpressure cannot be neglected and hence the
ow ceases to be self-similar.
The self-similar solution for the laser produced blast wave can be expressed in terms
f the non-dimensional velocity V, which varies in the range 2/5y ^ V < 4/5 (y + 1) where,
' = 4/5(y + 1) corresponds to the shock front, and the value V=2/5y corresponds to
le centre of the explosion. The complete solutions normalized with respect to the
alues behind the shock front are [9],
5(y+l)-2[2
2 + 3(y-l)
(14)
(15)
fr + 1) ^K-,
-DV2
(7-1)
5(7 + D
5(7
-2[2 + 3(y-
-f>
(16)
(7-1)
5(7 + 1)
1-
V
a 4 - 2a,
P_P2
Pi p
(17)
(18)
/here, o^ = 1-4565, a 2 = - 0-563, a 3 = 0-78947, a 4 = 12-1375 and <x s = - 3-333 if we
ssume air or nitrogen as the ambient gas surrounding the target (y = 1-4).
The distribution of velocity, density, pressure and the temperature behind the shock,
ont along the radial coordinate r computed from the above equations are shown in
gure 2. These results show that the velocity and density tend to zero near the centre of
ymmetry while the pressure reaches a constant value and the temperature tends to
ifinity. Thus large temperature gradients occur and the mass of the gas disperses near
ie centre of symmetry where the explosion takes place. The pressure is finite at the
entre but decays to zero as time increases. Hence a reverse gas motion towards the
entre of ablation must occur after a lapse of finite time when the etching is achieved by
TP irraHiatinn with laser nukes of ve.rv hiph fluence. This nrocess confines the elected
Figure 2. Distribution of the flow variables behind the shock wave in the laser
produced spherical blast wave along the radial distance r.
For the case of very intense laser pulse the analytic asymptotic expressions for the
velocity, density, pressure and temperature near the centre of the explosion can be
obtained from the above equations in the limits V-*2/5y and r-0 as
u
2_r
~5y t
F\-(3/5(v-l))
-
Po
- (t)- 615
Po/
E\< 2(v ~
Po/
(19)
(20)
(21)
(22)
where, /c lt /c 2 and /c 3 are the functions of the specific heat ratio y and c v is the specific
heat coefficient at constant volume.
A contact surface separates the shocked gas and the plume containing the fragments
of the ablated material which drives the blast wave. Hence the value of the specific gas
constant changes at the contact surface as we approach the centre of the explosion
starting from the shock front. Therefore appropriate value of y should be used to utilize
(20)-(21) to compute the flow variables. This values for the plume can be determined
experimentally.
4. Conclusions
The blast wave phenomenon produced due to the ablative photodeco.mposition
occurring during the photoetching of a polymer surface or a biological tissue using high
imensional gasdynamic equations are solved under the self-similarity condition for
le spherical blast wave. The similar solutions yield analytic expressions for the
istribution of flow variables. Asymptotic analytic expressions for the field variables
ear the centre of explosion where the laser pulse interacts with the target are obtained
om the general solutions. The analysis predicts the experimentally observed features
f the confinement of the plume closer to the target surface in presence of the blast wave.
eferences
[1] D Henderson, J C White, H G Craighead and I Adesida, Appl. Phys. Lett. 46, 900 (1985)
~2] R Srinivasan and B Braren, J. Polym. Sci. 22, 2601 (1984)
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oupSed scalar field equations for nonlinear wave
modulations in dispersive media
[NRAO
heoretical Physics Division, Physical Research Laboratory, Navrangpura,
hmedabad 380 009, India
[S received 18 February 1995; revised 29 January 1996
bstract. A review of the generic features as well as the exact analytical solutions of a class of
jupled scalar field equations governing nonlinear wave modulations in dispersive media like
.asmas is presented. The equations are derivable from a Hamiltonian function which, in most
ises, has the unusual property that the associated kinetic energy is not positive definite. To start
ith, a simplified derivation of the nonlinear Schrodinger equation for the coupling of an
nplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled
:ts of time-evolution equations like the Zakharov system, the Schrodinger- Boussinesq system
id the Schrodinger-Korteweg-de Vries system are then introduced. For stationary propaga-
on of the coupled waves, the latter two systems yield a generic system of a pair of coupled,
rdinary differential equations with many free parameters. Different classes of exact analytical
)lutions of the generic system of equations are then reviewed. A comparison between the various
:ts of governing- equations as well as between their exact analytical solutions is presented,
arameter regimes for the existence of different types of localized solutions are also discussed,
he generic system of equations has a Hamiltonian structure, and is closely related to the
ell-known Henon-Heiles system which has been extensively studied in the field of nonlinear
ynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized
[enon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative
roup dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which
structurally same as the generalized Henon-Heiles Hamiltonian but with indefinite kinetic
lergy. The above correspondence between the two systems has been exploited to obtain the
arameter regimes for the complete integrability of the coupled waves. There exists a direct
ne-to-one correspondence between the known integrable cases of the generic Hamiltonian and
le stationary Hamiltonian flows associated with the only integrable nonlinear evolution
^uations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of
ic generic system to other equations like the self-dual Yang-Mills equations, the complex
lorteweg-de Vries equation and the complexified classical dynamical equations has also been
iscussed.
keywords. Nonlinear equations; coupled scalar fields; dispersive media; modulational instability;
jlitons; plasma waves; NLS equation; KDV equation; Boussinesq equation; Henon-Heiles
[amiltonian; nonlinear dynamics; Hamiltonian flows; integrability; complexification; chaos.
ACS Nos 03-20; 03-40; 52-35
. Introduction
Coupled second-order nonlinear ordinary differential equations occur in many
ranches of physics. For example, in dispersive media like plasmas it is well-known
N N Rao
that an high-frequency wave with modulated amplitude can lead to the excitation of an
instability called the "modulational instability" [1-3]. The nonlinear development of the
instability is typically governed by a Schrodinger-like equation having a 'potential' which
depends on the associated low-frequency perturbations. The latter are governed by a linear
wave equation [4-9], or in some cases, by the nonlinear Korteweg-de Vries (KDV) [10, 1 1]
or the Boussinesq [12,13] equation, both of which are driven by the so-called pon-
deromotive force due to the high-frequency carrier wave [14, 15]. For stationary propaga-
tion of the coupled waves, the Schrodinger-KDV (or, -Boussinesq) system of equations
yields a coupled set of scalar field equations. These equations are generic in nature,
occurring for coupled wave systems such as Langrnuir and ion-acoustic waves [16-18],
upper-hybrid and magnetoacoustic waves [19-23], and electromagnetic and ion-acoustic
waves [24-27]. The equations are structurally very similar to the generalized Henon-
Heiles equations [28, 29] which are extensively studied in the field of non-linear dynamics
over the last couple of decades, but with one important difference: While the (generalized)
Henon-Heiles Hamiltonian for the classical dynamical systems has always associated
with it a positive definite kinetic energy, the generic Hamiltonian in most of the
modulational instability problems has indefinite kinetic energy; that is, the sign of the
kinetic energy can change as the independent coordinate varies.
The similarity between the generalized Henon-Heiles system in classical dynamics
and the generic system for modulational instability in dispersive media raises the
question about the complete integrability of the latter. In fact, for the case of the
coupled upper-hybrid and magnetoacoustic waves with negative group dispersion
[30], the generic system is exactly, the same as the generalized Henon-Heiles system,
and is, therefore, integrable for certain sets of parameter values. On the other hand, for
positive group dispersion there exists a novel Hamiltonian (with indefinite kinetic
energy) whose integrability is yet to be investigated.
The Schrodinger-KDV (or, -Boussinesq) system for the coupled Langmuir and
ion-acoustic waves in plasmas has two free parameters whereas it has been possible to
obtain [10-13] its exact (analytical) solutions valid only on a straight line in the
two-dimensional parameter space. For other wave systems, the associated equations
have more number of free parameters. Most of the analytical solutions that are
currently available have been obtained by using specialized boundary conditions
which lead to either the periodic or the localized solutions. Recently, different classes of
exact solutions of the generic equations valid in different regions of the parameter space
have also been reported [31,32]. An outstanding problem associated with such
coupled equations is to obtain their exact solutions in as much of the allowed
parameter space as possible and with wider boundary conditions.
Coupled scalar field equations with indefinite kinetic energy occur in other contexts
also. For example, for stationary solutions of the usual KDV equation with complex
dependent variable [33], one obtains a set of equations whose Hamiltonian has
precisely such property. This feature appears to be more general, occurring in all
one-dimensional classical dynamical systems in a conservative potential when the
deoendent variable is made onmnlex R41 The timp.-pvnlntinn rf curh
dimensions) which possess the soliton or the instanton type of finite energy solutions
[36-40]. Recently, there have been many attempts to consider higher dimensional
scalar field equations and to reduce them to known tractable equations by using the
inherent symmetry properties. In the last few years, much attention in this direction has
been focused on the classical Yang-Mills field equations which are known to yield as
special cases different types of scalar field equations. For example, it was shown [41, 42]
recently that the classical Yang-Mills equations which satisfy the self-duality condition
can be reduced to either the nonlinear Schrodinger or the KDV equation depending on
the choice of the available gauge degree of freedom. Since the nonlinear Schrodinger
equation is generically a subset of the coupled Schrodinger-KDV (or, -Boussinesq)
system, this opens up the possibility of reducing the Yang-Mills system to the latter
which is more tractable analytically. Furthermore, since the Schrodinger-KDV (or,
-Boussinesq) system reduces, for stationary solutions, to the generalized Henon-
Heiles system it may be possible to analytically investigate the regular or the chaotic
behaviour of the solutions of the self-dual Yang-Mills equations.
In this review, the aim is to present a discussion of the generic features as well as the
analytical solutions of the coupled scalar field equations mentioned above. In some
cases, it is possible to find exact analytical solutions of the generic equations obtained
from the Schrodinger-KDV (or, -Boussinesq) system. Wherever possible, examples
are chosen from the field of plasma physics where such equations are commonly
encountered. Unless otherwise stated, all the variables are assumed to be suitably
normalized so that the equations are dimensionless throughout. The manuscript is
organized in the following manner: In 2, a brief discussion of the modulational
instability of a high-frequency carrier wave is presented. Different sets of coupled scalar
field equations that govern the nonlinear development of the modulational instability
are presented in 3. The method of solution mentioned in this section can be used to
obtain exact analytical solutions of the coupled Schrodinger-KDV (or, -Boussinesq)
system of equations. Some discussion on the stability as well as the interaction
properties of such exact solutions is also presented. Section 4 deals with the generic
system of equations and its exact analytical solutions which are valid in different
regions of the parameter space. In 5, the relevance of the generic equations to other
systems such as the Henon-Heiles equations, self-dual Yang-Mills equations and
complex KDV equation are described. A discussion on the possible integrable par-
ameter regimes of the generic equations is presented by considering the concrete
example of the coupled upper-hybrid and magnetoacoustic waves in plasmas. Some
remarks on the close connection between the generic equations and the stationary
flows associated with certain types of nonlinear integrable evolution equations are also
made. A brief summary of the known results as well as the outlook for future work is
presented in 6.
2. Modulational instability
Consider the propagation along x-direction of a plane wave of frequency co and
wavenumber k represented by
X
Figure 1. High-frequency wave field whose amplitude is modulated.
where the wave amplitude E(x, t) is, in general, a slowly varying function of the space
and the time variables (figure 1). For the special case when the wave amplitude is
a constant, the propagation characteristics of the wave are completely determined by
the linear dispersion relation, CD = co(k) which is obtained by a normal mode analysis of
the relevant basic equations for the dispersive media. On the other hand, a governing
equation for the modulated wave amplitude can be derived by starting with the
nonlinear dispersion relation, namely, co = co(/c, || 2 ) which after Taylor expansion
around the carrier wave parameters (co , k ) yields [1]
(2)
where the partial derivatives are to be evaluated at k = k and \E\ = 0. In (2), only the
lowest order term in || 2 has been retained. Replacing the frequency shift (co co ) by
id/dt and the wavenumber shift (k k ) by id/cbc, one obtains the following
evolution equation for the slowly varying complex amplitude E(x, t)
dt
dx
~
(*) = 0,
(3)
where asterisk denotes the complex conjugate, V = dco/dk denotes the group velocity,
P = (d 2 co/dk 2 )/2 represents the group dispersion; and Q = dco/d(\E\ 2 ) is the nonlinear
coefficient. In view of its structural similarity to the Schrodinger equation of quantum
mechanics, (3) is called the "nonlinear Schrodinger equation", and is known to govern
the evolution of different types of high-frequency waves in plasmas. Examples are the
Langmuir waves, the upper-hybrid waves and the electromagnetic waves [14, 15].
It may be noted that for unmodulated linear normal modes, (3) is identically satisfied
trivially. On the other hand, (3) can be analyzed to determine the stability properties of
an high-frequency carrier wave when the wave amplitude is modulated with a lower
frequency. Depending on the relative sign between the dispersive and the nonlinear
terms, the modulation can become unstable (figure 2). Linear stability analysis shows
Coupled scalar field equations
Figure 2. Time evolution of the modulational instability of ion-cyclotron waves in
a magnetized plasma [Ref. 1]. The wave on the left-hand side has much larger
modulation of the amplitude than the one on the right side, and therefore becomes
unstable much faster.
that for PQ > the waves are unstable. Such an instability of the amplitude-modulated
high frequency waves is called the "modulational instability" of the carrier wave [1-3].
Physically, the instability arises due to the self-trapping of the wave field in the
"potential" which is determined by the wave itself [cf. the nonlinear term on the
left-hand side of (3)].
We shall now consider analytical solutions of (3) which are stationary in a frame
moving with a constant speed. The second term in (3) can be eliminated by going into
a Galilean frame defined by ( = x V & t and i = t. This yields
.dE _3 2 E
(4)
N N Rao
number" of the stationary frame. In order to allow for any possible shifts in the
frequency as well as in the wave number of the carrier wave due to the nonlinear
interactions, the amplitude field is represented by
where E a (r]) is the real, stationary amplitude of the modulated wave.
Substituting the solution (5) into (4), one obtains from the imaginary part,
X ((,)= M(/2P whereas the real part yields the following equation for the stationary
amplitude E a
2P-
(6)
where A = 2(dT/dt) + (M 2 /2P) is the nonlinear shift parameter. For localized bound-
ary conditions, (6) can be easily integrated to get the so-called "envelope soliton"
solution, namely,
\i/2
(C,T)=|-j seen
\*--/
1/2
2p) (C-Mz)
(7)
Clearly, the total high-frequency field s(x, t) of (1) has a structure wherein the amplitude
of the carrier wave field is modulated leading to a localized wave packet, and the
structure itself propagates with a constant velocity with respect to the laboratory
frame. The typical profile of an envelope soliton solution is shown in figure 3.
Before considering further generalizations of (4), let us briefly summarize the physical
mechanism that leads to such solutions. In plasma physics, the variable E can be taken
to represent the electric field of a suitable high-frequency wave. It can be easily shown
that when the amplitude of such a wave field is slowly modulated, the motion of
a particle of charge q and mass m can be decomposed into two parts: the first part
because of an average force due to the amplitude modulation, and is independent of the
sign of the charge. Such a nonlinear force is called the "ponderomotive force" and is, in
general, given by [14]
Clearly, the force is derivable from a potential ^ p = qE;/4mco 2 called the "pon-
deromotive potential". For electromagnetic waves, the ponderomotive potential repre-
sents basically the radiation pressure due to the wave fields. Due to the inverse
dependence on the mass, the ponderomotive force acts strongly on the electrons
pushing them away from the regions where the field is stronger. However, because
of the self-generated ambipolar field, the ions soon follow the electrons and
create a density trough which further traps the high-frequency field. The self-trapping
process continues till a dynamic balance between the nonlinear and the dispersive
effects is reached. The envelope soliton solution (7) is simply a representation of such
a state. A detailed review of the modulational instability, wave envelope self-focussing
and the consequent development of strong Langmuir turbulence together with the
theoretical analyses, numerical simulations, laboratory experiments and applications
to space plasma situations has been given by Thornhill and ter Haar [2] and by
Goldman [3].
3. Coupled systems of equations
The ponderomotive force due to a high-frequency field in a plasma drives a low-
frequency oscillation which may be a normal mode of the system. For example, the
amplitude modulated Langmuir oscillations are coupled to the wave-excited low-
frequency acoustic-like fluctuations called the "ion-acoustic waves". The nonlinear
Schrodinger equation derived above considers, however, only the static response of the
latter waves. Such an approximation can be justified when the envelope wave packet is
nearly static. However, when the envelope moves with finite, non-zero speed, dynamic
response of the low-frequency wave should be taken into account.
3.1 Zakharov system
For the coupled Langmuir-ion-acoustic waves, Zakharov [4] suggested the following
pair of (normalized) equations
3E 3f
+ ~ " {>
^ T* , (io)
at 2 dx 2 dx 2 \4
where m e (m,) is the electron (ion) mass, N^NJ is the perturbed electron (ion) number
density and = (mjm { ) 112 . The system of equations (9) and (10) is closed by means of the
quasi-neutrality assumption, N e = N { . Equation (9) is a Schrodinger-like equation with
Pramana - J. Phvs.. Vol. 46, No. 3, March 1996 1 67
The coupled set of equations (9) and (10) (together with N c = N i ) is called the
"Zakharov system", and has been very extensively studied in the literature in connec-
tion with the problem of strong Langmuir turbulence in plasmas [2-9]. It may be noted
that the nonlinear Schrodinger equation (4) follows from the Zakharov system for static
response. For, under this assumption, the time derivative term in the driven wave
equation (10) can be neglected. Substituting for JV e ( = NJ from (10) into (9), we readily
obtain the nonlinear Schrodinger equation (4). However, for dynamic response in the
stationary frame = x Mt, (10) gives
1 EE* \E\ 2
where "$" is the self-consistent ambi-polar potential. Thus, the solutions of the
nonlinear Schrodinger equation are valid when M 2 1.
In terms of the common perturbed number density N = N { = N e , (9) and (10) become
-"+-*"*
3 2 ,N d 2 N d 2 /I
which is the standard form of the Zakharov system of equations. The above coupled
equations can be derived from a variational principle by using the Lagrangian density
(&) given by [16]
= (EE* - * JE t ) 4- |,JE* + f + 2 Xt + (0 J 2 + X EE*, (14)
where subscripts 'x' and 't' denote the respective partial derivatives, i and are
auxiliary variables and & x is to be identified as N. From the invariant properties of the
Lagrangian density (J?), it follows that the coupled equations (12) and (13) have the
following integral invariants
-E*E X ) + 2QN. Ux, (16)
-CO (^ J
p + oo
I E = EE*dx, (17)
J CO
whereas (13) directly gives the invariant
JVdx. (18)
i
1 68 Pramana - J. Phys., Vol. 46, No. 3, March 1996
-0.3
-0.6
-45
-30
-15
I
Figure 4. Profile of the stationary solutions (19) and (20) of the Zakharov
equations (12) and (13) for the coupled Langmuir and ion-acoustic waves. The
envelope of the modulated carrier wave is shown by solid lines and the associated
low-frequency density perturbation by the dashed line. The entire structure moves
with a constant speed, namely, the Mach number M.
Physically, these integral invariants govern, respectively, the conservation of total energy
(I H ), total momentum (/ n ), Langmuir plasmon (I E ) number and the total perturbed
number density (I N ), and are useful in analyzing the allowed or the forbidden interactions
between the nonlinear entities obtained as solutions of (12) and (13) [cf. 3.8].
Equations (12) and (13) can be integrated for stationary solutions to yield
o)}> (19)
(20)
where = x - Mt, p. = (A/3) 1 ' 2 , is a constant which represents the initial "phase" and
the nonlinear shift parameter given by /l = 2A + (e 2 M 2 /3)2A is required to be
positive so that the solutions (19) and (20) satisfy the localized boundary conditions.
Equation (19) then requires M 2 < 1 for real a ; that is, the low-frequency density
fluctuations "loaded" with the high-frequency envelope wave packet can only travel at
sub-sonic (M 2 < 1) speeds. Without the loss of any generality, the constant f can be
taken to be zero.
Solutions (19) and (20) are plotted in figure 4. Note that these solutions imply the
following relation
1-M 2
M 2
(21)
This equation determines the ordering between the amplitudes of the wave fields and
the Mach number. Treating X as the smallness parameter, it follows that for E 2 = 0(A)
and N = 0(A), one needs M 2 1. The solutions of the nonlinear Schrodinger equation
(4) satisfy this ordering. On the other hand, for near-sonic propagations (M 2 ~ 1) there
Pramana - J. Phys., Vol. 46, No. 3, March 1996
169
exist two possibilities. First, for E a = 0(A) and 1 - M 2 = (9(X\ (21) requires W = 0(A)
and hence the linear driven wave equation (13) as well as the solutions (19) and (20) are
valid. This was the ordering used by Karpman [18] to describe the near-sonic
propagation of the Langmuir-ion-acoustic waves. Second, for El = (9(2.) and
1 _ M 2 = 6>(A), the low-frequency perturbations become finite and hence the linear
equation (13) is no longer valid but should be replaced by a suitable nonlinear
generalization. This is discussed in the next section.
3.2 Schrodinger-Boussinesq system
The nonlinear Schrodinger equation (4) as well as the Zakharov equations (12) and (13)
take into account only the linear response in the low-frequency dynamics. However, as
pointed out above, for the so-called 'near-sonic" (M 2 ~ 1) propagations, the amplitude
of the low-frequency density perturbation can be quite large requiring a nonlinear
dynamical equation in place of ( 1 3). For the coupled Langmuir and ion-acoustic waves,
Makhankov [12] suggested the following coupled Schrodinger- Boussinesq equations
as the appropriate set
a. ^ -a.. 2 2 v*r/~> \*"*-i
The latter equation (23) is called the (driven) "Boussinesq equation", and generalizes
the linear wave equation (13) to include the dispersive effects (third term on the
left-hand side) as well as the nonlinear effects (the fourth term). The right-hand side of
(23) which arises due to the ponderomotive force couples the two equations.
Equations (22) and (23) can be derived through the Lagrangian density
<e = (EEf - *,) + f ,* + X 2 '+ 2 Xt & + (0J 2
+ 2 x \l/ x + iA 2 + f(ej 3 + & X EE* (24)
where, as earlier, %, and \jj are the auxiliary variables, and x is to be identified as </>.
The integral invariants in the present case are given by (16) (18) together with
(25)
For stationary solutions of the form
(26)
where c; = x Mt, equations (22) and (23) yield
rl 2 F
(27)
me stationary, oi-airecuonai propagation or coupiea Langmuir ana ion-acoustic
waves in unmagnetized plasmas. Clearly, each of the terms on the right-hand side of
(27) and (28) are of the same order (A 2 ) provided E a = 0(0) = 0(1 - M 2 ) = 0(1). As we
shall see below, it is possible to obtain exact analytical solutions of (27) and (28) having
precisely this ordering.
3.3 Schrodinger-KDV system
For uni-directional propagation, Nishikawa et al [10] suggested a simpler set
which contains a driven KDV equation instead of the driven Boussinesq equation (23).
Note that the Boussinesq equation has fourth-order space and second-order time
derivatives whereas the KDV equation [43,44] has third-order space and first-order
time derivative terms. The latter equation can be systematically and rigorously derived
by using the reductive perturbation analysis [45,46] on the basic set of plasma
equations describing the low-frequency dynamics. However, it can also be directly
obtained from the driven Boussinesq equation (23) under uni-directional, near-sonic
approximation which allows us to use d/dt d/dx for propagation along the positive
x-direction. Equation (23) after one integration with respect to x then reduces to the
equation
d d\ 1<3 3 30 15 ( EE*\
dt dx J 2 dx 3 dx 23x1 4 /
/ \ /
which is the (driven) KDV equation. This equation is coupled, as in the case of the
Boussinesq equation, to the Schrodinger-like equation (22).
The Lagrangian density for the coupled equations (22) and (29) is given by
t>C \I2jt~i t -j JLj t } ~r* "T" X./ v JLJ .. t~ ZJ^J~,\*J* ' ^Iv^,,! ~T /
O ^ * I f J, X X XI ^ Jf ' "-
where x and are the auxiliary variables, and X = 0. The integral invariants in the
present are
/* + oo
oo
+ 00 Cjo
~'"dx, (32)
whereas the other two invariants are same as (17) and (18).
For stationary solutions, the Schrodinger-KDV set yields the equations
(33)
(34)
171
E a>*
-0.1 -
-0.2 -
-30
-15
Figure 5. Plot of the C-soliton solutions (35) and (36) of the coupled Schrodinge
KDV (or, -Boussinesq) equations for the Langmuir and ion-acoustic waves. The
solutions have only one free parameter (A) whereas the Mach number M
determined from M = 1 20A/3 for the driven KDV case and fro
M = (1 40A/3) 1/2 for the driven Boussinesq case.
which is very similar to the set of equations (27) and (28). Note that (34) can \
directly obtained, as expected, from (28) by using 1 M 2 2(1 M) which is val
forAf-l.
3.4 Exact analytical solutions
The coupled equations (33) and (34) can be solved for exact analytical solutions. FI
simplicity, we shall consider in the following localized solutions satisfying vanishii
boundary conditions. The equations have two free parameters, namely, M and A, ar
one would like to find solutions valid in the entire allowed regions of the paramet
space. However, it has been possible so far to obtain exact solutions valid only (
a straight line denned by the equation, M = 1 10/1/3 in the two-dimensional (M,
parameter space. The solutions are given [10, 11] explicitly by
= E =
</>()=- 6,1 sech 2 ^),
where /i = (A/3) 1/2 , and A2A. Note that (35) and (36) satisfy the orderii
E a = 0(<) = 0(4 Figure 5 shows the plot of () and <j>() obtained from (35) and (3
which are sometimes collectively referred to as the "C-soliton" solutions. It may
noted that (35) and (36) are also exact solutions of the stationary governing (27) and (1
provided the parameter M is determined through the relation, M 2 = 1 20/1/3.
It is interesting to compare the exact solutions of the Zakharoy set with those
the Schrodinger-KDV (or, -Boussinseq) set obtained above. The former have t\
free parameters (M and A) whereas the latter have only one free parameter since
and A are to be related by the equation, M = 1 20A/3. In the Zakharov case, t
hand, 'for the Schrodinger-KDV (or, -Boussinesq) set, the solution for the field
amplitude a (f ) is anti-symmetric with respect to = whereas the </> (f) is symmetric as
earlier. Clearly, in both cases, the solution for the Langmuir field intensity a () is
symmetric but with different shapes: For the Zakharov set, El () is bell-shaped with
only a single-hump whereas for the Schrodinger-KDV (or, -Boussinesq) set, it has
a double-hump structure with the local minimum at the centre (^ = 0) touching zero.
Both the solutions are localized, that is, the fields as well as their derivatives tend to zero
as |f | - oo.
In contrast to (21), the solutions (35) and (36) yield the relation
-7- = 8/.CJ) f </> 2 . (37)
4
Thus, the relative scaling of a and </> are different for the two cases. For (37), the fields
a and 4> satisfy the scaling mentioned earlier, namely, a = &(({)) (9(X\ This is to be
contrasted with the case of the Zakharov solutions where a scales as ^fi. In both
cases, the low-frequency perturbations have the same scaling, namely, proportional to
L Hence, for a given amplitude of the low-frequency density perturbation, the
Zakharov solitons have a higher loading of the high-frequency field than the C-
solitons. This is consistent with the fact that the latter move much faster than the
former. A detailed classification as well as the parameter values for the validity of the
various models for the one-dimensional propagation of coupled Langmuir and ion-
acoustic waves is given by Watanabe and Nishikawa [46],
3.5 Exact nonlinear equations
The equations describing the low-frequency dynamics in the three models discussed
above have been derived perturbatively using certain approximations. For the Zak-
harov set, it is a linear wave equation which is derived under the assumption of
quasi-neutrality. The Boussinesq as well as the KDV equations take into account the
effects due to charge separation through the Poisson equation, but only perturbatively.
Both are derived under the assumption of weak nonlinearity and, hence, are valid for
small amplitudes.
For large amplitude waves, one needs to take into account full nonlinearity as well as
charge separation effects by using the exact Poisson equation. The relevant fluid
equations for the low -frequency dynamics (namely, the continuity and the momentum
equations for the ions, and the Boltzmann distribution for the electrons, together with
the full Poisson equation) allow such a formulation of the problem [47, 48], yielding the
following set of exact stationary nonlinear equations for the coupled Langmuir and
ion-acoustic waves
A/f / l?2\
(38)
(39)
it is easy to veniy tnat (3%) contains, as limiting cases, tne low-irequency equations
used in the earlier models. For |^|, 2 1, we can expand the terms on the right-hand
side of (38) and keep only the most dominant nonlinear terms to obtain
d 2 tf> (1-M 2 )^ (3-M 4 )^ El
r~ ***./ ^ (u ~: (Jj , (T"\J)
The Zakharov case is obtained by neglecting (because of the quasi-neutrality assump-
tion) the left-hand side of (40) and dropping the nonlinear term 2 on the right-hand
side. This yields
S__OJ^!L Mn
4 - M 2 ' (41)
which is just (21). On the other hand, the Boussinesq limit corresponds to taking the
near-sonic limit M 2 -> 1 in (40) which becomes
which is same as (28). The KDV limit for the uni-directional propagation is trivially
obtained from (42), as earlier, by writing 1 M 2 %2(1 M) which is valid for
3.6 Approximate analytical solution
The coupled equations (38) and (39) are highly nonlinear and, as such, their exact
analytical solutions have not been obtained so far. While some numerical work on the
existence of regular as well as stochastic solutions of these equations has been carried
out [49], it is possible to find approximate analytical solutions by following a novel
method [47,48]. Such an analysis has the added advantage that the parameter regimes
for the applicability of the models based on the Schrodinger-KDV (or, -Boussinesq)
equations can be explicitly obtained. The method of solution is fairly straightforward.
We omit, therefore, the relevant details but indicate the main steps and discuss the
results.
Equations (38) and (39) can be derived from the Hamiltonian
(43)
where IT = d^/d(f and Il = (3/2)dE/dc!; are the canonical momenta conjugate to
and , respectively. The Hamiltonian (H) is an "integral of motion", whereas the
associated "kinetic energy" is not positive definite, but can change sign as varies. This
unusual feature is common to most of the problems dealing with the modulational
instabilities, and is further discussed in 5.
1 74 Pramana - J. Phys., Vol. 46, No. 3, March 1 996
Coupled scalar field equations
Using H, (38) and (39) can be combined to yield the "trajectory equation" [47,48]
9 [M(M 2 -
- exp(</> -
- exp(< - \l>]
\ U< /V
- >A)] = 0, (44)
where we have defined \]/ = 2 /4. Equation (44) admits a series solution of the form
00
<A= Z M"> (45a)
where 6 = 0/M 2 . Each of the coefficients b n can be explicitly and uniquely determined
by means of a first order algebraic equation. For localized solutions H ~ (1 + M 2 ),
the first coefficient b is zero whereas the next four coefficients are listed elsewhere [48].
To obtain explicit solutions which are of interest for the present discussion, consider the
two-term approximation to the expansion in (45a), namely
ij/ = b ,9 + b j6 2 . (45b)
E
Figure 6. Langmuir field intensity as obtained from the solution (46b) for the exact
coupled equations (38) and (39). For a given A, there is a transition from single- to
-0.3 -
-15
-10
-5
Figure 7. Low-frequency ion-acoustic wave potential from (46a) associated with
the high-frequency Langmuir field profiles shown in figure 6.
The analytical solutions are then given by
M 2
(46a)
(46b)
where /i = (/./3) 1/2 , and /? t and jS 2 are known functions of the free parameters M and L
Equations (46a) and (46b) can be analyzed for the existence of different types of
localized solutions. For a given A < 1, the solutions 2 () and <() have, for sufficiently
small Mach numbers (M 2 1), single-hump and dip structures, respectively, as shown
by the curves labelled 'A' in figures 6 and 7. This corresponds to the case of the solutions
obtained from the Zakharov equations. When the Mach number is increased (figure 6),
the solution for E 2 () flattens at the top till a critical Mach number M crjt is reached
(curve B). For further increase in the Mach number, 2 () develops a local dip around
the center ^ = (curve C). Thus, the Langmuir field intensity acquires a double-hump
structure which is symmetric with respect to its center. The depth of the dip increases
with further increase in the Mach number, and 2 ( = 0) becomes zero when M is equal
to a cut-off Mach number M cut (curve D). Beyond this value of the Mach number, E 2 ()
becomes negative around = which violates the boundary conditions, and hence the
solutions are not valid for M > M cut .
Figure 6 shows the plot of the profiles of 2 () for a typical value of A and for
different value of the Mach number (M). For a given A, the critical Mach number (M crit )
is calculated from the equation
2 = 0, (47)
b l + b 2 p 2 = 0, (48
as the equation for determining the cut-off Mach number (M cut ) for any given A. Th<
solutions for M = M cut correspond to the solutions obtained from the Schrodinger-
Boussinesq (or, -KDV) equations. In fact, for M = M cul and for sufficiently smal
values of A, the explicit solutions (46a) and (46b) can be shown to exactly yield (35) anc
36) [anti-symmetric () and symmetric (/>()] obtained for the Schrodinger-KDV (or
-Boussinesq) case. For all values of M, the solution for </>() has always a single-dip
symmetric structure whose amplitude increases with the increase in the Mach numbei
reaching the maximum value at M = M cut (figure 7).
Thus, the approximate solutions (46a) and (46b) of the exact coupled equations (38]
and (39) reduce, for near-sonic propagations, exactly to the exact solutions (35) and (36^
of the approximate equations (33) and (34) [or, (27) and (28)]. It should be remarked
here that there is no reason why such a complete equivalence between the two should
exist at all since the approximations involved in the solutions as well as in the equa-
tions are entirely of different nature. On the other hand, the very existence of such an
exact reduction lends support to the suitability of the method of solution used in
solving the coupled equations. The method of solution discussed in Refs [47, 48] is
fairly general and can be easily applied to any pair of coupled, nonlinear ordinary
differential equations wherein the independent variable does not explicitly occur, thai
is, for autonomous coupled equations. For example, it can be used to obtain analytical
solutions of the coupled equations (118) and (119) for quantized, charged solitons
discussed in 5-5.
M
A
A
Figure 8. Parameter values in the (M, A) space for the existence of different types
of coupled Langmuir and ion-acoustic solitons as obtained from the solutions (46a)
and (46b). For a given A, the lines marked 'B' and 'D' yield, respectively, the critical
(M crit ) and the cut-off (M cut ) Mach numbers. The line marked 'L 1 is the plot ol
the equation, M = 1 20A/3. The qualitative nature of the solutions for various
parameter values is shown in figures 6 and 7 by the corresponding letters. Solutions
(46a) and (46b) are not valid in the region 'E' above the line M = M cut .
M
0.75
0.50
0.25
.0.8
o.oq
0.04
0.08
0.12
Figure 9. Contours of constant Langmuir field amplitudes in the (M, A) parameter
space. For a = 0, the corresponding line degenerates into the line M = M cu( of
figure 8. For comparison, the critical Mach numbers (M crit ) are also plotted.
0.95 -
M
0.02
A
Figure 10. Comparison of the parameter values for the existence of the C-soliton
solutions [with anti-symmetric a (c) and symmetric </>()] for different cases. The
curve M = M cut is a plot of (48) for the solutions (46a) and (46b). The line labelled
"L" is a plot of M = 1 - 20A/3 for the driven KDV case, and that labelled "P" is of
M = (1 40A/3) 1/2 for the driven Boussinesq case.
The existence of different types of solutions for the coupled wave fields as given by
(46a) and (46b) can be conveniently summarized by means of the (M, A) parameter
space shown in figure 8. The lines marked 'B' and 'D' represent, respectively, the critical
and the cut-off Mach numbers. The straight line marked 'L' is the plot of the equation
M = 1 - 20A/3 which governs the exact solutions (35) and (36) of the coupled Schrodinger
-KDV system. In the region marked 'E', solutions (46a) and (46b) are not valid since
3 becomes negative. Typical structures of the solutions for parameters in the different
178
rnmann T Dhtrc.
regions A, B, C and D of figure 8 are indicated by the corresponding letters in figures 6
and 7. Figure 9 is another representation of the (M,A) parameter space showing
contours of constant a values. For the value E a = 0, the corresponding contour
degenerates into the curve M = M cut shown in figure 8. For comparison, the values of
the critical Mach number M crit are also plotted.
A comparison of the parameter values for the existence of the C-soliton solutions,
having anti-symmetric a (c^) and symmetric </>() is made in figure 10. The curve
marked M = M cut is a plot of the cut-off Mach numbers as obtained from (48). The
straight line marked "L" represents the equation M = 1 - 20A/3 which corresponds to
the Schrodinger-KDV case, and the curve marked "P" is a plot of Af = (1 40A/3) 1/2
which governs the Schrodinger-Boussinesq case. Clearly, for A ->0, the curves for the
latter two cases coincide with the curve M = M cut . However, for small, but finite, values
of A there are deviations from the M = M cut curve. Since the latter follows from the
solutions (46a) and (46b) which are obtained after retaining an higher-order nonlinear
term in the expanded form of (38) not included in the driven Boussinesq equation (28) or
the driven KDV equation (34), one can easily estimate the validity range for the latter
two cases. For example, it can be shown that the solution (46a) exactly reduces to
solution (36) of the driven KDV case when A 1/36, that is, when M 0-8 15. Similar
limits apply for the driven Boussinesq case also.
The above analysis which is based on the fluid equations shows the existence of a new
type of Langmuir soliton solution which consists of a double-hump structure for the
Langmuir field intensity and single-hump structure for the low-frequency electric
potential. Recently, Lin et al [50] have carried out detailed numerical studies on the
one-dimensional Langmuir solitons using the Vlasov-Poisson system. Their results
show that the Langmuir field intensity does undergo changes as the Langmuir soliton
speed approaches the sound speed. In particular, they have seen the formation of
double-hump structure for Mach numbers near the critical value M cril as predicted by
the theory [48]. For higher values of the Mach numbers, they observe the disappear-
ance of the double-hump structure in the near-sonic regime due to the wave-particle
interactions of the thermal ions with the soliton.
The coupled Schrodinger-Boussinesq (or, -KDV) system of equations occurs in
many different problems in plasma physics where a high-frequency wave is coupled
to a suitable low-frequency wave via the ponderomotive force. For example, it has
been shown that the coupled electromagnetic and ion-acoustic waves [24,25] as
well as the upper-hybrid and the magnetoacoustic waves [23] are also governed by
a Schrodinger-Boussinesq (or, -KDV) system but with different sets of free parameters.
In fact, the latter system with arbitrary free parameters for each of the terms in the two
equations constitutes a general set which, for stationary solutions, yields a generic
system of two coupled ordinary differential equations admitting wider classes of exact
analytical solutions than discussed above. This is discussed in 4.
3.7 Quasi-neutral, finite amplitude waves
The analysis presented above considers the governing equations or their solutions up
to certain order in the wave amplitudes and is, therefore, applicable for finite but small
amnlitude waves. For laree amplitude waves, the full set of nonlinear equations should
\.n\, iu LI^I mating, co-ais \ji laigw auipuiuu^ OLMIIVJUO wituiu 1110 i^uaoi-ii^uii ciuijr ajj|^ivjAi-
m&tion. Under the latter, the left-hand side of the Poisson equation (38) can be
neglected yielding
M(M 2 -2</> ) 1/2 = exp(< --2 =/V , (49)
where N is the common number density and the subscript "0' denotes respective values
at c = 0. Evaluating the "energy integral" (43) at <; =0 together with the conditions,
d/d = d^/dc = at c = 0, (46a) yields
E 2 / 2 \
1 + M 2 = M(M 2 20 ) 1/2 +(1 /.) + expl j. (50)
From (49) and (50), it is possible to determine /. and M 2 in terms of and N as
^o
(52)
These are the "existence relations" derived by Schamel et al [51] for the large
amplitude, quasi-neutral Langmuir solitons.
The effect of charge separation effects which appear through the Poisson equation
(38) can be estimated by considering (49) which provides a relation between E 2 and </>,
yielding
E 2
4
= (M 2 -1)0-|6) 2 . (53)
This can be compared with the two-term solution (45b) which becomes
E 2 f 4M 2 1
= j (M 2 - 1) - (2A + | 2 M 2 ) ^ 6 + M 2 . (54)
Comparing (53) and (54), it is clear that the quasi-neutral solution (53) departs from (54)
already in the linear term. Only in the limit M 2 -+0, that is, for the near-static solutions
does (54) reduce to (53). Thus, for near-sonic (M 2 ~ 1) propagations one should retain
the left-hand side of the Poisson equation (38), albeit perturbatively as done in 3.2
and 3.3.
In the various models for the Langmuir waves considered above, the basic nonlinear-
ity in the governing equations essentially arises due to the low-frequency dynamics. For
example, the Schrodinger-like equation (9) is linear in the field variable E whereas
nonlinearities arise depending on the model equation chosen for JV. On the other hand,
for wave energy density much larger than the electron thermal energy density, the
Langmuir waves become intrinsically nonlinear and, therefore, electron nonlinearities
should in general be included in the analysis. As shown by Zakharov [52], electron
nonlinearities vanish for one-dimensional case but can become important for the
1 80 Pramana - J. Phvs., Vol. 46, No. 3, March 1996
problem is given by Malkin [53].
3.8 Interaction and stability aspects
The various types of nonlinear entities (exact or approximate) discussed above are
stationary in character in which the temporal coordinate appears merely as a trivial
parameter. On the other hand, for realistic physical applications it is essential to
understand the non-trivial time evolution of these solutions as dictated by the basic
governing equations, that is, questions such as the interactions of these entities among
themselves as well as stability considerations become relevant.
Since the basic governing equations are coupled, nonlinear partial differential
equations the initial value problems associated with them become in many cases quite
intractable. In fact, there have been many attempts to prove their "integrability"; the
latter is used to imply that the equations are solvable by means of the inverse scattering
transform (1ST) techniques. The nonlinear Schrodinger equation is known to be
integrable [54]. On the other hand, the Zakharov system is generally believed to be
non-integrable except in two limiting cases: the nonlinear Schrodinger equation and
the Yajima-Oikawa equations [55]. The latest successful attempt in this direction is
the work of Kaup [56] who showed that a particular generalization of the Karpman
equations [18] are also integrable. However, nothing is known about the integrability
of the Schrodinger-KDV (or, -Boussinesq) system though there is an indication that it
may be integrable for the case when the high-frequency field governed by the Schrodin-
ger equation has negative group dispersion (cf. 5.4).
For the coupled Langmuir and ion-acoustic waves governed by the Zakharov
system, Gibbons et al [16] have developed an analytical method for studying
the interactions between the stationary wave solutions. The method is based on
the integral invariants (15)-(18) which the Zakharov equations possess. Since
these invariants have to be conserved during any interactions, it is possible to
obtain the selection rules for the allowed or forbidden nonlinear wave interaction.
For example, such an analysis shows that an initial state consisting of N solitons
cannot decay into ion-acoustic waves since it violates the total plasmon number
conservation. On the other hand, the interaction of a finite number of Langmuir
solitons among themselves yielding another Langmuir soliton and ion-acoustic
modes is allowed.
Numerical work on the Zakharov equations for studying the formation as well as the
interaction of Langmuir solitons and the consequent development of the strong
turbulence has been the subject of many papers [57-61]. Processes such as the fusion of
solitons in binary collisions, interaction of a soliton with a sound pulse and soliton
break-up during interactions have been investigated. Comparisons between the results
of particle simulations and those of the fluid Zakharov model have been discussed by
Pereira et al [59]. On the other hand, Degtyarev et al [57] have modelled strong
Langmuir turbulence as a gas of interacting Langmuir solitons and derived a kinetic
equation for such a gas. Their results from the latter approach are in qualitative
agreement with the predictions of the numerical simulations on one-dimensional
strong Langmuir turbulence. A very recent study by Wang et al [61] considers the
Vlasov simulation of the modulational instability and Langmuir collapse process,
to the so-called ionospheric modification phenomenon. On the other hand, Lin et al
[50] have carried out detailed simulation studies using Vlasov-Poisson equations.
Such simulations are essential to study the wave-particle interactions between the
solitons and the plasma particles. The simulation results show that there is significant
transfer of energy from the waves to the electrons during the heating process.
Furthermore, for larger ion temperatures, the double-hump structure for the Langmuir
field intensity disappears due to the nonlinear interactions of the thermal ions with the
solitons.
Appert and Vaclavik [17] have carried out detailed numerical work on the interac-
tions of the "C-soliton" solutions (35) and (36) of the Schrodinger-KDV system. Their
results show that the C-solitons are more fragile than the usual soliton solutions of the
Zakharov system. For example, two C-solitons strongly interact and destroy each
other fairly quickly during collisions, leading to the emission of low-frequency ion-
acoustic waves. However, the interaction between a Langmuir wave packet and the
C-soliton is relatively mu'ch weaker. Appert and Vaclavik [17] have attributed the
fragility of the C-solitons to the fact that for a given Langmuir field amplitude, the
number density depletion associated with a C-soliton is much larger than that in the
case of the usual Langmuir soliton (cf. 3.4). Thus, the C-soliton appears to behave like
high-frequency wave packet but carried around by a negative acoustic pulse. However,
the latter is not a stationary solution of either the Boussinesq or the KDV equation,
both of which only admit compressive solutions and hence the fragility of the C-soliton
solutions. Recently, Kuehl and Zhang [62] have numerically studied the interactions of
localized solutions of linearly polarized electromagnetic waves in a plasma. They find
that after the collision of the pulses, a trailing wake of plasma oscillations is created,
thereby modifying the original shapes of the solitary pulses. The formation and the
propagation of the plasma wakes is very important for the proposed, future generation
wake-field accelerators. This is a topic which is being increasingly studied in the field of
laser-plasma interactions [63,64].
Stability aspects of the stationary solutions obtained above have also been discussed
in the literature. For example, the Langmuir soliton solutions are unstable with respect
to perturbations in a direction perpendicular to the propagation direction [65,66].
This is to be contrasted with the well-known result that the usual ion-acoustic KDV
solitons are stable with respect to perpendicular perturbations [67]. Since the C-soliton
solutions (35) and (36) of the coupled Schrodinger-KDV system carry a negative
acoustic pulse and are relatively fragile during interactions, it is to be expected that they
may be unstable with respect to the perpendicular perturbations. Such an instability of
the C-solitons was proved by Appert and Vaclavik [68] who showed that the
associated self-focusing instability led to wave bunching in the perpendicular direction.
The growth rate of the instability was found to be relatively large, being proportional to
the square-root of the soliton amplitude. The stability of planar Langmuir solitons
using the full three-dimensional Zakharov equations has also been attempted [69, 70].
These analyses lead to the result that the transverse growth rate of soliton perturbation
is considerably less than the longitudinal growth rate. They also show that the soliton
solutions in cylindrical geometry when described by the full equations are also
unstable.
m me lurm
.dE , dE d 2 E
(55)
and which is coupled to either the driven Boussinesq equation
4 2 2
or, to the driven KDV equation
d d\ , d 3 6 86
(57)
where A t and ^,-, /= 1,2,3,4 are arbitrary free parameters which can take different
values depending on the problem at hand. In (55)-(57), E(x, t) is any (normalized) wave
field which is complex and 0(x, t) is a real, scalar field. In a stationary frame f = x Mt,
the above equations yield, for stationary solutions of the form (26) [cf. 3.2], the
following set of generic equations
PT^b^ + b^E, (58)
=d 1 <i)+d 2( t> 2 + d,E 2 , (59)
where A, /?, b li b 2 ,d 1 , d 2 and d 3 are free parameters suitably defined in terms of A t - and ^.,
i = l,2,3,4.
Equations (58) and (59) constitute a generic set of equations having seven free
parameters. However, by a proper rescaling of the variables, it is possible to reduce the
number of free parameters; this will be considered in a later section [cf. (81) and (82)].
For the present, it is of interest to find exact analytical solutions of the generic equations
valid in as much of the region in the parameter space as possible. This is done by
following the method of solution pointed out in 3.6. As earlier, it is convenient to try
a series solution of the form given by (45a) which, in general, does not truncate.
However, by properly choosing certain curves or surfaces in the parameter space, it is
possible to make the coefficients b n become zero for n greater than certain value, say, m.
The resulting polynomial relationship between E and <p thus becomes an exact solution
of the corresponding "trajectory equation". This relation together with the correspond-
ing Hamiltonian function for the generic system yield different classes of the exact
analytical solutions of (58) and (59). Omitting the details which can be found in [31], we
summarize below the various classes of explicit solutions.
4.1 Exact analytical solutions
The following classes of exact analytical solutions of the generic system of equations
(58) and (59) have been obtained so far [31]. [In all the solutions listed below, the
constant of integration representing the initial phase is taken to be zero.]
Pramana - J. Phys., Vol. 46, No. 3, March 1996 183
(A) Forpd l b 2 -2b 1 (3pd 2 -M> 2 ) = Q [with 31b 2 ^ fid 2 and
=
(61)
It may be noted that the exact solutions (35) and (36) which are valid for the coupled
equations (33) and (34) when M is determined by M = 1 - l(U/3 and for (27) and (28)
when M 2 = 1 20 A/3 are indeed just a special case of the solutions (60) and (61).
(B) For 3Ab 2 - /3d 2 = [with 4Abi
Jh ~H/2
sech(^), (62)
2b, /b,Y /2
= -I s ech 2 (^); A*=hr ' {63)
^2 V/V
(C) For jSdi + 2/.b i = a/id d 2 = [In this case, (59) is linear in 0]
= | 1 ) sech(/^^)tanh(^), (64)
(65)
(D) For /. = [In this case, (59) is singular and yields just an algebraic relation between
E and 0]
For this value of A, there are two cases:
(i) For &2^i 6&jd 2 =
1/7 /I \ I ~
(67)
-
(ii)
= 2 sechMtanh(/ia (68)
(E) Periodic solutions In addition to the above solutions, a new class of periodic
solutions were recently reported by Rao and Kaup [32]. These were found by trying
Q orvllltirtM r\f tho fnrm A\ f _L /"" 17 frvi- ttia "/-/->n-i t-vl am onto r-\r t-o ia^.t /-^i > ,1,-.,, <-J ^." fx-.^
(i) For C = 0, one requires, ).b l -^d l = 0;
(ii) For^! 4- d 2 C = one requires /^ + C (/J> 2 -^ 2 ) = 0.
In both cases, Cj is given by
The corresponding explicit solutions are given by
() = h, sn 2 0< k) + h 2 cn 2 (^, fe), (71)
(72)
where
-> ^2^1/1 , x , > h, h
-- 2
and, /i t , /i 2 and /i 3 are the three real roots of the cubic equation
- +M ^^-
In (71), "sn" and "en" are the Jacobian elliptic functions, and in (74), C is a constant of
integration. The solutions are, in general, periodic but for C - are localized, and given
by
(75)
= C + C 1 (c^); \i 2 = 1 2 . (76)
Note that the solution for 0() is truly localized only when C = 0.
5. Relation to other systems
The generic equations (58) and (59) have relevance to some other equations that are
commonly used in other branches of physics. We consider below some particular
examples.
5.1 Henon-Heiles system
The Henon-Heiles system has been extensively studied in the field of nonlinear
dynamics [71] since its first proposal by Henon and Heiles [72] in connection with
a model for the time-independent gravitational potential of a galaxy with axial
symmetry. It is also obtainable [73] as a cubic approximation to the 3-particle Toda
chain [74, 75] which is a completely integrable system for both the periodic as well as
the fixed-end boundary conditions. Interestingly, in the continuum limit, the n-particle
Toda chain yields the usual KDV equation [76]. Recently, Christiansen et al [77] have
shown that similar equations arise in the ultrasonic Davydov model with anharmonic
terms for phonon oscillations when travelling wave solutions are considered.
The generalized form of the Henon-Heiles equations with arbitrary parameters can
be obtained from the Hamiltonian [28]
H = i(P 2 + P 2 ) + \( A * 2 + By 2 ) + (iCy 3 + Dx 2 y), (77)
where A, 5, C and D are (real) parameters, x and j; are the spatial coordinates, and p x
and p y are the corresponding conjugate momenta. Clearly, H is simply the Hamiltonian
for a two-dimensional harmonic oscillator with certain specific nonlinear terms given
by the last two terms in (77). The standard form first investigated by Henon and Heiles
[72] corresponds to A = B = D = 1 and C = 1. The fundamental question of general
interest in such Hamiltonian systems is to identify the parameter regimes for which the
system is "completely integrable". Since the system is of two-degrees of freedom and the
Hamiltonian is an "integral of motion", the problem is equivalent to finding another
integral of motion with is in involution with the Hamiltonian. The existence of a second
(global) invariant of motion implies by Liouville theorem that the problem can in
principle be solved by quadratures.
By a proper renormalization of the variables, the Hamiltonian (77) can be re-written
in the canonical form, namely
(78)
where p t takes values +1 or 1, and d^ and d 2 are the two free parameters. The
corresponding equations of motion are given by
d 2 x
-- = Pi 3-2*3;, (79)
j,2 ~\j ~J ^ . \W
These equations are known to be integrable [28, 29] for the following three sets of
parameter values:
(B) any cfj, any Pi,d 2 = +6
(C) d t = I6p r ,d 2 + 16.
The question of the separability of the Henon-Heiles Hamiltonian has been discussed
by Ravoson et al [78]. Note that in all the cases d 2 is always positive which indicates
that for the known integrable cases, the nonlinear terms in the equations of motion,
namely, (79) and (80) have the same sign. On the other hand, it is now generally believed
that these are the only integrable cases of the generalized Henon-Heiles Hamiltonian
[rf.5.4].
Consider now the generic coupled scalar field equations (58) and (59) whose variables
can be suitably renormalized to yield the canonical set
d 2
- y = /?,-- 20 , (81)
df
and (82) is given by
3 ), (83)
where Y\ E = (dE/dc,) and n = p 2 d0/d^ are, respectively, the canonical momenta
conjugate to E and </>. Comparing the two Hamiltonians given by (78) and (83) [or, the
equations of motion, (79)-(80) and (81)-(82)] we note that they are structurally very
similar. In fact, for the case when p 2 = + 1, they are exactly same if we identify the field
variables (E, <j5>) with the spatial coordinates (x, y), and the stationary variable ^ with the
temporal coordinate r. However, as pointed out earlier, for most of the problems
dealing with the modulational instability, one finds p 2 = 1 and hence the "kinetic
energy" in (83) is not positive definite.
Thus, the stationary equations governing the nonlinear development of the modula-
tional instability of a high-frequency wave coupled to a suitable low-frequency wave in
plasmas are complementary to the Henon-Heiles equations, but seem to have funda-
mentally different qualitative features. For example, in the case of the usual Henon-
Heiles system (with positive definite kinetic energy), any minimum in the potential
guarantees local oscillatory motion of the particle. This need not be true of the generic
equations since the "kinetic energy" term can change its sign during the motion of the
"particle". Furthermore, integrable cases of the generic equations with indefinite
kinetic energy have not yet been identified except in some special cases [see, 5.3]. The
exact solutions obtained in 4.1 do not, however, guarantee the "complete integrabil-
ity" of the equations since they satisfy specialized boundary conditions and are valid in
restricted parameter regimes. On the other hand, even such specialized solutions which
are valid in the entire allowed regions of the parameter space have also not been
obtained so far.
5.2 Coupled upper-hybrid and magnetoacoustic waves
The Henon-Heiles Hamiltonian has served during the last three decades as the
canonical example of a rather simple looking dynamical system exhibiting a rich
variety of regular as well as chaotic behaviours. However, a direct physical realization
of this model Hamiltonian has not yet been well established. On the other hand, in
dispersive media like the plasmas there exists the interesting case of the stationary
propagation of coupled upper-hybrid and magnetoacoustic waves which for certain
frequency regimes behaves exactly like a Henon-Heiles system. Thus, it is possible to
have a direct physical realization of the latter system in terms of coupled scalar fields
and to determine the parameter regimes over which the coupled system is completely
integrable.
In. a magnetized plasma consisting of electrons and ions, upper-hybrid and the
magnetoacoustic waves are, respectively, the high- and the low-frequency normal
modes both of which propagate exactly perpendicular to the external magnetic
field, say, B = B z. The upper-hybrid modes are governed by the linear dispersion
relation [14,15]
22 1,2 ..2
ucmacuv^, ^ t \i e /iu e ; 10 iuu i^iwunuii 1.111^11110.1 o^v^vu, i^f-iO
ls tne upper-hybrid frequency, and all the other symbols have their
usual meanings [23]. In the long wavelength limit, (84) can be approximated by
where
3co 2 e0 u 2
denotes the group dispersion coefficient for the upper-hybrid waves. Clearly, the latter
have positive (negative) dispersion for plasma parameters such that co 2 e0 > 3Q 2
(co 2 e0 <3Q 2 ? ).
For nonlinear propagations, the slowly varying complex amplitude E(x, t) of the
upper-hybrid wave electric field is governed by a Schrodinger equation of the form [20]
dt ' ' 8 dx/ ' " ' '"--' (87)
where K g = dco /dk = k D denotes the group velocity, ju = i(co 2 c0 + 2Q 2 )/(co 2 e0 + Q 2 )
and the normalized low-frequency density perturbation N(=6n e /n Q ) of the magneto-
acoustic waves is governed by the driven Boussinesq equation [23]
ri 2 \
(88)
dt dx dx dx dx
where, K M = (V\ + C 2 ) l/2 is the magnetoacpustic speed, F A = (BQ/47i m.,) 1/2 is the Alfven
speed, C s = (r e /mi) 1/2 is the ion-acoustic speed, 9 = cVJco p( . Q , a 2 = (3K 2 +2C s 2 )/2, and
r l a} n C../co pe0 . It may be noted that for uni-directional propagation (.88) can be
reduced to a KDV equation of the type (29) but with different coefficients.
For stationary propagation of the wave fields, (87) and (88) yield the coupled
equation [30]
2 N, (89)
where A = 26 + (M 2 V 2 )/D is the nonlinear shift parameter, 6 = dT/dt denotes the
shift in the wave frequency, b 2 = 2jUco HO , and ^ = x Mi represents the coordinate in
the stationary frame whose speed is determined by the free parameter M, the Mach
number. Equations (89) and (90) can be reduced to a standard form by normalizing
N and E 2 , respectively, by - 2D D/b 2 and I6im T e (2\D \D/b 2 }. For the case of the
negative dispersion (Z) < 0) of the upper-hybrid waves, the coupled equations become
=-AE-2DEN, (91)
= - BN - CN 2 - DE 2 . (92)
d<r
d 2 /v
T DU,,o 17^1 A NT 1 IV /I ....^L. 1 nn/C
where,
Equations (91) and (92) can be derived from the Hamiltonian
H + = (Tl 2 + nj) + 1(^ 2 + 5W 2 ) + (|CN 3 + DiVE 2 ), (94)
where U E = d/d and U N = dN/dt, are, respectively, the "canonical momenta" conju-
gate to E and N. Clearly, the Hamiltonian H + is identically the same as the generalized
Henon-Heiles Hamiltonian (77).
On the other hand, for positive dispersion (D > 0), the coupled equations are
(95)
' =-BN + CN 2 + DE 2 , (96)
which can be derived from the Hamiltonian
H_ = i(H| - II 2 ) - i (AE 2 + BN 2 ) + (CN 3 + DNE 2 ), (97)
where the canonical momenta, for the present case, are given by U E = d/d and
TI N = dN/dt;. Apart from the trivial sign changes for the coefficients A and J3, the
Hamiltonian H + differs from H_ in an important way: the "kinetic energy" in H + is
always positive definite (like in all the Hamiltonians in classical dynamics) whereas in
H_ it is not.
It should be remarked here that exact analytical solutions of (91) and (92) as well as
(95) and (96) for certain parameter regimes and/or for specific boundary conditions can
be obtained as in 4.1. However, in view of the similarities of the Hamiltonians H + and
H _ to the generalized Henon-Heiles Hamiltonian (77), it is possible to enquire into the
more general question of their complete integrability which is discussed in the next
section.
We close this section by briefly discussing the case of wave propagation along an
external magnetic field. For quasi-parallel propagations, Alfven waves are governed by
derivative nonlinear Schrodinger (DNLS) equation which is known to be integrable.
On the other hand, for finite plasma-/? and for finite wave amplitudes, Hada [79] has
shown that the sound wave as well as the left- and the right-hand polarized Alfven
waves are governed by a new set of coupled equations of the form
8b .d 2 b d
dt dx 2 dx*
1 \
= 0, (99)
where b is the wave magnetic field and p is the perturbed plasma number density.
Clearlv. (98) is a generalization of the DNLS equation whereas (99) is driven KDV
N N Rao
above set of equations has been explicitly carried out by Hada [79], it may be possible
to obtain their exact stationary solutions as discussed in 4. The question of the
complete integrability of this set of equations may also be investigated in close analogy
with the discussion in the next section.
5.3 Parameter regimes for complete integrability
For complete integrability of the Hamiltonian H + or //_, there must exist a second
integral of motion (/) which is in involution with the Hamiltonian, that is, the Poisson
bracket {H + ,/} must vanish. For this purpose, it is convenient to consider a general
Hamiltonian (with arbitrary parameters A, B, C and D) of the form
(100)
where p = 1 and which contains the Hamiltonians H + and H_ as special cases. The
governing equations of motion corresponding to the Hamiltonian (100) are given by
^=-AE-2DEN, (101)
(102)
We have been able to obtain the second integral of motion for the following sets of
parameter values:
Case (1): Arbitrary A, B and C = 6pD
U N + AEN + ^DE 2 + DN 2 E\ - 4NY1 2
3 D
Case (2): B = pA and C = pD
(103)
, (104)
Case (3): B = l6pA and C = l6pD
(105)
(\
pU E U N + AEN + ^DE 3 + DN 2 E\,
where n == d/d^ and U N = pdN/d are the canonical momenta conjugate to E and N,
respectively. It may be verified by direct substitution that the Poisson bracket {H , /} is
identically zero in each case. For the case p = + 1, we recover, as expected, the well-
waves, the parameters C and D are positive definite by definition, and hence the above
results can be directly applied for the case of the negative group dispersion [30] which
corresponds to p = + 1.
To obtain explicitly the various plasma parameters for the complete integrability of
(91) and (92), it is appropriate to define the dimensionless parameters, a = co pe0 /Q e0 ,
/? = (C S /F A ) 2 and y-v te /c. In the parameter space spanned by (a, /?, y, A, M) the
integrable regimes are governed by the equations
3a 4 y 2 (3 + 2$ = v(2 + a 2 )(3 - a 2 ), (106)
and
3a 4 y 2 (l + 0)(1 - M 2 ) = vA(l + a 2 )(3 - a 2 ), (107) '
where A = /l/co HO is the normalized nonlinear shift parameter, M is the Mach number
normalized with respect to V M and v takes values 1, 6, or 16. Equation (106) is obtained
from the relation C = vD whereas (107) follows from B = vA. Note that the parameter
/? is essentially the usual plasma beta, that is, the ratio of the thermal pressure to the
magnetic field pressure.
For integrability of the coupled equations (91) and (92), both the conditions (106) and
(107) should be simultaneously satisfied for the case when v = 1 or 16, whereas only the
condition (106) needs to be satisfied when v = 6. Since f$ = <x 2 y 2 , (106) yields
3f . f. 8v(2 + a 2 )(3-a 2 )l 1 / 2 !
' 27 a 2
whereas (106) and (107) together yield
A __l-M 2 _(l+a 2 )(3 + 2fi)
A _ "7T~i 2\/i , n\ 009)
a
Figure 11. Parameter regimes obtained from (108) for the integrability of station-
ary, coupled upper-hybrid and magnetoacoustic waves with negative group disper-
sion in a magnetized plasma. For v = 6, (91) and (92) are completely integrable for
anv values of M and A. For v = 1 and 16, the latter parameters are determined from
A
Figure 12. Plot of A as a function of a 2 from (109) for the integrability of (91) and
(92) for the case of v = 1 and 16.
i.oo
0.75 -
0.00
a
Figure 13. Plot of y 2 = /?/a 2 as a function of a 2 for the three known integrable cases
of coupled upper-hybrid and magnetoacoustic waves with negative group disper-
sion. For a given a, value of /? is calculated from figure 1 1.
Figure 1 1 shows the regions for integrability in the (a, /?) parameter space for different
values of v. For v = 6, the system of equations (91) and (92) is completely integrable for
parameters given by the corresponding curve (v = 6) in figure 11. Note that for this
value of v, any arbitrary values of M and A are admissible. On the other hand, figure 12
gives a plot of A as a function of a 2 from (109) for v = 1 and 16. The corresponding
values of the parameter y for integrability are given by y 2 = j3/a 2 (figure 13) whereas the
parameters M and A are no longer arbitrary but are related by (109).
It follows from figures (!!)-( 13) that the coupled equations (91) and (92) are
completely integrable in relatively small regions of the entire allowed parameter space.
(negative) values of the frequency shift parameter A, the governing equations (91) and
(92) are integrable for sub-magnetoacoustic, that is, for M < 1 (super- magnetoacoustic,
M > 1 ) values of the Mach number M.
5.4 Stationary flows of nonlinear evolution equations
As discussed earlier, the classical generalized Henon-Heiles Hamiltonian is completely
integrable for three sets of parameter values. While this result has been known for quite
sometime [28, 29], later attempts have not led to any other integrable cases. On the
other hand, a detailed investigation of the generalized Henon-Heiles equations by
means of the Painleve analysis shows that their solutions would have the so-called
"P-property" only for these parameter values, thereby ruling out any additional
integrable cases. Recently, Fordy [80] has pointed out an interesting connection
between the known integrable Henon-Heiles cases and the stationary flows associated
with the only integrable nonlinear evolution equations belonging to a particular class.
This result further supports the observation that there are possibly no other integrable
cases of the Hamiltonian H + . Such an analysis can also be applied to the Hamiltonian
H _ , or, more generally, to the Hamiltonian H given by (100).
Following Fordy [80], we differentiate (102) twice with respect to , and use (100) and
(101) to eliminate the dependent variable E completely. This yields the fourth-order
equation
+ (4pDH) = 0. (Ill)
In order to compare the left-hand side of (111) with the integrable time-evolution
equations, it is convenient to define the notations
3U dN d 2 U d 2 N
U = N, ^- = ^ V = T7' -TT = U yy = -lK2<-~'
dy df dy 2 " d 2
Without loss of any generality, the coefficients A and B of the linear uncoupled terms
in (101) and (102) can be taken to be equal to zero [80]. Equation (HI) then takes
the form
(113)
where the definitions (112) have been used. This equation has the "scale symmetry" that
it is invariant (except for the constant term) under the transformations U -> U/a 2 and
y -> ay for any constant a [8 1 ] .
Pramana - J. Phys., Vol. 46, No. 3, March 1996 193
type with the above scale-symmetry have the form
V, = C^,,,, + ;-i VU yy + iU 2 - A, )(l/,,) 2 + i/ 3 L/ 3 ],, (114)
where / M , A 2 and / 3 are constants. Note that under the above scalings, the right-hand
side of (114) remains invariant except for an overall multiplying factor of a 7 ; that is,
(114) has a "scale- weight" of 7. Furthermore, there are only three equations belonging
to this class that are completely integrable [82]. These are given by
(1) Lax's fifth-order KDV equation
J7, = [C/ ywy + 10C/C/ yy + 5(l/,) 2 + 10t/ 3 ] v , (115)
(2) Sawada-Kotera equation
(3) Kaup-Kuperschmidt equation
(117)
For stationary flows, that is, for U, = 0, each of the above three equations yields,
[] = const. On the other hand, it is easy to verify that (1 13) reduces to each of these
equations for such values of C and D whose ratios are exactly same as those for the
integrable cases of the generic Hamiltonian H listed in 5.3. (Note that for integrability
only the ratios of C and D are important.) Thus, the known integrable cases of the
generic Hamiltonian (100) correspond precisely to the stationary flows associated with
the only integrable nonlinear evolutiorr equations (of polynomial and autonomous
type) with a scale-weight of seven. This correspondence seems to suggest that there are
possibly no other integrable cases of the generic Hamiltonian H given by (100).
The above result has relevance to the question of integrability of the coupled equations
(95) and (96) for the upper-hybrid and magnetoacoustic waves with positive group
dispersion. Tne corresponding Hamiltonian H. given by (97) corresponds to p = 1 in
the Hamiltonian H of ( 100), together with trivial sign changes for the coefficients A and B.
From 5.3 it follows that for complete integrability of the Hamiltonian H, a necessary
condition is that C oc pD which for p = 1 requires that C and D should have opposite
signs. However, for the coupled waves the parameters C and D are, by definition,
positive definite and hence the integrable cases of H obtained in 5.3 are not applicable
to the coupled waves having positive group dispersion. Thus, if the above result strictly
holds good, then, it appears that there are no integrable cases for the coupled
upper-hybrid and magnetoacoustic waves when the group dispersion is positive.,
It is of interest to discuss briefly here the connection between the integrability of
partial differential equations (PDEs) and that of ordinary differential equations
(ODEs) which are obtained from the former by an exact reduction. [As mentioned
earlier, an ODE is said to be integrable if it has sufficient number of involutive, global
invariants of motion whereas a PDE is integrable if it is solved by means of the inverse
scattering transform (1ST) technique.] In a series of papers, Ablowitz and co-workers
[83, 84] have discussed these aspects in detail. In particular, they have conjectured that
every nonlinear ODE obtained by an exact reduction of a nonlinear PDE belonging to
the 1ST class has the so-called Painleve property [85, 86]. [An ODE is said to have the
Painleve property if the critical points (namely, the branch points and the essential
singularities) of its solutions do not move in the complex plane, that is, they do not
depend on the constants of integration of the ODE. Thus, an ODE belongs to the
Painleve class if poles are the only movable singularities in its solutions.]
The importance of the above conjecture lies in the fact that it is generally believed that an
ODE having the Painleve property is integrable though there exists no explicit proof. As an
interesting consequence of this conjecture, it should be possible to test the 1ST nature of
a given PDE by analyzing the associated ODE for the Painleve property which is relatively
easier. Consider now the generalized Henon-Heiles system which is known to possess the
Painleve property for the three known integrable cases discussed earlier. On the other
hand, the coupled Schrodinger-KDV (or, -Boussinesq) system yields in the stationary
frame exactly the generalized Henon-Heiles system for the case when the high-frequency
carrier wave governed by the Schrodinger equation has negative group dispersion. Thus, if
the conjecture of Ablowitz et al [83, 84] is indeed true, then the Schrodinger-KDV (or,
-Boussinesq) system with negative group dispersion should belong to the 1ST class and
hence integrable. Further work needs to be carried out to check this possibility.
5.5 Self-dual Yang-Mills system
The existence of certain classes of coupled nonlinear field equations having exact
analytical solutions such as solitons, kinks and instantons has received much attention
in the field of particle physics [35]. Many attempts have been made to construct
quantum field theories wherein these classical nonlinear entities serve as the states
around which excited quantum states could be constructed [36-40]. For example, in
the context of quantizing charged solitons, Rajaraman and Weinberg [40] proposed
for 1 + 1 dimensions the following coupled system
A2~
(118)
r 2 -l)p, (119)
where cr() and p() are real scale fields, and /, X and d are the free parameters.
Equations (11 8) and (119) are structurally similar to the generic equations (58) and
(59), and admit an Hamiltonian of the form
Different classes of exact analytical solutions of (118) and (1 19) have been reported in
the literature [36-39]. In particular, they admit non-topological soliton solutions
which are static as well as uncharged. As earlier, most of the solutions obtained so far
are valid in restricted regions of the space spanned by the free parameters (/, A, d) and/or
satisfy specialized boundary conditions. The method of solution cited in 3.6 can be
applied also to these equations. On the other hand, for special choice of the orbit
equation like (45b) as well as the parameter regimes it may be possible to obtain other
: c A 1
In the last few years, there has been much interest in the reduction of the classical
Yang-Mills field equations to simpler nonlinear evolution equations such as the
nonlinear Schrodinger and the KDV equations by using certain symmetry proper-
ties.The classical Yang-Mills equations are written in the form [41,42]
D,,G /n , = ^G, /v + [/l,,G, v ]=0, (121)
where
G ftv = d tl A v ~d v A, + lA^A v l (122)
Here, [A lt , A v ~] is a suitable Lie bracket defined over the Yang-Mills field variables (AJ,
and the subscripts for "<T denote the respective partial derivatives. The Yang-Mills
fields are said to be self-dual if the condition
G, lv = iw G P* = + G /iv> ( 123 )
is satisfied where F,^, pa is the usual anti-symmetric tensor. Solutions satisfying the above
condition satisfy also the Yang-Mills field equations.
Using the gauge degree of freedom implied by the self-dual condition, it has been
recently shown [41,42] that the Yang-Mills field equations can be reduced either to
the nonlinear Schrodinger or to the KDV equation. Since nonlinear Schrodinger
equation can generally be thought of as a special case (namely, the static limit) of the
Zakharov or the Schrodinger-Boussinesq (or, -KDV) system, one would expect to be
able to reduce the self-dual Yang-Mills system to the Schrodinger-Boussinesq (or,
-KDV) system. Since the latter system is known to yield, for stationary solutions, the
generalized Henon-Heiles system, this would establish a cascading connection from
the Yang-Mills to the Henon-Heiles system via the Schrodinger-Boussinesq (or,
-KDV) system. It would also provide a good model to study the "nonlinear dynamical"
behaviour of classical Yang-Mills fields. While such a possibility is quite exciting, even
a formulation of the problem is yet to be carried out.
5.6 Complex KDV equation
The peculiar nature of the Hamiltonian (100) for the generic system of (101) and (102)
admitting the case when the "kinetic energy" term is indefinite is exhibited also by the
usual KDV equation when the dependent variable is made complex [33]. Consider the
KDV equation
du du
n n
+ au + = 0, (124)
dt dx dx 3 v '
where a and are the free parameters, and all the variables are real quantities. For the
stationary solutions depending on a single variable = x Mt with one free parameter
M,( 124) yields
d 2 u
j5^ TI = Mu-|aw 2 . (125)
The Hamiltonian for the above equation is
^04 (126)
V(u)= -iMu 2 + ^aw 3 . (127)
We now "complexify" the KDV equation by making the dependent variable u complex
and write, u = u l +iu- ) . Equation (125) then yields the set of equations
P = Mu l -^(u 2 l -u 2 ), (128)
(^ = (M-au l )u 2 . (129)
Accordingly, the potential becomes, V(u)-* K(u 1 ,u 2 )= K 1 (u 1 ,w 2 ) + iV 2 (u l ,u 2 )
where
M*) +ia(uj - Zu^l\ (130)
F 2 = - Muj u 2 -a(Hl - 3? 2 ), (131)
and the Hamiltonian H-^H l + \H 2 where
dtt
(132)
IM (133)
By identifying t^ with and u 2 with E, we note that (128) and (129) are structurally
similar to the generic equations (101) and (102).
Equations (128) and (129) can also be written in the form
MI 34)
'
which involve only the potential V^u^uJ given by (130). Note that unlike the
equations of motion for a classical particle with two degrees of freedom, the second
equation has a positive sign for the derivative on the right-hand side. This is a reflection
of the fact that the kinetic energy is indefinite.
Using the Cauchy-Riemann conditions, namely
^J: = ^2 ^JL = _^2 (135)
du^ du 2 ' du 2 du^
equation (134) can also be written in terms of the potential V 2 (u lt u 2 ) given by (131)
B^= ^, fl_!^=__-2. (136)
d Su 2 Q.C, ou i
While both the equations have now negative signs for the derivatives on the right-hand
sides (like in the case of the equations of motion in classical dynamics), the equation for
u l has derivative of V 2 with respect to u 2 , and vice versa for u 2 .
The above features of the "complexified" KDV equation are also exhibited by
equations obtained from classical dynamical systems with one-degree of freedom by
making the dependent variable complex [34]. Consider a one-degree of freedom
conservative system defined by the Lagrangian
L(q,q) = q 2 - V(q), ' (137)
where dot denotes the time derivative, and the corresponding Hamiltonian
H(q,p) = _p 2 + V(q\ (138)
where the conjugate momentum (p) is defined by p = q. Clearly, the equation of motion is
*--^. (139)
df 2 dq
Let us now make the dependent variable complex and write, q = qi+iq 2 - Accord-
ingly, the conjugate momentum becomes complex, that is p->p 1 -I- \p 2 . Then, (g^pj
and (q 2 ,p 2 ) constitute canonically conjugate variables. Under complexification, the
potential V(q) becomes complex, that is, V(q)-* V(q l ,q 2 )= V 1 (q 1 ,q 2
where V l and V 2 satisfy the Cauchy-Riemann condition
dVdV dV 8V
a ' 3
Wi <3<7 2 oq 2 dq l
Similarly, the Lagrangian and the Hamiltonian yield, respectively
(g^gj] +i[4 1 4 2 - V 2 (q l ,q 2 )'],
(141)
1 te 1 ,g 2 )] + i[p 1 /> 2 + V 2 (q l ,q 2 )~\.
(142)
Note that the kinetic energy term in L l and H l is not positive definite.
The Newton's equations of motion can be obtained from the usual Euler-Lagrange
equation using the Lagrangian L a . This yields
2
( }
dt 2 dqC df 2 dq 2
Note that here only the potential K t is involved and the second equation has a positive
sign before the derivative on the right-hand side. On the other hand, one can use the
Lagrangian L 2 to obtain the equations in terms of V 2 only
dV 2 d 2 q 2 dV.
dt 2 dq 2 dr 2 dq,' (144)
Here, both the equations have negative signs before the derivatives on the right-hand
sides. However, the equation for q\ is defined in terms of the derivative of V 2 with
respect to q 2 , and vice versa for q 2 . Similar features were also noticed in the previous
section for the complex KDV equation. As expected, the two sets of (143) and (144) are
identical in view of the Cauchy-Riemann conditions (140).
motion. Using Hj in the usual Hamilton's canonical equations of motion, we obtain
dV,
dV
4 2 = -P2i P2=-jrr-
uq 2
Note that if we eliminate p from the first set and p 2 from the second set of equations, we
recover the equations of motion given by (143). On the other hand, if we use, instead, the
Hamiltonian H 2 , we obtain the equations
dV,
\ y /
<? 2 =pi, p* = -ir- (148)
42
Unlike the previous case, here cross mixing of the equations is necessary in order to
eliminate p l and p 2 which leads to the equations of motion given by (144).
6. Summary and outlook
To summarize, we have presented a review of the generic features as well as some classes
of exact analytical solutions of coupled scalar field equations encountered in problems
dealing with the nonlinear development of the modulational instability of a high-
frequency carrier wave coupled to a suitable low-frequency wave in dispersive media
like plasmas. Depending on the strength of the wave amplitudes, the time evolution of
the instability is described in terms of various model equations such as the Zakharov
system, the Schrodinger-KD V system and the Schrodinger-Boussinesq system. These
equations are applicable to such coupled wave systems in plasmas as Langmuir and
ion-acoustic waves, upper-hybrid and magnetoacoustic waves, and electromagnetic
and ion-acoustic waves. For stationary propagation, the time-evolution equations
yield a generic system of coupled equations with many free parameters. These
equations admit different classes of exact analytical solutions which are valid in
different regions of the parameter space. We have also given a comparison of the
different model equations and their solutions by considering the specific example of
coupled Langmuir and ion-acoustic waves in plasmas. The parameter regimes for the
validity of the various model equations can thus be obtained.
The generic equations are derivable from a Hamiltonian which, in most cases, has the
unusual property that the associated kinetic energy is not positive definite. We have
then reviewed the nonlinear dynamics of coupled wave systems by taking the example
of upper-hybrid and magnetoacoustic waves in magnetized plasmas. For the coupled
waves with negative group dispersion, the generic Hamiltonian exactly reduces to the
classic generalized Henon-Heiles Hamiltonian and hence is integrable for three sets of
parameter values. On the other hand, for positive group dispersion, the equations lead
to a novel Hamiltonian of the generalized Henon-Heiles type but with indefinite
N N Rao
kinetic energy. We have also discussed an interesting connection between the known
integrable cases of the generic equations and the only integrable evolution equations of
a particular class. The existence of such a connection indicates that there are possibly
no other integrable cases of the generic equations. The relevance of the latter equations
to other systems such as the self-dual Yang-Mills equations, complex KDV equation
and complexified classical dynamical equations is also discussed.
The topic under review has many open problems which need further investigations.
First, the existence of other classes of exact solutions of the generic equations needs to
be explored. For example, in the simplest case of the coupled Langmuir and ion-
acoustic waves, there are two free parameters whereas it has been possible to only
obtain solutions valid on a line in the two-dimensional parameter space. Even though
the basic generic equations have symmetry properties which admit for the high-
frequency wave amplitude both symmetric as well as anti-symmetric solutions, only the
latter have been found in the case of the coupled Schrodinger-Boussinesq (or, -KDV)
system. Second, for the generic equations, it is necessary to look for solutions which
admit wider classes of initial/boundary conditions than has been used so far. Third, on
a more basic question, the existence of other integrable cases of the generic equations
needs to be explored, particularly for the case when the associated Hamiltonian has
indefinite kinetic energy. Fourth, the question of the integrability of coupled upper-
hybrid and magnetoacoustic waves with positive group dispersion is still unanswered.
There are, however, indications that there may not be any integrable cases at all. But,
more concrete work is needed before definite conclusions can be drawn. Fifth, the
existence of the three integrable cases with negative group dispersion seems to suggest
that the corresponding Schrodinger-Boussinesq (or, -KDV) system may belong to the
1ST class and hence integrable. Finally, there exists the possibility of reducing self-dual
Yang-Mills field equations to the coupled Schrodinger-Boussinesq (or, -KDV)
system and hence to the Henon-Heiles system. This may provide a model for studying
the nonlinear dynamics of the Yang-Mills fields.
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rRAMANA (Q Printed in India Vol. 46, No. 3,
journal of March 1996
Physics pp. 203-211
Self-interacting one-dimensional oscillators
MAMTA and VISHWAMITTAR*
Department of Physics, Panjab University, Chandigarh 160014, India
* Author for correspondence
MS received 26 October 1995
Abstract Energy eigenvalues and <x 2 > n for the oscillators having potential energy
V(x) = (co 2 x 2 /2) + A < x 2r > x 2s have been determined for various values of A, r, s and n using
renormalized hypervirial-Pade scheme. In general, the results show an improvement over the
findings of earlier workers. Variation of the evaluated quantities and of the renormalization
parameter with A, r, s and n has been discussed. In addition, this potential has been employed as
an illustrative example of the applicability of alternative formalism of perturbation theory
developed by Kim and Sukhatme (J. Phys. A25 647 (1992)).
Keywords. Self-interacting oscillators; energy eigenvalues; perturbation theory.
PACS No. 03-65
1. Introduction
There are a number of important and interesting situations in physical and biological
studies where the interaction between the system and its surroundings influence not
only the former but also the latter. This results in the modification of environmental
field so that the interaction potential acting on the system also depends upon its own
state \j/ n and is, therefore, written as V(\}i n }. Accordingly, the time-independent
Schrodinger equation for the system becomes non-linear and reads [1-3];
W.W, = [#<>+ WJ]^ = n A n (1)
with H as the Hamiltonian for the isolated system. A typical example of such a self-
dependent system is one-dimensional oscillator described by the Hamiltonian (h m = 1)
H= ~(!/2)(d 2 /dx 2 ) + (o) 2 x 2 /2) + A<x 2r >x 2i , (2)
where both r and s are integers. Obviously, when r = s = 1, (2) becomes Hamiltonian of
a self-interacting harmonic oscillator whose force constant is linearly dependent on the
mean square of displacement of vibration, and r^ 1, s^2 correspond to the self-
interacting anharmonic oscillators whose anharmonicity depends on <x 2r >. Also for
co = 0, r>l,s^2we have the self-dependent oscillators with extremely large magni-
tudes of anharmonicity - the so-called infinite-field limit of the oscillators. Further-
more, for r = and s = 2 or 3 we get the special cases of well-studied quartic or sextic
anharmonic (when CD ^ 0) and pure quartic or sextic (co = 0) oscillators.
perturbation theory (RSPT) has been put forward for this purpose and applied to the
problem of a molecule in a polarizable medium [1,3,4]. Surjan [3] also pointed out
limitations of iterative method, configuration interaction approach and variational
technique for solving the time-independent non-linear Schrodinger equation. How-
ever, since perturbation theory suffers from the uncertainty about its convergence,
particularly for large perturbations, Cioslowski [5] employed an extension of connec-
ted moments expansion to obtain the solution for (1) and illustrated it by rinding the
values of energy and mean square displacements of vibration <x 2 > for the ground state
(n = 0) of self-interacting harmonic oscillator for different magnitudes of L Vrscay [2]
analyzed the perturbation expansion for (2) using hypervirial and Hellmann-Feynman
theorems and performed numerical calculations to determine lower and upper bounds
to energy for different values of r and s = 2 corresponding to co = as well as <D ^
employing a renormalized RSPT wherein summation was carried out by the Fade
approximants method.
Killingbeck [6-8] presented a variational parameter-based renormalized hyper-
virial Fade scheme (RHPS) that yields very accurate energy eigenvalues (and also <x 2 >
values) in a simple manner for anharmonic oscillators. We examined the extent to
which this technique is successful in finding the energies and the expectation values of
x 2 for self-interacting harmonic and anharmonic oscillators and this communication is
an outcome of the effort for a wide range of values of r, s, 1 and n. We shall discuss in the
sequel the dependence of energy, {x 2 > and the variational parameter on various
quantities.
It is pertinent to mention that Kim and Sukhatme [9] have developed an alternative
formalism for Rayleigh-Schrodinger perturbation theory (ARSPT) for a one-dimen-
sional problem described by linear Schrodinger equation. Being an expansion in the
powers of perturbation parameter, this leads to the same results as RSPT but without
requiring cumbrous sums over intermediate eigenstates and also reduces to the
logarithmic perturbation theory of Au and Aharonov [10]. We have also obtained
expression for energy of the self-interacting oscillators in the framework of this
approach and compared the results with the findings of RHPS.
2. Renormalized hypervirial-Pade calculations
2.1 Theoretical framework
Following Killingbeck's [6-8] prescription, the Hamiltonian (2) is renormalized by
adding and subtracting (!Kx 2 /2) so that denoting (co 2 + 1K) 1/2 by co', we get
H = (- l/2)(d 2 /dx 2 ) + (a>' 2 x 2 /2) + A<x 2r >* 2i ' - (!Kx 2 /2). (3)
Using the hypervirial theorem together with the relevant commutation relations,
expanding E n and <X N > as
E = Y Etfti,
fi / -j n "
, we uuiam LUC iccuiiciiuc icia.uon
;=o
- [((2(AT + 5 + !))/((# + 2)co' 2 )] V Cj?:> fc _ j
k =
+ [(N - 1)N(N + 1)/4(JV + 2)co' 2 ]C ( p N - 2) . (6)
Here
C< 0) = <5o P (7)
and
'. (8)
The E ( ^ for; ^ 1 are related with C ( " } by using the Hellmann-Feynman theorem and
this gives us
(9)
Equations (8) and (9) when substituted in (4) give E n in terms of K and C( N) and the latter
are determined from (6) in a hierarchial manner. Similarly, substitution of (6) in (5) with
N = 2 leads to the formula for evaluation of <x 2 > n . It may be pointed out that the
product terms C ( fL } k _ j C[ N + 2s) in (6) and C ( k 2r} C<- 2 _ s) fc _ l in (9) allow the calculation to treat
the term /I<x 2r >x 2s in (3) directly, without any need for iteration of any kind.
2.2 Numerical results and discussion
In order to execute the computations the parameter K is determined in such a way that
maximum number of stable digits in the value is obtained from the final expression for
E n for a particular state. Also it is found that
for p, M = 0, 1, 2, 3, , which is a consequence of even power terms in the potential. The
computations for energy as well as <x 2 > have been executed with double precision and
by terminating the series summation in (4) at j = 32; though in some cases, particularly
those for which X is large, summation had to be extended up to j = 52 (or even 80) to
obtain proper convergence. However, in all cases the values being reported correspond
to a situation for which the maximum number of digits was stable. The energy values so
determined (EJ and those improved upon by finding Fade approximants, E n (P), to the
series expansion with chosen K parameters are listed in tables 1 -3 for different values of
r, s, n, a) and L Also projected in these tables are the relevant <x 2 > H obtained by
performing sum in (5) by the Fade approximants. With a view to compare our results
with those of other workers, their values have also been included in these tables. It may
be mentioned that the entries in the tables ; have been kept up to such an unrealistic
number of significant figures just to emphasize the degree to which results of the present
calculations can be trusted.
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the following conclusions:
(1) In self-interacting oscillators and the infinite-field limit oscillators (tables 1 and 3)
wherever comparison with the findings of other workers is possible, our results for
E n (P) as well as <.x 2 >,, are more accurate. This prompts us to infer that other values
found by RHPS are also quite reliable for these categories of oscillators. As far as
self-interacting anharmonic oscillators with s = 2 (table 2) are concerned, E (P) are an
improvement over the results of Vrscay for r = 1 -0 for A up to 1 0-0 and for r = 2-0, 3-0 for
A up to 1-0, in addition to being so for r = for all L However, in all other cases, the
convergence of E (P) is worse than the two bounds of energy reported by Vrscay.
Furthermore, such a comparison is not possible for <x 2 > as these values have not been
determined by Vrscay. Nonetheless, the extent to which these sums are convergent,
they too must be correct.
(2) For self-interacting oscillators, the renormalization parameter K is zero for A up to
0-1 and 0-5 for A values ranging from 0-5 to 10,000. Thus, for very low A values even
ordinary hypervirial-Pade or simple hypervirial method is successful. In the case of
anharmonic oscillators whether co is or not, K does not vary significantly or
monotonically with A, r or n. For a particular set of values of s and r, it decreases with
increase in n; and for specific values of r and n, K increases if s is increased.
(3) If we consider a particular type of oscillator with to ^ then the ground state energy
increases while <-x 2 > decreases with increase in the contribution of the perturbation
term, i.e., in the value of A. For a particular A, the values of E and <x 2 > for r = are,
respectively, higher and lower than the corresponding values for r^Q cases though for
r = 1, 2, 3 the variations of E as well as <x 2 > with r do not exhibit a well defined
trend. In the case of co = 0, A = 1 oscillators there is no definite trend in the variation of
H and<x 2 >,,.
3. ARSPT
In the framework of perturbation theory, the interaction
I/(<AJ = A<x 2 '>.x 2 ' . (11)
is treated as a perturbation and since it depends upon if/ tt (x), similar to if/ n and E n this too
is expressed as a power series in the perturbation parameter A. Thus, following Surjan
and Angyan [1], Surjan [3] and Kim and Sukhatme [9], we write
<M*) = W 0) (*)D + i n + 4/T + ], 02)
E B = J5i 0) + AEi 1 > + A 2 JE? ) +..., (13)
and
l/OAJ = A J/l/'Cx) + A 2 J/l 2) (x) + . (14)
Here, the wavefunctions are taken to be real because we are concerned with one-
dimensional bound state situation with local confining potential [11, 12]. From (11),
(12) and (14) we have
ijj ( n 0) ~(x')(x') 2r dx' (15)
nctmiHumun \^.j wiin r i U^LCIIHIU^U uj f \i\iji i ^Q;
and RHPS (P). All entries are in the units correspond-
ing to ft = m = o) = 1.
A
s ~
1
s
)
0-1
1-0
0-1
1-0
E Q (P)*
0-5225
0-5233
0-5000
0-6624
0-5361
0-5297
-0-5547
0-6491
*E (P) values have been rounded off to four significant
figures.
and
r *
(x') 2r iA (0) (x')/' fl) (x')dx' (16)
Following the customary procedure, we get
(17)
and
with
f.x fx'
^(i)(_ x ) = _2 [dx7^ (0)2 (x')] [ (1) l /(1) (x")]i// (0)2 (x")dx". (19)
" " " " "
J J ~ =0
With a view to compare the results obtained by using ARSPT with those of RHPS we
have determined ground state energy for the self-interacting oscillators with r= 1,
s= 1,2 keeping co= 1. We have for r = l,s= 1
(20)
and for r= 1, s = 2
= 0-5[1 +(3//4)-(183A 2 /64)]. (21)
The values found for A = 0-1 and 1-0 by two methods are compared in table 4. As
expected the values differ significantly from (P) and the deviation is higher for
higher s and /..
References
[1] P R Surjan and J G Angyan, Phys. Rev. A28, 45 (1983)
[2] E R Vrscay, J. Math. Phvs. 29, 901 (1988)
[3] P R Surjan, J. Math. Chem. 8, 151 (1991)
[4] J G Angyan and P R Surjan, Phvs. Rev. A44, 2188 (1991)
[5] J Cioslowski, Phys. Rev. A36, 374 (1987)
2 1 Pramana - J. Phvs.. Vol. 46. No. 3. March 1 996
L/J J Killmgbeck, J. Fhys. A2U, 6U1 (iyV)
[8] J Killingbeck, Phvs. Lett. A132, 223 (1988)
[9] I M Kim and U P Sukhatme, J. Phvs. A25, 647 (1992)
[10] C K Au and Y Aharonov, Phys. Rev. A20, 2245 (1979)
[11] L D Landau and E M Lifshitz, Quantum mechanics -non-relutivistic theory (Oxford,
Pergarnon, 1965)
[12] M Razavy and A Pimple, Phys. Rep. 168, 307 (1988)
HP-
Causal dissipative cosmology
N BANERJEE 1 and AROONKUMAR BEESHAM 2
Department of Applied Mathematics, University of Zululand, Private Bag X1001,
KwaDlangezwa 3886, South Africa
'Permanent address: Relativity and Cosmology Research Centre, Department of Physics,
Jadavpur University, Calcutta 700032, India
2 E-mail: abeesham^pan.uzulu.ac.za
MS received 15 November 1995
Abstract. The full version of the causal thermodynamics of non-equilibrium phenomena is
discussed in the context of the flat Friedmann -Robertson- Walker cosmological model. Power
law solutions for the scale factor are shown to exist. It is also shown that the temporal behaviour
of the temperature depends on the functional dependence of the coefficient of bulk viscosity on
density.
Keywords. Cosmology; bulk viscosity.
PACS Nos 98-80; 04-40
1. Introduction
The role of dissipative effects in the evolution of the universe, particularly during its
early stages, is a subject of growing importance. Although cosmological models with
a fluid with bulk viscosity have been well addressed (see Barrow [1] and references
therein), these models are not satisfactory for several reasons [2]. They violate
causality, there is a short wavelength secular instability inherent in them and perturba-
tions do not have a well-posed initial value problem. Most of these models consider
only the first order deviations from equilibrium. Earlier attempts to build up causally
well-behaved viscous fluid models, such as by Muller in 1967 [4] or Israel in 1976 [3],
included the second order terms (for an excellent review, see Maartens [5] and
references therein). Effects of these nonlinear theories in the cosmological expansion
were discussed by Zakari and Jou [6], Oliviera and Salim [7] and Chimento and
Jakubi [8]. Although these second order theories are causal and stable, they may lead
to some pathological behaviour during the evolution. As pointed out by Hiscock
and Salmonson [9], the drawback of these theories may arise out of the fact that most
of them drop certain divergence terms. Hiscock and Salmonson discussed a flat
Robertson-Walker cosmology with a viscous fluid where the divergence terms were
also taken into account. They integrated the equations governing the model numeri-
cally and obtained some interesting results. Subsequently, Zakari and Jou [10] and
Maartens [5] discussed this full theory and investigated the possibility of having
exponential inflation in this model. Recently Romano and Pavon [11,12] studied
dropped) in some anisotropic cosmological models.
In the present work, the possibility of having power law inflation in the flat
Robertson- Walker cosmological model in the full version of the theory is explored.
The notations used are those as in [5]. It is seen that if one assumes the standard
relations connecting the different thermodynamic variables and dissipative properties
with the energy density p ( [5], [10] ), a power law solution for the scale factor is possible
only when the coefficient of bulk viscosity is proportional to p 1/2 . If one allows different
behaviour for any of the thermodynamic variables, e.g. temperature, then power law
solutions may be obtained for other coefficients of bulk viscosity.
2. Cosmological solutions with a causal viscous fluid
The energy momentum tensor for a fluid with bulk viscosity is given by
TUV = (P + Pcff) Vv + Pefr0,<v. ( Z1 )
where p is the energy density, u fi is the velocity four vector and the effective pressure
p eff is given by
Peff = P + n, (2.2)
p being the thermodynamic pressure and n the bulk viscous stress. From the consider-
ations of energy-momentum conservation
T" v . v = 0,
number conservation
Af: M = 0,
Boltzmann H-theorem
and the Gibb's equation
where
JV" = MU", S 11 = sN tl - ( ^ ) u",
n is the number density, s the specific entropy, T the relaxation time for the bulk viscous
stress, c, the coefficient of bulk viscosity and T the temperature, one can arrive at the
evolution equation for the bulk viscosity
(2.3)
In the above equation, H = ^0 = ^u >l . ll is the Hubble parameter (for discussions in
detail, see [5, 9, 10]).
For T = one gets back the non-causal theory and the coefficient K = or 1 for the
"truncated" and the "full" causal theory respectively.
In a spatially flat Robertson- Walker cosmological model (k = 0), i.e. with the metric
ds 2 = - dt 2 + R 2 (t)(dx 2 + dy 2 + dz 2 ), (2.4)
Einstein's equations G MV = T flv (in units where 8?rG = 1 and c=l) become
3# 2 = p, (2.5)
2H + 3H 2 = -P-TT, (2.6)
where H = 3R/R and T^ v is given by (2. 1 ) and (2.2). The conservation equation of energy
momentum,
(2.7)
is not an independent equation as it follows from the Einstein equations as a conse-
quence of the Bianchi identity.
This system of equations is not closed as it has two independent equations (2.5) and
(2.6) and six unknowns namely p, p, R, c,, i and T. The popular practice is to assume the
ad hoc equations
-. (2.8)
P
Zakari and Jou [10] and also Maartens discussed exponential inflation with the choice
T = py. (2.9)
With the help of (2.3), (2.5) and (2.6), it is easy to construct the following evolution
equation for H,
^ 1 /* * T \ Q
+ ^T (2y + K) HH + H + - ex - - \ - - \H + -
2 2 V T C T J 4
(2.10)
which in view of (2.8) and (2.9), yields
H + -[e + (2 - e -
3 2 -' ? a- 1 7H 4 - 29 = 0. (2.11)
As discussed by Maartens [5], this equation is consistent with exponential inflation,
where H = H = constant.
But as we shall see this equation admits a power law solution for the scale factor
R only if q = \. If we choose
R = At (2.12)
where A and a are constants, then
a a 2fl -
A i t- 2 + A 2 t 2 < ] - 4 = Q< (2.14)
where
A , = [2fl - 1.{ + (2 - e - er)v}5 2 - e( 1 + r)fl + 1(7
and
Equation (2.14) is an algebraic equation in powers of r and the constant coefficients of
the different powers of r should be separately zero so as to make it valid for all t.lfq^ {,
both A ! and A 2 should be zero and it is easy to check that A 2 = yields 3yB = 2 while
this along with A l =Q yields 5 = which are clearly inconsistent. But if q = j, i.e.
c ~ p lj2 , (2.14) becomes
In this case A l + A 2 = and it will be possible to get a power law solution for JR.
In this connection, it is worthwhile to mention that the choices made for functional
dependence of c, T and T are ad hoc. If any one of them is left arbitrary to start with, the
power law solution leads to a different functional form of that variable. In what follows,
we shall leave the temperature arbitrary to start with and see what form it may take for
a power law expansion of the universe.
With the choice (2.12) for the solution of R, the energy density, given by (2.5) becomes
P = ^. (2.15)
Using this and the relation /? = (}' l)p, one can obtain the expression for n from
(2.6) as
fl(2-3y fl )
TI = - - 2 - , (2.16)
where 3ya ^ 2. For 3ya = 2, n becomes zero and we get the perfect fluid solution. Using
(2.16) and its derivatives in (2.3), one obtains the following equation
ln _
T 2a -- C - , (2.17)
where
fl = a (2 - 3ya) = constant (2.18)
and
H-M-
Using the expressions for c and T from (2.8) and that for p from (2. 1 5), it is easy to obtain
from (2. 17) the result
fl 1 f 2 i~ 2 + a 2 .x = fl 3 /f, (2.20)
where
") 1 A D.nn^onn I DU.,^ \/l ^^ VI^ 1 \ H . --- U 1
a 2 = a /2,
fl 3 = 2a -9 3 1-^ (2.21)
When <? = -|, (2.20) after integration will yield an expression for T as T ~ p r where r is
a constant. This is similar to the choice of T as given in [5, 10]. But if one has q^j,
(2.20) will yield a complicated expression for the temperature T,
T = r< 2a '- fl '/'exp(a 5 f 2 - 1 ), (2.22)
where a 4 is a constant of integration and a s = a 1 /(a 2 (2q 1)).
The temporal behaviour of the temperature will depend upon the values of the
constants a,, a 3 and a s . These values should be such that T actually decreases with
time.
3. Conclusions
In this work some power law solutions for the scale factor R have been found in the
Friedmann-Robertson- Walker cosmological model with a causal viscous fluid. It is
observed that the temperature is a simple power function of/? only when q=%, i.e. when
the coefficient of bulk viscosity is proportional to p 1/2 . But for other choices of as
{p), the temperature is a complicated function of time f.
After the present work had been carried out, we were made aware of similar
investigations by Maartens and Kgathi [13] and also by Coley et al [14].
Acknowledgements
One of the authors (NB) would like to thank the University of Zululand for hospitality
and the FRD for financial support.
References
[1] J D Barrow, Nucl. Phys. B310, 743 (1988)
[2] W A Hiscock and L Lindblom, Phys. Rev. D31, 725 (1985)
[3] W Israel, Ann. Phys. (NY), 100, 310 (1976)
[4] I Muller, Z. Phys. 198, 329 (1967)
[5] R Maartens, Preprint, Portsmouth University, UK (1995)
[6] M Zakari and D Jou, Phys. Lett. A175, 395 (1993)
[7] H P de Oliveira and J M Salim, Acta. Phys. Pol. B19, 649 (1988)
[8] L P Chimento and A S Jakubi, Class. Quantum Gravit. 10, 2047 (1993)
[9] W A Hiscock and J Salmonson, Phys. Rev. D43, 3249 (1991)
[10] M Zakari and D Jou, Phys. Rev. D48, 1597 (1993)
[11] V Romano and D Pavon, Phys. Rev. D47, 1396 (1993)
[12] V Romano and D Pavon, Phys. Rev. D50, 2572 (1994)
[13] R Maartens and A Kgathi (1994) Unpublished
[14] A A Coley, R\ J Van den Hogen and R Maartens, Preprint RCG 95/10, Portsmouth
University (1995)
An identity for 4-spacetimes embedded into E 5
JOSE L LOPEZ-BONILLA and H N NUNEZ- YEPEZ*
Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana-Azcapotzalco,
Apartado Postal 21-726, Coyoacan 04000 D.F., Mexico
* Departamento de Fisica, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal
21-726, Coyoacan 04000 D.F., Mexico
MS received 10 January 1995
Abstract. We show that if a 4-spacetime V 4 can be embedded into E 5 then, if b l} is the second
fundamental form tensor associated with K 4 , the quantity (trace b)-bi] l depends only on intrinsic
geometric properties of the spacetime. Such fact is used to obtain a necessary condition for the
embedding of a F 4 into E 5 .
Keywords. Embedding of Riemannian 4-spaces; local and isometric embedding.
PACS Nos 04-20; 04-90
1. Introduction
Let us consider a 4-spacetime local and isometrically embedded into E 5 . This means
that there exists the second fundamental form tensor b u = b ri which fulfils the Gauss
and the Codazzi equations [1-3]
K y * = *(**** -Mjk) (1)
and
where s = 1, R ijkc is the F 4 Riemann tensor and ; r stands for co variant derivative. It is
well-known that whenever det b( ^ then (1) implies (2); furthermore, it is not difficult
to obtain the following relationship [3-7]
- 24 det(b r c ) = K 2 = *R* iJkr R ljkr , (3)
K 2 being one of the Lanczos invariants [8, 9] defined in terms of the double dual [ 1 0] of
Riemann tensor
**'/ = l
^ ~
r l
where rj abcd is a Levi-Civita symbol. This work deals with the case in which K 2 ^ 0;
according to (3), this implies that the inverse matrix to the second fundamental form,
by 1 , exists. We offer a proof that it is possible to find a relation between *R* jkc and b~ r l
which is analogous to (1). Among other things this implies that (trace b)-b r x depends
only on the intrinsic geometry of K 4 .
For any class-one spacetime we can use (1) to obtain the identity [2, 1 1]
P b ^^J-\ R ^jG mn , (5)
where p = fib ar G flr /3, G ac = R ac - RgJI = *R* l aci is the Einstein tensor, R tj = R r ijr is
the Ricci tensor, and R = R c c is the scalar curvature. In [2] Gonzalez et al have shown
that p is determined by the intrinsic geometry of V 4 . Furthermore, when p = 0, using
(5) is trivial to establish that b is determined by the intrinsic geometry of the spacetime
and, therefore, that (trace b)!)" 1 is also an intrinsic quantity. The problem appears
when the case p ^ is considered [6, 12, 13] for then the previous statements become
non-trivial. It is the purpose of this work to show that (trace b)b -1 is in fact intrinsic
irrespective of the value of/?. To this end an equation of the type (1) but relating *R* jM to
b ~ r l is needed.
Yakupov [14,15] has shown that for every class-one V 4 the following equation
holds good
Substituting (1) into (6) we get
*jj(% 5 ? ~ % m ) = *R* ijmnj b an b jc ;
assuming K 2 ^ then, on multiplying (7) by b~ la b~ lc , we easily get
24
(7)
(8)
This expression has the same structure as Gauss equation (1) hence illustrating the
analogous role, the curvature tensor and its double dual play. The problem of
embedding for a class-one 4-spacetime is thus reduced to analyzing (1) or (8).
Gonzalez et al [2] have studied 'how (1) implies (5), in an analogous way from (8) we
may get
This exhibits that (trace bj-b" 1 is a quantity which only depends on the intrinsic
geometry of V 4 irrespective of the value of p.
Equation (9) is interesting since it allows establishing a new necessary condition for
the embedding of a F 4 (but with K 2 ^ 0) into E 5 . This condition can be obtained by first
contracting f withy in (5), in this way we get
(10)
220 Pramana - J. Phys., Vol. 46, No. 3, March 1996
^R^jR^R^G" + K 2 3 ( -1 + R 2 /2 - R^R^W- Wj + 2R;,R'l = 0,
(11)
where we used (10). We have established that every class-one spacetime with K 2 ^0
should comply with (11), which is thus a necessary condition for embedding that we
have not found previously published.
Acknowledgements
This work has been partially supported by CONACyT grant 4846-E9406.
References
[1] D Kramer, H Stephani, M Mac Callum and E Herlt, Exact solutions of Einstein's field
equations (Cambridge University Press, 1980)
[2] G Gonzalez, J Lopez-Bonilla and M A Resales, Pramana - J. Phys. 42, 85 (1994)
[3] J Lopez-Bonilla, J Morales and M A Resales, Braz. J. Phys. 24, 522 (1994)
[4] T Y Thomas, Acta Math. 67, 169 (1936)
[5] D Ladino, J Lopez-Bonilla, J Morales and G Ovando, Rev. Mex. Fis. 36, 354 (1990)
[6] R Fuentes, J Lopez-Bonilla, G Ovando and T Matos, Gen. Relativ. Gravit. 21, 777 (1989)
[7] D Ladino, J Lopez-Bonilla, J Morales and G Ovando, Rev. Mex. Fis. 36, 354 (1990)
[8] C Lanczos, Ann. Math. 39, 842 (1938)
[9] V Gaftoi, J Lopez-Bonilla, D Navarrete and G Ovando, Rev. Mex. Fis. 36, 503 (1990)
[10] C Lanczos, Rev. Mod. Phys. 34, 379 (1962)
[11] H F Goenner, Tensor New Series 30, 15 (1976)
[12] O Chavoya, D Ladino, J Lopez-Bonilla and J Fernandez, Rev. Colomb. Fis. 23, 15 (1991)
[13] J Lopez-Bonilla, J M Rivera and H Yee-Madeira, Braz. J. Phys. 25, 80 (1995)
[14] M Sh Yakupov, Sbv. Phys. Dokl. (Engl. Transl.) 13, 585 (1968)
[15] R Fuentes and J Lopez-Bonilla, Acta Mex. Cienc. Technol. IPN 3, 9 (1985)
PRAMANA Printed in India Vol. 46, No. 3,
journal of March 1996
physics pp. 223-227
A q deformation of GeSS-Mann-Okubo mass formula
B BAGCHI and S N BISWAS*
Department of Applied Mathematics, University of Calcutta, 92 APC Road, Calcutta 700009,
India
E-mail:bbagchi(fl.cubmb.ernet.in
* Department of Physics, University of Delhi, Delhi 110007, India
MS received 2 November 1995; revised 13 February 1996
Abstract. We explore the possibility of deforming Gell-Mann-Okubo (GMO) mass formula
within the framework of a quantized enveloping algebra. A small value of the deformation
parameter is found to provide a good fit to the observed mass spectra of the n, K and r\ mesons.
Keywords. Deformation; mass formula; chiral symmetry breaking.
PACS No. 12-70; 1 1-30; 02-20
The study of quantum groups has aroused [1] much interest of late. A glance through
the literature reveals [2] that several deformed algebraic structures have been develop-
ed to modify various physical systems. Since the idea of a quantum group is more easily
accessible via an enveloping algebra -the latter almost always corresponding to
a deformation [3,4] of a Lie structure of some sort, it is worthwhile looking for
deformations of those schemes in which Lie algebras are of potential relevance. In this
note we consider a deformation of the underlying SU(3) algebra of the quark model and
look for the consequences.
One can deform [4] the full SU(3) group as defined by say, Jimbo and Drinfeld or
Fairlie and Nuyts. Alternatively since SU(3) of the quark model is effectively described
by the constituent subgroups namely, the isospin, U spin and V spin, deformation of all
or any one of these subsectors may be carried out. In the following we enquire how
deforming a particular SU(2) influences the whole of SU(3) (minimal deformation).
SU(3) consists of two diagonal matrices A 3 and A 8 . Assuming A 3 = diag(l, 1,0)
along with m u - m d and noting that the members of a U spin multiple! enjoys the same
electromagnetic properties we perform, as a first step, deformation of the V spin sector
of SU(3) only. The idea is to retain the properties of the I and U spin invariances even
after deformation. The amusing point is that because of an interplay between the SU(2)
subalgebras, a q deformation of the V spin brings about modifications in the / spin and
17 spin sectors too. As a consequence we are led to a deformed GMO mass rule for the n,
K and r\ mesons involving a single deformation parameter.
In our scheme the deformation variable q plays essentially the role of a pheno-
menological parameter offering an extra freedom to fit the GMO formula with the
underlying aigeoraoi inequarK moueiproviues anomer possiomiy. usnouiu oesiaiea
that in principle we could have deformed all the SU(2) subsectors of SU(3). But this
would have only increased the number of deformation parameters corresponding to
each subsector (maximal deformation). In our simple-minded approach we have
avoided dealing with such extra parameters.
The framework of our analysis rests on the GMOR scheme [5] of chiral symmetry
breaking which, for concreteness, we deform according to the oft-studied [6a] quantum
algebra of Witten and Woronowicz [3,4]. It is needless to mention that we could
have adopted any other form of enveloping algebra yielding results [7] similar to the
present one.
Witten-Woronowicz SU(2) quantum algebra is given by
\W Q ,W + \ = W + ,\_W_,W\= W_,lW + ,W_^ /g ,= W (la)
for a triplet of operators W , W Q and the deformed brackets are to be read [A, B~\ q =
qAB q~ 1 BA for a pair of operators A and B. The q commutations (1) reduce to those
of the conventional Lie algebra SU(2) or SU(1, 1) in the limits q = 1 or 1 respectively
and so may be looked upon as a deformed map of either SU(2) or SU(1, 1).
The advantage in working with the algebra (la) is that the generators can be given
a matrix representation. Moreover, it has recently been shown by Lorek and Wess [6b]
that the deformed scheme (la) admits of a co-multiplication rule
= W 1 + T 1 3 12 W
) = W (g) 1 + T 3 W Q - qh" 2 W + W_ - -i\ 12 W_ W +
-fa + l/q) W Q - ~(q + l/q) W + W_
(Ib)
Further within the framework of (1 a) these authors have demonstrated that interacting
systems may be interpreted in terms of a free system based on g-deformed kinematics.
Curtright and Zachos [8] have worked out explicitly invertible functional of the
SU(2) generators which deform SU(2) continuously into some quantum algebra. For
the deformed brackets (1) they have found the following matrix representations
Since for q = 1, WQ -; and W -*j with the SU(2) algebra Q/ J ] = j , \J + J_ ] =;
holding good, it is clear that the deformed matrices (2) convert the SU(2) generators
into operators which obey the q-brackets of the quantized algebra (1) for all values of q.
Given this background we ask the question as to whether the underlying algebra of the
quark model can be mapped onto some quantum algebra say, the one provided by (1).
994 Pramana - .1 Phvc Vnl Jfi Mn
We shall presently see that even for an infinitesimal deformation Tr(/ k J / 3 ) ^ where / is
the deformed A 8 matrix. The interesting point is that an infinitesimal deformation (that is
in terms of O(logq)) does not affect the TC and K but alters only the >/ mass.
In analogy with (2) let us propose that the deformed V spin obeys the following
quantized algebra
d + ] = K d + ,[K d _,
(3)
where a superscript d indicates a deformed quantity.
It is straightforward to work out the representations of K d + and K d which satisfy (3).
These turn out to be
r/ d -
K +
1
000
000
000
000
1
K d =diag(^,0,
(4a,b,c)
Since /. 3 has not been deformed it follows from (4c) that the deformed /. 8 matrix is
4
If the deformation is taken to be infinitesimal q = 1 + <:, = log 4 then
;.J = diag(l _4e,l, -2-4s)/ v /3
along with
(5)
(6)
, 1 - 4e). (7)
Note that although Tr^/tg/ig) and Tr(/^A. 2 ) vanish, Tr(/,gA 3 ) does not vanish. Hence-
forth we shall work with infinitesimal deformation only.
We now turn to the pseudoscalar mass spectra and chiral symmetry breaking.
Because of the deformed /. 8 and A matrices the symmetry breaking Hamiltonian
density in the GMOR scheme reads
where q = (u, d, s)T and the symmetry breaking parameters c ( -(i = 0, 3, 8) in terms of the
conventional quark masses are
+
2 -3
(9)
m u = m tl m s [naive SU(3)J as a consequence ot deformation. 1 nus even an infinitesi-
mal deformation induces SU (2) and SU(3) breakings: in the / spin sector it is roughly of
the order of f(m m v ). This is reminiscent of Oakes observation [9] made years ago
that an isospin conserving Hamiltonian density when rotated about the 7th axis in the
SU(3) space picks an isospin violating piece in a natural way.
To obtain the pseudoscalar mass spectra we use the Heisenberg's equation namely,
8^ = i[H' d , J 5 ] and the PCAC relation 6 /t A j /t = fjm^j, j stands for TT, K and
r\ mesons respectively and//s are the corresponding decay constants.
From (8) we find
( fi2 ) ( lob )
/X = iCd - 4e ) w + 20 + 4eK]Z, 1/2 + (* 2 }- ( 10c )
In deriving (lOa), (lOb) and (lOc) we have, in the context of our approximation, identified
the meson particle states with those of the undeformed exact SU(3). The quantities
Z* /2 (p = TI, K and ?/) are the respective wave function renormalization constants.
Since the particle states are assumed to be eigenstates of exact symmetry we may
equate Z l n 12 = Z\! 2 = Z,J /2 and obtain from (10)
4/X - 3/X - /X = 4s(>" - 2m,)Z" 2 + 0( 2 ). (1 1)
We thus get a deformed GMO mass formula in the limit f n = f K = f
4m 2 - 3m 2 -m 2 = 4e M - 2 j m 2 + 0(e 2 ). ' (12)
The rhs of (12) gives the deformation corrections. Inserting the masses for n, K and 77 we
get e % 0-02 for mjm % 20-25 which shows that the deformation is truly infinitesimal.
It may be checked that the infinitesimal nature of deformation persists even for other
choices of deformed algebras enlisted in [8]. Indeed following the expressions in
(10)-(12) deformation may be looked upon [10] as a kind of perturbation.
Let us end by remarking that recently some attempts have been made to seek pheno-
menological applications [1 1-13] of q deformations. These include the study [1 1] of the
transition from SU(2) L x SU(2) K x (/(I) to SU(2) x 7(1) of the standard model by
^ deformation and an attempt [12] to understand the problem of the identity between the
superdeformed bands in neighbouring odd and even nuclei. The spirit of the present work
comes close to these: by treating the deformation variable as a phenomenological
parameter but confining ourselves to the constraints of an enveloping algebra we have
shown that a small deformation can account for the 20% discrepancy in the fit to the
GMO formula. Our scheme of deformation can be extended to other low-energy
theorems by quantizing not only SU(3) but also XU(3) x SU(3) algebra of currents.
Acknowledgements
We thank the anonymous referee for constructive criticism. This work was supported
by the Council of Scientific and Industrial Research, New Delhi.
[1] C Zachos, in Deformation theory and quantum groups with applications to mathematical
physics, contemporary mathematics edited by M Gerstenhaber and J Stashef (American
Mathematical Society, Providence, RI, 1992) Vol. 134
T Curtright, D Fairlie and C Zachos (eds) Proc. Aryonne Workshop on Quantum Groups
(World Scientific, Singapore, 1991)
J Fuchs, Affine Lie algebras and quantum groups (Cambridge Univ. Press, Cambridge, 1992)
[2] See for instance D V Boulatov, Mod. Phys. Lett. A7, 1629 (1992)
A Chodos and D G Caldi, J. Phys. A24, 5505 (1991)
M Kibler and T Negadi, J. Phys. A24, 5283(1991)
[3] E Witten, Comm. Math. Phys. 121, 351 (1989)
[4] S L Woronowicz, Comm. Math. Phys. Ill, 613 (1987)
V Drinfeld, Sov. Math. Dokl. 32, 254 (1985)
M Jimbo, Lett. Math. Phys. 10, 63 (1985)
D B Fairlie and J Nuyts, J. Math. Phys. 35, 3794 (1994)
[5] M Gell-Mann, R J Oakes and B Renner, Phys. Rev. 175, 2195 (1968)
[6] (a) See for example F J Narganes-Quijano, J. Phys. A24, 593 (1991)
(b) A Lorek and J Wess, Dynamical symmetries in q deformed quantum mechanics, Preprint
MPI-PhT/95-1 (February 95)
[7] It should be noted that in Drinfeld- Jimbo SU(2) q the generators T , T+ are related to the
ordinary SU(2) generators by = a.T , T Q = T where q = ^2(q + q~ 1 }. In the Cartesian
basis one therefore has T 3 = T 3 contrary to that of Witten's algebra (2).
[8] T L Curtright and C K Zachos, Phys. Lett. B243, 237 (1990)
[9] RJ Oakes, PAjjs. Lew. B30, 262 (1969)
[10] q deformation is to be distinguished from chiral perturbation theory which is an effective
theory.
[11] R Bonisch, Transition from SU(2) x SU(2) x U(l) representation to SU(2) x U(l) by
q deformation and the corresponding classical breaking term of chiral SU(2), DESY 94-129
Preprint (July 1994)
[12] A Abbas and P Behara, Quantum group SU(2) and identical deformed bands in proximus odd
and even nuclei, Inst. of Physics (Bhubaneswar) Preprint. IP/BBSR/92-39
[13] A S Zhedanov, Mod. Phys. Lett. A7, 507 (1992)
Pramona _ T Plivc Vnl 46 Mn T Mfltrh 1 QQfi 227
physics pp. 229-237
Scaling laws for plasma transport due to 17, -driven
turbulence
C B DWIVEDI and M BH ATTACH ARJEE*
Plasma Physics Division, Institute of Advanced Study in Science and Technology, Jawahar-
nagar, Khanapara, Guwahati 781 022, India
* Mathematical and Statistical Sciences Division, Institute of Advanced Study in Science and
Technology, Khanapara, Guwahati 781 022, India
*Present address: Department of Mathematics, Indian Institute of Technology, Institution of
Engineers Building, Panbazar, Guwahati 781 001, India
MS received 16 October 1995; revised 19 January 1996
Abstract. The scale invariance technique has been employed to discuss the f/ r driven turbulent
transport under a new fluid model developed by Kim et al [1]. Our analysis reveals that the finite
Larmour radius effect plays a decisive role to determine the scaling behaviour of the energy
transport under the new fluid model. However, the overall scaling of the transport coefficient
remains unchanged as compared to that derived by Connor [2] under the traditional fluid
model. The approximations considered by Connor [2] are qualified with additional require-
ments within the new fluid approach. In the dissipative case, which has not been discussed earlier,
additional constraints on the power scaling laws of the transport properties are imposed due to
the dissipative mechanisms in the basic governing equations.
Keywords. Scaling laws; invariance technique; similarity transformations; transport coefficient;
>7i -driven turbulence.
PACS No. 52-25
1. Introduction
Theoretical and experimental study of anomalous transport in magnetic confinement
systems (to determine the physical mechanism for the transport phenomena) has
become a subject of main concern for the plasma physicists. Turbulent transport due to
ion temperature gradient (or temperature drift) instability has been suggested as one of
the physical phenomena that leads to an anomalous ion thermal conduction. This has
been proposed as the cause of a deterioration in confinement in the AlcatorC
experiment at high density when gas puffing produces flat density profile [3]. Many
projects, theoretical [4-1 1] and experimental [12, 13] have been devoted to solving the
energy transport due to /7 r modes. Many of them are based on the fluid models
[5,6, 9, 1 1]. Nonlinear saturation level of the ^-driven fluctuations predicted by these
models are found to be much larger (by at least an order of magnitude) than the levels
predicted by the more sophisticated particle simulations [14]. This is generally
attributed to the lack of ion Landau damping in the conventional fluid models.
Accordinelv. Hammett and Perkins f~1 *H incornorated the anoroximated Landau
C B Dwivedi and M Bhattacharjee
terms into the basic equations to reduce the saturation level of the ^ ( -turbulence.
However, very recently Kim et al [1] have developed a new fluid model for the ^-driven
turbulence by incorporating the complete treatment of the polarization drift due to
finite Larmour radius effects in basic equations. Their general belief is that by just
including the damping effects into the basic equations may not be enough for bringing
down the saturated level for the ^-driven fluctuations to match with that predicted by
numerical simulations [14]. Rather one should very carefully treat the ion polarization
drift while working out the energy conservation property. In the existing fluid models
most of the authors either completely neglect divergence of polarization drift (V-v p )
term in the heat equation or include only part of it. Consequently, inconsistency arises
so far as the contribution of diamagnetic drift to the kinetic energy in conservation law
is concerned. They [1] considered this aspect and treated the ion temperature
fluctuation while deriving the expression for the polarization drift (v p ). Thus the new
set up of fluid equations for the description of ?/ r mode turbulence differs from the
others due to contribution of ion-diamagnetic drift to the kinetic energy in the energy
balance relation.
Once the responsible physical mechanism for the anomalous transport is estab-
lished, the nonlinear analytical calculation of the transport properties to describe their
scaling behaviour becomes quite a tedious and intractable job. To avoid this difficulty,
Connor and Taylor [16] were the first to suggest a technique more general than the
analytical calculation. It is based on the invariance principle of the basic governing
equations under a group of linear transformations which eventually describe the
scaling properties of the anomalous transport phenomena. This method has already
been successfully applied to various situations of physical mechanisms [2, 1 1, 16-19].
Based on the traditional fluid model of non-dissipative plasmas, scaling behaviour of
transport associated with ^-turbulence has been discussed by Connor [2]. However,
recent development of a new fluid model [1] incorporating the classical dissipations
due to collisions warrants a fresh look at the power scaling laws of the transport due to
^.-turbulence in the dissipative and non-dissipative cases. This paper considers various
cases of approximations and compares the findings with the earlier results [2] in the
non-dissipative domain of the basic equations. In this domain, the overall scaling
behaviour of the transport remains unchanged except that the approximations used by
Connor [2] are qualified with additional requirements in the light of the new fluid
model for the ;/, mode. Inclusion of the collisional dissipations and Landau dampings
introduce additional constraints on the power scaling laws of the ;? -driven turbulent
transport by increasing the number of free indices for describing the functional form of
the transport coefficients. Section 2 deals with the description of basic equations
developed by Kim et al [1] to describe the ^-driven turbulence. Section 3 includes the
derivations of the scaling laws for the diffusion coefficients under various possible
approximations of the dissipative and non-dissipative fluid models. Results and
discussions form 4 of this paper.
2. Basic governing equations
+ r^ ^
\_dy\ly
Parallel momentum equation
dv, ,
IT
A i * ji Y* ' L V ' *^ J * J_ T _l_ *"* i Y \ T i ^ 'i v/ ' \
Pressure equation
Q QQ
(p TFtt) -f- "cK. -j- r^J) (n TFtt)l y V p y V p == (3^
dt dy i i ii ii i
These are the normalized equations and their derivations and normalization constants
are described in paper [1]. T = TJT et K = K - F, K = rj l +l,r is the adiabatic gas
constant, \\t = 3? + p. The perpendicular dissipative coefficients are given by (.L L = rS/4,
Vj_ = O3T<5, x = t<5, where 5 = (v,./o; CI .)(L n //? s ) and L n , p s and v t are respectively Larmour
radius, density scale length and ion collisional frequency. For the parallel diffusion
coefficients, v,, and x\\ are chosen to model the ion Landau damping. Further
V |; = (d/d) + sx(d/dy), % z/L n with s( = LJL S ) as the shear parameter. Square bracket
[ ] enclosing within it the physical quantities denotes for Poisson bracket defined as
[/#] = ((df/dx}(dg/dy)) ((df/dy)(dg/dx)). Now these equations have been solved un-
der different sets of approximations in the dissipative and non-dissipative cases by
applying the concept of the invariance principle to establish the scaling behaviour of
thermal transport due to //,-driven turbulence.
3. Scale invariance
3.1 Non-dissipative case (^ = Vj_ = x x = v,, = ^ = 0)
Connor [2] has already discussed the power scaling laws of ^-driven thermal transport
under the traditional fluid model of collisionless plasmas. However, in the light of the
new fluid model [1], we applied the scale invariance technique to see the effect of the
structural changes in the new basic governing equations of j^-mode on the scaling
behaviour of thermal transport.
Under the fluid approximation Vjl and the assumption of weak potential
fluctuation <J>p, the relative dominance of the terms <E> and V^ (inside the para-
nthesis of (1)) suggests the possibilities of three different cases; cj> V^p, $ ~ V^p and
4> V^p. The self consistent validity conditions of these additional approximations
have been derived and verified by calculating the scalings of </>, p, V^p, d^/dt and d^jdy.
It is found that the first case is inconsistent whereas the remaining last two cases are valid.
Now under the approximations $ p, V^ 1 and </> ~ V^p equation (1) is rewritten as
^ ^r . -JL n i_ro v 2 z?i = o w
^ 3 ^ ~r" ^ 5 -\ | L 5 \_r J V i
ox ox
Now we seek all the linear transformations of the independent and dependent
variables
n -> an, - /?<>, p - yp, v -> fiv , K - vX, s -> 5s, x -> / 1 x, > - / 2 >',
which leave the basic equations (2-4) invariant. We find only one such transformation
T: n-+Pn,(j)-*l(l),p-^l 3 p,v ^I 2 v ,X->/ 2 X,s^r 2 s,x^/.x, }>-+/>',
C->/ 2 cr-+/r, for /! = / 2 = /.
Under this transformation, diffusion co-efficient D should transform as D -> /D, since
any transport coefficient must scale as (Ar) 2 /Af. Now if D is expressed as [2]
D=(D(s,K] (5)
where pi CJL n is a normalization factor and D(s, K) a normalized diffusion coefficient,
the functional form of D(s, X) can be determined as
D = s I 'K q .
The requirement that it remains invariant under the above transformation T, imposes
the following restriction on the exponents
T=-P+'<Z
so that the functional form of D is restricted to
I V /2 ( L 2
F
Following the same technique, the scalings of the normalized cj^p, V 2 p, d^/dt,
and V 2 are derived to reveal that 0-s /2 (sX>, p->s- 3/2 (sX) 9 , V^p-^s-^
d(f)/dt-^(sK} q , d^/dy-t^KY and V^sisX) 9 . Thus the self consistent validity of the
approximations requires s 1 for sX ~ 1 up to an unknown function of order unity.
The scaling of the diffusion coefficient in this case shows the s~ 1/2 dependence as
reported even by Horton et al [6].
Let us now consider the third case i.e. V 2 p under the approximations of weak
potential fluctuation $p and fluid regime V 2 1. In this limit again only (1) is
modified and given as
d
ot oy
dp 3<D
<5? ~dx
Now we seek again the linear transformations of the independent and dependent
variables
O->^O, p->yp, v ->/ILV, X-vX,
T,: n-+l 3 n,p^l 3 p,v^
^/ 3 ,r-*/ 2 t, for
Under the combined operation of these two transformations diffusion coefficient
D should transform as >-/?>, since any transport coefficient must scale as (Ar) 2 /Ar.
Now if D is again expressed by (5) with the functional form of D(s, K) as D = s p K q , the
requirement that it remains invariant under the above transformations T l T 2 ,
imposes the following restrictions on the exponents
1 = - p/2 + q, 0= -3p + 2q
so that the functional form of D is restricted to
o 2 C f L \ 3 ' 2
= ~^(T L ) (8)
L s \ L Tj
This functional form is the same as that derived by Connor [2] under the approxi-
mations O p, V 2 1 and (3<D/df) (3/3y). Again to verify the validity of the
approximations, the scalings of the normalized </>, p, V 2 p, d</)/dt, d(f)/dy and V 2 are
derived to reveal that </>->sK 3/2 , p^K 3 ' 2 , V 2 p-+K: 1/2 , d(j)/dt-*(sK) 2 , d(f)/dy^(sK),
V 2 -* K~ *. This is to note that self-consistent validity of the approximations requires
sK 1 for s 1 and K 1 up to an unknown constant of order unity. This implies that
s and K should scale as s ~ e 2 , K ~ e~ 1 for e 1. Physically it suggests a situation of
very weak magnetic shear. In this case the diffusion coefficient scales linearly with the
shear parameter 5 which is the same as reported by Connor [2]. However, the
approximation sK 1 has been questioned by Hamaguchi and Horton [5] based on
their numerical simulation. Their criticism lies on the positive footing and is based on
the linear property of the rj ( mode. In the light of their comments we would like to
add that the inertia term completely disappears in the continuity equation of the
traditional fluid model for sK 1 which poses a qualitative problem at the linear level
of the ^ mode. Nevertheless, in the new fluid model developed by Kim et al [1], the
inertia term now arising due to finite Larmour radius effect (polarization drift), still
survives for the approximation sK 1 and determines the explicit form of the diffusion
coefficient and other variables. As discussed above, the linear scaling of the D with
s holds good for very weak magnetic shear. This implicitly includes the approximation
d<j>/dt d(f>/dy.
Accordingly it may be argued that under the approximation of weak potential <t> p,
the dominance of finite Larmour radius correction term V 2 i// in (1) (arising due to the
polarization drift) can explicitly determine the explicit functional form of D. This
possibility is ruled out under the traditional fluid model within the validity of fluid
approximation. Furthermore, the anisotropic distribution of the turbulence
d/dxd/dy or d/dxd/dy does not affect the scaling properties of transport coeffi-
cients and other quantities. This is true even in the case of dissipative fluid model.
Our analysis predicted the same functional forms for thermal diffusion and other
quantities as those described by Connor [2]. Based on these conclusions we speculate
that the contribution of the ion-diamagnetic drift to the kinetic energy in the energy
balance relation, does not qualitatively alter the ^-driven transport mechanism and
consequently the power scaling laws of the transport coefficient associated with the
same. This is in good agreement with the analytical conclusions of Kim et al [1] which
report only quantitative changes (decrease) in the fluctuation levels of the /y,-driven
turbulence.
3.2 Dissipative case (^ , v x ,;Q , v , , ^ ; ^ 0)
Since the earlier description of the energy transport due to ^,-driven turbulence
[2] was limited to the non-collisional fluid model of plasmas, the restrictions on
the functional forms of thermal coefficients and other quantities could not be repre-
sentative of those plasma systems in which dissipations play an important role.
Equations (1-3) derived by Kim et al [1] include the classical dissipation due to
particle collisions and Landau damping terms. This section deals with the scaling
behaviour of thermal transport coefficients in the presence of dissipative terms in
basic governing equations. For simplicity we have approximated V^ ~ (d 2 /dy 2 )
and V,, sx(d/dy). This is to note again that the approximation <p V^p as discussed
in the nondissipative case is a non-valid approximation even in the dissipative
system. The self-consistency of this statement relies upon the assumption that the
scalings of <, p, V^p, d(f>/dt d(f>/dy and V^ are unchanged up to an unknown function
(due to dissipative effect) of the same order. Now under the approximations p,
V^ 1 and O ~ V^p, we again seek all the linear transformations of the dependent and
independent variables
n -+ an, <t> -> /?d>, p -> yp, u, - /lu,, , K -> vK, s -* y s, x - / 1 x, y -* 1 2 y , - / 3 ,
f->/ 4 t,^ 1 ->/ 5J u 1 ,v 1 -*/ 6 v i ,^ 1 ->/ 7 x i ,v || ^/ 8 v || ,^ || ->/ 9 ^ l| (9)
so that equations (1-3) be invariant. We find only one such transformation
f -^ /t ^-^^i 5 v J .-^/v 1 ,x 1 ^/Xi,v l| -^/ 3 v || ,^ ll ->/ 3 ^ || . (10)
To determine the functional form of D(s,K, n,v,x, V \\>X.\\) l et us express
s lX ' 1 v t ;^. (11)
The requirement of the invariance of D under the operation of scale transformation T 3 ,
imposes the following restrictions on the exponents
p = - 1/2 4- q + q'/2 + 3( + u)/2 where q' = r + s + t.
Now the general form of D is restricted to
o 2 C fl V' 2 / / 2 v I n V/ 2 f]
j) r^^ I I F f " _1. f Z!IL ) (z^
L n UJ 'UrL.' co,,.p,UJ ' P\L,
on t<5.
Now if we further specify the transport mechanism by applying the approximation
<t>V 2 p and retaining the other approximations as described above, only (1) is
modified to give
Again applying the invari.ance technique, we can find the transformations which will
leave (li), (2-3) invariant. Accordingly we get two such transformations
T 4 :
T 5 :
for / = / =
As described earlier the expression for D is restricted to
(13)
Note that in deriving the functional expressions for D, K has been approximated as
K& LJL r . Further, the expressions for other quantities can also be determined
following the procedure outlined here. This, being a trivial exercise, has not been
included in the paper. Furthermore, the dominance of the finite Larmour radius effect
determines the explicit functional form of the diffusion coefficients and other quantities
for the non-dissipative case and reduces the number of unknown free indices by one for
the dissipative case.
It is remarkable to add that the possibility of the approximation Ks 1 (i.e.
d<j)/dtd<f)/dy as discussed by Connor [2]) within the fluid model V 2 1 of weak
potential fluctuation p requires < ~ Vjp and 5 < K within the new fluid model [1]
of ^ r driven turbulence. Th? functional form of D in this case remains the same [2]. In
the dissipative case it reads as
(14)
4. Results and discussion
Invariance technique of theoretical analysis is complementary to analytical formula-
tion. It provides a more general framework for solving the basic equations governing
any physical mechanism to discuss the associated transport coefficients and saturated
C B Dwivedi and M Bhaitacharjee
fluctuation level. The more we specify the natural mode of transport mechanism, the
lesser the number of free exponents in the power scaling laws of the associated transport
coefficients and other quantities. The specification of the mechanism is correlated with
the approximations employed in describing the linear dispersion characteristics of the
mode responsible for the energy transport. However, only those approximations are
important which bring about the qualitative changes in the natural mode of transport
process. The non-determinism of the constant coefficients is a limitation of this
mathematical model of analysis.
From the analysis, one can notice that in the non-dissipative case as discussed in
3.1, the new model equations predict the explicit functional form of the transport
coefficients for longer wavelength (V 2 1) of the ^-driven fluctuation even without
imposing the condition (d(f)/ct)(c(f)/cy) as considered by Connor [2]. Possibly the
finite Larmour radius correction term in (1) (for (j>V 2 p and weak electrostatic
potential p) plays a predominant role to decide the scaling behaviour of the
transport in non-dissipative and dissipative cases of the /] ,-driven fluctuation. This puts
a restriction on the upper limit of the fluctuation scale size of the nonlinear spectrum
due to /7,-driven turbulence within fluid approximation (V 2 1). However, the inclu-
sion of the dissipative mechanisms in the basic governing equations allows further
restriction on the power scaling of the transport coefficients by introducing additional
free exponents. General forms of the diffusion coefficients for ^.-driven turbulence have
been derived which may hopefully provide an input to the tokamak physicists to
determine the scaling laws for energy confinement. Further extension of the analysis in
the toroidal geometry may be carried out to formulate more realistic power scaling
laws for the plasma experiments where dissipative mechanisms are supposed to affect
the energy transport. Finally, the validity of the approximation sK 1 demands very
weak magnetic shear within the new fluid model of the t] { mode.
Acknowledgements
Supports from members of the Plasma Physics Group and Mathematical and Statisti-
cal Sciences Group are thankfully acknowledged. The authors are extremely thankful
to the referee for his critical and helpful comments to improve the quality of the
manuscript.
References
[1] C B Kim, W Horton and S Hamaguchi, Phys. Fluids B5, 1516 (1993)
[2] J W Connor, Nucl. Fusion 26, 193 (1986)
[3] B Coppi, S Cowley ef a/, Plasma Physics and Controlled Nuclear Fusion Research, Proc.
10th Int. COM/:, London, 1984 (Vienna: IAEA, 1985) Vol. 2, p. 93
[4] B Coppi, M N Rosenbluth and R Z Sazdeev, Phys. Fluids 10, 582 (1967)
[5] S Hamaguchi and W Horton, Phys. Fluids B2, 1834 (1990)
[6] W Horton, R D Estes and D Biskamp, Plasma Phvs. 22, 663 (1980)
[7] W Horton, Phys. Rep. 192, 177 (1990)
r1 T <s Hahm cmH IW \A Tonn P/n-c F/,,,'^C. R1 ll
[12] D Scott. P H Diamond et al, Phys. Rev. Lett. 64, 531 (1990)
[13] M C Zarnstorff. C W Barnes et al. Plasma Physics and Controlled Nuclear Fusion
Research. Proc. 13th Int. Conf., Washington, DC 1990 (Vienna: IAEA, 1991) Vol. I, p. 109
[14] M Kotshenreuther, H L Berk et al. Plasma Physics and Controlled Nuclear Fusion
Research, Proc. 13th Int. Conf., Washington, DC, 1990 (Vienna: IAEA, 1991) Vol. II, p. 361
[15] G W Hammett and F W Perkins, Phvs. Rev. Lett. 64, 3019 (1990)
[16] J W Connor and J B Taylor, Nucl. Fusion 17, 1047 (1977)
[17] J W Connor, Plasma Phys. Control. Fusion 30, 619 (1988)
[18] J W Connor and J B Taylor, Phvs. Fluids 27, 2676 (1984)
[19] J W Connor, Plasma Phvs. Control. Fusion 35, 757 (1993)
Current algebra results for the B D systems
V GUPTA and H S MANI
The Mehta Research Institute of Mathematics and Mathematical Physics, 10, Kasturba Gandhi
Marg, Allahabad 21 1 002, India
MS received 17 February 1995; revised 12 February 1996
Abstract. Using the equal time commutation relations for the components of the vector and
axialvector currents and keeping single particle states we obtain relations for the weak form
factors for the B D systems. In the heavy quark effective theory (HQET) limit these relations
determine the Isgur-Wise function.
Keywords. Heavy quark effective theory; current algebra; Isgur-Wise function; weak form
factors of heavy quarks.
PACS Nos 11-30; 11-40; 13-20
1. Introduction
In the last few years a very interesting approach to physics of hadrons containing
a heavy quark has been developed [1-8]. Theoretically this has led to the formulation
of a heavy quark effective theory (HQET). In this approach new symmetries appear
which have led to interesting predictions. In particular, the 6 form factors, which in
general would determine the hadronic matrix elements in the semileptonic decays
B(p)-+D( P ')ev,B(p}-^D*(p')ev, (1)
get related. All of them can be expressed in terms of only one unknown scalar function
(v-v'), called the Isgur-Wise function. The argument of % is the Lorentz scalar co = vv'
where v^ and v'^ are the four velocities of the B and D (or D*) mesons.
In this paper, we use equal time commutators (ETC) for the time components of vector
(V ) and axialvector (A ) currents, keeping only single particle_states, to derive two
relations (see (14) and (20)) between the form factors entering in the B decays in (1). The eqs.
(14) and (20) so obtained are strictly speaking inequalities because of the contribution of the
multiparticle states. There results of current algebra in the HQET limit reduce to the result
a result noted earlier [5, 6, 7]. Though this result is quoted in the literature [7], the
derivation seems to use dispersion relations. In this, we note that the result follows from
all the equal time commutators (x = y ) [X (x), ^oGOl wnere X Y can ^ e either time
or space component of vector or the axialvector operator. For the spatial components
we assume that the Schwinger terms a complex c number.
relations a' la Gell-Mann [9].
(a) Consider the equal time commutator (ETC) (x = y ):
[f + (x\ Vj (y)] = 2 I/ 3 (.x)5 3 (x - y),
(3)
for the time components of the vector currents V ft for b and c quarks. In (3), VQ = r; />,
VQ = by Q c and VQ = \(cy Q c by Q b). The V Q V commutator is not expected to have
Schwinger terms.
Starting with (3), for an arbitrary q, we obtain [10]
~ Jq(x ~ y)
y).
(4)
Sandwich (4) between tiie states |B(p)> and<B(p')| with 4-momenta p and p'
respectively where B ~ bd.
The r.h.s. of (4) R then becomes (since Jd 3 .vKo(.x) is the generator of the 3rd
component of the heavy quark flavour symmetry)
= 2{B(p l )
(5)
Last equality specifies our normalization for the meson states.
In the l.h.s. insert a complete set of intermediate states and use translation invariance
V{x) = e~ lF ' x V(0)e ip ' x and perform the x and y integrations to obtain, .the l.h.s. of
(4) to be
(p'-q-P n )^ 3 (p-q-p n )]. (6)
Of course, L = R, because of (4).
We now approximate (6) by keeping single-particle states in the sum over n. Since
VQ ~ r/ i>, only the 2nd term will contribute and the possible states which can contribute
are D + (cd) and D* + (cd) with J p = 0~ and 1 ~ respectively. In this approximation
where
(1)
X(5 3 (p'-q-p D )<5 3 (p-q-p D )
(7a)
240
Pramana - J. Phys., Vol. 46, No. 3, March 1996
U AX
where p D = p q has been made explicit.
Similarly
(7b)
The transition amplitudes in (7a) and (7b) can be written in terms of form factors (in
usual notation) with our normalization as follows
(8)
and
(2n) 3 j4Jk\D* + (k)\ V;(0}\B(p}} = ig(Q 2 )B^ a e* v (p + kY(p-ky. (9)
Here, Q p k and the form factors f and g are real. Since, in (7a, b),
k p q,Q 2 = q 2 , one obtains
and
" ] 2 Z Wo.vX '(WOA W- (11)
A
Performing the sum over /I and D* polarization is straightforward and yields
**p (P q
To obtain a covariant result [10], we use the standard technique of going to the infinite
momentum frame in (10) and (12), namely
P , |p|-> GO, v = p-g fixed. (13)
To ensure fixed v we choose q such that p-q. This choice gives v = p q Q and implies
q Q ->0 and q 2 = q 2 . Putting together (4-12) and implementing (13) yields
I=fl(q 2 )-g 2 (q 2 )q 2 . (14)
This is the general basic result, for any finite q 2 . If we keep the multiparticle states this
becomes 1 ^/ 2 + (q 2 ) g 2 (q 2 )q 2 . Any estimation of the contribution from these
multiparticle states (such as D) would involve a detailed analysis of experimental data,
not all of which is available. We choose not to go into such details in this brief report.
In the HQET, the form factors f and g in (8) and (9) are expressible in terms of the
single function ^(co) and because of the spin symmetry the J p = 0~ and 1 ~ states are
degenerate in mass, so we take m D , = ra D below. In the leading order, HQET gives
,^_ SM
V Gupta and H S Mani
where co = vv' and p^ = m B u M and p^ = m t^ are the four momenta of B and D
( or D*). Also q 2 = (p - p') 2 = ml + m - 2m g m D ,,. Substituting (15) and (16) in the sum
rules (14) immediately yields (2).
(b) Next we consider the ETC for the time components of the axial vector currents,
viz.
[X + (x)Mo(y)] = 25 3 (x-y)7S(x). (17)
Any possible operator Sch winger terms, present additionally in the right hand side of
(17), would be neglected. These might be expected to involve long-range effects and
could go away in the heavy quark limit.
Following the same procedure as in (3), and keeping the single particle contribution
(viz. D* in this case) one obtains
(18)
The 3 real form factors/, a for the A^ transition amplitude are defined through
a_(Q 2 )(p - /c),J( *-p), (19)
where Q = p k. Substituting this in (18) and going to the infinite momentum frame
(see eq. (13) and following) gives the sum rule
4m 2 , = (TV)] 2 + 2f(q 2 )a + (q*)[m 2 B - mj. - g 2 ]
+ [fl + (q 2 }'] 2 [ - 4roX- + (ml + mj. - <? 2 ) 2 ]. (20)
As remarked above, (20) should read 4m, ^ -because of the multiparticle states.
This relation is expected to hold for a general .a + and/ for any finite q 2 .
In HQET, to leading order [4]
2 ], (21)
(22)
where, as before co vv' and v and v' are the 4 velocities of the B and D*. Using
(21-22) in the relation (20) again yields (2) for (co). It is gratifying that both the V - F
and AQ A ETC give the same result for (oj).
_ (c) If one uses the ETC between V G and A and evaluates the matrix element between
B* and D* states and saturate it with single particle states, one again obtains an answer
which is consistent with (2) in the HQET limit.
Finally we note that the same results follow on the use of commutation relation
\_A\(x\A\(y}'] = ie abc V c Q 8 3 (x - y) + S.T. or [V\(x\ V\(y)'] = ie abc V c S 3 (x -y} + ST.
i ci JLVJI 1 1 11^ w^-ark. ivji 111 iai/L\_ua \ji
provides a straightforward, simple and systematic approach to obtain constraints
among form factors which are more general than those obtained from HQET.
References
[1] M Voloshin and M Shifman, Soc. J. Nucl. Phvs. 45, 292 (1987); 47, 511 (1987)
[2] E Eichten and B Hill, Phvs. Lett. B234, 511 (1990)
[3] H Politzer and M Wise, Phvs. Lett. B206, 681 (1988); B208, 504 (1988)
[4] N Isgur and M Wise, Phvs' Lett. B232, 1 13 (1989); B237, 527 (1990)
[5] B Grinstein, Nucl. Phys. B339, 253 (1990)
A Falk and B Grinstein, Phys. Lett. B247, 406 (1990)
A Falk, H Georgi, B Grinstein and M Wise, Nucl. Phys. B343, 1 (1990)
[6] J D Bjorken, SLAC Preprint SLAC-PUB-5278 (1990)
[7] Eduardo de Rafael and Joseph Taron, Phys. Lett. B282, 215 (1992)
[8] T Manuel, W Roberts, Z Ryzak, Phvs. Lett. B254, 274 (1991)
J Rosner, Phys. Rev. D42, 3732 (1990)
[9] M Gell-ManrvP/iys. Rev. 125, 1067 (1962); Physics 1, 63 (1964)
[10] See for example, Current algebras and applications to particle physics edited by S L Adler
and R F Dashen (W A Benjamin Inc., 1968) Ch. 4
_ I, Phvc Vnl 4ft Nn T IVfarrh 1996 243
Waves with linear, quadratic and cubic coordinate dependence
of amplitude in crystals
G N BORZDOV
Department of Theoretical Physics, Byelorussian State University, Fr. Skaryny avenue,
4, Belarus
MS received 18 September 1995
Abstract. There exist inhomogeneous electromagnetic and elastic waves with linear, quadratic
and cubic coordinate dependence of amplitude, in addition to the usual plane harmonic waves in
crystals. This article is concerned with conditions for initiation of such waves and mathematical
techniques for their description. New types of electromagnetic waves with quadratic and cubic
coordinate dependence of amplitude on the transverse coordinate in a transparent biaxially
anisotropic plate are found. Any transparent biaxial crystal is demonstrated to have an infinite
set of cuts, each suitable for initiation of one or more such waves.
Keywords. Anisotropy; wave; operator; degeneracy; singularity.
PACS Nos 3-50; 41-10; 42-0; 43-0
1. Eigen waves
The plane harmonic vector wave (eigenwave)
W = W exp[i(k-r-Q)t)] (!)
is one of the primary and extremely fruitful notions in electrodynamics and elasto-
dynamics of anisotropic media. Electromagnetic and elastic eigenwaves resemble one
another in some respects and can be treated in the frame of similar mathematical
techniques [1-7]. Therefore, we shall specify below the physical meaning of the
oscillating quantity W only in those cases where it is essential. In particular, W can be
any of the following quantities: the electric (magnetic) field strength E(H), the electric
(magnetic) induction D(B), the six-dimensional vector such as col(E, H) and so on - for
electromagnetic waves; the displacement vector u - for elastic waves. In a homogene-
ous linear medium, substitution of W of (1) into the corresponding wave equations
[1-7] results in the eigenvalue equation of the form
C(k,ew)W = 0, (2)
where C(k,o>) is a linear operator (matrix) depending on the wave vector k, the
frequency co and material constants of the medium. If the determinant of C(k,co)
vanishes, i.e.
) = 0, (3)
245
normal n and to. Its solutions k m = /c m (n, oj)n together with the corresponding complex
amplitudes W m = W (k m , CD) completely define properties of all eigen waves possible in
the medium.
2. Voigt waves
The special place of eigen waves in electrodynamics, elastodynamics, magnetohydrody-
namics and many other branches of modern physics is fully deserved, since various wave
phenomena can be theoretically investigated or, at least, explained in main features by
making use of the eigenwave approximation or the Fourier expansion in eigenwaves.
However, in some anisotropic media the eigenwaves do not form the complete system of
plane-wave solutions. In particular, in some absorbing crystals there exist the so-called
singular axes [2,4, 7-15] along which Voigt waves [8] (see also refs [4, 7,9, 12-14]), i.e.
the plane light waves with linear coordinate dependence of amplitude:
D = [D + /(MDJexp^k-r - o)f)], (4)
can propagate. However, along an isotropic optic axis [4] any polarization state travels
unchanged through the crystal and along a singular axis only one polarization state can
travel unchanged. The polarization state of the Voigt wave D of (4), as it travels through
the crystal, gradually goes towards the only existing eigen-polarization, namely, the
one defined by complex vector D^
It was shown by Voigt [8] that the light normally incident on the crystal in the
direction of a singular axis excites the refracted wave of the form D of (4). Voigt has
obtained this result using the limit transition from the general case of light incidence,
when the total refracted wave consists of two eigenwaves, to the above-mentioned
special case, when this superposition of eigenwaves degenerates into the wave D of (4).
In the frame of this approach, the exact solution of the problem of reflection and
transmission for this special case has been found. However, for a long time Voigfs
paper was little known, and even now it is difficult to access and infrequently cited.
At a given frequency co, a non-gyrotropic absorbing crystal is described by two real
symmetric tensors, namely, the dielectric permittivity tensor and the electric conduc-
tivity tensor a, which can be united in the single complex tensor s' = & i4ncr/a). The
analysis of wave propagation in such crystals on the basis of the usual coordinate
approach is very laborious and involved. To facilitate this analysis, Pancharatnam
suggested [9] an elegant geometric approach based on the use of the Poincare sphere
and the infinitesimal operations of birefringence and dichroism. He showed [9] (see
also refs [11, 14, 15]) that various optical effects in absorbing crystals, including the
phenomenon of propagation along singular axes, can be interpreted due to the effects of
linear birefringence and linear dichroism superposed continuously along the depth of
the medium. Pancharatnam not only explained by his method [9] the existence of
singular axes as well as their main peculiarities concerning wave propagation but also
confirmed his conclusions experimentally [10].
In the last few decades, considerable progress has been made in the study of elastic
and electromagnetic waves in various anisotropic media by application of covariant
246 Pramana - J. Phys., Vol. 46, No. 4, April 1996
Cubic coordinate dependence of amplitude in crystals
(coordinate-free) methods developed by Fedorov [2-5]. For a plane wave propagating
along a singular axis, Maxwell's equations reduce to a system of homogeneous linear
differential equations with a degenerate matrix of coefficients [12, 13]. Fedorov and
Goncharenko [13] found the general solution of this system in the form D of (4) and
solved the problems of reflection and transmission for the light normally incident onto
a semi-infinite crystal, and a plate, cut normal to a singular axis. In the particular case of
semi-infinite monoclinic crystals, similar results have been obtained by Hapaluk [12].
In anisotropic and gyrotropic media, the plane monochromatic wave, propagating
in any direction n, can be written in the exponential form
(5)
where fc = co/c, c is the velocity of light in vacuum, N D is the tensor of refractive indices
[16-18], D(0) is an arbitrary complex vector satisfying the condition n-D(O) = 0. The
non-zero eigenvalues and the corresponding eigenvectors of N D specify the refractive
indices n (complex ones, in an absorbing medium) and polarizations (complex amplitudes
D , N D D = n D , N D n = 0) of isonormal eigenwaves. Thus, the non-degenerate N D
describes the superposition of two eigenwaves, i.e. D of (5) takes the form
D = D + exp[i(k + -r - cot)'] + D_ exp[i(k_ -r - cot)'], (6)
where k = /c n n.
The Voigt wave D of (4) is described by a degenerate tensor
JV D = n / + d(g)d, (7)
where n d = 0, d is a circular complex vector (d 2 = 0), i.e. its real and imaginary parts are
normal and equal in magnitude, / = / nn is the projection operator of phase
plane, / is the unit tensor, and is the tensor product. There is only one eigenvector
d corresponding to the double degenerate eigenvalue n (N D d = d). Substitution of
N D of (7) into (5) results in D of (4), where k = k n n, D = D(0), D, = d[d-D(0)]/n .
Hence, the magnitude of D! depends on the initial polarization. If D(0)~d, D 1
vanishes, in other words, only the circular polarization specified by the vector d travels
unchanged along the singular axis. At last, if d = 0, D : vanishes identically, i.e.
ND = n ol describes a wave propagating along an isotropic optic axis.
3. Fedorov-Petrov waves
In 1963, Fedorov and Petrov found [19, 20] a new type of waves with linear coordinate
dependence of amplitude. They showed that, in the case of oblique incidence onto
a transparent or absorbing uniaxial non-gyrotropic crystal, an eigenwave can excite
a non-uniform wave
E = [E + z/c (q-r)E 1 ]exp[f(kT-cor)], (8)
where q is the interface normal, E and E 1 are some vector parameters. Indeed, similar
relations can be written for other field vectors D and B = H as well as for the tangential
components E t and H, of E and H. It is essential that Fedorov-Petrov wave E of (8) is
non-uniform, i.e. k and q are always non-collinear, and it can be excited only at oblique
E t = exp[i(* b-r - a;0]exp[z7c (q-r)Ay E t (0), (9)
where
(10)
is the tensor of normal refraction [23,24], e'h = e-q = h-q = 0, / = ./ qq is the
projection operator of the interface, r\ = q-m and b = Im = m ^q are the normal and
tangential components of the refraction vector [2,4] m = k//c (m is the product of the
refractive index and the unit wave normal). There is only one eigenvector e correspon-
ding to the double degenerate eigenvalue 77 (N E e = rje). If E t (0) ~ e, E t of (9) reduces to
the eigenwave with the refraction vector m = b + rjq. The Fedorov-Petrov wave can be
considered as a limit case (tf -> r\) of a superposition of two eigenwaves with different
polarizations and refraction vectors m = b + ^q and in' = b + 77' q.
4. Degenerate evolution operators
In this article, consider the fields W = W(r, t\ whose dependence on the coordinates
and time is specified by some operator function (evolution operator [23-30] ) J*v(R, T),
i.e. the relation
W(r + R,r + T) = JV(R,T)W(r,r) (11)
is valid for any values of r, t, R, T. From the above discussion it follows that both Voigt
and Fedorov-Petrov waves can be treated as waves with degenerate exponential
evolution operators. It is essential that these operators are determined by non-
Hermitian tensors N D of (7) and N Ei of (10). This is a necessary condition for the
existence of linear dependence of amplitude on coordinates (see Berry's paper [14] ).
In the late 1970s, only a few types of waves with degenerate evolution operators were
found, which can be excited in an anisotropic or (and) gyrotropic medium by an
incident eigenwave. These rather unusual waves were expected to propagate only in
some media in some directions. In other words, the set of these degenerate superposi-
tions was believed to be not numerous as compared with the set of eigenwaves in the
medium. However, as it proved to be [28-35], the situation is just opposite. Waves with
linear dependence of amplitude on coordinates can propagate in any stationary
[29,35] or uniformly moving [33, 34] homogeneous linear medium (isotropic, anisot-
ropic or (and) gyrotropic) in any direction, and, in some media, waves with quadratic
[30,31] and cubic [28,32] dependence also can propagate. Moreover, for every
eigenwave, there exists an infinite set of waves having different degenerate evolution
operators, and yet reducing to the same prescribed eigenwave at the proper initial
amplitude (as with E, of (9)).
At oblique incidence onto a homogeneous layer with the unit interface normal q,
a plane harmonic wave W of (1) excites a field
W = W (z)expp(TT-a>t)], (12)
where z = q-r is the transverse coordinate, T = Ik = k - q(q-k) is the tangential compo-
nent of k. The dependence of the incident, reflected and transmitted waves as well as the
field excited in the layer on the longitudinal coordinates x and y (TT = T x x + Tj,y) is
T Tkl X7-I
\_i~ 'j uuii.1 uic necessity ui meeting uuum.icu}' uuuuiuuiib a.\. cac-ii JJUIUL ui uuiu me
interfaces. For W of (12), Maxwell's equations or, in the case of elastic waves, the
corresponding equations of elastodynamics reduce to [23-26, 28]
(13)
oz
where M is some linear operator depending on q, T and material parameters. Explicit
expressions for M are obtained in [23-25, 28], for electromagnetic waves in anisotropic
and gyrotropic media, and in [26], for elastic waves in anisotropic media.
The eigenvalues v\- } and the corresponding eigenvectors Wj of M(MV^j = rj j 'W j ,
j = 1, . . . , JV) specify the normal components rjj ~ q-k, of the wave vectors k,- = T + ^-q
and the polarizations of the eigenwaves which can be excited in the layer at given T. For
each of these waves (13) reduces to (2). Consequently, ^-(j = 1, . . . , N) are the roots of the
algebraic equation a(x + /?q, co) = which follows from (3) at k = T + ?yq. In the case of
electromagnetic (elastic) waves, it is a quartic (sextic) equation, which is why, in the
absence of degeneracy, the field in the layer consists of four (six) eigenwaves, i.e. N ^ 4
(N < 6). It is essential that the amplitude space ^ A of the total field in the layer, i.e. the
range of admissible W, is four-dimensional (JV A = 4) for electromagnetic waves [23-
25, 28], and is six-dimensional (JV A = 6) for elastic waves [26].
There are two subspaces, namely, the invariant subspace ^ and the eigensubspace
tfj, related to each eigenvalue rjj. Let Uj and Sj be algebraic and geometric multi-
plicities of r\j, i.e. the dimensions of ^ and y ^ respectively. One important point to
remember is that, by definition, MW belongs to ^ for every W in <? J5 and y ^ is the
subspace of ^., consisting of eigenvectors, i.e. Sj ^ ,-. The eigenvalues of M having been
calculated, one can find (see, for example, ref. [28]) the projection operators P ; -
(j = 1, . . . , N) of the invariant subspaces, which have the properties
Pj = P p P i P j = Q, i*j. (14)
The operator P j projects any vector W onto ^, in other words, P ; -W is the p } -
component of W, and P^W = W for every W from y r
Using the Cayley-Hamilton theorem [36] one can obtain the spectral expansion
M=Y J ( j Pj+T j ), (15)
j= i
where 7} = MPj r^Pj is a nilpotent operator [36] with index of nilpotency tj. By this,
it is meant that Tj satisfies the conditions
ry-^o, ry = o for^>i, (16)
and Tj = Q for t_. = l. This is best illustrated by a specific example. The index of
nilpotency of the matrix
/O 1 0\
0010
0001
O/
(17)
ana _ .T Phvs Vol. 4fi. Nn. 4. Anril 1996 249
G N Borzdov
is equal to 4, since
T
2 __
/O 1 0\
0001
0000
\0 O/
/O 1\
0000
0000
\0 O/
(18)
In view of (12) and the homogeneity of the layer, the general solution of (13) can be
written as
W = exp[i(t-r - o>r)]exp(izM)W(0),
(19)
where W(0) is an arbitrary complex vector from the amplitude space ^ A . By making use
of relations (14)-(16), this expression can be put in a more descriptive form
W=
(20)
,. = exp[f (k,T - cor)] Pj +
iz)
(21)
where W^O) = P_,W(0) is the ^-component of W(0). Thus, the field in the layer consists
of N partial waves W ; -, each related to one of the invariant subspaces, in the sense that
Wj belongs to <^ j at any r and t.
Iftj = 1(7} = 0), i.e. the eigensubspace ^ coincides with the invariant subspace ^., or
tj > 1 but W/0) belongs to 5^(7} W;(0) = 0), W 7 of (21) reduces to the eigenwave
W,- = exp[i(k/r - a>i)] W/0). (22)
In other cases the relation (21) describes a wave with polynomial dependence of
amplitude on the transverse coordinate z. If M does not have multiple eigenvalues
(N JV A , TJ = 0,y = 1, 2, . . . , N A ), the general solution W consists of N A partial eigen-
waves with different wave vectors k,..
Thus the structure of the evolution operator exp(izM) and, consequently, the
character of coordinate dependence of the field depend on the number N of different
eigenvalues Y\- } of M as well as their numerical values, algebraic u- } and geometric Sj
multiplicities, and indices of nilpotency t jt j = 1, 2, . . . , N. These parameters satisfy the
relations
I", = "A,
1
(23)
J=l
It makes sense to classify the operators M by values of the invariants N, u jt Sj and tj.
For electromagnetic waves in an anisotropic and gyrotropic layer, such classification
and explicit expressions for the projection operators Pj are obtained in [28]. From the
mathematical point of view, there are 14 types of operator M (see table 1 in ref. [28]),
but only eleven of them satisfy the physical restriction Sj ^ 2 stemming from the
n me ngnt normally incident on me plate cut normal to a singular axis, one has N = 2,
Uj = tj 2,Sj= 1,7= 1, 2. In this case, the field in the layer consists of two Voigt waves
travelling in the opposite directions. On the other hand, if the field consists of the Fedorov-
Petrov wave and two eigenwaves, then N = 3,u l = t l =2 ) s l = l t u j = Sj = tj =l,j = 2, 3.
Curiously, it was found [29] that waves with linear z-dependence of amplitude can
propagate even in an isotropic plate. The wave incident on such plate at the critical
angle excites the field W of (20) with parameters N = 1, Uj = 4, s 1 = t l - 2, ^ = 0. With
such four-fold degeneracy, the only eigenwave, excited in the layer, propagates parallel
to the boundaries (k t -q = 0) and has the two-dimensional amplitude space, its polariz-
ation being defined by the polarization state of the incident wave.
A completely different type of four-fold degeneracy (N = I,MJ = ^ =4,s l = 1), giving
rise to waves with cubic z-dependence of amplitude (see (21)), occurs [28,32] in a
transparent biaxially anisotropic plate cut parallel to the plane containing optic axes. Only
eight such waves, excited at different values oft, can propagate in this plate. However,
as will be shown below, similar waves can also be excited at some other orientations of
the interface normal q with respect to the optic axes. Moreover, for any transparent
biaxial crystal, there are infinitely many different cuts, each suitable for excitation of
one or more waves with quadratic or cubic coordinate dependence of amplitude.
5. Degeneracy conditions
At fixed co, (3) defines a surface in the wave vector space - the wave vector surface. Since
all wave vectors kj,j = 1, . . . , N in (21) have the same tangential component t, they can
be written in the form k } = k t + ,-q, where ,- = rj j ri i . Geometrically, these vectors
are specified by p'oints of intersection or (and) tangency of the wave vector surface and
the straight line k = k t + q, passing through the point k t in the direction of the
interface normal q. In the absence of spatial dispersion, when material parameters are
independent of k, ^(j 1, . . . , N} are the roots of algebraic equation
NA
/I, i s:~ \ V V-n i t\ ,\ A 11A\
#(.Kj -f- q, CO) = > d n C, + fll.Kj , CO) = (^)
n= 1
which follows from (3) at k = k x + q. Here a n is a homogeneous n-linear form with
respect to q, depending on k t , co and the material parameters for all n < N A , except that
a NA is independent of k^ This is true for both electromagnetic [29,35] (JV A = 4) and
elastic [37] (JV A = 6) waves.
It is obvious that, if ^ = is w r fold root of (24), i.e. kj and q satisfy the conditions
a(k l5 co) = 0, a,,(k 1 ,q,co) = 0, n=\,...,u 1 -l, (25a,b)
the normal component ?/ t = k t -q of k l is the w r fold degenerate eigenvalue of M.
Since a^ is linear in q, the condition of double degeneracy a^ = can be written as
A 1 -q = 0,
where
_da(k,co)
rV | _ _
t/K Ic ^ k
isonormal waves (K! ~ q), i.e. lor an optic (acoustic) axis, we obtain
A! -1^=0. (28)
It immediately follows that there are two types of such axes defined by the relations
Aj_ =0, and A t ^0, A^kj = 0, respectively. The distinction between them manifests
itself at oblique incidence. For an axis of the second type, there is a special case
(A^ki = 0, AJ -q , 0) when the total refracted wave consists of two partial eigen waves
with different vectors k t and k 2 = k t + 2 q, although one of these waves (with vector
kj) propagates along the axis. Such situation is impossible for an axis of the first type,
since (26) is satisfied identically for any q. It is worth noting that, in a transparent biaxial
crystal, there are two real optic axes of the first type and an infinite set of complex axes
[35] of the second type for non-uniform eigenwaves (Re(k 1 ) x ImtkJ^Q). In fact
complex optic axes exist [35] in any anisotropic or bi-anisotropic medium.
To excite a wave with degenerate evolution operator in some crystal, two main
questions must be answered: (i) How is a plate to be cut from the crystal? (ii) What
incident wave is to be used? Relation (25) supplemented, of course, with an explicit
expression for function a(k, co) (such expressions for various types of fields and media
can be found, for example, in refs [1-7,29,35,37]) provide the answers: the unit
interface normal q must satisfy the requirement (25b); the wave vector k in of the incident
wave must have the tangential component equal to T = k x q(q-k 1 ), where k t is any
solution of the dispersion equation (25a).
It is of first importance that, for any k t , (26) - the sufficient condition of double
degeneracy - has an infinite set of solutions, among which is, at least, one real unit
vector q. That is why every eigenwave is related with an infinite set of waves with
different degenerate evolution operators. There is a one-to-one correspondence bet-
ween these operators and solutions k t , q of the system of (25a) and (26). Although waves
W of (19) with complex z = q-r are also worthy of consideration [30], complex
parameter q cannot be interpreted as an interface normal. Our prime interest is in
seeking real solutions q making possible to excite wave W of (21) with t > 1 (T\ = 0), i.e.
a wave with z-dependence of amplitude. All one has to do is to satisfy the condition
n l >s li where s i =2 only for k x directed along an isotropic axis [4]; otherwise s^ = 1.
Let s x = 1, A! = 0. In this case, cuts of a crystal, suitable for initiation of waves with at
least linear z-dependence of amplitude (u i '^2) are determined by real solutions of (26).
If A! is complex (as in absorbing media), and its real and imaginary parts are not
parallel, the set of such solutions q is one-dimensional or more specifically, there is only
one suitable cut of the crystal. If A l is either a real vector or a real vector multiplied by
a complex scalar, there are an infinite number of such cuts, since this set - the real plane
tangent to the wave vector surface in the point k t - is two-dimensional.
Now let s i = 2, Aj = 0. In this case, (26) is satisfied identically, and any real solution
q of equation a 2 (k 1 ,q,o)) = 0, which is homogeneous and quadratic with respect to q,
determines a cut suitable for initiation of wave W t of (21) with parameters s t =2,
u l ^3,l<t i ^u i ,T i ^Q.
It is obvious that, with only a few exceptions (as in isotropic media, see the preceding
section), condition (25b) of three-fold (u t = 3) and especially four-fold (u^ = 4) degener-
acy can be met only for some solutions k t of (25a), if at all. In particular, at four-fold
T 4 A :i
degeneracy, the unit interface normal q, having only two independent components is
a solution of system (25b) consisting of linear, quadratic and cubic homogeneous
equations, and k l lies in its domain of compatibility. Such solutions do exist, for
example, the normal to the plane of optic axes in a transparent biaxial crystal is one of
them [28, 32]. In the succeeding section, we apply general relations (24)-(27) to find all
possible cases of three- and four-fold degeneracy of evolution operators in such
crystals.
6. Three- and four-fold degeneracy in biaxial crystals
Let e,. and ( - (i ==1,2,3) be the eigenvectors and the corresponding eigenvalues
(ej > 8 2 > e 3 ) of the dielectric permittivity tensor e of a transparent biaxial crystal. The
Fresnel equation for this crystal [2, 5, 6] written in terms of the refraction vector
m = k//c = m 1 e 1 + w 2 e 2 + w 3 e 3 , takes the form
e 1 (e 2 + e 3 )mj & 2 (&^ +e 3 )ml E Z ( E I + 2) m 3 + i 2 3 =0 -
In this case (/V A = 4), coefficients a n (24) can be written as
^ = ^-l/, 7 9/,..^, "=1,2,3,4, (30)
where A (n] (n= 1,2,3,4) are totally symmetric tensors, with the first three of them
depending on m, and the summation convention is applied to repeated indices. Explicit
expressions for A (n) can readily be obtained by expansion of a(m + q) in powers of <!;.
Let m l be a solution of (29). If m l is directed along an optic axis (binomial),
A (1) (m 1 ) = 0, i.e. a l = 0, and in consequence condition (26b) of threefold degeneracy
reduces to the single equation
Jq-Q (31)
defining a conic surface [30] in the g-space. Let a wave with refraction vector m in be
incident onto a plate of the biaxial crystal cut normal to an arbitrary generating line
of this conic surface (the corresponding solutions of (31) are obtained in an explicit
form in ref. [30]). If the tangential component of m in is equal to b = m t q(q-m 1 ), wave
Wj(21) with parameters u l =3, s l t l =2, i.e. a wave with linear z-dependence of
amplitude, arises in the plate. Unlike Voigt waves and Fedorov-Petrov waves for
which u, = r 1 = 2, s l = 1, it has a two-dimensional instead of a one-dimensional
eigensubspace.
If m 1 is not parallel to a binormal (A^^mJ ^ 0, s l = 1), to meet the condition of three-
fold degeneracy, q must satisfy the system of two equations: a l =Q,a 2 = 0. By calculating
A^'OnJ and A^^ntij), where m r satisfies (29), it can be shown that the system is
compatible if and only if the point defined by m { lies in one of the four domains cut from
the external segments of the surface of refractive indices by the cones of external conical
refraction. These domains are symmetric with respect to the principal planes of the
crystal, so that it is sufficient to consider one of them (see figure 1). For each interior point
of the domain (in differential geometry terms [38], the domain consists of hyperbolic
points of the surface), the tangent plane (a, = q- A (1) (nij ) = 0) to the surface of refractive
indices intersects the second degree surface defined by (31) in two straight lines. Their
4-
2-
0-
-2-
-4-
i i i r i r i i i | M i i i i i i i | i i i i i v i
4-20
i i '" i i i i | i i i i i i i i
4
Figure 1. The asymptotic curves (solid lines) of the surface of refractive indices in
a transparent biaxial crystal, their envelope circle (dots), and the intersection lines
(dashed lines) of this surface with the principal planes.
directions coincide with the so-called asymptotic directions [38]. For the whole
domain, the set of solutions q is described by two sets of asymptotic curves with
opposite directions of twisting around the binormal. By definition [38], the direction of
tangent to an asymptotic curve at its every point coincides with the corresponding
asymptotic direction issuing out of this point. Calculations show that these curves
(solid lines in figure 1) issue out of the point of self-intersection at the surface of
refractive indices at all possible azimuthals and then gradually approach the envelope
curve (dots). The latter is a circle; more particularly, it is the intersection line of this
surface and the cone of external conical refraction. This envelope circle can equally
be treated as the tangency line of a plane normal to a ray axis and the surface of
refractive indices.
For calculating asymptotic curves it is convenient to use spherical coordinates H, 9,
cp, where n is the refractive index (the magnitude of m), polar angle & is laid off the
binormal, and azimuthal angle cp is laid off the plane containing optic axes. The results
can be obtained in the form of functions m { =m 1 (<p, <p ) or n = n ((p,(p Q ),
& = i9 (<p, <p ), which describe two asymptotic curves with'the same initial azimuthal <p
and opposite directions of twisting around the binormal. Calculations show that total
angular twists of these curves are roughly equal to + K, i.e. as they approach the envelope
curves with cp = 0, n/2, n, 3n/2 are presented. To obtain an explicit figure, the
calculations were carried out for an invented biaxial crystal with s 1 = 25, 2 = 9, 3 = 4.
For most actual crystals, the apex angle of the cone of external conic refraction does not
exceed a few degrees.
Thus, for every m^ under consideration, there exist two different cuts of the crystal,
suitable for excitation of waves with quadratic z-dependence of amplitude (u t = 3,
5j = 1,^ =3). The corresponding interface normals q + ^j) and q_(m 1 ) are tangent to
the asymptotic curves passing through the point m^ The procedure of computing
parameter b of the incident wave remains as before.
The envelope circle and its tangent vector q are described by the relations
m i = 2 [h + Gs + G (sin 1/^62 -cos I/AS)], (32)
e 2 + smi/fs, (33)
where
h = c 1 e 1 + c 3 e 3 , s= 0^+0^, (34a,b)
P V/ 2 /P P \ 1/2
, (35a,b)
1 3/ l~~3/
1 / '/ ' F \f F \\ 1/2
- p-1 1-^ - (36a,b)
2\\e 2 J\ sJJ
For each i// vector m^ of (32) is directed along a certain generating line of the cone of
external conical refraction, and as \]/ varies from to 2n, the end of ra^O//) draws the
envelope circle of radius n 2 G, lying in the tangent plane normal to the ray axis vector
h of (34a). Substituting n^ of (32) and q of (33) in (29), (30) shows that they satisfy the
conditions of four-fold degeneracy a = a 1 =a 2 = a 3 ~Q for all if;. Consequently, there
exists an infinite set of cuts suitable for excitation of waves with cubic z-dependence of
amplitude (u 1 =4,s 1 = l,t 1 =4).To excite such waves in a plate with interface normal
q of (33), the incident wave must have the refraction vector with tangential component
b = n 2 [h + G(l cos \j/) (sin i^e 2 cosies)]. (37)
In this case, the total field W of (20) (N = 1, ^ = 4) in the layer contains only one partial
eigenwave. It has the refraction vector m i with normal component ^ = tt 2 Gsini/f
which is exactly the four-fold degenerate eigenvalue of operator M of (19). The
solutions obtained earlier [28, 32] for the cut parallel to both the optic axes result from
(32), (33) at \j/ = and ij/ = n. Naturally, similar relations can also be written for the
second optic axis.
To .calculate the evolution operators of waves under consideration and the reflection
and transmission operators of the layer, it is sufficient to substitute the corresponding
values of b and q in the general relations obtained in refs. [28-30].
7. Summary
The spectral analysis of exponential evolution operators of electromagnetic and elastic
waves reveals that, in any plate cut from a transparent or absorbing crystal, waves with
at least linear coordinate dependence of amplitude can be excited, and in some crystals,
there also exist cuts suitable for excitation of waves with quadratic and cubic coordi-
nate dependence. Taking into account waves with degenerate evolution operators is
necessary to obtain the complete system of basis functions in an anisotropic medium.
The method set forth in this article enables one to find the initiation conditions for all
types of such waves possible in a medium of interest as well as to obtain in-depth data
on geometry of the wave vector surface. The application of this method to a transparent
biaxial crystal has revealed new types of electromagnetic waves with (i) quadratic and
(ii) cubic coordinate dependence of amplitude. These waves can be excited in a plate cut
normal to
(i) an asymptotic curve at any hyperbolic point of the surface of refractive indices;
(ii) an envelope circle of a set of asymptotic curves.
There are four such sets, and their envelope circles coincide with the intersection lines of
the surface of refractive indices and the cones of external conical refraction.
Acknowledgements
The author thanks Prof. S Ramaseshan for sending copies of the collected works of
S Pancharatnam and the memorial issue of Current Science, and Dr E A Evdischenko
for sending a copy of Voigt's article.
References
[1] L B Felsen and N Marcuvitz, Radiation and scattering of waves (Prentice-Hall, New Jersey,
1973)
[2] F I Fedorov, Optics of anisotropic media, Izd. AN BSSR, Minsk, (1958) (in Russian)
[3] F I Fedorov, Theory of elastic waves in crystal^ (Plenum, New York, 1968)
[4] F I Fedorov, Theory of gyrotropy, Nauka Tekli., Minsk, (1976) (in Russian)
[5] F I Fedorov and V V Filippov, Reflection and refraction of light by transparent crystals,
Nauka Tekh., Minsk, (1976) (in Russian)
[6] L D Landau, E M Lifshitz and L P Pitaevskii, Electrodynamics of continuous media, Second
edition (Pergamon, Oxford, 1984)
[7] V M Agranovich and V L Ginzburg, Crystal optics with spatial dispersion and excitons
(Springer-Verlag, Berlin, 1984)
[8] W Voigt, Gott. Nachr. 5, 269-277 (1902)
[9] S Pancharatnam, Proc. Indian Acad. Sci. A42, 86-109 (1955); (reprinted in Collected Works
of S Pancharatnam, Oxford University Press, 1975), pp. 32-55
[10] S Pancharatnam, Proc. Indian Acad. Sci. A42, 235-248 (1955); (reprinted in Collected
Works of S Pancharatnam, Oxford University Press, 1975), pp. 56-69
[11] G N Ramachandran and S Ramaseshan, in Handb. Phys. (Springer-Verlag, Berlin, 1961)
25(1)
[12] A P Hapaluk, Opt. Spektrosk. 12, 106-110 (1962) (in Russian)
[13] F I Fedorov and A M Goncharenko, Opt. Spektrosk. 14, 100-105 (1963) (in Russian)
[14] M Berry, Curr. Sci. 67, 220-223 (1994)
[15] G S Ranganath, Curr. Sci. 67, 231-237 (1994)
[16] L M Barkovskii and G N Borzdov, Opt. Spektrosk. 39, 150-154 (1975) (in Russian)
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Sov. Phys. Crystallogr. 21, 245-247 (1976)]
[18] L M Barkovskii, Zh. Prikl. Spektrosk. 30, 115-123 (1979) (in Russian)
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[20] N S Petrov and F I Fedorov, Opt. Spektrosk. 15, 792-796 (1963) (in Russian)
[21] F I Fedorov, N S Petrov and V V Filippov, Zh. Prikl. Spektrosk. 42, 844-849 (1985) (in
Russian)
[22] F I Fedorov, V V Filippov and I M Gurevich, Izv. AN BSSR, Ser. Fiz. mat. nauk, 6, 61-66
(in Russian)
[23] G N Borzdov, L M Barkovskii and V I Lavrukovich, Zh. Prikl. Spektrosk. 25, 526-531
(1976) (in Russian)
[24] G N Borzdov, Dissertation, Byelorussian State University, Minsk, 1977 (in Russian)
[25] L M Barkovskii, G N Borzdov and A V Lavrinenko, J. Phys. 20, 1095-1106 (1987)
[26] L M Barkovskii, G N Borzdov and A V Lavrinenko, Acusticheskii J. 33, 798-804 (1987) (in
Russian)
[27] L M Barkovskii, G N Borzdov and F I Fedorov, J. Mod. Opt. 37, 85-97 (1990)
[28] G N Borzdov, Kristallografiya, 35, 535-542 (1990) (in Russian) [Soy. Phys. Crystallogr. 35,
313-316(1990)]
[29] G N Borzdov, Kristallografiya, 35, 543-551 (1990) (in Russian) [Sot). Phys. Crystallogr. 35,
317-321(1990)]
[30] G N Borzdov, Kristallografiya, 35, 552-558 (1990) (in Russian) [Soy. Phys. Crystallogr. 35,
322-326 (1990)]
[31] G N Borzdov, J. Mod. Opt. 37, 281-284 (1990)
[32] G N Borzdov, Opt. Commun. 75, 205-207 (1990)
[33] G N Borzdov, Opt. Commun. 94, 159-176 (1992)
[34] G N Borzdov, J. Math. Phys. 34, 3162-3196 (1993)
[35] G N Borzdov, in Proceedings of the Third International Workshop on Chiral, Bi-isotropic
and Bi-anisotropic Media, edited by F Mariotte and J-P Parneix (CEA/CESTA, Perigueux,
France, 1994), pp. 323-328
[36] R A Horn and C R Johnson, Matrix analysis, (Cambridge University Press, Cambridge,
1986)
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(in Russian)
[38] J Favard, Cours de Geometric Differentielle Locale, (Gauthier-Villars, Paris, 1957)
Dielectric behaviour of ketone-amine binary mixtures at
microwave frequencies
P J SINGH and K S SHARMA*
Department of Physics, Government, M S J College, Bharatpur 321 001, India
*Raj Rishi College, Alwar 301 001, India
MS received 26 July 1995; revised 21 November 1995
Abstract. Values of dielectric constant (e') and loss factor (e") have been experimentally determined
for binary liquid mixtures of ethyl methyl ketone + ethylenediamine and methyl isobutyl
ketone + ethylenediamine at 9-44 GHz microwave frequencies at 30C. The values of e' and e" have
been used to evaluate the molar polarization, apparent polarization and the excess permittivities.
Excess refractive index, viscosity and activation energy of viscous flow have also been estimated.
These parameters have been used to explain the formation of 1 : 1 complexes for both the systems.
Keywords. Polarization; excess parameters; dielectric behaviour; methyl isobutyl ketone
binary mixtures.
PACS No. 77-22
1. Introduction
When a binary mixture is formed, the refractive index, viscosity, thermodynarnic
parameters and dielectric parameters do not vary linearly. The deviation from linearity
of these parameters is termed as excess parameters and is helpful to understand the
nature of bonding between the two liquids. Recently several workers [1-7] have
studied the excess parameters in liquid mixtures.
The nature of complex formation between the molecules may be ascertained by studying
apparent molar polarization and is useful in determining the nature of molecular interac-
tions in the liquid systems. In the past, several workers [4, 8, 9] have made dielectric studies
of liquid mixtures taking amines as one of the constituent components in the binary
mixtures. Govindan and Ravichandran [3] have studied the excess viscosity of ethyl
methyl ketone + alcohol mixtures and Das and Swain [2] have studied the methyl isobutyl
ketone and monosubstituted benzene mixtures. Dielectric studies of ketone + amine
mixtures have not been carried out in the past. As such it was felt that the present studies
may provide useful information regarding the molecular interactions and the forma-
tion of complexes in the mixtures of ethyl methyl ketone (EMK) + ethylenediamine
(EDA) and methyl isobutyl ketone (MIBK) + ethylenediamine (EDA).
2. Experimental details
The dielectric constant (a 1 ) and the loss factor (e") were measured using Surber's [10]
technique of measuring the reflection coefficient from the air-dielectric boundary of the
259
Figure 1. Schematic diagram of the experimental set-up for the measurement of e'
and e" [1, microwave generator; 2, frequency meter; 3 and 6, directional coupler;
4, ferrite isolator; 5 and 8, slide screw tuners; 7, liquid dielectric cell; 9, mica window;
10 and 11, crystal detector and galvanometer].
liquid in the microwave X-band at 9-44 GHz frequency and at 30C temperature. The
experimental set-up is shown in figure 1. The dielectric cell has a movable short. To
hold the liquid in the cell, a thin mica window, whose VSWR and the attenuation were
neglected, was introduced between the cell and the rest of the microwave bench.
The power loss in dielectric may be expressed as a function of dissipation factor D,
defined by
tan 5
(1)
where A is the free space wavelength and A c is the cut-off wavelength for the waveguide.
The propagation constant for the dielectric filled guide may be written as
(2)
where d is the attenuation constant due to dielectric and A d is the wavelength of the e.m.
wave in the waveguide filled with the dielectric. Surber has derived the following
relations for the dielectric parameters D, e' and e"
i /o^n ' n^
i A A /2n)] (JJ
x d ) 2 [ 1 tan 2 (j tan ~ * D)] (4)
71
where a d A d is the attenuation per wavelength. Now the parameters to be measured
experimentally are A d and a d A d .
In order to determine the above quantities, the movable short of the liquid cell
(figure 1) was moved in and out and the corresponding reflection coefficient |F| was
measured by means of the crystal pick-up in the directional coupler. The relationship
between the reflected power and the depth of the liquid column is approximately given
by a damped sinusoidal wave. The distance between two adjacent minimas of this curve
gives 1J2.
260
Pramana - .1. Phvs.. Vol. 46. No. 4. Anril 1QQ6
Dielectric behaviour of binary mixtures
The dissipation factor D for the system may be computed analytically as follows.
We define a factor M by the relation
M n = ir n | 2 /irj 2 = /// (6)
where = 1,2,3,..., JFJ is the reflection coefficient by the liquid column of length
L = w(A d /2) and (r^ | is the reflection coefficient for the liquid column of infinite length.
/ and / w represent corresponding current values.
Let
* = V^ (7)
j/ = (l-jc)/(l+x). (8)
According to Surber, attenuation per wavelength is given by
Kj) 1 ' 2 }, (9)
where
and
(l-M n y^
Having determined a d A d ,A ,/l c and A d , the values of parameters D,e' and s" may be
calculated by using (3) to (5). The accuracy achieved in the present values of e' and e" are
1 and 5% respectively.
The density and viscosity of the pure components and their mixtures were measured
by using the pycknometer and Ostwald's viscometer respectively. The accuracy in the
viscosity measurements was estimated to be 0-1%. Refractive indices for sodium
/Mines were measured by using Abbe's refractometer having an accuracy up to the
third place of decimal.
Ethylenediamine (AG AR grade) supplied by M/s Riedel-Dehaen, Germany was
used as such, while ethyl methyl ketone and methyl isobutyl ketone, both AR grade
supplied by M/s E Merck, India were used after distillation. The two liquids according
to their proportions were mixed well and kept for 4 to 5 h in a well stoppered bottle to
ensure good thermal equilibrium.
3. Results and discussion
The values of viscosity (rj), square refractive index {n d ), dielectric constant (e'\ loss factor
(e"), loss tangent (tan d) and activation energy (E a ) for the viscous flow, with increasing
mole fraction (X) of EDA for the binary mixtures of EDA + EMK and EDA + MIBK
are listed in table 1 .
The variation of the dielectric constant (e') with molar concentration of EDA in the
.
activation energy (EJ: for the binary liquid systems at 30C.
x
rt(cp)
n D
c'
e"
tan b
a (Kcal/mole)
system: EDA + EMK
0-00000
0-403
1-90992
18-217
2-397
0-1316
2-7273
0-13393
0-595
1-98246
14-305
2-951
0-2063
2-9000
0-25599
0-999
2-04204
11-901
3-315
0-2786
3-1922
0-37243
1-870
2-07360
8-911
3-762
0-4222
3-5397
0-48129
3-935
2-11412
8-086
3-828
0-4734
3-9532
0-58215
4-519
2-12285
7-636
3-883
0-5085
4-0174
0-62827
4-078
2-12576
7-089
3-911
0-5518
3-9511
0-76277
2-974
2-12868
9-803
4-821
0-4918
3-7518
0-84674
2-374
2-13014
11-134
5-255
0-4720
3-6081
0-92485
2-126
2-13160
12-859
5-904
0-4591
3-5352
1-00000
1-653
2-13452
14-504
6-221
0-4565
3-3760
system; EDA + MIBK
0-00000
0-487
1-95720
11-671
4-799
0-4112
3-0390
0-17450
0-701
1-99092
10-993
5-017
0-4564
3-2072
0-34525
1-066
2-02493
10-613
5-109
0-4814
3-3875
0-44962
1-347
2-04776
10-427
5-212
0-4998
3-4895
0-58440
1-940
2-07360
10-356
5-222
0-5042
3-6560
0-65548
2-099
2-09670
10-100
5-240
0-5188
3-6678
0-74085
2-499
2-11994
8-868
5-344
0-6026
3-7426
0-81642
2-379
2-12576
10-124
5-577
0-5508
3-6799
0-89379
1-899
2-12868
11-142
5-749
0-5160
3-5098
0-94979
1-815
2-1316
12-538
5-921
0-4723
3-4558
1-00000
1-653
2-13452
14-504
6-621
0-4565
3-3760
19
EMK+EDA
\
i
I MIBK* EDA
15 \
i
\
-! -^
7
H
* *^
/
Ol2 0.4 0.6 0.8 1.0
Figure 2. Variation of e' versus mole fraction of EDA in the mixture.
Microwave absorption
It is seen from figure 3, that the absorption in the mixture is greater than that in pure
liquids, a maxima in the tan 5 curve occurring at 0-63 and 0-74 mole fraction of EDA in
Dielectric behaviour of binary mixtures
0.6
EMK*EDA
*MIBK*EDA
0-2 0.4 0.6 0.8 1.0
Figure 3. Variation of tan d versus mole fraction of EDA in the mixture.
5.0
EMK*EDA
AMIBK+EDA
0.2 0.4 0.6 0.8 1.0
X EDA >-
Figure 4. Variation of viscosity of the mixture versus mole fraction of EDA in the
mixture.
EDA + EMK and EDA + MIBK mixtures respectively. We may explain this, by
sidering the Debye's equation [12] for tan 5 for a dilute solution of a polar liquid in
3n-polar solvent
(e' + 2) 2 47i x NjU 2 COT
tan <5 =
2,.2\'
27KT (1+coV)
(12)
me loiiowing considerations.
In the complex, the dipole moment can be taken as (^ + ju 2 ), /^ and fi 2 being the
dipole moments of the constituent molecules. For n molecules of each liquid forming
the complex, the absorption would be proportional to n(/i 2 + /i 2 ) for pure liquids,
assuming no interaction. On the other hand, in the mixture the absorption would be
proportional to the greater term n(ji l + /i 2 ) 2 .
Regarding the role of T, it is reasonable to assume that at microwave frequencies
cor < 1 and due to the larger size of the complex molecules, T is expected to increase and
so would the term cor/ (I + co 2 i 2 ) leading to increase of absorption.
Maxima in the viscosity curve
When the viscosity (r\) is plotted against mole fraction, the curve shows a sharp
maximum (figure 4). The maxima for the EDA + EMK mixture occurs at 0-58 mole
fraction of EDA and for EDA + MIBK mixture it occurs at 0-75 mole fraction of EDA.
The maxima for the EMK is much pronounced than that for MIBK. Huyskens et al [9]
have explained the increase in Y\ for the acid-amine and phenol-amine mixtures due to
the formation of dissociated ions in the mixtures, which is exothermic and depends
upon the acidic strength of phenol. Since in the present case EMK reacts with EDA by
an exothermic reaction, the pronounced maxima for the EMK may be associated with
the formation of dissociated ions in the mixture and due to the more acidic character of
EMK than MIBK. The spectacular increase in rj may also be attributed to the mutual
viscosity of the ketone and amine molecules, as provided by the Andrade's [1 3] theory.
Molar and apparent polarizations
The values of polarization of the mixtures were obtained using the formula
g'-l X 1 M l +X 2 M 2 _
B' + 2 d 12
where M l and M 2 are the molecular weights; X { and X 2 the molar concentrations of
the constituents of the mixture; and d the density of the mixture. Then following Earp
and Glasstone [14] and assuming the formula
(where P 2 is the apparent polarization of each liquid in the mixture, if Pj is the
polarization of the other component of the mixture in the pure liquid state); values
of P l and P 2 were calculated. The values of molar polarization P 12 are plotted in
figure 5, as a function of mole fraction of EDA in the mixture.
In the present investigation of ketone-amine mixtures, the amount of complex
present is responsible for the shape of our polarization curves and the minimum in the
curve is caused by the presence of a complex (or complexes). We may determine the
presence of a maximum mole per cent complex by the method earlier used by Combs
etal[\\'\. We may assume that a single complex is formed in the complexation reaction
A + B^AB n = C (15)
r
110
EMK+EDA
AMIBK+EDA
0.2 0.4 0.6 0.8 1.0
X EDA >
Figure 5. Variation of molar polarization versus mole fraction of EDA in the
mixture.
where A represents the ketone, B represents the EDA and C represents the complex. On
imposing the extremum condition d[C]/dx = 0, we obtain the relationship
n = x/(\ -x)
(16)
where x is the mole fraction of B at the extremum.
We may interpret figure 5 as representing two regions (high and low EDA concen-
trations), the intersection of the straight lines representing these separate regions can
be interpreted as ideally representing the point of maximum concentration of
complex. The value of X EDA at this point may then be used in the above equation to
determine n. The point of intersection for both the systems occurs at about 0-55
X EDA which corresponds to a 1:1 complex for both the systems and this reflects
the acidic character of EMK and MIBK. The acidic character of ketones may be
attributed by the enolization of the carbonyl group. Thus these results regarding the
formation of complex are supported by our earlier conclusions made from the e' versus
mole fraction and tan<5 versus mole fraction curves for both the systems under
investigation.
The values of the apparent polarization for EMK and MIBK are presented as
a function of mole fraction of ketone in figure 6. The flat portion of these curves clearly
indicates the formation of complex in the ketone + amine mixtures. The more pro-
nounced the flat portion, the .more stable is the complex. This indicates that the
complex formed with MIBK is more stable than the complex formed with the EMK.
Similar results have been obtained by Combs et al [1 1] for the alcohol + o-dich-
lorobenzene mixtures.
Excess parameters
The excess values of permittivity AB', Ae", excess viscosity htj, excess square refractive
index An^ and excess activation energy AE a for both the systems are presented in
90
CL.
jj"50
30
0.2 0.4 0.6 0.8 1.0
X KETONE 1-
Figure 6. Variation of apparent polarization versus mole fraction of ketone in the
mixture.
X EDA
0.2 0.4 0.6 0.8 1.0
EMK+EDA
*MIBK*EDA
Figure 7. Variation of excess permittivity versus mole fraction of EDA in the mixture.
figures 7 to 1 1. The excess values were then calculated by using the relations of the form
y m -(x 1 y 1 + x 2 Y 2 ) (17)
where A Y is any excess parameter and Y refers to the above mentioned quantities. The
subscripts m, 1 and 2 used in the above equation are respectively for the mixture,
component 1 and component 2. X and X 2 are the mole fractions of the two
components in the liquid mixtures.
266
Pramana - J. Phys., Vol. 46, No. 4, April 1996
-1-2
A EDA -
0.2 (U 0.6 0.8 1.0
EMK*EDA
AMIBK+EDA
Figure 8. Variation of excess loss factor versus mole fraction of EDA in the mixture.
EMK+EDA
AMIBK*EDA
EDA
Figure 9. Variation of excess viscosity versus mole fraction of ED A in the mixture.
The free energy of activation a of the viscous flow for the pure liquid and, their liquid
mixtures is obtained by using the relation
~
(18)
where q and V are the viscosity and the molar volume of the liquids respectively and
other symbols have their usual meaning.
The excess values were fitted through least squares with all points equally weighted
by using the Redlich-Kister [15] equation,
X.-XJ < 19 )
AMIBK+EDA
80
x
Q ,~
<= AO
u 0.2 0.4 0.6 0.8 1.0
XEDA >-
Figure 10. Variation of excess square refractive index versus mole fraction of EDA
in the mixture.
Table 2. Values of coefficients A'.s and standard deviations (cr) in various excess
parameters for the two binary liquid systems at 30C.
Physical
parameter
(A Y) A Q A^ A 2 A 3 A^ A^
a
system: EDA + EMK
Ae'
-33-56
-11-96
-3-07
54-58
23-70
-61-45
2-3780
AE"
-2-37
-7-73
-1-34
17-91
2-87
-18-30
0-3646
Aty
11-69
14-84
-37-38
- 24-45
41-59
9-43
1-1876
AHJI
0-51
-0-29
-1-34
1-35
1-76
1-88
0-0260
A a
3564
2171
-7105
-3297
6626
2905
292-50
system; EDA + MIBK
AE'
- 10-64
- 12-35
-22-85
-1-01
2-58
2-91
0-9826
As"
-2-15
-4-60
-1-82
15-03
0-78
- 27-54
0-5075
Af/
2-05
5-09
3-21
6-09
-0-39
- 14-96
0-3700
AiiJ
0-05
0-12
0-11
0-11
-0-04
-0-31
0-0090
A a
1436'
1748
361
-1288
-455
-320
129
where Y is any physical parameter and X and X 2 are the concentrations of the two
constituents. Buep and Baron [7] have used three coefficients (from j = to 2) and
Fattepur etal [1,4] have used five coefficients (from ;' = to 4) in this equation.
However, the fitting achieved was not perfect in all the cases. It was therefore observed
that if we take more terms in the expansion and solve these equations by the method of
least squares, better curve fitting may be achieved. Therefore we chose six coefficients in
the expansion. The values of coefficients A'fi for j = to 5 are given in table 2 along
with the standard deviation cr. By using these Aj values, excess parameters (AY)
calculated are used as guidelines to draw smooth curves in figures 7 to 1 1. It has been
observed that the present fitting is better and the present values of the excess
parameters show deviations only up to 5% from the experimental data.
1000
EMK* EDA
X A MIBK*EDA
1.0
EDA
Figure 11. Variation of excess activation energy versus mole fraction of EDA
the mixture.
The excess permittivity (As') and excess loss (As") are negative for both the keton
amine mixtures. The minima in the Ae' and Ae" curves for the EDA + EMK mixtu
occur at 0-58 and 0-63 mole fraction of EDA respectively, which are close to the value <
rnole fraction of EDA at which we expect the formation of complexes on the basis of,
tan d and r\ curves given in figures from 2 to 4. Similarly for the MIBK + EDA mixtu
the minima in the Afi' and Ae" curves occur at 0-74 mole fraction of EDA, whic
supports our earlier conclusion of the complex formation based on figures 2 to 4. Tl
excess dielectric permittivity is associated with the polarization and loss is regarded di
to the molecular motions which are governed by the complex forces of molecul;
interactions. Thus the excess loss may be regarded as a parameter which reflects tl
entropy change in a binary system.
The excess viscosity, square refractive index and activation energy are all positrv
indicating strong interactions between the ketone and amine molecules. For all the
excess parameters the maxima for the EMK + EDA mixture occurs at about 0-55 mo
fraction of EDA and for MIBK + EDA mixture maximas occurs at about 0-75 me
fraction of EDA. The values of dipole moment fi obtained [16, 17] for EMK and MIB
are 3-13 Debye and 2-55 Debye respectively. The higher //. value of EMK indicates th
the dipole-dipole interactions in EMK are stronger than MIBK. This behaviour
EMK is supported by the higher values of activation energy and excess activati<
energy of EMK as compared to MIBK. The deviation of excess activation energy
viscous flow in these systems indicate the increase in the internal energy of the visco
flow, thus supporting the presence of strong interactions in the systems of ketone ai
P J Singh and K S Sharma
4. Conclusions
The molar polarization, apparent polarization and excess dielectric parameters ha
been reported for EMK -f EDA and MIBK + EDA binary mixtures at various conce
trations. These studies suggest the strong interactions between the ketone and ami
molecules. The molar polarization curves suggest the formation of 1 : 1 complexes in t
mixtures of EMK + EDA and MIBK + EDA systems. The pronounced peaks in t
viscosity and excess parameter curves suggest the more acidic character of EMK th
MIBK.
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[9] P Huyskens, N Felix, A Janssens, F V Broeck and F Kapuku, J. Phys. Chem. 84, 1387 (19?
[10] W H Surber, J. Appl. Phys. 19, 514 (1948) '
[11] L L Combs, W H McMahan and S H Parish, J. Phys. Chem. 75, 2133 (1971)
[12] C P Smyth, Dielectric behaviour and structure (McGraw Hill Book Co. Inc., New Yo:
1955)
[13] N E Hill, W E Vaughan, A H Price and M Davies, Dielectric properties and molecu<
behaviour (Van Nostrand Reinhold, London, 1968)
[14] D P Earp and S Glasstone, J. Chem. Soc. Part II, 1709 (1935)
[15] MI Aralaguppi, T M Aminabhavi, R H Balundgi and S S Joshi, J. Phys. Chem. 95, 52
(1991)
[16] P J Singh and K S Sharma, Indian J. Pure Appl. Phys. 31, 721 (1993)
[17] P J Singh and K S Sharma, Indian J. Pure Appl. Phys. (in press)
PRAMANA Printed in India
- jOU 7 lof Aprin996
Physics pp. 271-275
A model for the reflectivity spectra of TmTe
P NAYAK
P. G. Department of Physics, Sambalpur University, Jyotivihar, Burla 768019, India
MS received 3 September 1993; revised 3 February 1996
Abstract. A simple one dimensional diatomic chain model is proposed to explain the reflecti-
vity spectra of TmTe as observed by Ward et al. It is suggested that the system undergoes
a structural phase transition of order-disorder type at 4-2 K, where the new phase assumes an
anti-ferroelectric type of arrangement of the atoms. The results, we obtained, agree well
qualitatively with the experimental results.
Keywords. Lattice dynamics; reflectivity spectra; mixed valence system; structural phase
transition; anti-ferroelectric order.
PACS Nos 63-20; 75-30
1. Introduction
There exist a measurement of the infrared reflectivity study at different temperatures or
the sample TmTe by Ward et al [1]. Their measurements of the reflectivity spectra al
sample temperature 1-3 K show a single peak of optic frequency (<D TO ) at 1 1 5 + 2 cm ~ l
The increase of sample temperature to 4-2 K gives two additional reststrahlen peak ai
frequencies 173cm" 1 and 209cm" 1 . When the temperature is increased further tc
295 K, there is neither any change in the peak positions shown at 4-2 K nor appearance
of new peaks except slightly broadening of the two peaks. Not only that, it appears to be
the onset of a new structure with a dramatic 10 3 fold increase in the d.c. conductivity
from 1-7 x KT^Q-cm)" 1 at 1 -6 K to 2-5 x KT^Q-cm)" 1 at 7 K. They have accountec
for this increased infrared reflectivity and the increased d.c. conductivity due to the
existence of mixed valance state at 4 K, where some fractions of Tm ions are in the
trivalent state. According to them, this fraction is very small, since the infrarec
reflectivity spectrum shows a little change from the basic reststrahlen spectrum. Th<
additional structure has been explained qualitatively by considering the Tm 3 + ion a:
defect bound to the rest of the lattice by the larger force constant appropriate to th<
trivalant state. The resonant frequency of the trivalant ion will thus be higher than th<
coresponding frequencies of the divalent lattice in accordance with the observation
The larger width of these resonances may be due to the strong coupling between thi
vibrational motion and the temporal fluctuation in valence.
In this paper, we explain the reflectivity spectrum of the system TmTe, assuming i
. P Nayak
,n be given accurately by the sum of the ionic diameters of the constituent elements,
ot only that, the ionic diameters of ions of these compounds are intimately related to
eir valencies i.e. the number of 4f electrons in the rare-earth ions. Smaller valence
irger number of 4f electrons), means larger ionic diameters or larger volume,
tierefore, the fluctuating valence in a rare-earth ion, directly causes a fluctuation in the
nic diameter. The system TmTe, being a mixed valent one, the analysis of it can be
)ne in a similar way as described above. From the electronic configuration of Tm
om, it is seen that, it exists in two valence states Tm +2 and Tm~ 3 , which are
laracterized by the states (4f) 13 and (4f) 12 respectively. This fluctuation in valence
Jtween 2 and 3, fluctuates the volume which ultimately changes the force constants.
To incorporate the above idea, we choose here a one dimensional diatomic lattice
ith one species occupying one of the two off-centre lattice sites. In an earlier paper,
abaswamy and Mills [2] considered such a system as a model to study structural
lase transition of order-disorder type. Their analysis of the model has shown that the
indom force constant disorder result the ions of the system to rearrange themselves to
irm two types of arrangements (i) ferroelectric type where atoms of one sublattice
.ove rigidly with that of the other and (ii) antiferroelectric type, with a particular
>ecies of alternate cells move in opposite directions with respect to the equilibrium
Dsitions. The analysis of the temperature dependent effective exchange interaction,
lows that the antiferroelectric arrangement of atoms, have lower free energy com-
ared to ferroelectric arrangement. We have used this finding to explain the reflectivity
>ectra of TmTe observed by Ward et al. In doing so, we assign the Te and Tm atoms
> form the two units of the diatomic chain and for Tm we assume the existence of two
fF-centre sites. With this configuration, we have performed the lattice dynamical
dculation of this mixed valance system and found to give the correct behaviour as seen
i the above experiment.
The plan of this paper is as follows. In 2, we discuss our model. The results have been
iscussed in 3. Finally we conclude in 4.
Model
he model, on which our ana-lysis is based, is illustrated in figure 1. We consider
diatomic linear chain with masses M : and M 2 occurring alternately. We assume, each
f the alternate site a double well potential symmetrically situated at its equilibrium
Dsition. We assign the Tm(M 1 ) ions (because of both divalent and trivalent state) to
:cupy one of the equivalent site of the double well potential and the Te(M 2 ) ions to
Dcupy the single well potential sites of the unit cell. This is illustrated in figure l(a).
We confine our discussion on the nearest neighbour interaction. Let the force
>nstant be denoted by K. In the description of the lattice dynamical model, when the
m ions occupy the right hand side of the equilibrium site of the double well potential,
couples to the right hand side Te ions with higher force constant than to that of the
ft. Since the double wells are symmetrically situated at the equilibrium sites, there
i,-.ao*-o a tJrrJitoninfr rr inrrpzcp nf fh^ frtw r-nnQfant tnwflrHs rieht hand side and
(b)
Figure 1. (a) The diatomic linear chain with single- and double-well potentia
occurring alternately. The masses M { (Tm) is restricted to the double well ar
M 2 (Te) to single well potentials. K is the nearest neighbour force constant, (b) In tl
changed phase at 4-2 K, diatomic linear chain model, with antiferroelectric arrang
ments became a four atomic linear chain. A is the force constant change.
Similarly, occupancy of Tm ions on left side corresponds to a coupling of K + A on le
hand side and K A on right side of Te ion. Moreover, the Tm ions have the equ;
probability of occupying any one of the equilibrium sites of the double well dependin
on the correlation that exist between different ions. Since the findings of Subaswam
and Mills [2] lead to an antiferroelectric arrangement of the atoms for a stab
structure having lower energy compared to a ferroelectric arrangement, we assurr
here the system to suffer a phase transition at 4-2 K with ions to arrange themselv(
through an antiferroelectric way i.e. if ions of even cells move to right, say, then th
masses of the odd cells move to the left without changing any lattice symmetry. Wit
this arrangement, the diatomic system acquires a character of four atomic linear chai
[Nayak [3] as depicted in figure l(b)].
As in figure l(b), the diatomic chain becomes a four atomic chain of mass sa
M ta with nearest neighbours separated by a distance of a/2 where a is the lattic
constant. Now it is straightforward to write the equations of motion. If U la be th
displacement from the equilibrium of ions of masses M la on the sublattice a in the /t
unit cell, the equations of motion is given by
A n , ai is the force constant matrix and a,/? takes values from 1 to 4. Definin
n , afi
M n = M, 3 = M t (mass of the Tm ion) and M 12 = M /4 = M 2 (mass of the Te ion) we ca
write the dispersion relation which is given by
+ 2K) 2 (-M 2 co 2 + 2v)(-M 2 co 2 + 2v')- 2(- M,co 2 + 2K)
x [(-
+ 2v)v' 2 4- (-
+ 2v> 2 ] + 2v 2 v' 2 (l -cosqa) = 0, (:
where v = K + A and v' = K ~ A. Equation (2) is a quartic equation in co 2 and gives tr
dispersion relation. For a general wave vector q, this equation can be solved numer
["41 However, we shall examine the value for a = 0. i.e. in lone wave length limi
The non-zero values in this limit give rise to optical phonon modes, which ultimal
gives the strong reststrahlen absorption peak in the reflectivity spectra of the h
crystal. In the limit q = and using the notation
= X- = X and a) 2 = X
Mj ~ " M 2 ~ 2
equation (2) takes the form of
x .MY x \f x V / x
~ 2{ K
Equation (4) can be written in a more convenient form as
Y(Y- l)(m 2 7 2 - m(m + 2) Y + (1 + m)(l - r] 2 }) =
where
m = -, Y=- and ri = .
X 2 ' X, K
Equation (5) correspond to the solutions of
and
m 2 Y 2 - (m(m H- 2)) Y + (1 + m)(l - /y 2 ) = 0.
Equation (7) gives two roots one at Y = and other at Y = 1 which are independer
r], while (8), a quadratic equation give two ^-dependent roots. Thus as expected, on
the two roots correspond to the acoustic mode as the frequency of it goes to zer<
q = 0. The other three roots, which have finite values, correspond to the three o
mode and exhibit strong absorption in the reflectivity spectra.
3. Results and discussion
In the last section we obtained four solutions to the quartic equations (eqs 7 and 8'
the frequencies. Equation (7) gives two roots which are ^-independent and the
roots of the quadratic equation (8) are ^-dependent ones. One of the jy-indepem
roots of (7) gives zero frequency at q = which correspond to the acoustic mode.
other ^-independent root i.e. 7=1 correspond to a peak in the reflectivity spectra
appears at the same frequency for both 1-3 K and 4-2 K as observed in the experin
[1]. The additional two peaks which appear at phase transition temperature 4
correspond to the two ^-dependent roots obtained from (8). The values of the two n
are calculated for different values of ^, which is the free parameter of the model. In d<
so, we consider the quadratic equation (8) and evaluated the roots for different valui
. .^/-,to 1 n-^A A TU Q f/
Values of
n
Root 1
Root 2
Root 3
Root 4
0-0
1
1-3239
2-3239
0-1
1
1-2940
2-3537
0-2
1
1-2131
2-4347
0-3
1
1-0980
2-5497
0-4
1
0-9623
2-6854
Table 2. Values of optic frequency at q = for r\ and 0- 1 .
Frequency in
cm" 1
Values of
roots
Calculated
frequency
Values of
root
Calculated
frequency
Expt. values
(cm- 1 )
W I( W TO)
W 2
W 3
1-0
1-3239
2-3239
115
152
267
1-0
1-2940
2-3537
115
148
270
115
173
209
tabluated in table 1. Assigning the frequency value 1 15 cm l , observed in the experi-
ment, to the second root i.e. 7=1 which appear at 1-3 K and 4-2 K, we evaluated the
frequencies of the other two peaks. It is found that for small values of rj between and
0-1 the calculated values agree well with the observed values. These are given in table 2.
4. Conclusion
In this paper, we have used a simple one dimensional diatomic chain model to explain
the observed reflectivity spectra of TmTe. This, we do essentially by assuming
structural phase transition of order-disorder type at 4-2 K. The results so obtained
agree well with the observed spectra. The discrepancy with the position of two peaks
can be attributed to the fact that, the double well potential used for Tm, is more likely to
have some asymmetry, which has not been considered here to avoid complications.
However, we intend to explore this possibility in future.
References
[1] R W Ward, B P dayman and T M Rice, Low Temp. Phys. LT143, 480 (1980)
[2] K R Subaswamy and D L Mills, Phys. Rev. B18, 1446 (1978)
[3] P Nayak, Promana-J. Phys. 19, 467 (1982)
[4] M Abramowitz and A Stegun Irene, Handbook of mathematical functions (Dover, New
York, 1968) p. 17
Pramana - J. Phvs.. Vol. 46, No. 4, April 1996 275
Evidence for superconductivity in fluorinated La 2 CuO 4 at 35 K:
Microwave investigations
G M PHATAK, K GANGADHARAN, R M KADAM*, M D SASTRY* and
U R K RAO**
Chemistry Division, Bhabha Atomic Research Centre, Bombay 400085, India
*Radiochemistry Division, Bhabha Atomic Research Centre, Bombay 400085, India
**Applied Chemistry Division, Bhabha Atomic Research Centre, Bombay 400085, India
MS received 13 July 1995; revised 28 February 1996
Abstract. In the fluorinated La 2 CuO 4 _. v prepared using a solid state reaction with NH 4 HF 2 as
a fluorinating agent at 550 K at ambient pressure, superconductivity was detected by microwave
and EPR techniques with a T c of 35 K.
Keywords. Superconductivity; microwave absorption; EPR; La 2 CuO 4 .
PACS Nos 74-1 0; 74-60; 74-70; 76-30
It is widely accepted that the superconductivity in YBa 2 Cu 3 O 7 _ 5 is related to the
mixed valence of copper, which can be varied by varying the oxygen content [1].
Recently a partial fluorination of non-conducting cuprates was found to be a conve-
nient route to introduce mixed valence of copper. Fluorination resulted in the
formation of both n-type and p-type superconductors. Nd 2 CuO 4 _ x F, ( was found to be
an n-type superconductor below 27 K [2] whereas fluorinated Sr 2 CuO 3 was thought
to be a p-type superconductor below 46 K [3, 4]. In the latter case the superconducting
phase was believed to be Sr 2 CuO 2 F 2 + x (x = 0-2-0-6). The presence of interstitial
fluorines in this structure is expected to produce p-type charge carriers. Therefore, the
fluorination of cuprates has the ability to convert insulating cuprates into supercon-
ducting products with different possibilities of charge carriers. We have shown that
a simple solid state reaction between Sr 2 CuO 3 and ammonium hydrogen fluoride
yields the required superconducting phase. The additional advantage of using am-
monium hydrogen fluoride for fluorination of the oxide lies in the fact that different
amounts of fluorine can be incorporated into the product. A case in point is fluorina-
tion of Sr 2 CuO 3 using this route in which we [5] have shown that one can tune T c of the
fluorinated product from to 53 K by changing the fluorine content in the product.
Slater et a\ [6] have shown that Sr 2 _ x .Ca v /Ba v CuO 3 type of compounds can be
fluorinated by NH 4 F to give superconducting products, one of which exhibited a T c of
64 K. These successes with regard to fluorination led to renewed attempts for fluorinat-
ing other perovskites. In the early days of cuprate superconductors, Tissue et al [7]
reacted La,CuO 4 with F 2 gas and prepared a superconducting compound with T c
of 34 K. However, their compound was inhomogeneous. Subsequently no reports
077
fluorination of La 2 CuO 4 using ammonium hydrogen fluoride route, which is a much
simpler method than reacting with F 2 gas, does induce superconductivity below 35 K.
Microwave technique is best suited for investigating fluorinated cuprates [8]. This
method is contactless. Therefore, there is no need to sinter the pellet at high tempera-
ture which leads to-decomposition of the superconducting compound. In this prelimi-
nary note we give microwave evidence for superconducting transition at 35 K in
La 2 CuO 4 fluorinated by this route.
The starting material La 2 O 3 (99-9%) was preheated in air at 1270K to remove
carbonate. The oxides in appropriate stoichiometry were mixed and heated in air at
1150K for 3h followed by heating at 1270 K for 80 h in air with three intermittent
grindings and furnace cooling to room temperature. The resulting La 2 CuO 4 was
mixed with NH 4 HF 2 in 1 : 1 ratio and heated in air at 550 K for one hour. The X-ray
diffraction pattern was recorded on a Ni filtered CuX a radiation on Philips PW 1729
wide angle goniometer. There is no difference observed in X-ray diffraction pattern
between parent and the fluorinated compound. If oxygen atom is simply replaced by
fluorine atom without change in symmetry of the lattice, no significant change in X-ray
pattern is anticipated on fluorination [8]. This is because the anionic sizes of oxygen
and fluorine are identical and therefore no change in cell dimensions will take place.
Further the atomic numbers of these two atoms are close and hence the scattering
powers for X-ray are also very close.
Estimation of fluorine in the sample was done by heating the sample with SiO 2
followed by steam distillation of the H 2 SiF 6 formed into alkali solution and estimating
the fluorine content in the resulting solution by ion selective electrode. Based on the
analysis, the fluorinated compound may be represented as La 2 CuO 4 _ a F 2(5 , where
<5 = 0-005.
The superconducting transition was monitored using EPR and direct microwave
absorption technique. The sample was loaded as a pellet on a sapphire rod. The
temperature was varied by using helium closed cycle refrigerator supplied by M/s APD
cryogenics. Low field microwave absorption studies were conducted using an X-band
Bruker ESP-300 spectrometer, whereas the direct microwave absorption (without field
modulation) was monitored using a Varian V-4502 EPR spectrometer having a home
built microwave bridge. The direct microwave absorption as a function of temperature
was monitored by monitoring the temperature dependent changes in the microwave
power reflected from the sample loaded cavity. This is shown in figure 1. Below 35 K,
the microwave absorption has shown continuous fall. The changes are not as sharp as
those observed in Y-123 and other high temperature superconductors [9, 10]. This
broad transition is in conformity with that reported by Tissue et al [7]. The onset of
superconductivity was further confirmed by monitoring the low field out of phase line
which is characteristic of microwave losses at the weak links [11]. This is shown in
figure 2. The typical hysteresis loop of the low field signal obtained at 10 K for
fluorinated La 2 CuO 4 is shown in figure 3. The field increasing and decreasing cycles
are shown by the arrows. It is seen that in the fluorinated La 2 CuO 4 sample, below
T c the signal appears at slightly higher fields during the field decreasing cycle compared
to the field increasing cycle. In the present case the extent of hysteresis is less which is
probably associated with low superconducting volume fraction and also due to large
7 380
s 340
<
=>
O
o
UJ
ui 260
o
220
200
10 20 3O 40 50
TEMPERATURE (K)
60
70
80
90
Figure 1. Temperature dependence of the microwave absorption of fluorinated
La 2 CuO 4 .
15 K
-25K
31 K
I8K
32 K
20K
- 35K
-20
H (GAUSS) +40
-20
H(GAUSS) +4O
Figure 2. Temperature dependence of the low field signal in the EPR measurement
of fluorinated La,CuO,.
-15
JO 20 30
H (GAUSS)
40
Figure 3. The hysteresis of the low field signal of fluorinated La 2 CuO 4 at 10 K.
The field increasing and decreasing cycles are shown by the arrow.
width of superconducting transition. This kind of hysteresis is expected for supercon-
ducting compounds [12]. Ji et al [1 3] have shown that at temperatures below the array
transition temperatures, the surface resistance during decreasing field is somewhat
smaller than that at the same external field during increasing fields. Further they have
shown that, when the field is decreased, a minimum resistance point is reached before
the field goes to zero. This minimum point in the derivative presentation corresponds
to the "EPR line position". Our results are consistent with that expected for the
superconductor below the array transition temperature, as this measurement was made
at 10 K which is much below the transition temperature. Further it may be noted that
a- sharp in phase line also appeared below 20 K. This line was found to be highly
sensitive to field cooling and/or exposure to magnetic field in superconducting phase.
Similar effects were observed in superconducting phase of Y-123 [14]. This is asso-
ciated with the trapping of field inside the sample. As the sharp in phase line appears
only at zero field, its intensity would decrease as the increase in volume fraction in
which the flux got trapped. These results clearly show that the product of fluorination
of La 2 CuO 4 by NH 4 HF 2 is superconducting with T c = 35 K. This gives an alternative
and a lot more convenient way of fluorination instead of hazardous F 2 gas route.
References
[1] R J Cava, A W Hewat, E A Hewat, B Batlogg, M Marezio, K M Rabe, J J Krajewski,
W F Peck Jr and L W Rupp Jr, Physica C165, 419 (1990)
[2] A C W P James, S M Zahurak and D W Murphy, Nature (London) 338, 240 (1989)
[3] M Al-Mamouri, P P Edwards, C Greaves and M Slaski, Nature (London) 369, 382 (1994)
[4] R M Kadam, B N Wani, M D Sastry and U R K Rao, Physica C246, 262 (1995)
[5] B N Wani, S J Patwe, U R K Rao, R M Kadam and M D Sastry, Applied Superconductivity
(1995) (Communicated)
[6] P .R Slater, P P Edwards, C G Greaves, I Gamson, M G Francesconi, J P Hodges,
M Al-Mamouri and M Slaski, Physica C241, 151 (1995)
[7] B N Tissue, K M Cirillo, J C Wright, M Daeumling and D C Larbalestier, Solid State
[8] U R K Rao, A K Tyagi, S J Patwe, R M Iyer, M D Sastry, R M Kadam, Y Babu and
A G I Dalvi, Solid State Commun. 67, 385 (1988)
[9] M D Sastry, R M Kadam, Y Babu, A G I Dalvi, I K Gopalakrishnan, P V P S S Sastry,
G M Phatak and R M Iyer, J. Phys C21, 1607 (1988)
[10] M D Sastry, R M Kadam, Y Babu, A G I Dalvi, I K Gopalakrishnan, P V P S S Sastry and
R M Iyer, Physica C1667, 153 (1988)
[11] S V Bhatt, P Ganguly and C N Rao, Pramana-J. Phys. 28, L425-427 (1987)
[12] M D Sastry, K S Ajayakumar, R M Kadam, G M Phatak and R M Iyer, Physica C170, 41
(1990)
[13] L Ji, M S Rzchowski, N Anand and M Tinkham, Phys. Rev. B47, 470 (1993)
[14] M D Sastry, K S Ajaykumar, R M Kadam, G M Phatak and R M Iyer, Proc. of DAE Solid
State Physics Symposium, Varanasi (1991)
" "".' n-n . ^ A HULGU in mum VO1. to, 1NO. "
journal of April 1996
Physics pp. 283-288
Harmonic generation studies in laser ablated YBCO
thin film grown on <100> MgO
NEERAJ KHARE* 1 , J R BUCKLEY 2 , R M BOWMAN 2 , G B DONALDSON 2
and C M PEGRUM 2
Superconductivity Group, National Physical Laboratory, New Delhi 110012, India
2 Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 ONG,
Scotland
* Author to whom correspondence should be addressed.
MS received 17 February 1995; revised 16 January 1996
Abstract. The generation of harmonics in a laser ablated YBCO film deposited on a <100>
MgO substrate is reported. Higher odd harmonics appeared when the film was subjected to an ac
field. The presence of a dc field induces only the second harmonic with a small value of slope of
V 2 ~ H d c curve (d V 2 18H &I .} compared to bulk YBCO. The variation of the amplitude of third
harmonic (F 3 ) with H ac and temperature was studied. These results are explained in terms of
a critical state model. The observation of only a small amplitude of second harmonic (V 2 ) with
a small 8 V 2 /SH^ is explained in terms of a special kind of clean grain boundary present in YBCO
laser ablated films on <100> MgO.
Keywords. Harmonic generation; thin film YBCO; laser ablation.
PACS Nos 74-60; 74-75; 74-70
1. Introduction
Magnetic harmonic generation in bulk high-jT c superconductors has been observed by
many workers [1-10]. A small amount of this material, when subjected to an ac field
(H ac > H cl ) of frequency /, generates odd harmonics (2n + I)/. The addition of a dc field
induces the generation of even harmonics. The amplitude of both families of harmonics are
found to modulate with the variation of dc field. Even harmonics are odd functions of dc
field whereas odd harmonics are even functions of dc field. Grain boundaries in bulk
YBCO acting as pinning sites are responsible for the generation of harmonics.
Study of harmonic generation in films would be useful for understanding the quality and
nature of grain boundaries. However, although there have been many reports on harmonic
generation in bulk YBCO only a few are available on YBCO thin film [11-13]. Yamamoto
et d [1 1] reported the observation of third harmonic in YBCO film prepared by evapora-
tion technique. Shaulov etd [12] showed the use of third harmonic to investigate the
multiphase nature of YBCO thin films and Revenaz and Dumas [13] reported the harmo-
nic generation in YBCO thin films due to magnetically modulated microwave absorption.
In this paper, we report studies of harmonic generation in a laser ablated YBCO film
on <100> MeO with and without a dc field. Variation of the third harmonic with ac
YBCO film [tf ac = 8-30e p k - p k
/=10kHz,H dc =
Harmonics Amplitudes
1
lOmv
2
3/<v
3
31/<v
4
5
10 /iv
6
7
4//v
8
9
2//v
2. Experimental
Epitaxial YBCO films were deposited in situ on (100) MgO substrates by a laser
ablation technique using the third harmonic (355 nm) of a Nd-YAG laser. Details of the
deposition techniques are described elsewhere [14]. The films were highly oriented with
the c-axis perpendicular to the substrates and had J c > 10 s A/cm 2 at 77 K. The film
used in the present study had T^R = 0) = 80 K and thickness % 200 nm.
To study the harmonic response two fiat spiral copper coil often turns were used.
The primary coil was glued on the back side of MgO substrate while the pick up coil
was glued on top of the film. This configuration measures shielding. The field was
applied perpendicular to the surface. An HP synthesizer (HP 3325 A) was used to apply
the ac signal and a dynamic signal analyzer (HP 3561 A) was used to observe the
frequency spectrum of the signal in the pick up coil. A lock-in amplifier was used to
record the variation of the second harmonic with dc field. A variable temperature
cryostat was used in the study of the temperature variation of the third harmonic.
3. Results and discussion
Table 1 shows for a typical film the amplitudes of various harmonics induced in the
detection coil, as recorded by the dynamic signal analyzer when both ac and dc field
were present (H ac = 8-3Oe,/= 10 kHz, # dc = lOOe). With bulk YBCO the second
harmonic appeared only when a dc field was present is a point to be noted.
In bulk YBCO the amplitude of even and odd harmonics had been found to
modulate with the variation of // dc showing maxima and minima [5, 7]. In these
experiments on YBCO films different results were found: (i) Application of a dc field
cause the appearance of the second but no higher even harmonics, and (ii) the variation
of H dc results in a very small modulation of the harmonics.
Figure 1 shows the variation of the amplitude of second harmonic (V 2 ) with H dc for
three different values of H ac as recorded using the lock-in amplifier. This shows that
slope of V 2 H dc curve (&V 2 /6H Ac ) increases as // ac is increased. This behaviour is the
same as was observed for bulk YBCO [5], However, 8V 2 /dH dc is two orders of
> 1-60
(K 1-20
<
O
O 0.80
O
UJ
(/>
0.4O
00
O*
8
i
i
H^-4.0
00
2-00
4.00
H 4e (0)
6-00
8-00
Figure 1. Variation of second harmonic with dc field at three different ac fields.
magnitude smaller than the bulk YBCO. The application of dc field also causes the
variation in the third harmonic amplitude. The change in the amplitude of third
harmonic when dc field is changed from to 10 Oe is only 6% of the value at zero dc
field. These changes in the third harmonic in film are much smaller than what has been
observed in bulk YBCO.
The observed features of higher harmonics generation and very small change in
harmonics amplitude due to application of dc field for the YBCO films can be
understood in terms of a critical state model in which grain boundary weaklinks
provide the pinning as flux sweeps in and out of the material, between the grains. The
Kim model for the critical state, which assumes the dependence of J c on local field
explains the appearance of even harmonics on application of a dc field. However, if J c is
independent of the local field (Bean model) then even harmonics will not appear. Since
the appearance of higher harmonic is a bulk effect, the amplitude will be proportional
to the number of pinning sites. In laser ablated YBCO films on (100) MgO substrates,
pinning sites are clean low angle grain boundaries. Such films have a high degree of
crystallographic orientation, but in addition possesses a larger number of grain
boundaries due to the lattice mismatch between YBCO and MgO basal planes [15].
These grain boundaries had been observed to have only some specific low angle
orientations and appeared to be clean, (absence of any secondary phases at the
interfaces) unlike the grain boundary usually found in bulk YBCO [15,16]. It is
therefore reasonable to assume that for most of the grain boundaries J c will be
independent of local field and only for a few grain boundaries J c will depend on the
local field. In such a case the application of a dc field will result in second harmonic with
small 6V 2 /6H dc as observed experimentally. J c for the film was 10 5 A/cm 2 which was
about two order higher than bulk YBCO. The observed two order of magnitude lower
value of 8V 2 /SH dc in film compare to that of bulk is due to larger J c in the film.
Pramana - J. Phys., Vol. 46, No. 4, April 1996
285
-390
-4.00-
- 490
- -50O
o*
-3. SO
-.oo
-0.60
0.00 0.80
Log (Hc/0,)
1.20
Figure 2. Variation of third harmonic with ac field (H dc = 0).
Figure 2 shows the variations of the third harmonic with ac drive field (H dc = 0) at
three temperatures. At 77 K, V 3 was initially negligibly small, and for H ac > 2-5 Oe, it
increased with no sign of saturation up to 1 1 Oe. In this region V 3 a (H ac ) 3 . At 78-2 K no
substantial growth of V 3 was observed up to 0-82 Oe. At higher H ac we found
V 3 a (Hj,,.) 1 ' 9 with a sign of saturation at 7 Oe. At 79 K growth of third harmonic was
observed even from H ac as small as 0-39 Oe and saturation was reached at H ac = 3-5 Oe.
These observations can also be understood in the framework of Bean's critical
state model. The third harmonic will appear only when flux starts penetrating into
the film. With the increase of temperature, J c and H cl decreases. Thus, the value of
H ac where the third harmonic starts growing will shift to lower values with the increase
in temperature. The increase of H ac increases penetration and so V 3 increases to a
point of saturation when ac field approaches to the saturation field (H*), which is
given as [10]
H = KJ c a (I)
where K is a geometric constant, J c is the critical current and a is the dimension
perpendicular to the field. The saturation field depends on J c , thus with the increase
of temperature the saturation of the third harmonic is expected to occur at lower
value of # ac .
Figure 3 shows the variation of the third harmonic with temperature (H ac = 8-3 Oe,
/= 10 kHz, H dc = 0). At lower temperatures (T< 74 K) no harmonic generation was
observed. Third harmonic appeared only after 75 K and then it continued to grow up to
78 K, after which it began to decay rapidly. At low temperatures, H cl is large so that no
penetration of flux occurs and therefore the third harmonic does not appear. The
increase of temperature started penetration of flux in the film through the grain
Harmonic generation in YBCO thin film
240
, -,
5. 190
2O
V
O
>
^v^.
^
z
^-v^^^
i \
o
2
r
1 ^~^^^-
of N ;
0.4O 79.10 79
a: 120
T (K) i \
^
1 \
I
> >
tr
I )
i
i *,
H- 6O
1 '
' \
j ^
/ \
70 73 80 85
TEMPERATURE (K)
Figure 3. Variation of third harmonic with temperature (H ac = 8-3 Oe,
/= 10kHz, H dc = 0). Inset shows fitting of decrease of V 2 to (1 - T/T e ) 3/2 near T c .
boundaries and V 3 started growing and became maximum at the temperature when the
flux fully penetrated the film. As the flux started penetrating into the film the pickup coil
received signals from the applied field coil and also due to the circulating super currents
on the film flowing through the various grain boundaries. The amplitude of K 3 is
proportional to the pinning potential at the grain boundaries. As the temperature
reaches near T c , the pinning potential which is proportional to J c decrease very fast. We
have found that near T c the variation of K 3 follows (1 - T/T C ) 3/2 . For the YBCO laser
ablated film, the decrease of J c with T near T c has been found [17] to follow
(1 r/T c ) 3/2 . This supports the view that K 3 is proportional to the pinning potential
and hence ./ of the film.
4. Conclusion
The generation of higher harmonics was observed in YBCO films deposited by laser
ablation on <100> MgO substrate. Grain boundaries originating due to the lattice
mismatch between YBCO and <100> MgO are responsible for harmonic gener-
ation.The presence of a dc field caused the appearance of small magnitude of second
harmonic with a value of 5V 2 /8H dc which is small compared to what has been seen in
bulk YBCO. The change in V 3 with dc field has also been found to be very small
compared to the bulk YBCO. These features are due to the special nature of grain
boundaries present in the YBCO films. These grain boundaries are clean with high
References
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B Dutta, T Venkatesan, and X D Wu, Appl. Phys. Lett. 56, 2243 (1990)
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A Compton profile study of tantalum
B K SHARMA, B L AHUJA*, USHA MITTAL, S PERKKIO**, T PAAKKARI**
and S MANNINEN**
Department of Physics, University of Rajasthan, Jaipur 302004, India
*Department of Physics, M Regional Engineering College, Jaipur 302017, India
** Department of Physics, University of Helsinki, Siltavuorenpenger 20D, Helsinki, Finland
MS received 12 December 1995; revised 26 March 1996
Abstract. We report the results of Compton profile study on polycrystalline tantalum.
Measurements have been made using 59-54 keV gamma-rays. The results are compared with the
APW band structure calculations of Papanicolaou et al and other available data. In contrast to
the work of Chang et al the overall agreement is better with the APW band structure which
worsens on incorporating the electron correlation correction. Estimates of the errors due to the
contribution from bremsstrahlung, non-validity of impulse-approximation and anomalous
dispersion are also briefly discussed.
Keywords. Compton profile; electron momentum density; APW band structure calculations;
electron correlation effects.
PACS Nos 71-25; 78-70; 51-00
1. Introduction
1.1 Compton scattering
In Compton scattering experiments, the quantity measured is basically the spectral
distribution of the Compton scattered gamma radiations. In the last two decades,
Compton scattering has emerged as a powerful tool for the investigation of the
behaviour of valence electrons [1]. A few years ago, theoretical Compton profile of Ta
along the three principal directions was reported by Papanicolaou et al [2], This work
was improved later on and extended to include several other cubic metals [3]. They have
used scalar-relativistic APW method to compute the electron momentum densities as well
as directional Compton profiles together with the isotropic Lam-Platzman (LP)
electron correlation correction and X-ray form factors. Encouraged by these results, we
had measured Compton profiles for a number of cubic 5d metals to enable a compari-
son with the above calculations [4]. A preliminary report of our work on Ta was
reported earlier [5],
A couple of years ago, Chang et al [6] also reported isotropic Compton profile of
tantalum measured by using 59-54 keV gamma-rays but no comparison was made with
our earlier work (Ref. [5]). A critical comparison of these results revealed that unlike us
[5], they observed relatively good agreement within RFA model for 5d 4 6s 1 configur-
ation as compared to the APW calculation of Papanicolaou et al [2]. An examination
289
(a) Poor statistics: Only 10,000 counts per channel (channel width 32 eV) at Compton
peak were accumulated whereas in our work over 50,000 counts were collected at the
Compton peak.
(b) Under estimation of multiple scattering: These authors have found the intensity of
the second scattering to be less than 0-2% of the first scattering for a sample of 0-2 mm
thickness which is unusually small for the thickness used by them. The appropriate
parameter for describing the intensities of the various orders of scattering is the dimension-
less quantity optical thickness (fit) [7]. It was 1-04 in the measurement of Chang et al [6]
which suggests about 7% of total intensity of double scattering [7] - an order of
magnitude larger in comparison to the value (0-2%) reported by Chang et al [6].
1.2 Review on related studies
Considerable theoretical and experimental work has been done on Ta with a view to
determine its electronic band structure and Fermi surface to explain the characteristic
properties of the metal. An earlier review up to 1970 can be found in the work of
Cracknell [8]. Among the later work, Boyer et al [9] have reported self-consistent
APW band structure calculations using X a exchange approximation. Lytle [10]
estimated the number of unfilled ^-states in Ta using X-ray absorption measurements.
Later on, Wei and Lytle [11] studied the shape of resonance absorption peak at the
L edges of Ta. Kane and Babaprasad [12] studied .K-shell Compton scattering
cross-sections for H2MeV gamma-rays. Singh et al [13] measured the spectral
distribution of 0-279 MeV gamma-rays scattered from K-shell electrons and compared
their results with relativistic theories due to Schumacher [14]; Pradoux et al [15] and
Whittingham [16]. Raju et al [17] studied angular distribution of incoherent scattering
of 279-2 keV photons. Davenport et al [18] have determined the electronic band
structure of 5d transition metals by applying the linear augmented Slater type orbital
method. Hamalainen et al [19] have measured resonant Raman scattering cross-
section from Ta at incident photon energies close to L 3 absorption edge.
In this paper we compare our more precise Compton profile results with available
theoretical and experimental data. The effects of continuous spectrum of brems-
strahlung emitted by photo electrons, non-validity of impulse approximation and
anomalous dispersion are also examined and discussed in these measurements.
2. Experiment
We present here a brief summary of the experimental procedure, as the details of the
experimental set-up have been published earlier [20]. Gamma-rays of 59-54 keV energy
from a 5 Ci annular 241 Am source were scattered by a thin sheet (0-1 mm) of poly crystalline
Ta metal held vertically in a vacuum chamber kept at 0-01 torr. Over 50,000 counts per
channel (channel width ~ 60 eV) were accumulated at Compton peak during the measur-
ing time of 48 h. The stability of the system was checked several times during the course of
measurement. The momentum resolution of the spectrometer was about 0-55 a.u.
The profile was corrected as usual for the various effects [21] such as background,
instrumental resolution, sample absorption, energy dependence of the Compton
profile for Ta was normalized to have the area of 25-67 electrons being the area of free
atom profile (core + 5d 3 6s 2 ) in the momentum range to + 7 a.u. [22]. In determining
these values, the contributions of K, L 15 L 2 shells have not been included and that from
L 3 shell was taken only up to 1-9 a.u. on account of the binding energy considerations.
However, in the low momentum side (from 7-0 to a.u.) L t and L 2 electrons also
contribute between - 7-0 and - 0-9 a.u. and 7-0 and 0-5 a.u. respectively. The
contribution of L 3 electrons would, of course, be present over the entire range. This
produces asymmetry in the Compton profile and yields different values of the area
corresponding to the low and high energy sides of Compton peak.
3. Calculations
As mentioned before, Compton profiles for band electrons in Ta along (100), (1 10) and
(111) directions are available in the literature [2, 3]. We have calculated the spherical
average of these profiles (Ref. [3]) using the standard averaging formula [23]. The
spherically symmetric correlation correction which models exchange and correlation
effects in an interacting electron gas is also taken directly from this work. After proper
normalization, to obtain the total Compton profile the contribution of inner electrons
[22] was suitably added to it.
As already reported in our earlier papers [5], the Compton profile of Ta was
computed for the various 5d-6s configurations using the RFA model approach of
Berggren [24] and 5d band occupancies have been estimated. We consider here only
one configuration which showed relatively the best agreement with the data and refer
the reader to more details in ref. [5].
4. Results and discussion
Table 1 presents the experimental Compton profile together with the unconvoluted
theoretical values for the APW models with and without the LP correction. The RFA
values for 5rf 3 6s 2 configuration which showed the best agreement among RFA models
are also included in the table. Further details of the RFA model can be seen in Refs
[5, 24]. First, we compare the experimental data before and after double scattering (DS)
as given in columns 5 and 6 of the table. It can be seen that correction due to DS is not
negligible although the sample was only 0-01 cm thick. This correction increases the
J(0) values by 4-5% while in the high momentum region (say 5 to 7 a.u.) the value is
decreased by about 5%. The ratio of double (elastic + inelastic) to single scattering was
found to be about 4% whereas this figure was reported to be 0-2% by Chang et al
despite the fact that their sample was of double thickness as compared to ours and in
both the cases 60keV gamma-radiations were employed. As pointed out earlier, our
ratio is very close to the value expected for this thickness.
Next we consider the comparison of theoretical results (columns 2-4) with experi-
ment. In the high momentum region (i.e. p z > 4 a.u.), it is seen that all theoretical values
are nearly equal and close to the experiment. This was expected since core contribution
which dominates in this region was same in all the cases and inner core electrons are
reasonably described by the free-atom Compton profiles. A good agreement in the high
between and + 7 a.u. DS means double scattering and BS means bremsstrahlung.
Statistical error (3cr) is also shown at a few points.
Experiment
P 2
APW
APW -f LP
RFA
Before
DS
After
DS
After DS
andBS
0-0
9-284
9-207
9-392
9-056
9-460
9-463
0-072
+ 0-072
0-1
9-309
9-232
9-348
9-031
9-414
9-417
0-2
9-272
9-196
9-275
8-949
9-310
9-313
0-3
9-116
9-042
9-093
8-811
9-145
9-148
0-4
8-881
8-810
8-889
8-617
8-923
8-926
0-5
8-586
8-520
8-538
8-371
8-650
8-653
0-6
8-289
8-290
8-182
8-081
8-334
8-337
0-7
8-008
7-957
7-648
7-755
7-981
7-983
0-8
7-685
7-646
7-190
7-405
7-600
7-603
1-0
6-696
6-708
6-618
6-692
6-812
6-814
+ 0-058
+ 0-058
1-2
6-042
6-064
6-015
6-037
6-114
6-116
1-4
5-400
5-413
5-472
5-484
5-535
5-537
1-6
4-868
4-874
4-989
5-016
5-041
5-043
1-8
4-436
4-440
4-592
4-601
4-605
4-608
2-0
4-149
4-152
4-197
4-233
4-225
4-227
3-0
3-298
3-301
3-313
3-280
3-240
3-241
+ 0-040
+ 0-040
4-0
2-639
2-641
2-644
2-650
2-597
2-597
5-0
2-004
2-005
2-007
2-033
1-945
1-944
6-0
1-499
1-500
1-502
1-565
1-482
1-480
7-0
1-139
1-140
1-141
1-202
1-120
1-117
0-020
+ 0-020
momentum region suggests that the impulse approximation (IA) can be considered
valid for all the electrons that are contributing in the Compton scattering. We examine
this in terms of binding energy criterion. As pointed o'ut earlier, in this data the
electrons from K, L l and L 2 shells do not contribute to single Compton scattering.
They can contribute via double elastic scattering for which a suitable correction has
been included through the multiple scattering correction [21]. Also the L 3 electron
contributes only up to 1-9 a.u. and beyond 1-9 a.u. the condition of the IA is satisfied by
all the electrons because their binding energies are much smaller than the recoil energy.
To study the behaviour of valence electrons, we have plotted in figure 1, the
difference between the various RIF convoluted theoretical values and experiment. It is
worth mentioning that the convolution by RIF changed the theoretical values mainly
up to 4-0 a.u. of momentum. Figure 1 depicts that near p z a.u. all the theoretical
values are close to the experiment. However, between 0-4 and 1-0 a.u., the APW profiles
are in better agreement than the RFA, while the trend is reversed between 1-4 and
2-0 a.u. For p z larger than 2-0 a.u., the APW values are again close to the measurement.
292
Pramana - J. Phys., Vol. 46, No. 4, April 1996
Experimental error
Figure 1. Difference (A J) profile for polycrystalline tantalum. Theoretical profile
have been convoluted with the residual instrumental function (RIF).
As is known, the effect of LP correction in theoretical results is to shift electron
across Fermi surface from low to high momentum states [25]. It can be seen fron
figure 1 that the effect of the LP correction is to decrease the APW values up to 1 a.u
and increase these values up to 1-8 a.u. The agreement, however, in this case worsens ii
the low momentum region (with LP correction), but there is a slight improvemen
between 1 and 2 a.u. In order to find overall agreement between theory and experimenl
we have calculated % 2 which was found to be the lowest for APW values without LI
correction. This is somewhat surprising because the agreement between theory an<
experiment is expected to improve on incorporating LP correction in one electroi
band calculations. We cannot assign any particular reason for this but wish to point ou
that it perhaps suggests some shortcoming in the calculation. As for the dip seen i;
figure 1 between 1 and 2 a.u., we cannot provide any specific explanation but wish t
mention that such a dip was also seen for W [4]. It is worth noting here that th
measurements on Ta and W were made at Helsinki and Jaipur respectively. In spite c
different Compton spectrometers the same trend in difference Compton profiles is seer
We, however, wish to point out that in heavier elements the influence of spin-orb
coupling is known to be important in determining the electronic states, and therefore
should be included in proper treatment of the band structure and related propertie
For W, Rozing et al [26], through a detailed calculation of two photon momentui
distribution, have observed the fact that spin-orbit coupling may affect the Fern
surface and the electron momentum distribution. Also, as shown by Bacalis et al [27
the effect of spin-orbit interaction will increase with the mass of the element and henc
can be expected to be significant for the case of 5d metals only. In the band calculatio
considered here, Papanicolaou et al have neglected spin-orbit coupling. It is likely thi
the disagreement seen here might be related to this shortcoming. It would, therefore, t
Pramana - J. Phys., Vol. 46, No. 4, April 1996 2S
mention that the count rate in our set-up for polycrystalline Ta is 0-3 counts/sec at the
Compton peak. Due to sample and geometrical restrictions the count rate will further
decrease in single crystal studies making anisotropy measurements extremely difficult.
Now we compare our data with the experimental values reported by Chang et al [6].
We have seen that valence profiles, determined by subtracting core contribution from
total profiles, before double scattering correction is close to the duly corrected
(including multiple scattering) data of Chang et al. This, in fact, serves to confirm our
possible apprehension about underestimation of multiple scattering correction in the
data of Chang et al. Further, Chang et al have estimated incorrectly the localization
trend which is a key point in their discussion. They have quoted that 86-3% of the 3d
wave function in vanadium atom, 81-3% of the 4d wave function in the niobium atom
and 78-1% of the 5d wave function in the tantalum atom are contained in the
Wigner-Seitz sphere, while the corresponding values reported in the literature are
around 96% for V[24], 90% for Nb[28] and 90% for Ta [4]. In fact, the RFA profile
for 6s electrons is very sensitive to charge localization, for which they have not
mentioned any numbers. Moreover, in their paper no reference of the wavefunction is
given to enable a check on the RFA Compton profile reported by them. In a separate
calculation we have seen that a difference of 2-3% at J(0) can be obtained in profiles
computed using different available free-atom wave functions.
However, in view of the points discussed in 1 . 1 concerning the work of Chang et al,
probably, their conclusions need further examination along the lines discussed here.
Scl-band occupancy: It is worthwhile to mention that a number of other workers have
studied Ta experimentally as well as theoretically with a view to estimate the '5<f band
occupancy. In table 2 we have compiled these values in which it is clearly seen that the
results of the various workers do not agree and the same is true for our results based on
simple RFA model. However, our data show relatively better agreement with the values
inferred from the band structure calculation [29]. Some improvement can be expected
if one uses relativistic wavefunctions for 6s electrons but, to the best of our knowledge,
no such data are available in the literature.
Now we examine the possible causes for the small discrepancy in the high momen-
tum region. Some of these could be (a) effect of bremsstrahlung, (b) non-validity of
impulse approximation, (c) anomalous dispersion and (d) self-scattering within the
source.
Table 2. Occupied charges in Ta.
References Q 6s Q 6
Lytle* [10]
4-2
Davenport et al [18]
0-82-0-89
0-67-0-92
3-51-3-19
Papaconstantopoulos [29]
0'85
0-36
3-78
Present work (RFA model)
2-0
3-0
*Reported only 5d occupancy.
294 Pramana - J. Phys., Vol. 46, No. 4, April 1996
(a) Effect of bremsstrahlung (BS): As pointed out by several workers [30], the photo
electrons produced during the interaction process produce BS, which will also be
measured along with the Compton scattered photons. We have estimated the BS
contribution in this case using the procedure of Mittal et al [31] which is given here
briefly for the sake of completeness. First of all, the ratio (/ BS // c ) total was determined for
which the photoionization cross-section was obtained by extrapolating the values
given by Rakavy and Ron [32]. This ratio was found to be 0-06. Thereafter we
calculated the spectral distribution of BSas shown in figure 2 alongwith the corrections
due to Elwert factor and form factor. This figure shows an abrupt decrease around
44keV which arises due to the fact that the contribution of L-electrons vanishes beyond
this value and only outer electrons produce the BS. It also shows that the BS
contribution is flat around the centre of the Compton profile (i.e. /? 2 = 0) which
corresponds to an energy ~ 48-5 keV.
1000
Elwert correction to Born app
Elwert and form factor corr.
20 30
Energy 63
Figure 2. Theoretical spectral distribution of bremsstrahlung calculated for
i i t * i i* 1*"P *- *"'"*' ** A * "* r
B K sharma et al
To find the ratio / B s//Bs al ^- e - intensity of BS in the Compton profile region relative to
total BS intensity), we have calculated the ratio of the areas in the energy region of
Compton profile and total, since the BS intensity is proportional to the BS cross-
section. This intensity ratio was found to be 0-09. This gives the ratio / BS // C to be
equal to 0-0054 and if this ratio is multiplied by the total Compton contribution in the
range 7 a.u. to 7 a.u., the contribution of BS(/ BS ) comes out to be 0-28 e. Therefore, the
area under the BS curve of figure 2 was normalized to 0-28 e. to get the contribution of
BS in conventional atomic units at different points of the profile. At p z 0, this
contribution, was found to be 0-017 e/a.u., being maximum (0-037 e/a.u.) at 7 a.u. The
data was corrected for BS and normalized to 25-67 electrons again which provided
only a small correction, within the statistical errors of our data, as can be seen in
table 1.
(b) Non-validity of impulse approximation: We have also estimated the correction due
to non-validity of impulse approximation (IA) for the L-shell electrons for the present
experiment following the prescription of Holm and Ribberfors [33]. At J(0), the first
order correction was found to be < 0-02 e/a.u., which is very small in comparison to the
experimental error. It may be mentioned that one should also calculate the second
order correction whenever the condition e/<j < 1 is not satisfied. This may, however, be
much smaller as compared to first order correction due to the presence of the (l/<? 2 )
term [33]. Further, it has been shown that for 2s and 2p electrons there is some
cancellation of the effects due to non- validity of I A in 2s and 2p profiles [33]. Thus, it
seems that the disagreement in the profile rrfay not be due to the non-validity of impulse
approximation.
(c) Anomalous dispersion: Next, we consider the effect of anomalous scattering which
may occur in this case since the incident energy (59-54 keV) is close to the X-shell
binding energy of Ta. It is known that anomalous scattering may change the normal
scattering factor of the metal and that could introduce an error in the estimation of
absorption correction, double scattering contribution and in the Compton profile
through normalization. We have calculated the contribution of anomalous scattering
to the normal scattering factor using the formulation of Parratt and Hampstead [34]. It
was found that in this case, the anomalous scattering decreased the scattering factor by
about 2% at zero scattering angle which did not produce any significant effect on the
double scattering correction.
(d) Self-scattering within the source: A possible cause of disagreement could be the
presence of low energy tail in the primary y-spectrum which is known to arise due to
inelastic scattering within the source material. The effect of such a contamination
would be to produce Comtpon spectra shifted in energy and weighted by the intensity
of the low-energy tail [35]. In heavy metals, such as in the present case, this will produce
some contribution through large elastic scattering. For the case of Am source, this
problem has not yet been fully solved, because enough details of the source are not
available. However, in the high energy side of the profile, as has been considered here
also, these effects are not expected to be very severe [35]. A novel scheme has been
The experimental Compton profile for Ta presented here show relatively good overall
agreement with the band structure calculations based on augmented plane wave
method. The Lam-Platzman electron correlation correction somewhat worsens the
agreement. In the present measurement there exists a disagreement, similar to that seen
for W, between theory and experiment from 1-2 a.u. which cannot be explained on the
basis of bremsstrahlung contribution, non-validity of impulse approximation and
anomalous dispersion within the sample. Self-scattering within the source may affect
the profile to a larger extent and hence suitable correction might be needed for its
contribution in case of heavy metals such as 5d transition metals. Neglect of spin-orbit
coupling in the calculation could possibly be one of the main causes. Our data do not
show agreement with the work of Chang et al mainly due to underestimation of double
scattering and relatively larger statistical errors in their work. Experimental data
particularly on directional Compton profiles are needed so that when one considers
anisotropy, problems such as those arising due to bremsstrahlung self-scattering within
the source, multiple scattering and isotropic part of LP correction would cancel out
automatically. Also, theoretical calculation using fully relativistic formulation will be
important to determine the effect of spin-orbit coupling on electron momentum
densities in Ta and thereby help to understand its electronic structure.
Acknowledgements
This work is supported partially by the Department of Atomic Energy, India. One of
the authors (UM) is thankful to the UGC for granting a Teacher Research Fellowship.
References
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[2] N I Papanicolaou, N C Bacalis and D A Papaconstantopoulos, Phys. Status Solidi B137,
597(1986)
[3] N I Papanicolaou, N C Bacalis and D A Papaconstantopoulos, Handbook of calculated
electron momentum distributions, Compton profiles and X-ray form factors of elemental solids
(CRC Press, London, 1991)
[4] U Mittal, Ph.D. thesis (unpublished) University of Rajasthan, Jaipur, (1993)
Also see a review article; B K Sharma, Z. Naturforsch. A48, 334 (1993)
[5] B K Sharma, U Mittal, B L Ahuja, S Perkkio, S Manninen and T Paakkari, in Positron
annihilation and Compton scattering edited by B K Sharma, P C Jain and R M Singru
(Omega Scientific Pub., New Delhi, 1990) p. 261
[6] C N Chang, Y M Shu, C C Chen and H F Liu, J. Phys. Condens. Matter 5, 5371 (1993)
[7] Compton scattering, edited by B Williams (Me Graw-Hill, New York, 1977) Ch. IV
[8] A P Cracknell, The Fermi surface of metals (Taylor and Francis, London, 1971)
[9] L L Boyer, D A Papaconstantopoulos and B M Klein, Phys. Rev. B15, 3685 (1977)
[10] F W Lytle, J. Catal. 43, 376 (1976)
[11] P S P Wei and F W Lytle, Phys. Rev. B19, 679 (1979)
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[28] M Tomak, H Singh, B K Sharma and S Manninen, Phys. Status Solidi. B127, 221 (1985);
Also H Singh, Study of electron momentum distributions in some 4d metals by compton
scattering technique (unpublished) Ph.D. thesis, Univ. of Rajasthan, Jaipur (1986)
[29] D A Papaconstantopoulos, Handbook of the band structure of elemental solids (Plenum
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[30] N C Alexandropoulos, T Chatzigeogiou, G Evangelakis, M J Cooper and S Manninen
Nucl. Instrum. Methods A271, 543 (1988) and references therein
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A distributed feedback dye laser based on higher order
Bragg scattering
S SIVAPRAKASAM, Ch SARADHI BABU and RANJIT SINGH*
School of Physics, University of Hyderabad, Hyderabad 500046, India
*Author to whom all the correspondance should be addressed
MS received 1 1 September 1995; revised 22 January 1996.
Abstract. A distributed feedback dye laser based on second order Bragg scattering due to
a sinusoidal susceptibility modulation is reported. Rhodamine 6G dye solution in three different
solvents; methanol, ethanol and benzyl alcohol is pumped by interference fringes produced by
two beams from the second harmonic of Nd: YAG laser. Output power is plotted as a function of
the pump power. The spectrum of dye laser shows a new type of modulation.
Keywords. Distributed feedback lasers; Bragg scattering; dye lasers.
PACSNos 42-60; 42-55
1. Introduction
Laser oscillators generally consist of a laser medium which produces gain and
a resonator structure which provides the necessary feedback for build up of oscillations.
A" conventional resonator is formed by two (or more) mirrors at the ends of the
gain medium. In 1971, Kogelnik and Shank [1] demonstrated that lasing can be
achieved without the external cavity mirrors also. In such a device the feedback
mechanism is distributed throughout and integrated with the gain medium. To be
specific the feedback is provided by Bragg-back-scattering from a periodic spatial
variation of either the refractive index of the gain medium or the gain itself or both. This
type of lasers is very compact and has mechanical stability which is intrinsic to
integrated optical devices. Also, frequency selective nature of Bragg scattering allows
the oscillations to build up over a narrow spectral band. In addition, self cavity
dumping takes place in the case of gain coupled distributed feedback (DFB) lasers
leading to pico-second pulses. Such a mechanism is applicable to different types of
lasers like dye lasers, semiconductor lasers, parametric oscillators and doped optical
fibre lasers.
Following their first observation, Kogelnik and Shank gave a coupled wave theory
of such a laser [2]. Following this, many theoretical and experimental studies [3-18]
were reported on this type of lasers because of their importance mainly in semiconduc-
tor lasers. Bjorkholm and Shank [4] showed that even the higher order Bragg
scattering also can form the basis for such a laser. However, the exact mechanism of
a higher order DFB laser is not well understood. Bor [12] in 1986 gave the first
experimental measurement of the temporal profile of a distributed feedback dye laser
S Sivaprakasam et al
(DFDL) pumped by nitrogen laser and showed that a gain coupled DFDL is capable of
giving short (pico seconds) pulses. He also gave the rate equation formulation of his
laser. Recently, Dutta Gupta and Agarwal [19] showed that such a structure, used
along with the standard Fabry-Perot cavity, can lead to a further narrowing down of
the resonances. However, till today there is no report of a precisely measured spectral
profile of such a dye laser.
In the present work, we have taken the Rhodamine 6G solution as active medium
pumped by two beams from Nd:YAG second harmonic interfering at the surface of
a dye cell. We have chosen the dye medium as it is convenient to work with and
provides the same information as far as physics is concerned. In the following we briefly
describe the relevant theory followed by the experimental details, discussion of results
and the conclusions.
2. Theoretical approach
The basic functioning of a distributed feedback laser can be understood in terms of the
coupled wave theory [2], This theory is based on the scalar wave equation for the
electric field
3 2 F
a^ + ^-o. (i)
The spatial modulation of refractive index n(z) and the gain coefficient a(z) is
assumed to be of the form
n(z) = n + rtj cos (2/? z) (2a)
a (z) = a + a t cos (2 jS z) (2b)
where "n and a are the average values of the parameters of the medium and, n t and a t
are the amplitudes of the spatial modulation. The Bragg condition is given by
/? = no> /c = nco/c. For higher order Bragg scattering /? = nco/mc, where m is the order
of scattering.
The constant k 2 in (1) is given by [2]
k 2 = p + 2ja(3 + 4jtf cos(2j? z) (3)
where the coupling constant % is defined by
7m, 1 . '
X defines the energy exchange between the two counter propagating waves. The modes
of the structure can be calculated from the relation
xjSjxcosh(yL) = ycoth(yL) f (5)
which gives the intermode spacing approximately equal to c/2nL, where L is the total
length of the gain medium. The threshold gain coefficient increases as we move away
from thp. Rraocr
Output
Figure 1. Experimental set up: L-c.ylindrical lens, BS-beam splitter, M ,
M 2 -mirrors, C-dye cell.
the deviation of the spatial modulation from the perfect, sinusoidal type due to saturation
and thermal effects. In other words, one can say that spatial modulation has higher
harmonics of the fundamental periodicity which leads to a scattering at the other
frequencies. Since the magnitude of such harmonics will be very small the coupling will be
very weak. This is also supported by the fact that threshold is higher for such a laser [4].
3. Experimental details
The main experimental set up used in this study is shown in figure 1. Pump beam is
a frequency doubled Q-switched Nd:YAG laser (Continuum, USA, model no 660B-
10), repetition rate 10 Hz and pulse width of 6ns. Pump beam is focussed by a cylin-
drical lens and is split into two beams which interfere at an angle 26 on the surface of the
dye cell. The periodicity of spatial modulation is determined by the angle between the
two beams and the corresponding lasing wavelength is given by
= n/l JmsinO
(6)
where A p is the wavelength of the pump laser.
If the angle 6 is large, the alignment becomes very difficult. However to overcome this
difficulty, one can use a small prism fused with the dye cell in order to increase the
effective angle of interference as shown in figure 2(a). The laser can be tuned either by
changing the angle 9 or by changing the refractive index of the medium. We have also
used another way of achieving the gain modulation [7] in which two parts of a single
beam interfere. This is shown in figure 2(b). In our case A p = 532 nm, therefore even at
the largest possible angle, and smallest possible refractive index, the lasing wavelength
will be larger than the red end of the gain curve of Rhodamine 6G, if we work on the
fundamental Bragg scattering. However we can safely work in the second order which
does not create any such problem.
The gain medium is the solution of Rhodamine 6G dye in different solvents viz.
ethanol, methanol and benzyl alcohol or any proportionate mixture of these. The
typical concentration of the dye used is 3 mM. The output of the DFDL is recorded
using a monochromator (Jobin-Yvon, HRS2) and a strip chart recorder.
Output
Dye cell
Pump
(b)
Dye cell
Figure 2. (a) Using a prism to increase the effective angle between the interfering
beams 9 = n4 s'm~ 1 ((l/n)~ sin(7r/4- 9')). (b) Single beam pumping method,
g = ( n /4) + sin' > ((1/n) - sin(i)).
SM M7
$70 S71
WAVEIENOTH <nm)
S72 S7)
Figure 3. Spectrum of the dye laser output. Spacing between two adjacent peaks is
92 GHz.
302
Pramana - J. Phys., Vol. 46, No. 4, April 1996
1.5
1
(0 1.0
o.
o
0.5
0.0
Benzyl alcohol
fx*- ;
Methanol
246 8 10 12
Pump Energy (mJ)
14
Figure 4. Variation of dye laser output as a function of pump power for dye
solution in two solvents; benzyl alcohol and methanol.
4. Results and discussion
A typical spectrum of the output is shown in figure 3. The spectrum shows a new type of
modulation which is observed for the first time to the best of our knowledge. The
various peaks observed are not the longitudinal modes of the cavity as one would think.
In the coupled wave theory [2], the longitudinal modes are given by (5) which shows
that intermode spacing is approximately c/2nL where L is the width of the gain
medium, n its refractive index. In our case the length of the gain medium is 1 cm,
refractive index is 1438 which gives the expected intermode separation to be 10-5 GHz,
whereas the observed spacing between the two successive peaks is ~ 92 GHz. If the
spectral modulation is due to the cavity dimensions then the width of the gain medium
has to be =1 mm. The peak-to-peak separation does not depend on solvent, dye
concentration or the angle at which the pump beams interfere. This further rules out the
possibility that these may be attributed to the longitudinal cavity modes because the
refractive index depends on the solvent. For example the refractive index of methanol is
1-54 while that of benzyl alcohol is 1-329. The same type of modulation is observed for
a prism DFDL (figure 2b) also. Confirmation for lasing is accorded firstly by destroy-
ing the interference by blocking one of the pump beam and no output is recorded, and
secondly the tunability is observed by varying the angle from 39 to 44. The width of
the overall spectral profile appears to be little large. This may be due to the fact that we
are working at higher order scattering which is less frequency selective [4]. To see the
effect of pumping we have plotted the output power as a function of the pump power
(figure 4). For dye solution in benzyl alcohol the output DFDL energy increases with
In conclusion, we have shown a dye laser with a feedback mechanism based on the
higher order Bragg-scattering. The spectrum of the dye laser shows new type of peaks.
Further experiments aimed at precisely measuring the spatial modulation in the gain
medium are being carried out to understand the origin of the spectral modulation.
References
[1] H Kogelnik and C V Shank, Appl. Phys. Lett. 18, 152 (1971)
[2] H Kogelnik and C V Shank, J. Appl. Phys. 43, 2327 (1972)
[3] C V Shank, J E Bjorkholm and H Kogelnik, Appl. Phys. Lett. 18, 395 (1971)
[4] J E Bjorkholm and C V Shank, Appl. Phys. Lett. 20, 306 (1972)
[5] I P Kaminow, H P Weber and E A Chandross, Appl. Phys. Lett. 18, 497 (1971)
[6] R L Fork, K R German and E A Chandross, Appl. Phys. Lett. 20, 139 (1972)
[7] S Chandra, N Takeuchi, and S R Hattmann, Appl. Phys. Lett. 21, 144 (1972)
[8] J E Bjorkholm and C V Shank, IEEE J. Quant. Electron. QE-8, 833 (1972)
[9] S R Chinn, IEEE J. Quant. Electron. QE-9, 574 (1973)
[10] K O Hill and A Watanabe, Appl. Opt. 14, 950 (1975)
[1 1] M Sargent III, W H Swantner and J D Thomas, IEEE J. Quant. Electron. QE-16, 465 (1 980)
[12] Z Bor, IEEE J. Quant. Electron. QE-16, 517 (1986)
[13] Irl N Duling III and M G Raymer, IEEE J. Quant. Electron. QE-20, 1202 (1984)
[14] A Flusberg and M Rokni, IEEE J. Quant. Electron. QE-22, 7309 (1986)
[15] M Terada and J Muto, Opt. Commun. 59, 199 (1986)
[16] A A Spikhal'skii, Opt. Commun. 60, 23 (1986)
[17] P K Milsom, A Miller and D C W Herberi, Opt. Commun. 69, 319 (1989)
[18] G Hasnain, K Tai, L Yang, Y H Wang, R J Fisher, James D Wynn, B Weir, N K Dutta and
A Y Cho, IEEE J. Quant. Electron. 27, 1377 (1991)
[19] S Dutta Gupta and G S Agarwal, Opt. Commun. 103, 122 (1993)
304 Pramana J. Phys., Vol. 46, No. 4, April 1996
Nonlinear Schrodinger equation for optical media with
quintic noniinearity
G MOHANACHANDRAN, V C KURIAKOSE and K BABU JOSEPH
Department of Physics, Cochin University of Science and Technology, Kochi 682 022, India
MS received 4 October 1995; revised 19 December 1995
Abstract. A nonlinear quintic Schrodinger equation (NLQSE) is developed and studied ii
detail. It is found that the NLQSE has soliton solutions, the stability of which is analysed usin
variational method. It is also found that the soliton pulse width in the materials supportin
NLQSE is small compared to soliton pulse width of the commonly studied nonlinear cubi
Schrodinger equation (NLCSE).
Keywords. Nonlinear quintic Schrodinger equation; optical solitons; nonlinear fibre optic;
variational method; pulse width; stability; critical energy.
PACS Nos 03-40; 42-50; 42-65; 42-81
1. Introduction
The possibility of effecting optical communication through fibres, in the form c
solitons was theoretically predicted by Hasegawa and Tappert [1, 2] in early 1970s, bu
it took about a decade for the experimentalists to observe solitons in fibres [3]. Sine
then this field has been in constant focus of both experimental and theoretical activities
Solitons are supported in an optical fibre by the mutually compensating presence c
dispersion and noniinearity in the medium. Such solitons are generally called envelop
solitons which form a class of solutions to nonlinear Schrodinger equation [2]. In mos
of the earlier experimental and theoretical considerations a Kerr-type noniinearity am
anomalous dispersion was matched. In Kerr type media third order polarization terr
X (3) is responsible for the noniinearity and the resulting nonlinear equation is usual!
called nonlinear cubic Schrodinger equation (NLCSE). Because of their uniqu
property of propagation without distortion, optical solitons have attracted intens
experimental and theoretical studies.
Recently, thrust has been put on developing materials with non-Kerr like nonlinearit
[4]. Success has been achieved in developing materials like semiconductor doped glasi
organic polymers, etc. that higher order nonlinearities come into play at not too big
intensity of light, which is a necessary requirement for preventing dielectric breakdowi
Kaplan [5-7] considered a more generalized nonlinear equation and showed that fc
a certain class of noniinearity bistable or more generally multistable soliton solutions ca
exist. Pushkarov et al [8] and Cowan etal [9] modified the NLCSE by includin
a quintic term and obtained a solitary wave solution to NLCQSE. Cowan et al [9] use
numerical methods to study the stability of the soliton solution to NLCQSE.
G Mohanachandran et al
Ajit Kumar et al [10] have approached the problem of the stability of the solitary wave
solutions of NLCQSE using analytical methods and they have observed that inclusion of
fifth order nonlinearity in the usual NLCSE considerably modifies the pulse propaga-
tion. Recently Angelis [11] has also studied the stability of the solution of NLCQSE
using variational approach of Anderson [12].
In this paper we develop a nonlinear Schrddinger equation by considering the effect of
quintic non-linearity alone. This situation can be achieved in a fibre by doping it with
proper materials [13]. In 2 the nonlinear quintic Schrodinger equation [NLQSE] is
derived. The solution to NLQSE is obtained in 3 and it possesses soliton behaviour.
In 4 the stability of the solution is studied using the variational method [12] and
the present investigation reveals the existence of a critical energy for the soliton solution
to exist.
2. Nonlinear quintic Schrodinger equation
The basic wave equation for a wave propagating parallel to the z direction (one
dimensional wave propagation) is given by [14] .
where P (NL) is the nonlinear polarization, E(z,t) is the macroscopic electric field.
Expanding P (NL) as a series in powers of the macroscopic field E(z, t) associated with the
incident laser radiation, we find
<2) [ J Z > t)e i( * z - w + E*(z, tje-^-""] 2
+ X (3) [ Jz, t)e' (fcz ~ wt) + E*(z, ^e-'-C"-'")] 3
+ ....... (2.2)
where the quantities % (2 \ % (3 \ etc are the higher order susceptibilities, defined as
aflyd
In perfectly isotropic medium (possessing centre of inversion) the even terms, of
susceptibility vanish.
Considering the contribution from # (3) alone we can write
CY * * (3) l w (z, Oe'* 8 -" + E*(z t fle-"**-" 03 ]. (2.4)
On expanding, terms in e~' 3<0( appear which represent the third harmonic generation. If
proper phase matching is not achieved the intensity of third harmonic wave will be very
weak. Assuming that proper phase matching is not achieved, the terms representing third
harmonic generation may be neglected. Thus,
Pg"j> = % (3) 3 |E ffl (z, t)| 2 EJz, fle 1 **-""] + cc (2.5)
where cc denotes the complex conjugate term. Including x (5) ,
p(NL) = T9<3)IF l 7 t\\ 2 R 1 7.
neglected and thus
(z t) ~ /C I co ^ * / co > 5 ^/C ~T~ CC, \^" ' )
Substituting (2.7) alongwith the values of second order derivatives of and D in (2.1)
we find
2i|
[~ 3 w co / 1 SeXS-E^, d 2 E m if 5e 1 2 5 2 e~]
|_ dz c 2 \ 2 8a)J dt _ dz 2 c 2 |_ da) 2 dot 2 J
^^ 2
2 r;(5)
(2-8)
The group velocity and the phase velocity are respectively given by
dco
v * (m) =jk and
V ~ (29}
Kp . \^"^}
The propagation constant k is a function of frequency.
k 2 (a)) = ^8(a)) (2.10)
c
where s(co) is the permittivity of the medium which is numerically equal to the square of
the index of refraction. Using the first and second derivatives of (2.10) with respect to CD,
we get the following identities
?H"K
and
1 . 5(1/K) 1 / ds 1 , d :
r~> ' v i ? I ' ^ ' o a j
'g 5co c V dco 2 do)
,_..
(2.12)
Substituting (2.9), (2.11) and (2.12) in (2.8) and after simplification we get,
dz.v g dtj < 2k\dz 2 v
g
* lLcolLt - ( J
We have the identity [4]
(2A4)
Under the slowly varying envelope approximation (SVEA) the square bracket on the
right hand side of (2.14) may be replaced by unity since the envelope function E(z,t)
varies little over a spatial region of the size of a wavelength and varies little over one
cycle of the carrier wave frequency a>.
Hence the term
inside the square bracket makes only a small correction to unity and therefore may be
neglected. Then (2.13) becomes,
d I
dz ' V.dtJ"' ' 2 dco dt 2
= C 2 T'l^xfi.. (us)
Defining
, 207TCOK, .,.
3 p o' 3 ) /i 1 \
A C 2 X (2.16)
and
d(l/V g ) __ i dV s _
=> 77? "^ ^/^ (2.17)
oco Kg dco
where p is always a positive number and a= 1. If 5F g /3<D is positive, a = + 1 and if
dK g /dco is negative a = - 1. Substituting (2.16) and (2.17) in (2.15) and rearranging,
(2.18)
The variables (z, t) may be changed to (, T)
Then (2.18) is transformed into
rrn "> ]IF 1 4 17 ; /T 1 Q\
~2 ~th r ~ w co= ~d?~' ^ *
If 1 = 0, this has the form of the ordinary Schrodinger equation for a free particle,
whose mass is inversely proportional to ^. The variable T is an effective spatial
coordinate and is an effective time. The term A |E J 4 represents a potential energy, the
form of which depends on L Assuming % (5} to be positive, the parameter A > 0, and
similarly for dVJda> > 0, a = + 1, then (2.19) becomes
1 u. a JIF I 4 F - i d ;
2^ Qr 2 A l 'wl
Writing T =
, ,4
'ai + 2a? + ]uf <Z22)
Equation (2.22) is the nonlinear quintic Schrodinger equation.
3. Soliton solutions of the NLQSE
A solution to (2.22) may be sought of the form
U(y,Q = <l>(y)e. (3.1)
Then (2.22) becomes
1/dcA 2 1 1
4\fy) ~2^ +6^ COnStant (3<2)
Applying the boundary conditions that (j>(y) and its derivative d$/dy vanish as
y- oo, the constant in (3.2) vanishes. Then we find
(3-3)
\ \ J / /
Hence we find
deb
= y- (3.4)
The solution is given by
or
U(y, = (3 K ) 1/4 [sech( N /(8K)>')] 1/2 e iK . (3.6)
Going back through all the previous transformations, we find,
r, / ^ F 3 *C 2
E <Z, t) = ;
\_2QncoV f x
This represents a pulse of stable shape which propagates through the medium with
the group velocity V K .
The solution to the NLQSE may be compared with the solution to the NLCSE given
by [14]
1/2 piKZ
(3.8)
1/2 (z-V)]
In both these equations K is the parameter which decides the width of the pulse with
the only condition that K> 0. Figure 1 shows the wave profile in the two cases against
the same normalized values of the parameters.
T -
Figure 1. Pulse profile in the cubic and quintic materials.
4. Analysis using variational approach
Using the variational formulation [12] an approximate analytical expression for the
self-trapped solutions to the NLQSE can be obtained.
Consider the NLQSE
i=0 (4.1)
(4.2)
where the suffixes z and t denote differentiation with respect to them.
The Lagrangian density corresponding to (4.1) is
^
= -\ E
21 dz
M
*
dz
dE
~dt
Since we are dealing with a one-dimensional confined pulse, a simple ansatz is
E(t,z) = A(z)e- (t2 i 2a - 2) e iat2 (4.3)
where A(z\ a(z) and a(z) are parameter functions to be determined from the reduced
variational problem.
The reduced Lagrangian is then obtained by inserting the trial function into the
Lagrangian density and integrating from oo to +00.
Substituting (4.3) in (4.2)
L=
- A Z A*) + \A\
Defining
Ldt
(4.4)
(4.5)
310
Pramana - J. Phys., Vol. 46, No. 4, April 1996
+ \A\ 2 [_4a 2 +(l/a 4 )]( v /7i/2)a 3 -(A/3)|X| 6 (V*A/ 3 )- (4.6)
The reduced variational problem is
S <L>dz = 0. (4.7)
J
Using the variational principle [12] with the reduced Lagrangian <L> given by (4.6),
the following variational equations are obtained:
a<L> d(taX)
6A* ^ dz
= - iv.A z + a 3 Aa z + a 3 A[4a 2 + (I/a 4 )] - (2/j3)al.\A\ 4 A. (4.8)
8<L> = d(-i*A*)
5A ^ dz
~ - ~ X\A\* A*. (4.9)
a
<K L>
+ \2a 2 \A\ 2 a 2 - \_\A\ 2 , /a 2 ] - - l\A\*. (4.10)
da. dz
= 8<x 3 |A| 2 a. (4.H)
Multiplying (4.8) by A*, (4.9) by A and then subtracting and adding we get the following
equations
-(a^| 2 ) = (4-12)
dz
and
i(A*A 2 -AA*) = \A\ 2 2a 2 z + 2a 2 (4fl 2 + [^n-(-^ U|4fJ. (4.13)
Equation (4.12) implies a constant of motion;
I 2 -a \A \ 2 -E (4.14)
JCrt /!/-> t-T\ \ '
where E is the initial energy of the pulse which does not change.
By comparing (4.10) and (4.13) we get
(a 3 |,4| 2 ) = --a|^4| 2 . (4.16)
dz dz
Thus we find
0/V^V M \ A\
_ ZiCXW.- UC /I
'da
= 2aa z . (4.17)
Comparing (4.17) with (4.11) we get
2aa,E = Sa 3 !^! 2 ^ = 8a 2 E a (4.18)
i.e.
(4.19)
Combining the derivative of (4.19) with (4.15) we get
dP 12 401E
l j
dz 2 a 3 3V3 a 3 '
Equation (4.20) may be considered to be derived from a potential such that
d 2 _ dV
dz 2 da
where
^ . (4.22)
Self-trapped solutions of (4.1) correspond to extrema of the potential, i.e., they
correspond to and a values such that
. (4.23)
da
Applying this condition we find
36^3 -40 IE 2 = 0. (4.24)
This implies
En =
10A
9V3
(4.25)
It appears that there is a critical value for the energy for a self-trapped solution
of (4.1) to exist. The critical value of energy is found to depend on the fifth order
susceptibility of the fibre material. Figure 2 shows the variation of the critical energy
with L
312 Pfflmanfl .T. Phvs.. Vnl. 4fi. Nn. d Anril
Nonlinear quintic Schrodinger equation
0-2 0-4 0-6 0-8 1-0
Figure 2. Variation of critical energy with the parameter L
From the Lagrangian formulation the stability analysis of the solutions can be
carried out. Stable solutions correspond to local minima of the potential function
d 2 V 36 120 IE 2
dor
36
a 4
3^/3 a
40 AE 2
(4.26)
For a minimum, (d 2 K/da 2 ) is positive which implies that A must be less than
(9,/3/lOE 2 ).
5. Conclusions
A remarkable result of considering the effect of (5) alone to the soliton propagation in
optical fibres is the reduction of pulse width and an increase in the peak value of
intensity which is useful in optical communication.
The present study reveals the existence of a critical energy for soliton solutions. This
fact may be conveniently used for any application where a cut-off is desirable in terms of
energy of the incident radiation.
Recently Herrmann [13] studied the coefficient of cubic term which has a small value
compared to that of the quintic term. In the present work we considered a situation
thp rnhir tprm ic ahpnt anH nnintir tprm alrmp is nresfint We. found that hv
active, more advantageous soliton propagation may be possible.
Acknowledgements
One of us (GMC) thanks the UGC for the award of a Teacher Fellowship under the
faculty improvement programme. VCK and KBJ thank the DST, Government of
India, for financial assistance under a research project.
References
[1] A Hasegawa and F D Tappert, Appl. Phys. Lett. 23, 142 (1973)
[2] A Hasegawa, Optical Solitons in Fibres, Springer Tracts in Modern Physics (Springer, Berlin,
1989) Vol. 116
[3] L F Mollenauer, R H Stolen and J P Gordon, Phys. Rev. Lett. 45, 1095 (1980)
[4] S Gatz and J Herrmann, IEEE J. Quantum Electron. 28, 1732 (1992)
[5] A E Kaplan, Phys. Rev. Lett. 55, 1291 (1985)
[6] A E Kaplan, IEEE J. Quantum Electron. 21, 1538 (1985)
[7] R H Enns, S S Ranganekar and A E Kaplan, Phys. Rev. A36, 1270 (1987)
[8] Kh I Pushkarov, D I Pushkarov and I V Tomov, Opt. Quantum Electron. 11, 471 (1979)
[9] Stuart Cowan, R H Enns, S S Ranganekar and Sukhpal, S Sanghera, Can J. Phys. 64, 3 1 1
(1986)
[10] Ajit Kumar, S N Sarkar and A K Ghatak, Opt. Lett. 5, 321 (1986)
[11] C De Angelis, IEEE J. Quantum Electron. 30, 818 (1994)
[12] D Anderson, Phys. Rev. A27, 3135 (1983)
[13] Jaochim Herrmann, Opt. Com/mm. 87, 161 (1992)
[14] D L Mills, Nonlinear Optics, Springer International Student Edition (Springer, Berlin 1991)
physics pp. 315-322
Geometric phase a la Pancharatnam
VEER GRAND RAKHECHA and APOORVA G WAGH
Solid State Physics Division, Bhabha Atomic.Rescarch Centre, Mumbai 400085, India
MS received 27 September 1995
Abstract. In mid-1950s, Pancharatnam [1] encountered the geometric phase associated with
the evolution along a geodesic triangle on the Poincare sphere. We generalize his 3-vertex phase
and employ it as the fundamental building block, to geometrically construct a general ray-space
expression for geometric phase. In terms of a reference ray used to specify geometric phase, we
delineate clear geometric meanings for gauge transformations and gauge freedom, which are
generally regarded as mere mathematical abstractions.
Keywords. Geometric phase; Pancharatnam triangle; parallel transportation.
PACSNo. 03-65
While studying evolutions of optical polarization states, Pancharatnam [1] made three
seminal contributions, about 30 years ahead of their time, to the understanding of
phases between distinct states. His deceptively simple, yet incisive physical observation
that two states are in phase when the intensity of their superposition is maximum, led to
a completely general phase definition [1-5], now known as the Pancharatnam
connection. Secondly, Pancharatnam made the first explicit recognition of geometric
phase by deriving the solid-angle expression for the invariant phase associated with
a geodesic triangle on the Poincare sphere. Pancharatnam's third contribution, which
is not so well known as the first two, consisted in showing that if two states l*^ > and
|*P 2 > are subjected to an analysis along a ray |^ > prior to superposition, the resultant
interference pattern will display an additional phase equal to the geometric phase for
the geodesic triangle formed by the three rays on the Poincare sphere.
Just over a decade ago, Berry provided a general quantal framework [6] for
geometric phase, independently of Pancharatnam's earlier work [1], for an eigen state
of a Hamiltonian whose parameters are cycled adiabatically. This paper triggered an
intense activity [7-9] in the field. Geometric phase is now recognized to be the
Hamiltonian-independent, nonintegrable component of the total phase, depending
exclusively on the geometry in the ray space. Geometric phase identifies with the phase
anholonomy of a parallel transported [4,9-11] quantal system and is manifested in
a wide spectrum of physical phenomena [12-16]. Recently, geometric phase has been
subjected to a group theoretic [17] and a kinematic [18] treatment. Here, we opt for the
natural, i.e. geometric, treatment of geometric phase. We reword Pancharatnam's
results in the modern quantum physics language [3-4, 9] and show that his triangle
phase is the only fundamental input needed to obtain the most general expression for
geometric phase purely geometrically.
Figure 1. Pancharatnam geodesic triangle in the ray space. Two successive filter-
ing measurements on the ray \\I/ Q > along rays \\l> l > and |^ 2 > yield an invariant phase
dependent solely on the geometry of the geodesic triangle. The surface S spanned by
the triangle is the solid angle Q for the special case of a two-sphere.
On a polarization state represented by a ray \\j/ Q >, Pancharatnam considered a pair
of successive filtering measurements carried out along rays \\l/ 1 > and |^ 2 > (figure 1), the
three rays being mutually nonorthogonal. These operations are represented by shorter
geodesies joining \\j/ o y to \^ l > and \\l/ 1 > to |i^ 2 > respectively on the Poincare sphere.
Pancharatnam realized that a filtering measurement is a phase-preserving projection.
He however recognized that the two successive projections yield a state which has
a pure geometric phase <D = Q/2 with reference to the initial state, Q denoting the
solid angle spanned by the geodesic triangle completed by joining \\l/ 2 > to \\l/ > with the
shorter geodesic. The phase <DQ resulting from the two successive phase-preserving
operations brings out the non-integrability of geometric phase, recognized by Pan-
charatnam [1] as 'an unexpected geometrical result'. It is remarkable that Pancharat-
nam derived his result for a nonadiabatic, nonunitary and noncyclic evolution, never
invoking any specific equation or Hamiltonian to effect the evolution. The result
therefore has a completely general applicability.
Pancharatnam's results for a two-state system can be extended to a general quantal
system. The filtering measurement on the initial state I^Q) made along the ray \\]/^
yields the state |<Ai><^ 1 | v F ) = Pil x ^o)' v i z - tne component of the initial wavefunction
along !//!>. The pure state density operator p = | X F>< X P|/< V F| X F> used here, with the
familiar properties: p f = p,Trp = 1, p 2 = p, is a ray space quantity, i.e. it is uniquely
determined by the ray | if/ > regardless of the phase or norm of the wavefunction | ^ > . The
second projection then yields the state p 2 Pil l f'o) w i tn a phase
fl = arg< |p 2 p 1 |'F > = argTrp p 2 p 1 , (1)
with respect to the initial state |*F >, according to the Pancharatnam connection
[1, 3-5]. The two-state geometric phase Q/2 thus generalizes to the argument of the
Bargmann invariant '[i9']Tip p 2 p l , a. pure geometric quantity associated with the
geodesic triangle in the ray space of a general quantal wavefunction |*F>.
Since Trp p 2 p 1 = Trp 1 p p 2 = Trp 2 p 1 p ,the same geometric phase is obtained no
matter which vertex of the triangle one begins with, as far as the sequence of filtering
measurements is maintained. In an actual evolution of a system, represented in the ray
space by the geodesies |(A >- > 'l ) Ai> and l^i>~H^2> produced by an appropriate
Hamiltonian, the total phase in general has a dynamical component which depends on
the actual Hamiltonian. The remaining, i.e. geometric phase however is invariant
irrespective of the Hamiltonian [7], regardless of adiabaticity or unitarity of the
evolution and whether or not the evolution includes [18] the geodesic \\j/ 2 > - |^ >. If
any two of the projection operators coincide, the three-vertex geometric phase equals
zero as the triangle collapses to a single geodesic and its area vanishes.
We now make a simple physical observation which will yield a crucial ingredient of
geometric phase. If the sequence of the filtering measurements is reversed, the phase
acquired just changes sign, i.e.
$G A = argTrp oPl p 2 = - <X>. (la)
This observation readily leads to the expression
,
Trp {p l9 p 2 } l
in terms of the commutator (square) and anticommutator (curly) brackets between p\
and p 2 . The three- vertex geometric phase (D (2), originating directly from the
commutator between the projection operators, depends exclusively on the ray space
geometry and is gauge independent.
If the rays (i/^) and |i// 2 > are separated infinitesimally, so that p 1 = p and
p 2 = p + dp, say, the infinitesimal 3-vertex geometric phase becomes [20]
iTrp [p,dp] _ .Trp (p-jl)dp
I
Trp {p,p + dp} 2 Trp p Trp p
since pdp + (dp)p = d(p 2 ) = dp. Here 1 signifies the unity operator. The infinitesimal
3-vertex phase (2a) forms the smallest possible building block with which we will
presently build the general geometric phase.
Before harnessing (2a), we highlight the special physical significance of the commuta-
tor between p and its differential appearing in (2a). As noted earlier, geometric phase is
the phase acquired by a parallel transported [4,9-11] state. The specific Hermitian
Hamiltonian which parallel transports [21-22] a state |*F>, is ift [dp/dt, p]. Over each
infinitesimal step |^>-(exp[dp,p])|i/0 = (cos(dp) + sin(dp))|^> (since pdpi v P> = 0,
cf. [22] ), in the state evolution effected by this Hamiltonian, the phase is preserved, as
cos(dp)|^>isalongandinphase with|^> whereas sin (dp) |^> is orthogonal to |^).To
the first order, this step in the evolution is equivalent to a projection (p + dp)| v P>.
Parallel transportation along a geodesic produces an identically null phase [21-22]
until a ray |i^ >, orthogonal to the initial ray, is reached. In the parallel transportation
along the two geodesies |iAo>~H l A> an d !<A>-Kexp[dp,p])|iA> therefore, the phase of
the final state (exp^pjp])!^}, as prescribed by the Pancharatnam connection [1-5],
is arg< v P j(l + [dpjp])!*?). This equals d<D G (2a) obtained above geometrically, since
IMP) is in phase with |Y ) Ki/'ol'A) rea l) due to the geodesical parallel transportation.
The operator ih [dp/dt, p] was introduced 45 years ago as a generator of adiabatic [23]
evolutions.
We now derive the geometric phase in terms of the infinitesimal 3-vertex phase (2a).
Consider a completely arbitrary evolution of a general quantal system, represented in
the ray space by an open curve C (figure 2) from \i]/ i > to \\l/ 2 >. The acquired phase <D can
be measured from the interference pattern obtained by superposing the initial and final
Figure 2. The curve C in the ray space representing an arbitrary evolution from
(I/T! > to |i/r 2 > is divided into infinitesimal geodesic segments. The ray |i/f > on the
shorter geodesic G, joining the ends of C, is chosen as the reference. The sum of
geometric phases associated with the geodesic triangles having the infinitesimal
segments of C as bases and \ij/ > as the vertex, equals the geometric phase acquired
along C. If the reference is shifted to |i// R >, an additional geometric phase arising
from the geodesic triangle |^ R >->|i/' 1 >-H^2)> w ^ accrue - S is the surface spanned
by the closed curve C + G.
states. Pancharatnam showed that if the states are passed through an analyzer oriented
along the ray | // R >, say, prior to the superposition, the resulting interference pattern (cf. eq.
(10) in [1]) displays a phase (<t> + <D G ) where <I> G is the 3-vertex geometric phase associated
with the geodesic triangle |i^ R > - \\l/ i > -> |t// 2 ) This can be understood from figure 2. The
projection of the final state on the initial state represented by the shorter geodesic G joining
\\l/ 2 > to li/^) does not alter the phase [1]. Hence the |i// R > analysis of the two states,
represented by the shorter geodesies joining them to |t// R >, effectively corresponds to an
additional evolution along the geodesic triangle |i/' 2 )- ) 'l 1 /'R>- > ! l /'iX generating the
extra phase.To obtain the correct phase <D, therefore, the analyzer state should be
selected such that <D G vanishes, e.g. by making the geodesic triangle collapse to a single
geodesic. This is achieved with any arbitrary |i^ R > if the evolution is cyclic
(!'Ai> = I'Aa/O- For a non-cyclic evolution however, one can select |i// R > to lie anywhere
on the shorter geodesic G joining \\}/ 2 > and it/^ >. We therefore choose such a reference
ray |^ > on G (figure 2).
We divide the curve C into infinitesimal geodesic segments and join their ends to i^ >
with shorter geodesies to form infinitesimal geodesic triangles. Along each side shared by
two such contiguous triangles (figure 2), the virtual evolutions to and fro |i/^ > contribute
equal and opposite phases arg {^\\I/ Q ) and arg (i// \\l/y. The successive triangle evolutions
thus yield <X> G for the evolution along C + G, which is identical to that along C (cf. also
[18]). The infinitesimal extent of each segment implies a sequence of infinitely dense
filtering measurements [9,24] along C, yielding an effective unitary transformation of
purely geometric nature. Hence the geometric component <E> G of the phase <D acquired over
the evolution along C just equals the sum of the individual dO G (2a), i.e.
J Pl
C
3 1 8 Pramana - J. Phys., Vol. 46, No. 5, May 1996
If C is a single (shorter) geodesic between | if/ 1 > and |i// 2 >, the numerator of the integrand
in (3) vanishes [20] identically and a null geometric phase results. This is geometrically
clear as G then retraces C.
The phase <D G is determinate if the rays )>//!> and |i^ 2 > are not mutually orthogonal. If
the curve C passes through a ray |i// > orthogonal to |iA >, the denominator of the
integrand in (3) vanishes, but so does the numerator. The contribution from the
corresponding infinitesimal triangle however never diverges (eqs. (1), (2)). If the
infinitesimal segment of the triangle is centred at the orthogonal ray \\j/ >, the triangle
approaches a finite slice between two geodesies of length n each between 1 i// > and | \j/ Q >,
as the segment length tends to zero. We can then write without any loss of generality,
A: =1,2, (4)
so that the 3-vertex geometric phase (1) for the slice tends to
AO G Wo Xlfc>>) = 0i-&, (4a)
as <5->0, i.e. as the segment i/^ -i// 2 approaches zero. Thus unless the curve retraces
itself at |$o> (/? 2 =_/?!), the contribution from this slice to <X> G (p ) is finite, viz. a jump
(4a) as C crosses |(/^ >.
If the geodesic continues through the orthogonal ray |i// > without a kink, a switch-
over to a distinct geodesic with the label /J 2 = /? t + n occurs and a + n jump (4a) in
O G (p ) results. This can be understood from the Pancharatnam triangle phase (1).
When 1^) and |i// 2 > lie on the same side of |0- > on the geodesic |<A )~*! l Ao)' tne
shorter geodesic G (figure 1) between |i/^ 2 > and |i/f > closing the triangle just retraces
the geodesic | ^ > -> 1 1{/ 1 > -> 1 1// 2 >, enclosing a null area and yielding <J> G = 0. However if
li/^ > lies on the other side of |iA >, the shorter geodesic G continues in the same
direction as C and the closed curve C + G encloses a finite slice generating a geometric
phase jump of + n. Thus while an evolution along a geodesic [1, 5, 18] yields a null
geometric phase until the ray orthogonal to the initial ray is reached, a geometric phase
jump of + n occurs just when the geodesic crosses the orthogonal ray without a kink.
Such n phase jumps for a 2-state system have been discussed [5, 25] previously.
For a 2-state system such as a spin- 1/2 particle, the ray space is a 2-sphere and the labels
jS fc in (4) are the azimuthal angles (j) k of the geodesies, measured in the plane transverse to the
direction of the ray |i^ >. Therefore <D G (p ) for a 2-state jumps by ^ (j) 2 = ~~i^ s iice as
C crosses |<A >- A case in point is the first experiment [26,27] clearly demarcating
dynamical and geometric phases. Here an interferometer is illuminated with a beam of
neutrons in the [ j > state. Two identical spin flippers F 1 and F 2 placed in the two anus of the
interferometer take the neutron state to |J,>. A relative translation between F l and F 2
generates a pure dynamical phase, while their relative rotation <5/? about the ID-direction
produces a pure geometric phase <D G = 6 ft (d. figure 2b in [26] ). If | \f/ Q > is selected to be the
initial, i.e. the | f > ray, the net geometric phase G (3) is just the phase jump dp caused by the
kink 8P of the curve at |J,>. This experiment, inclusive of a direct verification [27] of Pauli
anticommutation, has been performed [21,28].
We now redefine the ray |i/f> with reference to a fixed ray |i/r R > as
which is in phase [1] with |i^ R >. The representation |$> of the ray differs from |i//> only
by the shown phase factor. Equation (5) represents a gauge transformation, which we
associate here geometrically with a change of the reference ray |i// R >. Since the inner
products of both the rays |^> and \\f/ + di//> with |^ R > are real, for the Pancharatnam
triangle | t/^ R > - | i- |i/r +
dd> G = arg<< + d> = arg(l + <d| = - i<d>. (6)
This can also be directly verified by differentiating (5) and taking the inner product with
|i//> to get the purely imaginary quantity
-(6a)
(cf. eq. (2a)). Hence the integral
yields the geometric phase that would be observed if the states |*P> were analyzed [1]
along the reference ray |i// R > (figure 2) prior to superposition. Since all \j/ are in phase
with i/> R due to the gauge transformation (5), the total phase (7) is just the integral along
C of the relative phase between neighbouring ^ rays. As discussed earlier, this would
yield, apart from the phase <E> G (3) acquired along C, an additional phase < G associated
with the geodesic triangle |i^ R ) -> \\j/ 1 > -> |i^ 2 )- If the evolution is cyclic, i.e. the curve
C is closed (p^ = p 2 ), GtPi( equals the correct phase <> G (3) regardless of the choice of
|^ R >, as discussed before, even though the integrand in (7) is a gauge-dependent, i.e.
|^ R >-dependent, connection 1-form. Equation (7) then reduces to the nonadiabatic <D G
derived for the special case [7] of a cyclic and unitary evolution. Thus for a cyclic
evolution, the reference ray |i^ R > may be selected anywhere in the ray space, i.e. the
gauge freedom is complete. For an open curve C however, the relations (7) and (3)
become identical, if |i// R > is confined to the shorter geodesic G (figure 2) joining the ends
of C. Thus even for noncyclic evolutions, the gauge freedom survives, albeit in
a restricted form and the integral (7) over the open curve C yields the integral over the
closed curve C + G, i.e. the correct <D G .
Whether the evolution is cyclic or not, the integral (3, 7) of the connection 1-form for
<D G transforms to the integral
A|d> = i TrpdpAdp, (8)
Js
of the gauge invariant curvature 2-form (for a normalized state |*F over the surface
S spanned by C (+G, if necessary), vide a Stokes-like theorem. The relation (8) is
known [29, 24] for cyclic evolutions.
Nonintegrability along a general curve in the ray space is a salient feature of
geometric phase, yet results (3) and (7) show that geometric phase with respect to a fixed
reference ray is triangle-integrable, in the spirit of the surface integral (8). This
trianele-inteerabilitv of <t>^ was recngni7eH earlier Cen ("Iftt in I" 171 anH pn (A T>1\ in
(jeometnc phase a la Pancharatnam
[18]). Relation (18) in [17] was derived group theoretically for d<l> G , which coincides
with eq. (6) read with (5) above. In eqs (3.17) and (4.34) of [18], only a formal irreducible
expression
0> G = argTr! piPexpl dpi),
\ \ Jpj / /
was obtained in terms of the ordered sequence of noncommuting operators. As
discussed, this expression originates from the unitary operation Pexp(J^ 2 [dp, /?])
effected by the parallel transport Hamiltonian, if each infinitesimal step of the evolution
is evaluated only to the first order. The noncommutation between the successive
infinitesimal operators makes the geometric phase nonintegrable. We have exploited
the triangle-integrability of O G by considering virtual evolutions to and fro the
reference ray |tA )> a f ter eac h infinitesimal step of the actual evolution to arrive at
a simple closed-form integral (3) for <D G , in contrast to [17,18]. Further, we have
explicitly identified the integrands in (3) and (7) with the invariant 3-vertex phase of the
associated infinitesimal Pancharatnam triangle. A special version of (3), with the reference
ra y I Ao ) fi xe d at the initial ray | ij/ i >, applicable to any general evolution,has been obtained
previously [20] by invoking the Pancharatnam connection continuously.
The key requirement for generating geometric phase has also emerged here. During
the formative years of geometric phase immediately following the publication of [6],
geometric phase was believed to arise only in very special evolutions obeying several
constraints. It is now realized that for producing geometric phase, the state evolution
need not be adiabatic [7], cyclic [4], unitary [8] or be governed by a specific equation
[18]. Does geometric phase then have any prerequisite? The relation (2) for the
Pancharatnam triangle phase implies that a ray space characterized by noncommuting
density operators is necessary for producing geometric phase.
In conclusion, we have presented a geometric formalism for the general geometric
phase using Pancharatnam's ideas [1]. The 3-vertex invariant phase for the infinite-
simal Pancharatnam triangle is the only basic input here. This has obviated the need to
invoke any Hamiltonian or governing equation for effecting a state evolution and
hence to make reference to dynamical phase, enabling us to confine the discussion of
geometric phase strictly to the ray space. We have thence arrived at a general, yet
simple, closed-form expression for geometric phase in terms of just the density
operator.
References
[1] S Pancharatnam, Proc. Indian Acad. Sci. A44, 247 (1956) .
[2] S Ramaseshan and R Nityananda, Curr. Sci. 55, 1225 (1986)
[3] M V Berry, J. Mod. Opt. 34, 1401 (1987)
[4] J Samuel and R Bhandari, Phys. Rev. Lett. 60, 2339 (1988)
[5] A G Wagh and V C Rakhecha, Phys. Lett. A197, 107, 112 (1995)
[6] M V Berry, Proc. R. Soc. London A392, 45 (1984)
[7] Y Aharonov and J Anandan, Phys. Rev. Lett. 58, 1593 (1987)
[8] R Bhandari and J Samuel, Phys. Rev. Lett. 60, 1211 (1988)
[11] A G Wagh and V C Rakhecha, Phys. Rev, A48, R1729 (1993)
[12] A Shapere and F Wilczek (editors), Geometric phases in physics (World Scientific, Singapore,
1989)
[13] M V Berry, Sci. Am. 259, 46 (1988)
[14] J W Zwanziger, M Koenig and A Pines, Ann. Rev. Phys. Chem. 41, 601 (1990)
[15] J Anandan, Nature (London) 360, 307 (1992)
[16] A G Wagh and V C Rakhecha, in Recent developments in quantum optics, edited by
R Inguva (Plenum Press, New York, 1993) p. 117
[17] EGG Sudarshan, J Anandan and T R Govindarajan, Phys. Lett. A164, 133 (1992)
[18] N Mukunda and R Simon, Ann. Phys. 228, 205 (1993)
[19] V Bargmann, J. Math. Phys. 5, 862 (1964)
[20] A G Wagh and V C Rakhecha, Pramana - J. Phys. 41, L479 (1993)
[21] A G Wagh, Indian J. Pure Appl. Phys. 33, 566 (1995)
[22] A G Wagh and V C Rakhecha, to be published
[23] T Kato, J. Phys. Soc. Jpn. 5, 435 (1950)
[24] J Anandan and A Pines, Phys. Lett. A141, 335 (1989)
[25] R Bhandari, Phys. Lett. A157, 221 (1991)
[26] A G Wagh and V C Rakhecha, Phys. Lett. A148, 17 (1990)
[27] A G Wagh, Phys. Lett. A146, 369 (1990); Solid State Phys. C34, 8 (1991)
[28] A G Wagh et al (BARC- Vienna-Missouri Collaboration), to be published
[29] J E Avron, A Raveh and B Zur, Rev. Mod. Phys. 60, 873 (1988)
pnysics pp.
Time dependent canonical perturbation theory III: Application
to a system with nonconstant unperturbed frequencies
MITAXI P MEHTA and B R SITARAM
Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India
MS received 2 November 1995
Abstract. In this communication, we report the results of the application of time dependent
perturbation theory to a non-integrable Hamiltonian which is a perturbation on a Hamiltonian
with nonconslant frequencies. The theory provides good time dependent local constants
of motion and also gives good approximation for mapping of solutions for a time limit
determined by the nearest singularity in complex K plane for fixed real time and the order of
calculation.
Keywords. Canonical perturbation; Hamiltonian systems; KAM.
PACSNos 05-45; 03-20
1. Introduction
Canonical perturbation theory is one of the main mathematical tools to which
classical physicists look forward to when a nonlinear Hamiltonian system is encoun-
tered. If the given Hamiltonian system has chaotic dynamics, the theory fails in the
sense that the generator of the canonical transformation calculated from the theory
turns out to be singular [1]. To overcome this problem, use of time-dependent
canonical perturbation theory (TCPT) was suggested [2, 3], where time was considered
as a new degree of freedom and a canonically conjugate variable T was introduced.
It was shown that TCPT removes some of the singularities of the canonical perturba-
tion theory. Our work also suggested existence of natural boundary in complex e
plane for a class of Hamiltonian systems [3]. In this paper we apply the TCPT to
the Hamiltonian,
/2COS0,). (1)
In this paper only the application part is considered, details of TCPT formalism can
be found in [2]. The plan of the paper is as follows. Some important aspects of the
chosen Hamiltonian system are discussed in 2. Section 3 discusses the results and
conclusion are indicated in 4.
There has been a lot of progress in the analysis of non-integrable Hamiltonian systems
which are of KAM [1] type. One important property of these systems is
where (o t are unperturbed frequencies and /,- are unperturbed action variables. For
comparison of TCPT results with KAM results, we applied TCPT to the Hamiltonian
ofeq.(l).
This Hamiltonian was chosen because, (1) it is a Hamiltonian on which KAM theory
can be applied, (2) the simplicity of the solutions of unperturbed equations of motion
makes the application of TCPT simpler. Usually integrable Hamiltonian systems with
phase-space dependent frequencies have solutions which are Jacobi-elliptic or related
functions of time. TCPT requires integration of K i over unperturbed orbits. Integra-
tion of the functions of Jacobi-elliptic type is difficult to do analytically whereas for the
chosen Hamiltonian system integration of H t over unperturbed orbits is easier to
calculate.
The Hamiltonian also has scaling property, which can be used to relate the
complex-time properties of solutions of equations of motion of H, with complex-fi
properties of canonical transformation equations [2], which takes Hamiltonian with
e = 1 to Hamiltonian with some other &(&^ 0). The scaling equations are
which yields
H = 8 2 /i (2)
where
IOS 02 ~t~ J 2 COS 1 ) (3)
Thus, as in the case of the Henon-Heiles system [3], studying the system at a fixed
energy and different e values is equivalent to studying the system at a fixed e value and
different energies. Decrease of energy for a fixed value of s increases the mixing and so
gives rise to increase in chaotic behavior. The Hamiltonian has chaotic dynamics at
small energy values. As can be seen from the Poincare sections in figures 2a and 2b
regular and chaotic dynamics co-exist at the same energy E = 2-0. In the Poincare
sections the variable # t was set to n/2 on the section and the condition on 9 2 was
mod(0 2 , 2n) < n at intersection. The plane of the section is the (I : , / 2 ) plane.
3. Results
We used Mathematica programs for calculation of generating functions, invariants and
mapping results. All calculations were done with = 0-15. There are terms in the
generator and the invariants with denominators of the form (nco) and its powers, which
vanish in certain regions of phase-space. The apparent singularities (resonant terms)
Canonical perturbation theory III
0.08
0.06 -
2 0.04 -
0.02 -
0.00
-0.02
-0.10
Figure la. Relative variation in predicted Figure Ib. Same as figure la, for a chaotic
time-dependent constant of motion l\ for orbit,
a regular orbit at = 2-0 and 8 = 015. C^
and O 2 represent first and second order per-
turbation theory results respectively.
can be removed by taking the limit (no)) -> in the appropriate regions of phase-space.
As expected, the limit turns out to be finite as can be seen from expressions for the first
two-order calculation of the generating function and two of their limits are given in
appendix A. The limit calculations were also done on Mathematica. The expressions
for invariants and mapping of solutions up to nth order are given by the following
formulae. Invariants I[ can be calculated from
/'. = exp(8"F,,).--exp(fiF 1 )/ i (4)
where I t represent solutions for equations of motion for H. Mapping from solutions f |
of H to solutions t of H is given by
e"F U- (5)
n/S>i* ^ '
Note that the transformation appearing in (5) is inverse of what appearing in (4). But in
mapping the action-angle variables appearing in RHS are to be evolved using H
equations of motion, whereas in (4) the RHS evolves according to equations of motion
forH.
For calculation of invariants a Runge-Kutta fourth order algorithm was used.
Figure la shows, the relative variation in I\ for a regular orbit at energy = 2-0 (the
relative variation is defined as 2*(/' 1 (t)-/' 1 (0))/(/' 1 (t) + /' 1 (0))). Figure Ib shows the
same for a chaotic orbit at the same energy E = 2-0. First and second order results of
perturbation theory are shown in figures which are denoted by O x and O 2 respectively.
To calculate /' at different times, the expression for /', is calculated analytically in terms
1.94
1.92
1.2
J I L
1.30 1.35 1.40 1.45 1.50 1.55
I,
Figure 2a. Poincare section for the regular
orbit considered in figure la.
1.90
J L
0.25 0.30 0.35 0.40 0.45
Figure 2b. Poincare section for the chaotic
orbit considered in figure Ib.
0.002
_- 0.000
-0.004
-0.006
0.0
0.5
1.0
time
1.5
2.0
0.04
0.03
0.0
0.00
-0.01
0.0
0.5
1.0
time
1. 5
2.0
Figure 3a. Relative error in predicted sol- Figure 3b. Same as figure 3a, for the orbit
ution for the regular orbit shown in figure 2a. of figure 2b.
O l5 O 2 and O 3 show first, second and third
order calculations repectively. The dotted
curves are numerical prediction and the solid
curve is the analytical prediction.
Figure 3a shows mapping of solution for regular orbit of figure 2a. The connected
line is the relative error in mapped solution with respect to numerically calculated exact
solution at first order of calculation. The dotted curves show the relative error at higher
orders. (The relative error in mapping is defined as 2(I lp (t) - I in (t))/(I lp (t) - I la (t))
where I lp is the predicted solution and I in is the numerical solution.) Calculation for
mapping at higher orders (shown by dotted curves in the graph) was done numerically
using a program that calculates derivatives of the perturbed solution with respect to e at
s = at given time and initial conditions. The program uses contour-integrals to
326
Pramana - J. Phys., Vol. 46, No. 5, May 1996
calculate derivatives. As figure snows, with higher order calculation in perturbation
theory predictions become better. To show that the numerical calculation gives the
same result as the analytical one, the first order mapping is calculated using both the
methods. Figure 3b is the same as figure 3a, for initial conditions corresponding to the
orbit of figure 2b.
We also calculated position of blowing-up singularities (where solutions become
infinite) in complex e plane for real time, which gives the radius of convergence of the
perturbation series in & if there are no other finite singularities (where solutions have
finite values). The NAG program d02baf was used with 256 points equally spaced on
a circle with centre (0,0) and radius 0-15 in complex-e plane, to get evolution of given
initial conditions in real time and to find the smallest value of time at which one of the
points on the circle has a singular solution. The initial conditions of figure 2a has
a singularity at t ^ 22-6 and for figure 2b there is a singularity at t ^ 11-3.
4. Conclusion
From the analytical and numerical studies of the Hamiltonian system, we can conclude
the following:
1. One can use KAM approximation only in the case of irrational tori, whereas our
analysis can be used irrespective of the unperturbed frequencies being rational or
irrational. It is well-known that KAM predicts total breakdown of perturbation theory
even under a very small perturbation but TCPT predicts breakdown of perturbation
theory only when a singularity in the complex-s plane for real time is encountered. It
can be easily seen from figures la and Ib that TCPT converges for both regular as well
as chaotic orbit (broken tori) for small time. At the same time, in TCPT results time
appears algebraically and so for convergence at large time, very high order calculations
are needed.
2. With the use of TCPT it is possible to establish a relationship between the time
dependence and g dependence of the solutions of a chaotic Hamiltonian for a class of
Hamiltonian systems, as shown for the Henon-Heiles system in [2] and for the
Hamiltonian studied in this paper.
Appendix A: Generating functions
/ sing
I 2
/^ t T^t J^tr>nc(T t\ T trr\v(J t\
1 L lil J. 1 1 L/Uol i ill * i t/wai i } 1 1
p - 1 I z I _ > J. / i ^ -a
-*? ^r9~r ...9^ TTo r , -.9
2 4/ 2 4/J 4/5 4/J
7^008(20! -J^) /5^cos(20 2 -/ 2 f)
sin^ -0 2 )
-0 2 ) /?sin(20 2 )
-4/J/2 + 4/I 8/
in^ + 2 ) / 2 sin(0!
/ 1 sin(0 1
2/
f?sin(/ 2 t) ^
2/ 8/
^!-^-/^) sin^H- 2 ~ A
4
sin(0 1 + 2 -/ 1 f) /;sin(20 2 -2/ 2 t) 8^(0! + 2 - J 2 t)
4/2" 8/1
sin(0 1 + 2 -/ 2 t) sin(0 1 + 2 -/ 1 t-/ 2 r)
/ 2 sin(0 1 +0 2 -/if-/ 2 / 1 sin(0 1
sin(0 1 -0
sin(0 1 -0 2 -/ 1 t + / 2 t) / 2 sin(0 1 -0 2 -/ 1 t
lim F-L = / 2 t 008(0!).
r.-^o
lim J 7 1 = / 1
lim F 2 = ~- x (12cos(0 2 ) - 12cos(0 2
1,^0 1 ^2
+ 6/ 2 f sin(0 2 ) + 6/ 2 tsin(0 2 -I 2 t
sin(0,)
lim ^ =^x (12008(0^ -12cos(0!
References
[1] M V Berry, in Topics in Nonlinear Dynamics: A tribute to Sir Edward Bullard, AIP
Conference Proceedings edited by S Jorna (AIP, New York, 1978)
[2] B R Sitaram and Mitaxi Mehta, Pramana - J. Phys. 45, 141 (1995)
[3] Mitaxi Mehta and B R Sitaram, Pramana - J. Phys. 45, 149 (1995)
physics pp. 331-339
Cosmic strings in Bianchi II, VIII and IX spacetimes:
Integrable cases
L K PATEL 1 ' 2 , S D MAHARAJ 1 and P G L LEACH 1
Department of Mathematics and Applied Mathematics, University of Natal, Private Bag XI 0,
Dalbridge 4014, South Africa
2 Permanent address: Department of Mathematics, Gujarat University, Ahmedabad 380009, India
MS received 1 September 1995
Abstract. We investigate the integrability of cosmic strings in Bianchi II, VIII and IX space-
times using a Lie symmetry analysis. The behaviour of the gravitational field is governed by
solutions of a single second order nonlinear differential equation. We demonstrate that this
equation is integrable and admits an infinite family of physically reasonable solutions. Particular
solutions obtained by other authors are shown to be special cases of our class of solutions.
Keywords. Cosmology; strings; integrable.
PACSNos 04-20; 98-90
1. Introduction
Topologically stable defects such as vacuum domain walls, strings and monopoles are
produced during phase transitions in the early universe [1]. Domain walls and
monopoles are not important in the study of cosmological models at later times. On the
other hand strings can lead to many interesting astrophysical consequences. Strings
may be one of the sources of density perturbations that are required for the formation of
large scale structures in the universe ([2], [3]). They possess stress energy and hence
couple to the gravitational field. Various features of cosmic strings have been discussed
by Vilenkin [1], Gott [4] and Garfinkle [5].
The general relativistic treatment of strings was initiated by Letelier [6] and Stachel
[7]. Subsequently many relativistic exact solutions were found which describe homo-
geneous string cosmological models with different Bianchi symmetries. Krori et al [8]
and Chakraborty and Nandy [9] have considered models with Bianchi types II, VIII
and IX spacetimes. Bianchi type I string based models are studied by Banerjee et al
[10]. Tikekar and Patel [1 1] have discussed some Bianchi type VIo string models with
and without magnetic fields. More recently a number of exact solutions, in the presence
of a magnetic field and also with vanishing magnetic field, in Bianchi type III
spacetimes were obtained by Tikekar and Patel [12]. Maharaj et al [13] investigated
the integrability of cosmic strings in Bianchi type III spacetimes using a symmetry
analysis and extended the class of solutions studied in [12].
Tikekar etal [14] have obtained a new class of physically relevant inhomo-
solutions for string cosmnloev endowed with cylindrical svmmetrv on the
background of singularity-free cosmological spacetimes. Patel and Beesham [15] have also
obtained a new class of plane symmetric inhomogeneous string cosmological models.
The purpose of the present paper is to study the integrability of cosmic strings in the
context of Bianchi types II, VIII and IX spacetimes, Essentially the solution of the field
equations reduces to integrating a single second order nonlinear ordinary differential
equation. We show that this equation has a rich structure and admits many solutions, some
of which may lead to new physically significant string models.
2. The field equations
We consider the general Bianchi type II, VIII and IX spacetimes given by the line element
ds 2 = dt 2 - A 2 (dr + 4m 2 d<) 2 - B 2 K 2 (d9 2 + sin 2 0d< 2 ), (2.1)
where A and B are functions of time, t, and m and K are functions of 6 satisfying the
differential equations
K 2 sm6 d0 [
and
i\?-V 1 / r\V\l i\V
(2.3)
Here /1 1 and /t are constants, /x being proportional to the curvature of the two-
dimensional surface with the metric
dZ 2 = K 2 (d0 2 + sin 2 0d</> 2 ). (2.4)
The metric (2.1) with 1 1 ^ represents
(i) a Bianchi type II spacetime if ^ = and K = cosec0,
(ii) a Bianchi type VIII spacetime if n = 1 and K = tan0 and
(iii) a Bianchi type IX spacetime if ^ = 1 and K = l.
The energy-momentum tensor is given by
T ik = pv i v k -^w i \y kt v^ = - w,w* = 1, i>X = (2.5)
for a cloud of strings, In (2.5) p, the proper energy density, and A, the string tension
density, are related by
P = P P + A, (2.6)
where p p is the particle density of the configuration. We use comoving coordinates and take
the string fibres along the r-direction. One can easily check that the Einstein field equations
(2.7)
corresponding to the string distribution for the metric (2.1) reduce to the system
A B AB 11 A 2
Here and in the sequel an overdot indicates differentiation with respect to t. The
particle density is given by
^AB 21 2 A 2
= - 2 n+ 2 T-5 + -i4-- (2.H)
We generate many solutions to (2-8-2-10) in subsequent sections.
Here it should be noted that the expansion scalar and the shear scalar a for the
velocity field u ; have the general expressions
2B A
3. The model equation
We have three equations (2-8-2-10) for four unknown functions p, A, A and B. In order to
obtain explicit solutions of the system we must impose one additional constraint. We
assume that
A = B", (3.1)
where n is a real constant, so that (2.10) becomes
(n + l)^ + n 2 ^+A 2 5 2 <- 2 >==0. (3-2)
Chakraborty and Nandy [9] have provided solutions of (3.2) for n = and n = 2. Our aim
here is to find all possible solutions of the differential equation (3.2).
Equation (3.2) can be written as the simpler form
/ + / = (3.3)
by means of the transformation
y = B* x = 0t, (3.4)
where
2 2 2 --
l J
and we take /? ^ 0. In (3.5) we require that n i- - 1 and ^ 9^ 0. From (3.2) we see that
n = 1 leads to the degenerate case
U+A 2 B- 6 = (3.6)
B
with solution
(3-7)
where K t is the single arbitrary constant of integration. If A 1 = 0, the solution of (3.2)
would be
B = (K 1 + K 2 ( " +1)/( " 2+n+1) . (3.8)
Henceforth we consider the general case n ^ 1 and A t ^ 0.
Equation (3.3) is an Emden-Fowler equation [16-19] of index v. Analysis of the Lie
point symmetries of (3.3) using Program LIE [20] shows that for general v there are two Lie
point symmetries
G^ (3.9)
There are three particular values of v for which the number of point Lie symmetries is
greater than two.
When v = 1, (3.3) is linear, has eight Lie point symmetries and is trivially integrable. For
v = 1, n = 2, 1 the second of which values has been already treated. For n = 2 the solution
of (3.2) is
B(f) = (K 1 swyt + K 2 cosyt) 3n , (3.1 1)
where
J 2 = ^l (3.12)
When v = 0, (3.3) is also linear and has eight point symmetries. For v = 0, n takes the values
(1 N /37)/6 and the corresponding solutions of (3.2) are
1 F 296 + 27 \/37 o]] 3 *"-^
25211 J "J 2|_ 252AJ _|j
(3.13)
i.e. only one solution is obtained. When v = - 3, (3.3) is a special form of the Ermakov-
Pinney equation [21, 22] and has the three point symmetries
d d
7 =2x + y ,
2 dx y dy
^ d
the Lie algebra of which is well-known to be s/(2, R). For v = - 3, n = 0, - 1/3 and the
solution of (3.3) is [22]
11 IcT _l_ V v J_ V v-2 V V f2 _ 1 /-> 1 c\
forn = (3.16)
\i \i
and
26 / 7
for n=- 1/3. (3.17)
That covers the algebraically special values of v (and so n).
For general v the algebra of 0^3.9) and G 2 (3.10) is A 2 in the Mubarekzyanov
classification scheme [23-27]. Since A 2 is a solvable algebra and Gj oc G 2 , the algebra is
that of Lie's type IV [28, p. 424]. However, we do not use the standard representation of
a second order equation invariant under a type IV algebra since the form (3.3) is more
suitable for the purposes of the present discussion.
Chakraborty and Nandy [9] have presented some particular solutions to (3.2) in
Bianchi II and VIII spacetimes. Their n = solution corresponds with our (3.16) and
their n = 2 to our (3. 11).
4. General treatment of (3.3)
For general v (3.3) possesses the two symmetries (3.9) and (3.10) so that its solution can
be reduced to an algebraic equation and a quadrature. The normal subgroup, G v , is
used to reduce (3.3) to a first order equation. The zeroth order and first order differential
invariants are obtained from the solution of the associated Lagrange's system
dx dy dy'
T = T = ~b~ ( j
and are
u = y, v = y'. (4.2)
The reduced equation is
yy' + w v = 0, (4.3)
where now ' denotes differentiation with respect to u. In the new coordinates G 2 is
X' = 2w + (v+l)y (4.4)
2 du dv
(up to a constant multiplier). Its invariants are found from the solution of
du dy dy' ,. -^
^^ - \J~Tt*sj
and are
g = 1/w -(v-l)/2 (4.6)
so that (4.3) is reduced to the algebraic equation
=-l. (4-7)
dy
-(2/(v +!))/+ 1]*/ 2 (4g)
J[/-logy] 1/2 '
in which x and 7 are constants of integration and (4.8b) corresponds to the special case
v= 1 (n = 2~ 1/2 ). The integral in (4.8b) is related to the exponential integral, Ei(ax),
which cannot be expressed in terms of a finite number of terms [29, p. 93]. Hence it does
not lead to a closed form solution of (3.3). In the cases v = 2, 3 (n = (3 + ^29)72, - 3/2)
the integral in (4.8a) can be evaluated as an incomplete elliptic integral and the solution
of (3.3) (and so (3.2)) is given in terms of elliptic functions.
5. General results
Apart from the special case of the degenerate solution (3.6) the field variables can be
written directly in terms of y(x) with x being replaced by /?~ 1 1. Whatever the outcome
of the quadratures in (4.8) we can write down some general results. (We omit the special
case v = 1 to avoid what is virtually repetition.) The energy density (2.8) is
the string tension (2.9) is
8rcA= ^ v " 1+ ^( / -7fr J ' v+1 )~ A;j ' 2( "" 2)/8 ~' iJ '" 2/ " (5 ' 2)
and the particle density (2.1 1) is
The expansion scalar is
ay \ v +
and the shear scalar is
22
3 a 2 y
f cc\
(5 ' 5)
The expressions (5.1-5.5) give the physical parameters of interest once y is known.
We note the appearance of the constant of integration, /. In (4.8) we can take / = as
a special case when v ^ - 1 which occurs for - 1/^/2 ^ n ^ 1/^/2. Then (4.8a) is easily
336 Pramana - J. Phys., Vol. 46, No. 5, May 1996
integrated to
(Y_Y \2/(l-v)
l^ *o;
2
(That (4.8b) cannot be evaluated in closed form has already been noted.) This gives
a range of solutions for the field variables for n in the interval specified albeit with only
one parameter present.
6. Solutions of th,e governing equation
The solutions of the field equations (2.8-2.11) and the evaluation of the field variables,
etc (4.2-4.6) have been reduced under the assumption (3.1) to the evaluation of the
integral (4.8),
dy
|WV+ ]1/2 (6.1)
It is well-known that (6. Ib) cannot be evaluated in closed form under any circumstances
since it is a variant of the exponential-integral function [29, p. 93]. However, (6. la) is
known to be evaluable as a standard integral for v = 3, 2, 0, 1, 2, 3.
There is a sequence for which (6. la) can be evaluated in closed form [13].
Let
(6.2)
m + 2
Them (6. la) is
dy
(6.3)
-(m + 2)y 2 ' (m+2 y 2 '
For m + 2 ^ the first nontrivial value of m = 1. The appropriate substitution is
,1/2
sinw (6.4)
y-
so that (6.3) becomes
sin m+1 MdM (6.5)
which can be evaluated in closed form for all integral m ^ 1. Inversion is not generally
possible apart from locally. The one exception is m = 2. However, (6.4) with (6.5) does
define a parametric solution. It is a simple matter to express (4.2-4.6) in terms of the
parameter u through (6.4).
+,'"]'<> (6 - 6)
We rewrite (6.6) as
_ /'My
" 22 ( '
and the integral is evaluated in closed form by the substitution
r/nY/2 >
r= (7) sinht/ . (6.8)
Finally we note that there is another set of values of v for which (3.3) is integrable. If
v = ^, p 6 3r + , (6.9)
(3.3) possesses the Painleve property [30]-and is integrable in the sense of Painleve [31].
Unfortunately the evaluation of the quadrature is by no means obvious.
7. Conclusion
In this paper we have extended our previous analysis [13] of Bianchi type III cosmic
strings to cosmic strings in Bianchi types II, VIII and IX spacetimes. The procedure
followed here is similar to our analysis in [13]. The evolution of our models is governed
by a single nonHnear ordinary differential equation. On utilizing the Lie symmetry
analysis we reduce the behaviour of the gravitational field to the quadrature (4.8).
A detailed investigation of (4.8) shows that it may be evaluated as a standard integral
only for certain values of v which contain, as a proper subset, the cases considered by
Chakraborty and Nandy [9]. In addition we present a particular sequence for which
the integral may be evaluated in closed form; in general the solution can only be put
into parametric form and inversion is only possible locally. Our analysis is an attempt
to obtain more exact solutions of cosmic strings so that our understanding of these
objects may be improved. It is hoped that some of the solutions presented here will
prove helpful in building physically reasonable models of cosmic strings in the early
universe.
Acknowledgements
LKP thanks Prof. S D Maharaj and the Hanno Rund Fund for their hospitality while
this work was undertaken. The Foundation for Research Development of South Africa
and the University of Natal are thanked for their continued support.
References
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(1990)
[9] S Chakraborty and G C Nandy, Astrophys. Space Sci. 198, 299 (1992)
[10] A Banerjee, A K Sanyal and S Chakraborty, Pramana-J. Phys. 34, 1 (1990)
[11] Ramesh Tikekar and L K Patel, Gen. Relativ. Gravit. 24, 397 (1992)
[12] Ramesh Tikekar and L K Patel, Pramana - J. Phys. 42, 483 (1994)
[13] S D Maharaj, P G L Leach and K S Govinder, Pramana - J. Phys. 44, 51 1 (1995)
[14] Ramesh Tikekar, L K Patel and N Dadhich, Gen. Relativ. Gravit, 26, 647 (1994)
[15] L K Patel and A Beesham, Hadronic J. (to appear)
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meteorologische Probleme (Leipzig, Teubner, 1907)
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A spherically symmetric gravitational collapse-field with
radiation
P C VAIDYA and L K PATEL
Department of Mathematics, Gujarat University, Ahmedabad 380009, India
MS received 16 March 1995; revised 17 January 1996
Abstract. An interior spherically symmetric solution of Einstein's field equations correspond-
ing to perfect fluid plus a flowing radiation-field is presented. The physical 3-space t = constant of
our solution is spheroidal. Vaidya's pure radiation field is taken as the exterior solution. The
inward motion of the collapsing boundary surface follows from the equations of fit. An
approximation procedure is used to get a generalization of the standard Oppenheimer-Snyder
model of collapse with outflow of radiation. One such explicit solution has been given correct to
second power of eccentricity of the spheroidal 3-space.
Keywords. General relativity; collapse with radiation.
PACSNo. 04-20
1. Introduction
Gravitational collapse is one of the important problems in which general relativity can
play a significant role. The problem has many interesting astrophysical applications. It
is well known that the formation of compact stars is usually preceded by an epoch of
radiative collapse. In the collapse problems, the surface of the star divides the entire
space-time into two different regions: the region inside the surface of the star, called the
interior region, filled with matter and flowing radiation, and the region outside that
surface called the exterior region which will usually be filled with pure radiation. These
two regions must be matched smoothly across the surface of the star.
Historically Oppenheimer and Snyder [1] were the first to discuss the gravitational
collapse of dust ball with static Schwarzschild exterior. Since then the study of relativistic
models describing collapsing bodies has received considerable attention. Vaidya [2, 3] and
Lindquist et al [4] studied outgoing radiation from collapsing bodies. Many attempts have
been made to formulate and solve the relativistic equations for collapse [5]. Misner [6]
obtained the basic equations of spherical collapse allowing for a simplified heat transfer
process in which internal energy is converted into an outward flux of neutrinos. Santos and
his collaborators [7-10] have carried out a detailed analysis of non-adiabatic collapse of
spherical radiating bodies and have used this analysis to propose models for radiating
collapsing spherical bodies with heat flow [11]. Vaidya and Patel [12] have presented
a radiating collapse solution based on Schwarzchild interior solution.
In the present paper we discuss a new spherically symmetric collapse solution with
radiation whose physical 3-space t = constant is spheroidal. The space-times with
(see also Tikekar [14]).
2. The interior space-time
Vaidya and Tikekar [13] have shown that the metric
r 2 - r 2 (d6 2
can represent the interior of a superdense star with total mass of about 3-5 M . If the
mass exceeds that limit equilibrium is not possible and gravitational collapse must
follow. It is our aim to study this collapse.
Put r = R sin A and rewrite the metric as
ds 2 = e v dt 2 - R 2 [{cos 2 A + (1 -k)sin 2 A}dA 2 + sin 2 A(d0 2 + sin 2 ed< 2 )]
with k = 1 b 2 /R 2 where the 3-space dt = is
x 2 + / 4- z 2 w 2
R~ 2 + F
For a contracting situation we assume R and b to be functions of t such that
b 2 /R 2 = 1 fc is a constant. So if e is the eccentricity of the spheroidal 3-space, our
assumption implies that during gravitational contraction this eccentricity of the
spheroidal 3-space remains constant.
We introduce a new co-ordinate r by setting sin A = r and choose e v = 1 . We therefore
consider the spherically symmetric space-time given by the line-element
'n _ kr 2 }
V^ n/ ' / j_.2 , _.2/J/l2 , _:.-2/lJJ.2\ I {-l\
l-U
(1-r 2 )
where R is an arbitrary function of time t and k is a constant. The metric (1) is an
obvious generalization of the Oppenheimer-Snyder metric. We name the coordinates
as x 1 = r, x 2 = 9, x 3 = 0, x 4 = t. It is a routine matter to compute the Einstein tensor G l k
for metric (1). The surviving components of G^ are listed below for ready reference
1= 1-fc R R^
1 _ Ir
_/^2_
3
1
k
."*!
R 2
2(1
R 2 (l
-k)
-kr 2 ) 2
1
i
D
k
R 2 '
4 ~R 2 (1 -kr 2 } 2 -r R
Here and in what follows, an overhead dot indicates differentiation with respect to time t.
Einstein's field equations are
(3)
where T l k are the components of energy momentum tensor.
witn
u l Uj = l, w i w' = 0, u l Wj=l, (5)
where p, p, o- are respectively the fluid pressure, the matter density and the density of
flowing radiation. We take v l and w' in the form v l = (u 1 , 0, 0, v 4 ) and w 1 = (w 1 , 0, 0, w 4 ).
Then the condition (5) imply
- eV) 2 + (w 4 ) 2 = U - e'V) 2 + (w 4 ) 2 = 0,
-e a u 1 w 1 +u 4 w 4 =l, e a = R 2 (l-kr 2 )/(l-r 2 ). (6)
Equation (6) can be used to find e a/2 y 1 ,i; 4 ,e a/2 w 1 and w 4 in terms of a single parameter
n. Thus we get
w 1 e a/2 = w 4 = cosh n sinh n, (7)
where n is a function of co-ordinates to be determined from the field equations. Using
(2) and (7) we have seen that the field equation (3) give a system of four non-trivial
equations. These four equations are sufficient to determine four physical parameters p,
p, a and n. They are given by
- 2feJ) h3* (8)
R 2 (l-kr 2 )
R 2 (l-kr 2 ) 2 R R
/c(l-/c)r 2
R 2 (l-kr 2 ) 2
,
-r 2 ) 2 R ^ 2
It can be seen that
T[ - T 2 , = - ((p + p) sinh 2 n + <7(cosh n - sinh n) 2 )
which is negative. But using the expressions (2) one can see that
k(l-k)r 2
l - T 2 )
R 2 (l-kr 2 ) 2 '
Pramana - J. Phys., Vol. 46, No. 5, May 1996 343
T\ T 2 being negative implies that k(\ k) is positive. Therefore we must have
0<fc<l. (12)
When k = 0, we get a = 0, n = and the above solution reduces to Oppenheimer-
Snyder solution. When k 1, then a = 0, n = 0. In this case we get Einstein-de Sitter
universe.
3. Equations of fit
We take the contracting boundary of sphere to be r = a(t). For r ^ a(t) we have Vaidya's
radiating star metric [3].
ds 2 = [1 - 2m/S + 2S/u]u 2 dt 2 - [1 - 2m/S + 2S/ri] w' 2 dr 2
(13)
where m is an undetermined function of u and S is an undetermined function of t and r,
and
Sii = -S'. (14)
An overhead dash denotes differentiation with respect to r.
We shall use the standard system of equations of fit viz. at r = a(t)
(i)p = 0, (u)v 1 /v 4 = a, (iii) g ik continuous. (15)
For r ^ a(f) our interior metric is (1). Let us put S = rR(t) so that the external metric is
ds 2 = jl - + 1 (u 2 dr 2 - u /2 dr 2 ) - R 2 r 2 (d6 2 + sin 2 0d0 2 ). (16)
I rR u }
From (1) and (16) it is clear that g 22 and all its derivatives are continuous over r = a(t).
We use the notation [J] to denote the value of X on the boundary r = a(t). We now
consider the continuity of # u and # 44 . That leads to
W, (17)
r u (i-fl~)
and
We rewrite (18) as
o "1
[tf] l-~ +2Jifl[u]=:l (19)
L ^j
For the external metric (13) we have g il u' 2 + g 44 u 2 = 0. Using the continuity of g l x and
g 44 we have [u' 2 ] = [e a/2 u]. Taking the boundary values of the relation (14) and
using [u'~\ [e* 12 u] we get
^fi-^1
*** \1.J , tmu \t-V) WW ,*.
and
The result (15(ii)) gives
a = - [e a/2 tanh n]. (23)
Also vanishing of pressure p at r = a(t) gives
1-fc _R R 2 _
The function u satisfies (14). Taking the boundary values on both sides and substituting
[ii] = |Y]e~ a/2 weget
lS]e al2 =-R. (25)
Using (22) in this equation we get
Finally we have
~2m
(26)
5 is the radius of the sphere as seen by an external observer and S 2 2m/ S - (function
of time f). Thus equations (14), (21), (22) and (26) among themselves give us the
boundary values of S', S, u' and u. The functions S = rR(t) and u are continuous across
the boundary. Therefore we have, on the boundary, the values of u and S and their first
derivatives. This will enable us to write the march of functions u and S in the external
solution. The function m is arbitrary.
Equation (8) shows that p is always positive. Differentiating (9) we find that Snp' is
always negative. Clearly at the centre Srcp is positive. As Srcp' is negative p continuously
decreases from the origin to the boundary r = a.
It can be verified that
4
0/^2 /^>i /->
2Cr 2 G^ Cr
Since p is positive throughout, G\ is positive. We have verified that G\ G\ is negative
and G\ G\ is positive. The denominator is 87i(p + p) and hence is positive. This shows
that the radiation density a remains positive throughout the distribution.
We have seen that if e is the eccentricity of the spheroidal 3-space, then our parameter
k is e 2 . In what follows we shall try to integrate eqs (23) and (24) and get solutions
correct up to e 2 .
We have already obtained in (8), (9), (10) and (11) expressions for p, p, a and n. We
now regard k to be a small parameter and so rewrite these expressions correct to the
first power of fe. They are
2R R 2 1 k(l-2r 2 }
(27)
(28)
R2 n2
t\
(29)
kr 2 /R 2
_2R 2R 2 2_'
~~R + ~R 2+ ll 2
Vanishing of the pressure p at boundary r = a(t) will now give
2R , R 2 1 fc(l-2a 2 )
(31)
which is (24) when k 2 and higher powers are neglected. Here a = a(t) is given by (23).
Neglecting k 2 and higher powers of k (23) becomes
._(-ka 2 /R 3 )(i-a 2 Y/ 2
( j
2 '
__ i __ I __
R R 2 ^ R 2
When /c = 0, we have Oppenheimer-Snyder solution, a vanishes and a becomes
a constant. To the first power of k we take
(33)
where c is the constant value of a when k = 0. Then (31) becomes
2R R 2 1 fc(l-2c 2 )
which admits a first integral
(35)
where B is a constant of integration. Therefore we can rewrite p, p, a and n, using (33)
and (35). They are given by
= 2k(c 2 - r 2 )/R 2 ,
(36)
From (32) and (33) we find that
a(t) = c + kF(t)
(37)
where c l is another constant of integration. We must now find R as a function of
t correct to first power of k. We have already obtained the first integral (35). Let
R = R Q (i) be the solution of (31) when k = 0. Then (31)'can be integrated up to first
power of k. The solution is
(38)
where D is a constant of integration. From (20) and (21) it is easy to see that
Using (33) and (35) one can verify that
2[m] = Be 3 - kR Q c 5 + IkBc 2 F(t) (40)
with F(t) given in (37).
One can now show that a is continuous across the boundary. On the interior side
(41)
On the exterior side 87i[o-] e is given by (Vaidya [3]).
~ 2 (42)
We can find [m u ] from (40) and we have already found [ii 2 ]. So we can find [cr] e . Up to
the first power of k, 8?r [a] e is given by
e = kc 2 /R 2 . (43)
The result (41) and (43) establish the continuity of the radiation density a across the
boundary r = a(t). Though we have proved this continuity using approximation - neg-
lecting k 2 and higher power oik, we have verified that this continuity does hold good in
the general solution discussed earlier.
Lastly since we are working in co-ordinates which are co-moving in the limit k = 0,
we can find the finite co-ordinate time in which the radius Ra of the distribution would
tend to zero. We have
+ k(l-2c 2 ).
Therefore we have
R=-(B-RnY' 2 /R 112 ,
Ai=l-/c(l-2c 2 ),
makes jR zero to the value R = is given by
3fc(l-2c 2 )/2}. (45)
5. Conclusion
In the above analysis a model describing a radiating collapsing sphere is studied. Vaidya's
radiating star solution is taken as the exterior solution. The equations of fit are explicitly
derived. An approximate solution corresponding to small values of the parameter k is
presented. This approximate solution represents a radiating generalization of the well-
known Oppenheimer-Snyder solution. This solution has an interesting property that the
radiation density is continuous across the moving boundary of the sphere.
References
[1] J R Oppenheimer and H Snyder, Phys. Rev. 56, 455 (1939)
[2] P C Vaidya, Proc. Indian Acad. Sci. A33, 264 (1951)
[3] P C Vaidya, Astrophys. J. 144, 943 (1966)
[4] R W Lindquist, R A Schwartz and C W Misner, Phys. Rev. B137, 1364 (1965)
[5] C W Misner and D H Sharp, Phys. Rev. B136, 571 (1964)
[6] C W Misner, Phys. Rev. B137, 1360 (1965)
[7] N O Santos, Mon. Not. R. Astron. Soc. 216, 403 (1985)
[8] A K G de Oliveira, N O Santos and C Kolassis, Mon. Not. R. Astron. Soc. 216, 1001 (1985)
[9] A K G de Oliveira, J A de Pacheco and N O Santos, Mon. Not. R. Astron. Soc. 220, 405
(1986)
[10] A K G de Oliveira and N O Santos, Astrophys. J. 312, 640 (1987)
[11] D Kramer, J. Math. Phys. 33, 1458 (1992)
[12] P C Vaidya and L K Patel, J. Indian Math. Soc. 61, 87 (1995)
[13] P C Vaidya and R Tikekar, J. Astrophys. Astron. 3, 325 (1982)
[14] R Tikekar, J. Math. Phys. 31, 2454 (1990)
\MAINA (0 Printed in India Vol. 46, No. 5,
Durnal of May 1996
Physics pp. 349-355
itic and dynamic properties of heavy Sight mesons in
inite mass limit
: CHOUDHURY + and PRATIBHA DAS*
ipartment of Physics, Gauhati University, Guwahati 781 014, India
partment of Physics, Nalbari College, Nalbari 781 335, India
received 28 November 1995
ract. We summarize the consequences of the infinite limit of heavy quark mass in the
Its of form factors, charge radii and decay constants of heavy light mesons within a QCD
ired quark model recently reported.
vords. Heavy light mesons; form factors; decay constants; charge radii.
:SNo. 12-40
ntroduction
i recent communication [1] referred as I, we have reported the results of form
ars, charge radii and decay constants of both light and heavy flavoured pseudo-
ar mesons in a QCD inspired quark model. The technique used was the quantum
hanical perturbation theory [2] with plausible relativistic corrections [3,4].
he present paper reports the results of the same model in infinite mass limit
-* co, m Q being the heavy quark mass). It is well-known that in the limit of infinitely
e quark mass, additional symmetries beyond the ones of QCD arise [5-7] which
Me one to obtain model independent information on the weak decay matrix elements
savy mesons. Indeed, in heavy to heavy transitions like b -> c decays, all heavy quark
lear current matrix elements are described in terms of only one form factor - the so
id Isgur-Wise (IW) function in leading order. This result is phenomenologically
ill since it allows a model independent determination of the CKM matrix element
'] \V bc \ for semileptonic B-+D and B-+D* decays.
henomenological utility of such an infinite mass limit in the study of static and
amic properties of mesons as a low energy phenomenon is perhaps not so much, as
number of form factors involved are small. However, it is still meaningful to study
consequences of such an extra symmetry as a low energy approximation even for
i familiar quantities. It will at least allow one to see if such a limit is close
roximation to reality.
he aim of the present paper is to study the static and dynamic properties of the
yy light mesons in the infinite mass limit and estimate the percentage of deviation of
2. Theory and results
2.1 Analysis with Coulomb potential
The Coulombic wavefunction in the ground state is given by [2]
(1)
where a is given by eq. (19) of I with
being the reduced mass of the heavy light mesons. Here m q and m Q are the masses of
light and heavy quarks respectively.
In Isgur-Wise limit (m Q -^ oo) [5-7]
/zm q , (3)
and
I
mq-'oo 3 q s
The relation between a and a^ is
(5)
With the introduction of spin effect, a changes to af corresponding to the modifica-
tion of a s to oCg for pseudoscalar mesons given by eq. (38) of I.
With the introduction of relativistic effect, the wavefunction eq. (1) modifies to
where is defined in eq. (26) of I.
The elastic charge form factor, eq. (42) of I with eq. (6) yields
for Q 2 m 2 3 , where n = 2, 1-25, 1-1 and 1 corresponding to 8 = 0, 0-25, 0-4 and 0-5
respectively. In eq. (7) above, e q and e Q are the charges of the light and the heavy quarks
respectively.
For Q 2 mL on the other hand,
\
_
(1 + K 3 Q 2 /4)) n
Equation (7) suggests that in low energy limit Q 2 m^, heavy quark component of the
form factor is approximately Q 2 independent and measures its charge.
F(Q 2 )
0.25 0.5 0,75 1.0
Q 2
Figure 1. D and D s meson form factors in the infinite mass limit (solid lines) and
without infinite mass limit (dashed lines) taking n = 1 in eq. (7) of the text.
F B (Q 2 )
0.25 0.5 0.75 1.0
Figure 2. B meson form factor. It shows no appreciable difference in the infinite
and finite mass limits.
Using eq. (7), we plot F(Q 2 ) vs Q 2 for D and D s mesons (solid lines) in figure 1, while
in figure 2, we plot the same quantity for B meson, using 1/Q 2 behaviour corresponding
to n = 1 in eq. (7). In the same figures, we also plot (dashed lines) the corresponding
functions taking into account the finite quark masses m c = 1-55, m b = 4-95GeV. The
two results for D and D s differ by ~ 7-6% to ~ 10% but the difference is negligible for
B meson.
2.2 Charge radii
Using eq. (7), the average charge radius defined by eq. (51) of I yields
lim
m Q -oo
(9)
where n has values as in eqs (7) or (8). In table 1, we record (r 2 ) 00 for different pseudoscalar
Pramana - J Phvs Vnl. d6. Nn. 5. Mav IQQfi
351
laoiei. V / in im lor neavy iigm mcsuna using
eqs (9) and (10) of the text.
Eq. (10) Eq. (9) Change in
Particles (finite m Q ) (infinite m Q ) percentage
D + 0-25 0-23 8
D -0-44 -0-46 4-5
D s 0-13 0-11 15-4
B 0-46 046 ~0
B d -0-23 -0-23 ~0
5 S -0-11 -0-11 ~0
1.2
I.I
(r 2 )
tf
(r 2 ) 00 1
oD' d
o.q
0.8
C
1
) 50
Figure 3. <r 2 >/<r 2 > vs m Q for charmed and bottomed mesons.
mesons for n = 1 and compare with the values obtained with the expression
(l+K/mJ)'
(10)
for the same mesons with finite m Q . Equation (10) corresponds to eq. (52) of I with
9 conf = 1-
From eq. (9), the following symmetry relations are obtained:
and
2 \ /r 2 \ co 1
/D + ~ V /B~~2
/,,2\oo _ / 2\co
\' /Z), ~ \' /B s -
1 />- 2 \ co
(11]
(12;
(13)
In figure 3, we plot <r 2 >/<r 2 > 00 vs m Q for various heavy light mesons. Table 1 and
figure 3 suggest that infinite mass limit m Q -> oo is nearly true for b quark but not so for
c quark.
2.3 Analysis with confinement
(i) Form factors: The form factor eq. (44) of I in the infinite mass limit can be rewritten as
1 f / 1
eF(Q 2 ) =
45
(l-Q 2 /4))
(l+Q?/4)r
x 3-
(14)
for Q 2 m Q where Q, is defined in eq. (22) of I. With relativistic effect (e = 0-5, a s = 0-65),
eq. (14) becomes
(1-
8(H-(a - 2 Q 2 /4))
(15)
In figure 4, we plot the representative form factor for D meson using (15) with b =
and 0-05 GeV 2 (solid lines). For comparison, we also plot for the same meson without
infinite mass limit (dashed lines).
(ii) Charge radii: The expression of charge radii with confinement effect in the m Q -> co
limit is
>y_2W e
/ ~ 2 i/conf^q
where the confinement factor g^ is defined in eq. (53) of I, with
c onf = lim 0co nf (/*, fl ) = 0conf
WQ-*00
(16)
(17)
b=0
^0.05
0.25 0.5 0.75 1.0
Q 2
Figure 4. Representative form factor for D meson in infinite mass limit (solid
lines) and without infinite mass limit (dashed lines) for confinement parameter b =
and 0-05 GeV 2 .
Table 2. Decay constants in MeV for heavy light
pseudoscalar mesons in infinite mass limit.
Mesons
Results with Results with Change in
finite M
I Q -* co percentage
D
^209
^ 124 40-67
D s
^237
^262 10-55
B
^107
< 104 2-8
Unfortunately, unlike pion or kaon, there is no data on charge radii of heavy light
mesons and hence eq. (16) or its counterpart eq. (52) of I without infinite mass limit do
not yield any phenomenological information on string constant b.
(iii) Decay constants: Using eq. (4) in (37) and (54) of I, we obtain values of decay
constants of heavy light pseudoscalar mesons in infinite mass limit and record them in
table 2. In the same table, we also compare the results without such infinite mass limit.
For charmed mesons, they differ by ~40%, while for bottom, the difference is
negligible.
3. Conclusion
In this paper, we have made an analysis of a QCD inspired quark model for heavy light
pseudoscalar mesons in the limit of infinite heavy quark mass.
The simplicity of the infinite mass limit does not guarantee that it is a close
approximation to reality. This should be determined by an analysis of the correction to
the limit which is the motivation of this paper. Our analysis has shown that while for
mesons with b quark, the infinite mass limit is nearly exact, for c quark, it can differ even
by ~ 40%. For top quark (m t ~ 174 GeV) the limit would have been almost exact, but
absence of bound states of top quark makes this observation rather academic.
References
[1] D K Choudhury, Pratibha Das, D D Goswami and J K Sarma, Pramana - J. Phys. 44, 519
(1995)
[2] A K Ghatak and S Lokanathan, in Quantum mechanics: theory and applications (Macmillan,
Madras, 1986) p. 255
[3] J J Sakurai, in Advanced quantum mechanics (Massachusetts, Addison- Wesley Publishing
Company, 1967) p. 128
[4] C Itzykson and J Zuber, in Quantum field theory (McGraw Hill, Singapore, 1986) p. 79
[5] N Isgur and M B Wise, Phys. Lett. B232, 113 (1989); B237, 527 (1990)
[6] H Georgi, Phys. Lett. B240, 447 (1990)
[7] E Eichten and B Hill, Phys. Lett. B234, 511 (1990); B243, 427 (1990)
[8] M Kobayashi and T Maskawa, Prog. Theor. Phys. 49, 652 (1975)
[9] D Griffiths, in Introduction to elementary particles (John Wiley and Sons, New York, 1987)
p. 321
Effective potentials and threshold anomaly
S V S SASTRY and S K KATARIA
Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400085, India
MS received 22 September 1995; revised 16 January 1996
Abstract. The strong E and L dependence of the effective elastic channel potentials is shown to
be an implicit radial kinetic energy (e) dependence. It is also shown that this effective potential
satisfies the dispersion relation in K variable at the strong absorption radius. Further, the
experimental data for both elastic and fusion channels are consistent with this L-dependence of
the corresponding effective potentials. The effective transfer channel potentials derived using
CRC code FRESCO are shown to exhibit strong energy dependence as a result of couplings. The
energy dependence of effective transfer strength for 16 O + 208 Pb and 16 O -f 232 Th systems is
determined using the experimental transfer angular distributions.
Keywords. Effective potentials; TELP; threshold anomaly; dispersion relation; CRC calcula-
tions; fusion.
PACS No. 24-10
1. Introduction
The optical model (OM) study of elastic scattering of heavy ions near the Coulomb
barrier energies has resulted in many interesting observations. At energies well above
the Coulomb barrier, it is well known that the scattering process is much simpler as in
the case of point particles and the corresponding optical model potentials (OMP) are
static or vary weakly with energy. In depth OM studies around the Coulomb barrier
energies and below, have shown that the OMP parameters strongly vary with energy.
This phenomenon is known as threshold anomaly (TA) [1]. In optical model analysis,
where the potential strength parameters alone are varied, it was shown that the real
part of the nuclear potential becomes more attractive around the barrier energies and
decreases on either side, thus resulting in a bell-shaped curve. The imaginary part which
remains more or less constant at high energies, decreases sharply with decreasing
energy below the Coulomb barrier as observed in OM analysis of several heavy ion
systems like 16 O + 60 Ni, 16 O + 208 Pb, 32 S + 64 Ni, 32 S + 40 Ca, 16 O + 63 Cu, [2, 3] and
ICQ _j_ 209gj j-^ The rea d er ma y re f er to the detailed review by Satchler [3] and the
references therein. In a nonrelativistic formalism, it is known that the causality
condition on potential and the quantum mechanical wave function results in Cauchy's
integral relation between the real (V) and imaginary (W) parts of the OMP [1, 3], as
stated by
357
u r LJ uujtf y unu ij J.\. ^vutuviu
where P stands for the principle part of the integral. Equation (1) is also known as
dispersion relation (DR). Therefore, the energy dependence of imaginary part implies
energy dependence in real part and vice versa. This contribution to the OMP is also
called dynamic polarization potential. By normalizing the strength of the real potential
at a high energy point E s , (1) can be generalized as [5],
V(r,E)=V(r,E.) + W(r,E) (2)
with,
w w d -. (3,
t ~* ~ ~< \ f
Equation (3) suggests that the energy dependence of real and imaginary parts of
OMP is related but does not rule out any other energy dependence besides that coming
through eq. (2). For example, the effective momentum and energy dependence can arise
from non-local part of the interaction or intrinsic energy dependence and it is not
obvious that such a potential will obey a dispersion relation (DR). However if eq. (3) is
satisfied, then the effective nuclear potential becomes more attractive at around the
barrier energies and results in energy dependence of the barrier height. This energy
dependence of OMP is only qualitatively in conformity with the conclusions drawn
from the studies of fusion channel. In fusion studies, the experimental data can be
summarized as (i) the cross section enhancement at sub-barrier energies and (ii) the
broadening of the spin distribution around and below the barrier energies. It was
proposed that these results can be accounted for in the barrier penetration model
(BPM) if the fusion barrier height is made energy-dependent. However, the energy
dependent part, derived from the elastic channel scattering data does not reproduce the
fusion cross section enhancement quantitatively [2, 5]. Further, the energy dependence
of the effective barrier height even when adjusted to account for the fusion cross-
section, fails to account for the fusion spin distribution. Therefore, the effective barrier
height and hence the effective heavy, ion potential is known to be both E and
L dependent in order to be consistent with experimental fusion spin distribution. It was
shown earlier [6] that the fusion data can be well described in the BPM if the fusion
barrier height depends implicitly on E and L through a variable e, the radial kinetic
energy at the interaction barrier. Similar conclusions were also drawn for the OM
effective. fusion potential [7-9]. Therefore, it is expected that the effective heavy ion
potential for the elastic channel also be L dependent and this dependence must be
consistent with DR for effective potentials. In contrast to this, the OM analysis of
elastic angular distributions does not demand a serious need for this L dependence,
even though strong energy dependence is found to be necessary. Hence there is an
anomaly between the elastic and fusion channel potentials with respect to L depend-
ence. In order to understand this anomaly one needs to study this L dependence of
effective elastic channel potential used in OM analysis. It should be noticed that OM
analysis of elastic scattering data is not a very sensitive test for the L dependence since
the angular distributions around the barrier energies are featureless and do not vary
strongly with angle. Therefore, we have studied the L-dependence of effective potentials
based on the coupled reaction channel (CRQ results
The CRC formalism has been very successful in describing the heavy ion reaction
mechanism [2]. In this formalism, all the important reaction channels are coupled
together and the CRC solution is obtained by iteration. Fusion is estimated as the flux
removed from the coupled channels system by the use of OM imaginary potential (W)
in each channel or by means of ingoing wave boundary condition. We have studied the
i6Q _j_ aospk svs t em w ith tne coupling scheme as well as the coupling parameters taken
from [2]. The energy dependence of effective potential and their dispersive nature are
also discussed in detail in [2]. These calculations show that the fusion and reaction
excitation functions are well described, in addition to inelastic and transfer cross
sections. The method gives reasonably good predictions for elastic angular distribu-
tions and the observed TA of effective potentials. However, there are still some
important shortcomings as discussed below:
(i) the fusion mean square spin does not agree with the experimental data at low
energies,
(ii) the elastic angular distributions predicted are not in very good agreement with the
experimental data for near barrier energies and in addition there is a backward
angle enhancement of elastic cross section compared to experimental data [2, 10].
(iii) the polarization potentials derived are not in good agreement with dispersion
relation over a wide energy range [2].
(iv) the derived energy dependent effective barrier, when used in the standard barrier
penetration model, does not reproduce the CRC fusion excitation function at low
energies [2].
In the present work, we study the energy dependence of effective potentials as a result
of couplings for elastic, fusion and transfer channels. In the first section, the E and
L dependence of the effective elastic channel potential obtained using CRC wave
functions and the dispersion relation for these potentials are presented. Further, -the
E and L dependences of these potentials are shown to be consistent with the implicit
dependence through a single variable 8 to a good approximation. In the second section,
the dependent potentials are shown to reproduce experimental results for both elastic
and fusion channels. In the third section, the effective one channel transfer strength
parameter derived from CRC results is shown to be energy dependent similar to
effective elastic channel potentials. Use of this effective particle transfer strength in the
CCFUS code yields better estimates for the fusion cross section and its mean square
spin values.
2. Effective elastic channel potentials
The CRC calculations have been performed over a wide energy range and the elastic
wave functions are obtained for each partial wave. In order to understand the TA, one
needs to construct local polarization potentials using this CRC wave function. There
are many ways of deriving the local polarization potentials [2,10,11]. In the first
method, one obtains the local effective elastic channel potential from the condition that
it must reproduce the CRC elastic wave function at every point (r) as given by
f T + V. ,,M 4- V~(r\ + Vr Ml i/ RC M = E\ls c (rl (4a)
V^(r}=V^(r)+V pol (r). (4b)
This is equivalent to the potential derived from the OM fits of elastic angular
distributions as given in eq. (2),
V eff (r)=V(r,E). (4c)
Here, V N is the bare nuclear part of heavy ion potential for elastic channel, as used in
the CRC calculations. The second term K po , includes all the effects of the channel
couplings on the elastic channel wave function. The F cff defined in this way is known as
trivially equivalent local potential (TELP) which is wave function equivalent [2] and
can be obtained numerically using (4a), provided the wavefunction at the point r, does
not vanish. In the second approach, F pol is obtained from the effective potentials that
reproduce the 5-matrix elements for each partial wave [10]. There are also attempts
to obtain F pol by inverting the fusion cross section data at each energy [1 1]. However,
one should notice that such an inversion gives effective fusion barrier and hence the
derived polarization contribution will not be in agreement with the elastic channel
potential. This discrepancy between the effective elastic channel potential and the
fusion barrier has been discussed earlier in the introduction. In addition, such
a polarization potential may not be consistent with the fusion spin distributions.
The TELP strongly depend on E and L and especially the imaginary part can be
emissive as well as absorptive as a function of r, destroying the flux conservation for
each channel. This is.because the flux from one channel at a point r and the partial wave
L can be removed and added at a point r' at L' to the same or different channel. This
rearrangement of flux takes place for a given energy and total angular momentum
consistent with the flux conservation over all the channels in the CRC calculations. It is
not obvious that such a nonlocal effect is reducible to an effective local and dispersive
potentials and therefore, the DR is a stringent condition to be verified in these
calculations. For the purpose of studying DR, there are attempts to obtain the weighted
L-averaged polarization potential (which depends only on energy) using the TELP.
However as discussed earlier, this energy dependent potential when incorporated in
a BPM, could not reproduce even the CRC fusion cross sections at low energy [2].
Thus, any choice of L-averaging may not be good enough. In order to establish a DR,
(3) should be applied to the imaginary part of TELP under the assumption that it is
applicable to each partial wave (total J). The applicability of DR to weighted
L-independent potential requires explicit justification since only one weighting pro-
cedure is likely to satisfy DR as the weight factors involve energy dependence. Here we
present the results for TELP at a fixed radial separation.
Figure 1 shows the TELP as a function of L for five different bombarding energies
[10] at R s = 12-5 fm, which is the strong absorption radius. The radial distribution of
flux in the reaction channel peaks at this radius and therefore, the effective potential
strength is important around this radial separation. As seen in figure 1, the real and
imaginary parts of TELP show rapid variation as a function of L in addition to
undulations which are difficult to eliminate. The L-weighted procedure can be seen to
be averaging over many important features of the TELP and results in a mean potential
which varies smoothly with energy. For example at 96 MeV, (see figure 1) the real part
uq
cxi
-O.2
-0.4
-0.6
-o.e
0.0
-0.2
-O.4
-0.6
-0.8
-l.O
Figure 1. L versus real part of TELP (solid curve) and imaginary part of TELP
(dashed curve), obtained from the CRC elastic wave function at R s = 12-5 for
different incident energies as mentioned in the plots. This and all subsequent figures
refer to the system 16 O + 208 Pb.
becomes repulsive for L-values around 21h to 30# and becomes attractive beyond. The
imaginary part also exhibits emissive and absorptive behaviour and is strongly
absorptive for L&25h 3Qh. It is to be noticed that the hump seen in the L-
distribution for 96 MeV is absent in the case of 90 MeV. At this energy the TELP is both
increasing and decreasing as a function of L, whereas at 80 MeV the potentials vary
monotonically with L. One interesting point to observe in figure 1 is that the large
L behaviour of TELP at 96 MeV is similar to the low L behaviour at 90 MeV. Similar
observation can be made for any two sets of neighbouring plots of figure 1 and the most
prominent are the cases of 80 MeV and 86 MeV. These correlations suggest that the
TELP may not depend explicitly on E and L but have implicit dependence through e,
similar to the case of effective fusion potentials.
Figure 2a shows the L-dependence of imaginary part of TELP for all the cases of
figure 1, plotted as a function of s. The rot is evaluated at R s = 12- 5 fat. As seen in the
figure, the widely different cases of figure 1 merge into a single curve on e scale showing
the validity of the proposed E and L dependence through e. It also suggests that there is
no need to construct L-independent potentials in order to study their dispersive nature.
This is a total departure from the methods followed in earlier studies. This e dependent
effective elastic channel potential gives rise to E dependent effective barrier and thus
relates well with the e dependence of the potentials derived from fusion studies [7], This
procedure eliminates the need of arbitrary weight factors. However, this e dependence
of effective potentials does not smooth out the observed strong oscillations in the
0.5
o.o
-l.Q
folynomial-
IdMeV
AAAAA 102
xxxxx 86
80
50 60 70 80 90 100 110
6 (MeV)
Figure 2a. Imaginary part of the TELP versus e for different symbols as described
in the figure. The continuous curve is a polynomial fit to the data. The dashed curve
represents the weighted mean energy dependent potential.
L-dependence but rather transforms into a general behaviour at high energies. Gener-
alizing the DR to each partial wave (of elastic channel), from (3) we get,
A7(r,,L) =
(E-E s )
dE'.
(5)
_(-')(.-')
As a consequence of dependence of W i.e., W(E, L) = W(e, 0), eq. (5) can be
transformed into & scale. The strong absorption radius, R s (which is used to define e\
depends weekly on energy and to a first approximation it can be neglected. From this it
follows
W(r,e)
_ , (e e )
-de'.
(6)
It is to be noticed that s is a fixed quantity in eqs (3, 5) and similarly e s is also taken to
be a fixed quantity in (6). Thus (6) is not simply a transformation of (5) but in a way
e replaces in it the dynamical role of E of (5). The LHS of (6) is the dispersion
contribution to the real part induced by the energy dependence of imaginary part of
TELP. Therefore the validity of DR for effective potentials can be verified by checking
the equality of AF(r,s) and the energy dependence of real polarization part of the
TELP,i.e.F pol ofeq.(4).
Figure 2b shows the L-dependence of real part of TELP (represented by various
symbols) for different cases of figure 1, plotted as a function of e. The TELP for different
energies and L values of figure 1 merge into a single curve on e scale, similar to the case
in figure 2a. The dispersion result obtained by applying (6) to the imaginary part of
TELP is shown by solid curve. The agreement between these two cases (symbols and
solid curve) implies that the g dependence of real part of the TELP is accounted by DR
to a good approximation and thus verifies the validity of DR on e scale. The L-averaged
p.nfircrvHp.np.nHent notftntialxarp. nhtainprl frnm tVipTPT P-nntVi
T _/-1icti-iKn+i/-ii-ic
80 90 100
6 (MeV)
110
Figure 2b. Same as figure 2a, but for the real part of TELP. The solid curve is the
dispersion contribution corresponding to the solid curve of figure 2a. The dashed
curve is the weighted mean energy dependent potential.
as weight factors. This averaging gives rise to smooth energy dependence and is also
shown in figures 2 (a, b) as dashed curves. The imaginary part of TELP as a function of
s is small at high energies. Therefore at this radial separation, the absorption of flux into
reaction channel is smaller for low partial waves as compared to larger partial waves.
The large backward angle enhancement predicted by CRC calculations for elastic
channel may be understood in terms of this weak imaginary potential for low partial
waves at high energy. The oscillations in W(s) at high energy are due to the strong
oscillations in the asymptotic wave function with energy at a given radial separation
(.R s = 12-5 fm). The imaginary potential shows maximum absorption around the
barrier energies and the corresponding real part shows maximum attraction at these
s values (figure 2b).
Thus, it is shown that the energy dependence of real and imaginary parts of TELP are
consistent with DR at strong absorption radius. However, (6) implies that the DR is
also valid at other radial separations. In order to study this, the TELP were obtained at
10 and 1 1 fm for 1 10 and 102 MeV bombarding energies.
Figure 2c shows the real (bare potential subtracted) and imaginary parts of TELP at
these radial separations represented by symbols. The E rot is evaluated at the corre-
sponding radii. The solid curves shown in the real potentials are obtained by applying
(6) to the solid curves in imaginary parts (polynomial fits). The agreement of symbols
and the solid curves in the real potentials shows the approximate validity of the DR at
these radial separations. To these dispersion corrections at a given radius, one must add
a static potential to match with the real part of TELP (see (2) and (4c)). The static
potentials required at 10, 11 and 12-5 fm are respectively 35-1, 5-2 and 1-25 MeV
and the total potential compares well with the phenomenological potentials of Kim
et al [7], The imaginary part of TELP in figure 2c shows rapid variation around
90 MeV and in this energy region, the real part also shows strong energy dependence. It
can be seen from figure 2c that the imaginary potential at 90 MeV is absorptive for
1 1 fm whereas it is emmissive for 10 fm.
Pramana - J. Phys., Vol. 46, No. 5, May 1996
363
-12.5.
30 40 50 60 70 80 90 100 110
-15.0.
30 40 50 60 70 80 9O 100 110
15
10
> 5
a>
s
^
a
*-
o -5
*-10
-15
anano HOMeV
++-*++ 102MeV
polynomial fit
30 40 50 60 70 80 90 100 110
e (MeV)
-3
90 100 1 10
Figure 2c. Imaginary and real parts of TELP versus e at the radial separations of
10 and 11 frn. The symbols used are explained in the figure. The solid curves in the
imaginary potentials are the polynomial fits. The solid curves in the real potentials
are obtained by applying the dispersion integral to solid curves of the imaginary
potentials.
At large distances it is difficult to obtain the TELP from CRC wave functions reliably
using present method. At large radii, the wave function is large in magnitude, the
expected polarization potential is small and numerical accuracies pose a problem.
Therefore, this study has not been extended for radii beyond 13 fm. At small distances
the wave function decreases in magnitude and obtaining effective potentials which are
of large strength also is difficult. Further, for the radial separations for which the
validity of DR has been presented, the dispersion integral is applied in a limited energy
region where the imaginary part of TELP is obtained from CRC calculations. In both
cases (of small and large distances), one needs a coupled channel wave function at a very
small step size to determine the kinetic energy accurately in order to obtain correct
effective potentials. This requires enormous computation time and memory.
The TELP are explicitly dependent on radial separation, energy and angular
momentum. This method of analysis of studying the DR at fixed radial separation
assumes that the imaginary potential can be factorized into radial and energy depen-
dence (for example, see (7b)). This is commonly used and is a good approximation for
energies around the barrier region, where the polarization effects are known to be
maximum. However, the disagreement of dispersion correction with the real part of
Effective potentials and threshold anomaly
TELP shown in figure 2b (see beyond 96 MeV) may be suggesting that this factoris
ation may not be valid at high energies and large radial separations. The use of e scale t
study the dispersion relation instead of explicit E and L dependence is for its simplicit
and it may be a good approximation only around the Coulomb barrier. At hig
energies, inversion gives potentials that do not converge.
3. Dynamical potentials for fusion
In the previous section, it was shown that effective elastic channel potentials obtainei
from CRC calculations are dependent on a variable a and that the derived real an<
imaginary parts obey DR. However, as stated in the introduction, the stronj
L dependence was not realised in the OM analysis of experimental elastic data
Therefore it is necessary to show that experimental elastic angular distributions as wel
as the fusion data are consistent with the & dependent effective potentials using th<
optical model analysis. The optical model code SNOOPY was modified to incorporati
the proposed e dependence (defined at the strong absorption radius) as given in (6) t(
the real and imaginary parts of OMP. Following [7], the dependence of W i:
factorized into inverse Woods-Saxon form and the r dependence of OMP is a Woods-
Saxon type, as given by
W(r,e)=f(r)g(e) (7b
with,
_/ Here, F p (r,e) is obtained by using (6) and W(r,s). It is not obvious that a strong
''j: L-dependent potential (through variable) can give good fits to elastic angula:
distributions. Figure 3 shows the OM fits to elastic data by this method represented fr
circles, from very high energy of 129-5 MeV to a low energy of 78 MeV (up to 80 MeV it
the figure). The best OM fits to experimental data as reported in [12, 13, 14] are showi
by solid curves. In trying to obtain good fits, jR s was varied with energy. The variation o
R s for different bombarding energies is given in table 1. This dependence compares wel
with the findings of Videbaek et al [12] only at low energies. However at high energies
the reaction is dominated by fusion which takes place around the barrier radius (fusioi
radius), and therefore, the strong absorption radius is also expected to approach fusioi
radius.
The reaction cross-section obtained from the present method is given in table 1 alon]
with the results of simple energy dependence of OMP as reported by Kim et al [5]. Thi
CRC results [21 and the experimental data [12, 13, 141 are also shown. As shown in tb
180
Figure 3. Elastic to Rutherford ratio as a function of cm for different energies for
the 16 O + 208 Pb system. The solid curves represent the best OM fits to experimental
data as reported in refs [12, 13, 14]. The plus symbols represent the results of [5].
The circles represent the fits by present method and the OMP parameters used are
as follows: V = 100 MeV, W = 70 MeV, r = r = r c = 1-23 fm, a = a t = 049 fin,
e s = 102 MeV, e = 78MeV and a e = 4MeV. The plots at different energies are
successively scaled down by a decade.
Table 1. Reaction excitation function by various
methods.
MeV
fm
mb
mb
mb
<7 B (expt.)
mb
78
12-5
37-5
49-4
41
45-7
80
12-5
99
123
101
100 10
82
12-5
195
83
12-5
248'
290
259
237 20
84
12-5
301
86
12-5
406
461
425
440
88
12-5
507
568
529
570 + 58
90
12-5
603
669
629
578 + 55
96
12-5
867
940
899
904
102
12-5
1099
1173
1134
1147 + 95
104
11-0
1117
1244
110
11-0
1300
1440
129-5
10-5
1724
1943
A = present method using e dependent potentials;
B = method of Kim et al [5] using energy dependent
potentials; C = CRC method using FRESCO; expt.=
Experimental data [12, 13, 14].
MeV
'i.
<L 2 >
A
B J
B
ff f
C
expt.
<L 2 >
expt.
78
10-96
173
9-0
109
6
104
5-6 + 0-6
170 + 30
80
34-3
186
39
138
37
107
36 4
200 20
82
77-3
229
83
105
259
127
224
157
195
108 10
270 + 40
84
135
291
86
205
361
225
336
297
314
235
88
283
435
290
415
385
400
350 + 40
90
367
512
352
496
466
488
377 + 50
430
96
634
765
534
746
683
754
685 + 70
102
888
1044
725
1004
884
1000
844 90
104
825
999
110
1009
1248
920
1346
1060 + 50
1275
129-5
1376
1945
1353
2234
131565
2085
A, B, C and expt. are as in table 1; o f = fusion cross section in mb; <L 2 > = L(L + 1)
in units of h 2 .
In direct reaction theory, the fusion cross-section is estimated as the overlap integral
of the elastic wave function with the fusion potential [5,7]. The radial and energy
dependence of the fusion potential was factorized as W F (r,s)=f(r)g F (e). The radial
dependence /(r) is taken to be same as that of reaction potential (eq. (7c)). The
parameters of F (e) are determined by fitting the fusion excitation function by varying
only one parameter ( ). The optimum value s of # F is 84 MeV and a e is same as that of
reaction potential. As listed in table 2, the fusion excitation function is well reproduced
showing the quality of the fit with one parameter. The resulting fusion mean square spin
is also listed in the table. It can be seen that the fusion mean square spin agrees well with
the experimental data at all energies, especially at low energies. At low energy of
78 MeV, the present method gives higher fusion cross section compared to experimen-
tal data. However when the parameters of # F are adjusted to reproduce the data at this
energy, the fusion <L 2 > still remains the same. The experimental data, CRC fusion
results [2] and the results of Kim et al [5] are also listed in table for comparison. In
other words, the L dependent potentials are in agreement with the experimental data
for both elastic and fusion channels.
4. Dynamical potentials for transfer channel
The elastic and fusion channel effective potentials have been shown to be energy
dependent as a result of couplings. Similarly, the effective potentials that describe
transfer angular distributions for different transfer channels in a heavy ion collision are
also expected to be energy dependent. The imaginary ptoential for direct reactions
consisting of inelastic and transfer channels can be obtained as the difference oi
potentials for total reaction and fusion ( 3) and this will also depend on B. However, it is
difficult to decompose this part into the components responsible for individual inelastic
and transfer channels. In addition, such a partition of imaginary reaction potential intc
Y O LJ14JI.I y
UllU O IV JVLtlUf
10
10
"0 +" M Pb at E L =BO UeV
Sum of six channels of
^O and
1.5 1.6 1.7 1.8 1.9 2.0
d (fm)
10
OTlO
P?10
10
10
"o+ 20<> Pb at E,=80 MeV
CRC data for M N (a , + )
d tt C (A)
(b) :
1.5 1.6 1.7 1.8 1.9 2.0
d (fm)
Figure 4. P tr /sin(0/2) versus semi-classical distance of closest approach par-
ameter, d, for neutron transfer (a), proton and alpha transfer channels (b) obtained
from CRC transfer results at 80 MeV. For details see text.
its constituent channels only gives the respective partial wave distributions for various
channels, whereas for the transfer angular distributions one needs phase information.
Therefore, we adopt a semiclassical approach to estimate the effective one channel
particle transfer strength function. In this formalism, the transfer angular distribution
as a function of semi-classical distance of closest approach [15] is given by
with
and the transfer form factor (strength function) is given by
(8a)
(8b)
(8c)
From these relations, P tr (0) can be obtained by integrating over the Q variable, with
<2 opt and a as defined in [15]. Therefore, from the log plot of P tr /sin(0/2) versus d, the
transfer strength (/ ) and slope parameters (a) can be determined. The transfer
probability for the system of 16 O + 208 Pb is calculated using the transfer angular
distributions obtained from the CRC code. The coupling scheme is given as in [2].
Figure 4a shows a plot of this transfer probability versus d at 80 MeV for the neutron
transfer channel. The neutron transfer to the ground state of 209 Pb and to the various
excited states are combined into one effective neutron transfer probability. Figure 4b
shows the proton and alpha transfer probabilities at 80 MeV. The proton transfer
probability to the ground state (squares) and the excited state (plus symbols) of 209 Bi
are shown seoaratelv in fieure4b. As seen in the figure, fnr nrrvtrm transfer rapt tVip
65 70 75 80 85 90 95
E lab (MeV)
Figure 5. Transfer strength function at closest approach (F(D<.)) versus lab energy
for different transfer channels. The symbols represent energies where the calcula
tions were performed and intrapolated by the respective curves.
Therefore, the strength of a given particle transfer to ground state and various excitec
states are combined into effective one channel transfer strength.
Figure 5 shows the variation of effective one channel transfer strength as a functior
of incident energy evaluated at the distance of closest approach (D c ). The strong
r dependence of eq. (8c) is cancelled at D c and therefore the effects of channel couplings
on transfer strength can be seen at D c . It is seen from the figure that the effective transfei
channel strength evaluated along a Coulomb trajectory, shows rapid variation as
a function of energy. The strength attains maximum at around the barrier energies
depending on the Q-values of the channels and decreases on either side of the barriei
energies, as a result of couplings. However, at high energy the transfer strength
parameters will be modified owing to nuclear branch of semiclassical deflectior
function [16].
Following this method, the effective transfer strength consistent with experimenta
transfer angular distributions can be determined. This effective transfer strength car
then be used in the simplified coupled channel codes like CCFUS in order to estimate
approximately the fusion enhancement due to transfer couplings [15]. The experimen-
tal data of 16 O + 208 Pb system for nitrogen and carbon channels [12] was analyzed al
80, 88 and 90 MeV and the transfer strength at the barrier position was estimated
Similar calculations were performed for the 16 O + 232 Th system from the available
experimental data at various energies [17, 18]. For this system, the transfer strength
was combined into two effective one and two proton transfer channels. The parameters
thus estimated are listed in table 3. The transfer strength functions at R b for various
cases are seen to be strongly energy dependent. This is expected and as discussed in the
previous section, the channel couplings renormalize the transfer strength (refer tc
figure 5).
The predictions for fusion cross section and mean square spin from the CCFUS
code following this method are listed in table 4 for 16 O + 208 Pb system. The otha
parameters used in CCFUS code are also given. The experimental data are tabulatec
for comparison. It is seen that the results of the present method for fusion <L 2 ]
following semi-classical analysis of experimental
data [12, 17, 18] and used as input parameters to
CCFUS.
(a) 16 O + 208 Pb system; A7=22-0,
V h = 75-92 MeV, R b =ll-7fm, ftco = 4-78 MeV, Q g
(nitrogen) = 0-658 MeV and Q g (carbon) = 4-7 MeV.
E lab (MeV) F(R b ) (nitrogen) F(R b ) (carbon)
80 1-26 1-93
88 0-6 0-62
90 0-48 0-37
(b) 16 O + 232 Th system; A7=22-0,
V b = 81-70 MeV, K 6 = 11-96 fm, fcco = 4-7MeV, Q g
(nitrogen) = 2-9 MeV and Q g (carbon) = 9-0 MeV.
lab (MeV) F(R b ) (nitrogen) F(R b ) (carbon)
80
0-52 .
0-49
83
0-59
0-52
86
0-64
0-41
92
0-545
0-43
105
0-417
0-46
Table 4. Results for fusion of 16 O + 208 Pb system (For
comparison with CRC results using FRESCO refer to
table 2).
lab ff/ (mb) <L 2 >(ft 2 ) <r /iexp (mb) <L 2 > exp (ft 2 )
80
37
191
36 + 4
200 + 20
88
312
426
350 + 40
90
397
425
377 + 50
430
are in agreement with experimental data. However the estimates of the present
method for fusion <L 2 > are higher than FRESCO estimates, especially at low
energy (see table 2). The corresponding L-distributions for 80 MeV case are shown in
figure 6. It is seen that the present method using CCFUS code predicts much broader
fusion L-distribution compared to both 1DBPM and FRESCO. This discrepancy
between the present method and the FRESCO at 80 MeV is not expected because
the CRC transfer form factors were adjusted in FRESCO at 80 MeV in order to
match the CRC integrated transfer cross sections with experimental data. However,
it was observed that though the integrated cross sections at this energy match, the
predicted transfer angular distributions are not in good agreement with the experi-
mental data of [12]. In the present method, we are extracting the transfer strength
consistent with experimental data. Therefore this effectively includes all significantly
10 15 20
L (h)
Figure 6. Fusion L-distributions from the present method using CCFUS (solid
curve), FRESCO (long dashes) and one dimensional BPM results (short dashed
curve).
contributing channels in addition to all higher order coupling effects. Further,
this discrepancy may also be due to the reason that the CCFUS calculations treat
the coupling form factors in constant coupling approximation, evaluated at the
barrier position. In the case of 16 O + 232 Th, the transfer couplings do not appreciably
effect the fusion results as reported by Esbensen et al [19] and the enhancement
in fusion excitation function is dominantly due to permanent deformation of the
target ( 232 Th).
Summary
The E and L dependence of effective elastic channel potentials implicit through the
variable e is shown to be a good approximation. Further, the dispersion relation in
e variable is shown to be valid at the strong absorption radius. The DR is also studied at
other radial separations. The s dependence for elastic channel potential is shown to be
consistent with the experimental data for both elastic and fusion channels. The
variation of effective transfer strength as a function of energy is obtained using the
transfer angular distributions of the CRC code. It is observed that the variation of
effective transfer strength, estimated along a Coulomb trajectory, is similar to the TA
seen in the elastic channel case. It is seen that the effective transfer strength derived from
experimental transfer angular distributions can account for both the fusion cross
section enhancement and its mean square spin.
Acknowledgements
The authors are thankful to Dr I J Thompson for providing the FRESCO code, and
Drs S S Kapoor, A K Mohanty, D M Nadkarni, A K Jain, R S Mackintosh and
M A Nagarajan for very fruitful discussions during the course of this work. The referee
is acknowledged for important suggestions.
Pramana - J. Phys., Vol. 46, No. 5, May 1996
371
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physics pp. 373-380
Depopulation of Na(8s) colliding with ground state
He: Study of collision dynamics
A A KHAN 1 , K K PRASAD 2 , S K VERMA 3 , V KUMAR 4 and A KUMAR 5
Department of Physics, Z.A.I. College, Siwan 841 226, India
2 Department of Physics, D.A.V. Inter College, Siwan 841 226, India
3 Department of Physics, Jagdam College, Chapra 841 301, India
4 Department of Physics, Rajendra College, Chapra 841 301, India
5 University Department of Physics, J.P. University, Chapra 841 301, India and Department of
Physics, D.A.V. College, Siwan 841 226, India
MS received 9 February 1995; revised 17 October 1995
Abstract. A systematic study of the collison dynamics associated with depopulation of Na(8s)
atom colliding with ground state He has been made by applying the semi-classical impact
parameter method using molecular orbital (MO) basis sets of different sizes. The cross-sections
for total depopulation of the parent atom as well as those for individual transitions have been
calculated. It is shown that the basis set must be large enough so as to include not only the
immediate adjacent states coupling with the parent state but also other nearby states, which can
affect the overall flux distribution in the reaction.
Keywords. Rydberg atoms; depopulation; collision dynamics.
PACSNo. 34-60
1. Introduction
Quenching of low-lying Rydberg states due to the impact of neutral atomic perturbers
at very low impact energies (thermal energies) has been subject of both theoretical
[1-4] and experimental studies [5-7] in recent past. It has been established beyond
doubt that the free-electron model [8] is not suited for such investigations where
detailed interactions need to be considered. An impact parameter method using
molecular orbital (MO) basis sets of appropriate size has been successfully used by
Lane et al [3,4], for the study of depopulation of low-lying Rydberg states of Li and Na
colliding with the ground state He. We have also successfully employed the same
method for calculating cross-section for depopulation of low Rydberg states of
Rb(n = 8 to 10) colliding with neutral He perturber at thermal energies [9]. Here we
arrive at a conclusion that each such pair of Rydberg atom-structureless perturber
needs to be investigated separately, specially in the case of low lying states where the
energy defects involved are comparatively large. Gallagher and Cooke [7] have also
emphasized individual analysis of such interactions. They have observed that the
upward transition [ns -* (n l)d] for Na(ns) + Ar is possible even for 8s state for which
the available thermal energy /cT(T = 425 K) is just sufficient to overcome the energy
defect Casvmntotic valuel However, similar transition is not reported for Na(ns) + He
before n - 9. Obviously, along with this asymptotic value of the energy defect the actual
shape of different adiabatic states will have to be considered in explaining these
interactions. Complete details of total interactions between the colliding system must,
therefore, be considered to account for these findings (see also [3, 4, 9]).
Such a detailed study is helpful in not only calculating the cross-section for total
depopulation but also in understanding the importance of the individual transitions
responsible for quenching the parent Rydberg state. An interesting finding of these
studies has been the development of s -> / propensity rule A/ = 3 as one goes from low
ns-state of Na to high ns-state in Na-He collision.
These developments prompted us to undertake a systematic and detailed study of
collision dynamics of Na(8s)-He interactions in the thermal energy region. The amount
of energy available at a collision temperature 425 K for this pair is 1-36 x 10~ 3 a.u.,
which is slightly more than the asymptotic energy defect &E(R co) for upward
transition of Na(8s) to Na(7d). Hence the probability of excitation to higher states is
also expected to make a small but finite contribution towards depopulating the parent
Rydberg Na(8s) state. This also makes the Na (8s)-He pair worth investigating. Another
advantage of taking He as a perturber is its compact structure. Its orbital electrons are
tightly bound to the nucleus and hence He atom as a whole acts as a mere core in our
MO calculation.
2. Calculation
For our calculation we make use of the previously discussed semi-classical MO method
[3, 4] whose suitability for such studies has been discussed in detail by Kimura and
Lane [10]. In this approach, the relative nuclear motion is described by a classical
trajectory and the active electron by a time dependent wave function expanded in terms
of electronic states represented by molecular orbitals (see ref. [10] for details). The
molecular orbitals and the corresponding electronic energy curves of the interacting
system are calculated by a standard variational procedure. The Rydberg electron's
binding in the combined (Na + -He) core is accounted through the standard
pseudopotential approach [11]. In the present calculation we make use of the
pseudopotential parameters given by Bardsley [11] and Pascale [12] for Na + and He
cores respectively. In the expansion of the system's wave function the coefficients (a n ]
satisfy a set of linear first order coupled differential equations
^ n W= Z V-(P + A) kn a k exp[-ij<(E n -E k )dt], (1)
where only the first order terms in V (relative nuclear velocity) have been considered. In
this equation P and A represent the non-adiabatic coupling (radial and rotational)
matrix and its ETF (electron transfer factor) correction terms respectively (see ref. [3]
for details). These equations are solved numerically for a sufficiently large number of
impact parameters (b) employing the initial boundary conditions
a k (-co) = <5 {k , (2)
where i represents the initial state and <5 ik is the Kronecker delta. The asymptotic values
aic omamcu uy mi.egra.ung me square 01
the respective transition amplitudes over b.
(3)
3. Collision dynamics
As stated above, in the present method a set of coupled equations, each representing
a particular molecular state, are numerically solved for each contributing impact
parameter to obtain the probability for transition from the initial state to various final
states. This obviously poses a few questions about the convergence of the cross-sections
calculated in this way. Since the range of impact parameter required for solving these
coupled equations depends on the radius of the Rydberg atom as well as the effective
size of the perturber, the calculation is usually carried out up to a certain maximum
value of impact parameter. This in turn, requires that all important R dependent
couplings are taken into account so that a realistic spectrum of probability is obtained.
Hence, the calculated cross-sections are usually checked for convergence with respect
to inter-nuclear separation R. By doing calculations up to different R values we have
arrived at the conclusion that a maximum limit of R 30a is sufficient to obtain the
needful convergence and to provide a realistic picture of the process of depopulation of
the parent Na(8s) state due to impact of thermal He. This choice ensures that all important
couplings are incorporated when the coupled equations are being solved. It is worth
mentioning at this juncture that Kumar et al [3] have discussed the convergence of the
calculated cross-section with respect to the inter-nuclear separation in a somewhat detailed
manner. By repeating their previous calculations and going up to /? = 120a they have>
concluded that no significant change is caused in the magnitude of the total depopulation
cross-sections; only the details of some individual transitions are found to change.
Another equally important aspect of such a calculation is .the convergence of the
estimated cross-section with respect to the number of molecular states coupled
together. An ideal case should include as many states as possible to have a good
convergence. The other choice can be coupling of only immediate neighbouring states
of the parent state, because transitions to these states alone are expected to make
appreciable contribution towards depopulating the initial Rydberg state. It must be
emphasized here that larger the number of coupled equations, greater is the computa-
tional effort needed to solve them. This obviously puts a limitation on the number of
states one can couple together. But at the same time the calculated individual transition
cross-sections must project a realistic picture of the process of depopulation. The above
mentioned aspect of convergence of the calculated cross-sections has been the main
motivating factor behind the present study of collision dynamics.
In the following section, we present our calculated cross-sections.
4. Results and discussion
The calculated adiabatic potential energies for Na(8s)-He pair is presented in figure 1;
for clarity only states have been included in these figures. The calculated adiabatic
mental** Na energies (a.u.).
nl Present cal. Experiment Difference (%)
3s
1-8885 E-l
1-8879 E-l
0-03
3p
H156E-1
1-1140 E-l
0-14
4s
7-1579E-2
7-1561 E-2
0-03
3d
5-5937 E-2
5-5936 E-2
0-00
4p
5-0939 E-2
5-0888 E-2
0-10
5s
3-7585 E-2
3-7579 E-2
0-02
4d
3- 1442 E-2
3- 1423 E-2
0-06
4/
3-1261 E-2
3- 1268 E-2
0-02
5p
2-9197 E-2
2-9169 E-2
0-10
6s
2-3132 E-2
2- 3 129 E-2
0-01
5d
2-0106 E-2
2-0046 E-2
0-30
5f
2-0011 E-2
2-0003 E-2
0-04
6p
1-8920 E-2
1-8892 E-2
0-15
Is
1-5663 E-2
1-5659 E-2
0-03
6d
1-3953 E-2
1-3869 E-2
0-60
6/
1-3895 E-2
1-3844 E-2
0-36
IP
1-3254 E-2
1.- 3222 E-2
0-24
8s
1-1306 E-2
1-1295 E-2
0-09
Id
1-0246 E-2
1-01 56 E-2
0-88
If
1-0208 E-2
1-0072 E-2
1-34
8p
9-8000 E-3
9-7627 E-3
0-38
9s
8-5437 E-3
8- 5223 E-3
0-25
*The STO's supplied by Kumar et al [3] have been used.
**Fromref. [13].
energy values are also compared with the experimental results in table 1 and the two
are found to agree each other.
It may be pointed out that the present representation of the electronic states is
expected to yield good results except for R < 2a . At these internuclear separation the
charge clouds of the Na core and He atom begin to overlap. But they are of no
importance because various states couple strongly only at much larger internuclear
separations (see also Saha et al [4]). We start with a 3-state calculation in which only
8sZ, 7pS and Ipn states are included. This is because the transition Na(8s)-(7p) is
expected to be the most important direct mechanism for depopulating the parent state
(see figure 1). The calculated cross-sections is found to have a maximum (figure 2)
around V = 0-00075 a.u. (3-76 x 10~ 2 eV) and nearly the full contribution comes from
population of the 7pS state (the rotational coupling between 8p and Ipn is very small).
However, since thermal impact energies are just sufficient to excite Na(8s) to the next
higher state Na(7d), it is appropriate that the basis set should not exclude these states.
Even if the possibility of upward transition at such low energies is very small (due to the
threshold effect), virtual (transient) excitation of these states could be important.
Indeed, we have found (in a 5-state calculation) that the molecular states IdL and Idn
influence the cross-sections for Na(8s) depopulation, changing it significantly in
magnitude from that obtained from a 3-state calculation (figure!), although the
fcp _
. o
00
0. I
?5
10 15 20
Inter Nucl. Sep. ( in QU )
Figure 1. Adiabatic potential curves of 14-state calculations for the Na(8s) + He
collision. Various states are labelled through the corresponding atomic states of Na.
individual cross-section for excitation to Na(7d) remains very small (less than 1 % of the
total). The 3- and 5-state cross-sections have nearly the same energy dependence except
that in the former the cross-section shows a comparatively sharper peak in the
investigated energy range. Also when the molecular states correlating with Na(6/) (i.e.,
6/S and 6fn) are included in a 7-state calculation, the calculated cross-sections (also
shown in figure 2) remain nearly unchanged compared with those of the 5-state
calculations. The presence of a strong avoided crossing between lpl< and 6/E around
R = 25a , which is manifested by a strong radial coupling between the two in that
region, suggests that there should, however, be a probability redistribution in the
7-state calculation with majority of the probability transfer to IpL eventually passing
through to 6/S. Here it is found that this holds only at lower energies; for V> 0-00070
a.u. (3-28 x 10~ 2 eV) the cross-sections for deexcitation to Na(7p) again prevail over
those for deexcitation to Na(6/).
We next enlarged the basis set by including the molecular states 6dL and 6dn,
correlating with Na(6d). As shown in figure 2, this 9-state calculation results in
a significantly different depopulation cross-section not only in magnitude but with the
peak shifted to still lower energies (0-00068 a.u. = 3-09 x 10~ 2 eV). There is also
a change in the relative importance of the individual cross-section in that the cross-
section for transition to Na(7p) is larger than that to Na(6/) at low impact energies,
whereas the order reverses at high energies. No marked difference is observed when the
lower state 7sE is included in a 10-state calculation since there exists no significant
coupling (radial or angular) between the molecular states 6dE, 6dn and 7sE). We expect
// /
I T
0.00062 0.00066 0.00070
Impact vel. ( in QU )
0.00074
Figure 2. Cross-sections (in a^) for total depopulation of Na(8s) colliding with He.
Curves labelled as 3, 5, 7, 9, 12 and 14 are the cross-sections obtained through 3-, 5-,
9-, 12- and 14-state calculations. The 8s-7p and 85-6/ cross-sections calculated with
14-states are also shown for comparison.
that the molecular states originating from still lower atomic states of Na, such as
Na(6p), will play essentially no role in depopulating Na(8s).
Next, recognizing that the 7/L and Ifn states correlating with Na(7/) couple
strongly with IcEL and Idrc, which can be populated from the 8sZ parent state at
thermal energies, we carried out a 12-state calculation including these states as well.
Although there is only a slight change in peak position of the total depopulation
cross-section the transition to 6/Z is found to dominate transition to IpL throughout
the energy range investigated. Thus, the redistribution of the probability in the 12-state
calculation evolves during the collision in such a way that probability transferred to
6/S is permanently trapped there, making the corresponding cross-section the largest
at all impact energies studied. The inclusion of two higher states, 8pZ and 8p7t does not
cause much further change in the depopulation cross-section (figure 2). Since the
molecular states correlating with Na(7/) and Na(8p) interact with one another at large
R values their presence probably serves merely to redistribute the small probability
transferred to states correlating with Na(7d) and Na(7/) at smaller R. The inclusion of
9sS and higher states is expected to have no significant effect.
From the aforesaid study of collision dynamics we find that a realistic picture of the
process of depopulation emerges only if a fairly large number of molecular states are
coupled together. Hence, not only those states that couple directly with the initial
channel, but also the neighbouring states that couple strongly with one of the adjacent
states should be grouped together in this type of study. By increasing the number of
378
Pramana - J. Phys., Vol. 46, No. 5, May 1996
depopulation cross-section of a parent state is far less sensitive to the choice of the basis
set size, significantly different results are obtained for various individual transitions
contributing towards complete annihilation of the initial state. This also helps us in
understanding the propensity rule that led to selective population of a particular state.
This aspect of collision study can hardly be explained in a simplified model. For
example, we found in the present study (Na[8s] + He) that all molecular states
correlating with various atomic levels of Na lying between Is and 8p should be taken
into consideration. This not only ensures convergence of the calculated cross-section
for total depopulation of the parent Rydberg state, but also explains the propensity
rule: ns -> (n 2)/(A/ = 3) in downward transition. Obviously as we go up in the
Rydberg series individual characteristic of the initial Rydberg state may corne into
picture. As the energy gaps between adjacent states go on decreasing with increasing
n value, more and more molecular states will have to be coupled when considering
depopulation of the Rydberg state with larger n value. Beyond a certain limit the MO
method will no longer be appropriate and an AO (atomic orbital) method, or even a free
electron model can be applied to study their depopulation.
In the end we also compare the present results obtained from a 14-state calculation
for Na(8s) with the previously reported results on depopulation of Na(9s) under similar
circumstances [3, 4]. The process of deexcitation rather than excitation is found to be
the most important factor in quenching the parent state in both cases. Also for both the
systems, the deexcitation to (n-2)l = 3 [6/ for Na(8s) and If for Na(9s)] attains
maximum probability in the investigated (thermal) energy range. However, the two
systems differ significantly with regard to the contribution to the total depopulation
that comes from upward transitions. In the present Na(8s) case, thermal energies are
just sufficient to excite Na(8s) to the immediate higher state Na(7d). However, the
cross-sections for upward transitions are very small (less than 10% of the total
magnitude). In contrast, for the case of Na(9s) thermal impact collision energies are
a factor of 2 to 3 above the threshold for excitation and the cross-sections for upward
transitions are quite significant and amount to nearly 25% of the total depopulation
cross-sections at high collision velocities (V = 0-00075 a.u.). Since a fully quantum
mechanical treatment would suppress the excitation cross-sections near threshold, the
semi-classical calculations are likely to overestimate these cross-sections and hence,
give too large a total depopulation cross-sections for Na(9s). In spite of the difference in
magnitudes of the contributions from upward transitions in Na(8s) and Na(9s), we do
observe one significant similarity, namely that two-step upward transition is present in
both the systems. The probability transferred to (n - l}d from the parent ns-state finally
ends up in (n - 1)/ Thus, both for upward and downward transitions the Rydberg
atom tends to occupy the / = 3 final substate. We would like to emphasize at this point
that exclusion of higher m-states from the basis set of the present calculation is not
expected to cause any significant change in the magnitudes of the calculated cross-
sections. Through a test calculation Kumar et al [3] have shown that the flux does not
flow primarily to higher m-states; only about 15% of the probability flux is found to pass on
to these states through long-range rotational couplings within the nearly degenerate
H-manifold. We can, therefore, safely conclude that cross-sections for transition to
both 6/ (downward transition) and If (upward transition) are representative of
and deexcitation we expect that a propensity rule A/ ^ 3 should hold good.
It may be pointed out that a 14-state calculation for depopulation of Na(8s) colliding
with ground state He has also recently been carried out by Saha et al [4] over a large
span of collision energy. Thereafter, they calculated the quenching rate by taking
a Boltzmann average of relative collision velocity times the cross-section at a tempera-
ture of 425 K. Their calculated rate is found to agree with the measured value of
Gallagher and Cooke [7] within a factor of 3. We, however, do not attempt to carry out
a similar calculation for the reaction rates from our present calculations for basis sets of
different sizes. This is because our calculations have been done in a very limited region
of impact velocity and therefore it is not advisable to estimate the reaction rate from
them. Still we can compare our results with the experimental measurements of
Gallagher and Cooke [7] at an impact velocity corresponding to 425 K. The mean
Maxwellian velocity corresponding to this temperature [u m = (S/cT/Tr//) 1 ' 2 ] turns out
to be 7-4 x 10 ~ 4 a.u. Assuming the depopulation cross-section to be energy indepen-
dent in the thermal region Gallagher and Cooke [7] have estimated its magnitude by
taking a ratio of the measured reaction rate and the mean Maxwellian velocity
corresponding to 425 K. The cross-section for total depopulation of Na(8s) due to the
impact of ground state He, estimated in this way, turns out to be 19-64 a^ which agrees
within a factor of 2 to 3 with our calculated cross-sections at the same temperature. This
agreement is more or less same as has been achieved by Saha et al [4] through their
detailed investigation. A 14-state calculation at an impact velocity 7-4 x 10 ~ 4 a.u.
employing the present method estimates the total quenching cross-section to be
33-71 a^ and its agreement with the experimental observation of Gallagher and Cooke
(<r ex = 19-64 ajj) is within a factor of 2.
In the end we would like to emphasize that the present study has been undertaken
with the sole intention of investigating the collision dynamics of the pair: Na(8s) + He.
Conclusions drawn from such a study are expected to help probe other similar low
Rydberg atom - structureless perturber pairs.
References
[1] Y Sato and M Matsuzawa, Phys. Rev. A31, 1366 (1985)
[2] E de Prunele and J Pascale, J. Phys. B12, 251 1 (1979)
[3] A Kumar, N F Lane and M Kimura, Phys. Rev. A39, 1020 (1989); Errata Phys. Rev. A49,
1514(1994)
[4] B C Saha, A Kumar, M Kimura and N F Lane, Phys. Rev. A (to be published)
[5] M Hugon, B Sayer, P R Fournier and F Gounand, J. Phys. B15, 2391 (1982)
[6] F Gounand, P R Fournier and J Berlande, Phys. Rev. A15, 2212 (1977)
[7] T F Gallagher and W E Cooke, Phys. Rev. A19, 2161 (1979)
[8] E Fermi, Nuovo Cimento 11, 157 (1934)
[9] S K Verma, A A Khan, V Kumar and A Kumar, J. Phys. B (Communicated)
[10] M Kimura and N F Lane, Adv. At. Mol. Opt. Phys. 26, 79 (1989)
[11] J N Bardsley, Case Stud. At. Phys. 4, 299 (1974)
[12] J Pascale, Phys. Rev. A28, 632 (1983)
[13] C E Moore, Atomic Energy Levels, National Bureau of Standard (US) Circular No. 467
(USGPO, Washington, DC, 1949) Vol 1
Multiconfiguration Hartree-Fock calculations
inCr 5+ ,Mn 6+ andFe 7+
S N TIWARY, P KUMAR and R P ROY
Department of Physics, L.S. College, BRA Bihar University, Muzaffarpur 842001, India
MS received 4 April 1994
Abstract. The multiconfiguration Hartree-Fock (MCHF) method is used to calculate the
excitation energies and oscillator strengths, of both the length (/ L ) and velocity (/ v ) forms, for
Is 2 2s 2 2p 6 3s 2 3p 6 3d 2 -ls 2 2s 2 2p 6 3s 2 3p 5 3d 2 2 P, 2 D, 2 ,F transitions in Cr 5+ , Mn 6+ and
Fe 7 + ions of the potassium isoelectronic sequence. Comparison is made with our earlier relevant
results obtained by employing the configuration interaction (CI) method which is closely related
to the MCHF method. Our present investigation demonstrates that the MCHF method is more
accurate than the CI method in all ions of present consideration.
Keywords. Multiconfiguration Hartree-Fock; configuration interaction; excitation energies;
oscillator strengths.
PACSNo. 31-20
1. Introduction
The study of ionized atoms has attracted special interest in modern atomic physics and
other related fields such as astrophysics and plasma physics. Lines emitted from ionized
atoms such as the first transition elements immersed in solar, stellar and laboratory
plasmas have been playing a very important role in modelling these matters [1].
Development of experimental techniques such as electron-beam ion sources or ion
accelerators has helped to study the spectroscopic properties and scattering cross-
sections of various ion-collision processes with high accuracy [2]. In the analysis of the
spectra observed in these experiments, accurate level structures and optical oscillator
strengths for an atom in various charge states are required.
There has been growing interest in the inner-shell excitation of alkali metal atoms
and alkali-like ions from both experimentalists [3-5] and theorists [6-16], because
inner-shell excitation may lead to autoionization which has an important role in
explaining the structure observed in the integrated ionization cross-section curves for
electron impact. Consequently, the reliable theoretical calculation of position of the
autoionizing level and hence the theoretical estimate of the excitation threshold, which
is used in the calculation of the oscillator strengths, of both the length and velocity
forms, is of special interest. The oscillator strength information is important to know
the electronic probabilities for both valence and inner-shell excitation and ionization
processes in many areas of application including plasmas, fusion research, lithography
aeronomy, astrophysics, space chemistry and physics, laser development, radiation
a crucial requirement for the development and evaluation of quantum-mechanical
theoretical methods and for the modelling procedures used for various phenomenon
involving electronic transitions induced by energetic radiation [17].
Recently, Tiwary [18] calculated the excitation energies and oscillator strengths, of
both the length (/ L ) and velocity (/ v ) forms, for the inner-shell excitation
Is 2 2s 2 2p 6 3s 2 3p 6 3d 2 D e -+ls 2 2s 2 2p 6 3s 2 3p 5 3d 2 2 P, 2 D 2 F transitions using the
configuration interaction (CI) method. There is a considerable discrepancy between the
CI /L and / v which suggests further detailed calculation. In general, the multiconfigura-
tion Hartree-Fock (MCHF) method yields better results than the CI method. The
MCHF method has, over the years, been refined and generalized to treat a large variety
of systems and it is also a very efficient method for treating correlation.
To test the accuracy of the MCHF method, we have performed the calculation for
the inner-shell excitation Is 2 2s 2 2p 6 3s 2 3p 6 3d 2 D e -> Is 2 2s 2 2p 6 3s 2 3p 5 3d 2 2 P, 2 D 2 F
transitions, employing the same configurations as in our earlier CI method, using the
MCHF method.
2. Method
The most widely used techniques to study the electron correlation in atoms, molecules
and ions are the multiconfiguration Hartree-Fock (MCHF) method [19] and the
closely related, but in general less accurate, configuration interaction (CI) method [20] .
The basic assumption of both methods is that the atom is represented by an atomic
state function (ASF), ^(yLS), which is a linear combination of configuration state
function (CSF), <D(a,.LS)
NCSF
(yLS) = c f ^(iLS). (1)
i = l
Each CSF is constructed as a coupled, antisymmetric sum of products of one particle
functions, $(nl), called spin-orbitals,
(2)
according to the standard notations.
Equations (1) and (2) have two sets of unknowns, the c t coefficients and the radial
functions P nl (r). The difference between the MC and CI methods is now basically just
the way the last set is obtained. In an MC calculation, the variational principle is used
to derive a set of coupled integro-differential (ID) equations, one for each radial
functions, while coefficients are obtained by solving a secular equation of the form
(3)
where the matrix H has the elements
(4)
where H is the Hamiltonian operator.
the present calculation.
2 D
2nO 2)0 2pO
3s 2 3p 6 3d
3s 2 3p 5 3d 2
3s 2 3p 6 4d
3s 2 3p 6 np;n = 4,7
3s 2 3p 4 3d 3
3s 2 3p 6 nf;n = 4,7
3s3p 6 3d 2
3s 2 3p 5 3dns;n=4,5
3s 2 3p 5 3d4f
3s 2 3p 5 3d4d
3s 2 3p 5 3d5f
3s 2 3p 3 3d 4
3s3p 5 3d 3
3s 2 3p 4 3d 2 4f
3s 2 (3p 4 3d 2 ) 1 Snp;n = 4,7
3s 2 3p 5 4fnp;n = 4,7
3s3p 6 3dnp; = 4,7
In CI calculations, on the other hand, the radial functions are predetermined and
also the secular equation is solved for the coefficients.
We have used a nonrelativistic notation throughout, but the same discussion applies
to a relativistic case. The main difference is that the good quantum number is a total J,
and the ASF and CSF should be labeled ^(yJ) and 0(^.7), respectively.
Once the MCHF initial state wavefunction ^ and the final state wavefunction Tf
are determined, with energies E { and E f respectively, they can be used to obtain
absorption oscillator strengths. We have two equivalent forms for the absorption
oscillator strengths.
The length and velocity forms of the electric dipole oscillator strengths, for transition
between initial and final states % and f respectively (assuming e = h = m = 1) are
F, =
*
En
f - 2 J V
Jv >
k=l
(5)
(6)
Important configurations used in the present MCHF calculations are given in
table 1 for 2 D e , 2 P, 2 D and 2 F states.
3. Results and discussion
Tables 2-4 display the Hartree-Fock (HF), our present multi-configuration Hartree-
Fock (MCHF) and our earlier configuration interaction (CI), excitation energies (AE)
as well as optical oscillator strengths (OOS), of both the length (/ L ) and velocity (/ v )
forms, of the inner-shell excitation Is 2 2s 2 2p 6 3s 2 3p 6 3d 2 D e -+ls 2 2s 2 2p 6 3s 2 3p 5 3d 2 2 P,
2 D and 2 F transitions in Cr 5+ , Mn 6+ and Fe 7+ ions of the potassium iso-electronic
sequence.
"">+*
^J- n rs N
IN (N
'*-' C
II II
1 1
' o **
^
rj- m in co en oo
2S
o +
IT-
OOO OOO
ooo
CJ
+
it 0) 0) CS
IN <N
T3
t-.
II II
1 1
in
3 S
IU
ON t" 1 ""* ON O ON ^"
cs o r-
J
03
fe
^
So ^ n in 5^ ro
in S^ en
o
~1
OOO OOO
OOO
X
So ^
r^ rn -, m oo Tf
^t ON ^
S
~
<
O CO OO O ^^ vS
c^ en ON
o
"^
04 (N -H <N M -i
rs <N i-<
Cfl
rCjT3
tf fS (S 1-1
It II
<N -i
i i
S
O I/-*
^
^ VO ON C^ CO i*
OO CN CS
JD &i
^^J^
'O ^ <N OO O O
1
^"^
<d- cS i i vo (N r-l
t>- CS CS
^~l
'ON
OOO OOO
000
\>o
^<en
o
O^
vo
II II
r~- Tf <N i i t~- o
(S ~H
1 1
3
" x Cfl
G
^"v
vo oJ f^ in i ( I-H
*n I-H I*H
-* >
W ^""^
^^
^
oo m 11 **o i~-i cs
v *O 11 CS
S N e
^
OOO OOO
ooo
17
a
_o
(S
Kl
SSI iSS
oo m os
H
O T3
CS CM ^ CM CM -H
CSCS-H
1
S^D,
m *
JD
'o co aj
1 1 1
\ 1
^
, ro cj
^>
ON >n CN oo os i i
ON vo "4" ro in ON
0^ oo
J3
j3
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OOO OOO
000
vS-
"-J2 cS QJ
^_,
o ^ o
'S
T3 r* G
m (S mm
m m
3
c ^ o
1 I II
1 1
_O
Oi PI
Ei^ O *^
*L
U
^
<* i i rn m vo oo
en' ^O oo
O
<1 '^3 "fl3
OOO OOO
ooo
t5
' +S o
q
u '3 .S3
-H
'& S s
^
OS i^- rj- \O OO Tf
CS OS S
5
G 13 '5
^
CS 1-1 i-l CN 1-1 1-1
CS i i i i
<3
--S S
2
C i *^
i
t-H <U
"o
|i
^F5 m^F5
^??
3 "^ * j
x S3 <
"S
pq JG o
c
C/3
o
**^ c
*3
<s .2
ri
tu,
cd
4> o?
ff
ffi
3
X
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rr
K S
1 (
O
W
. -, vo
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~~~^ Q
.2
1 1
r q
S T
CL,
o
'E3b o
<u &
x
w
.<*-*_
Hoo
u
ex
II
II
II
oo
in
IT!
< < rn in
ooo ^ o o 'HOO
o
OO -HOO 'HOO
X
o
o
II II
O\ IS 00 "H Tf O . . _ -
OOO ^H O O r^H O O
o
jo
II I rn \ V 3
co * i in Tt~ m in ctf
oo oo CN ' "
r^- oo i> XT in <N 'sf in rs Q
OO '-HOO i-HOO
&
o
_ ~^r r^i v~t r-t r-< oo
oovorO' lO'H-CNO '^i
in O op a
<N
o
* m m *) m rt
ll'll II
<N oo m
o o o A
'q
ro ro m ot rt (S
II II O
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^ O
o o A o o
jv, OOCJCA O^CN ^Hro-^-
~H rpqpin roopvp rococo
^J V -,..,. , , ,
'o
ffi
lable 4. hxcitation enegies (AE) and optical oscillator strengths (OOIS), ol both
the length (/ L ) and velocity (/ v ) forms, of the inner-shell excitation
Is 2 2s 2 2p 6 3s 2 3p 6 3d 2 D e ^ls 2 2s 2 2p 6 3s 2 3p 5 3d 2 2 F Q transition in Cr 5+ , Mn 6+ ,
Fe 7+ ions of the potassium iso-electronic sequence.
Cr s+ Mn e+ Fe 7 +
3p 5 3d 2 A"' / L f v AE / L / v AE / L / v
HF ( 3 F) 2-128 1-429 0-908 2-333 1-317 0-843 2-533 1-261 0-875
CD) 1-689 0-029 0-029 1-859 0-028 0-028 2-034 0-027 0-029
CG) 1-607 0-012 0-013 1-769 0-011 0-012 1-939 0-011 0-013
MCHF ( 3 F) 2-048 0-772 0-773 2-259 0-786 0-779 2-458 0-756 0-756
CD) 1-705 0-040 0-045 1-873 0-040 0-043 2-037 0-036 0-039
CG) 1-629 0-010 0-012 1-806 0-008 0-010 1-960 0-008 0-009
CI ( 3 F) 2-063 0-774 0-775 2-274 0-787 0-758 2-477 0-757 0-758
CD) 1-708 0-041 0-048 1-881 0-040 0-044 2-051 0-037 0-042
CG) 1-642 0-010 0-013 1-808 0-009 0-011 1-971 0-008 0-011
a) Excitation threshold AE is in atomic unit (au).
Tables 2-4 have several important features. First, we notice that the HF / L and / v
differ by about a factor of two which indicates that the HF description is not adequate
for the determination of the oscillator strengths of the complex inner-shell excitation.
Second, our earlier CI calculation has reduced the disagreement substantially but
considerable discrepancy exists. This suggests further accurate calculation. Third, our
present MCHF / L and / v are in agreement better than our earlier CI / L and / v which
shows that the MCHF method is more accurate than CI method. However, it is also clear
from tables 2-4 that there is disagreement between the MCHF / L and f v . This may be
probably due to the neglect of the effect of relativity. Finally, our extensive investigation
shows that in order to obtain the excellent agreement between / L and / v , it is
indispensable to include the correlation and relativity simultaneously in heavy ions.
4. Conclusion
We have demonstrated that (1) the inclusion of electron correlation is necessary but not
sufficient for obtaining the accurate energy levels and oscillator strengths, (2) the
incorporation of relativity is indispensable for reliable results, (3) the MCHF method is
more accurate than the CI method, and (4) reliable theoretical predictions of energy
levels and oscillator strengths require method that accounts for electron correlation,
relativistic and quantum electrodynamic corrections simultaneously. We hope that our
present theoretical investigation will stimulate interest for more accurate theoretical
predictions and reliable experimental observations.
Acknowledgements
Part of the work was done at ICTP, Trieste when one of the authors (SNT) was there
and he thanks Swedish Agency for Research Cooperation with developing countries for
and Denardo for discussion. UGC, New Delhi, is also acknowledged for the Major
Research Project.
References
[1] B C Stratton, H W Moss, S Suckewer, U Feldman, J F Seely and A K Bhatia, Phys. Rev.
A31, 2534(1985)
[2] Review Articles in Physics of Highly-Ionized Atoms, Vol. 201 of NATO Advanced Study
Institute, Series B: Physics, edited by R Marrus (Plenum, New York, 1989)
[3] R S Peterson, W W Smith, H C Hayden and M Furst, IEEE Trans. Nucl. Sci. NS-28, 11 14
(1981)
[4] R S Peterson, W W Smith, H C Hayden and M Furst, Bull. Am. Phys. Soc. 25, 1 125 (1980)
[5] P Dahl, M Rodbro, G Herman, B Fastrub and M E Rudd, J. Phys. B9, 1581 (1976)
[6] Kh Reazul Karim, M H Chen and B Crasemann, Phys. Rev. A28, 3555 (1983)
[7] S N Tiwary, A E Kingston and A Hibbert, J. Phys. B16 2457 (1983)
[8] A Hibbert, A E Kingston and S N Tiwary, J. Phys. B15, L643 (1982)
[9] S N Tiwary, P G Burke and A E Kingston, XIII I C P E A C, Berlin, West Germany, 1983,
p. 20
[10] S N Tiwary and A P Singh, VI National workshop on atomic and molecular physics, B.H.U.,
Varanasi, p. 139 (Dec. 8-13, 1986)
[11] S N Tiwary, A P Singh, D D Singh and R J Sharma, Can. J. Phys. 66, 405 (1988)
[12] A E Kingston, S N Tiwary and A Hibbert, J. Phys. B20, 3907 (1987)
[13] S N Tiwary, Astrophys. J. 269, 803 (1988)
[14] S N Tiwary, Chem. Phys. Lett. 96, 333 (1983)
[15] S N Tiwary, Astrophys. J. 272, 781 (1983)
[16] S N Tiwary, The Fifteenth Annual Meeting of the DEP 1984 Storrs, Connecticut, USA
[17] Proceedings of Workshop on Electronic and Ionic Collision Cross Sections Needed in
the modelling of Radiation Interactions with Matter, Argonne, Illinois, 1983 (Argonne
National Laboratory, Argonne, 1984) Report No. ANL-84-28
[18] S N Tiwary, Chem. Phys. Lett. 93, 47 (1982)
[19] C Froese Fischer, Comput. Phys. Commun. 64, 399 (1991)
[20] A Hibbert, Comput. Phys. Commun. 9, 141 (1975)
LAM AN A Printed in India Vol. 46, No. 6,
journal of June 1996
physics pp. 389-401
near periodic and quasiperiodic anisotropic layered
edia with arbitrary orientation of optic axis A
imerical study
VIAHALAKSHMI, JOLLY JOSE and S DUTTA GUPTA*
lool of Physics, University of Hyderabad, Hyderabad 500046, India
uthor for correspondence
! received 27 December 1995; revised 18 January 1996
itract. We study numerically the linear properties of periodic and quasiperiodic anisotropic
sred media. Each anisotropic slab can have arbitrary orientation of optic axis. We apply the
eral numerical code to recover the known results for sole filters. We propose novel periodic
ictures where the location and width of the gaps can be controlled easily. We also study the
ismission properties of a Fibonacci sequence of anisotropic layers and show the interesting
;ures like self-similarity and scaling.
'words. Layered media; birefringence
CSNo. 42-25
Introduction
the past few decades there has been considerable interest in layered anisotropic
jctures [1-7]. This is because of the various applications these structures have.
sse include narrow band sole filters, polarizers, etc. The study of anisotropic layered
dia is of utmost importance from an altogether different angle. Most of the second
ler nonlinear materials are anisotropic in character. In order to have a theoretical
lerstanding of second order nonlinear optical processes in layered configuration, it
mperative to know the properties of its linear analogue. The knowledge of the
ward and backward fundamental wave amplitudes in each layer is necessary to
:ulate the source polarization in each layer [5]. Thus the knowledge of linear
>perties is a first step to explore the nonlinear properties.
fhe general theory of anisotropic layered media is well understood [2-7]. It is well-
>wn that the theory for anisotropic layers with arbitrarily oriented optic axis
olves 4x4 matrices and in general a computer program is needed to calculate the
nsmission and reflection coefficients. Our interest in linear anisotropic layered
dium is motivated by its importance in the context of nonlinear studies. We
r eloped a verv general comouter code which can cone with anv number and
revealed the self-similarity and scaling features. To the best of our knowledge, such
studies for uniaxial crystals have not been carried out in the past. All through our paper
we have used the same anisotropic material and just varied the orientation of the optic
axis from layer to layer. Thus, all the gaps and other features reported, in this paper
appear as a consequence of anisotropy.
In 2 we recall the essential mathematical background for the matrix formalism. In
3 we present the numerical results for both periodic and quasiperiodic structures and
in 4 we conclude the paper.
2. Mathematical background
In this section we present the essential mathematical background for calculating the
reflection and transmission coefficients of layered anisotropic media. The theory is
applicable to any (uniaxial or biaxial) anisotropic layers with arbitrary orientation of
optic axis and for arbitrary angle of incidence. It was mentioned earlier that a general
theory is well understood. We will follow Yeh [3] in recalling the essential steps though
there are equivalent methods by other authors [2, 5].
Consider the system shown in figure 1, consisting of N anisotropic layers with plane
of stratification along the xy plane. For any specific jth layer the dielectric tensor? in
the xyz basis can be expressed as follows:
= A
3!
e 2
(1)
where e k (k = 1,2,3) are the components of the tensor for the jth layer along the
principal axes, and A is the general rotation matrix which can be expressed in terms of
the Euler angles 6, cp and i//. In choosing the form of A we stick to the x-convention [13].
It may be noted here that simple rotations in the coordinate planes can be affected as
follows:
yz rotation: by varying 9 with <p = \l/ = 0,
xy rotation: by varying <p with = \\i = 0,
zx rotation: by varying 9 with (p = i// = n/2.
It may be noted here that the method of Yeh [3] can be generalized to include the
description of Faraday rotation and gyrotropic media. However, it lacks the generality
Y 1 2 3 j N
Figure 1. Schematic view of the layered medium.
3 of media.
or fields having the dependence exp i[k Q (<xx + f$y + yz) cat] with k = (o/c, the
e equation takes the following form
kx kx E + /c5a*-E = 0, k = fc (a,0,y). (2)
homogeneity of the layered medium in the xy plane implies that a and ft remain the
e throughout the layered medium. Equation (2) represents a set of three homogcne-
squations with respect to the three cartesian components of E. The nontriviality of
iposes the following condition on the allowed values of y:
o = a ( ** - **, - zz.x-x + + >) + ** V==
+ 2e je ,e yz fi sje - &l x & yy - el y & gz - 2 yz E xx , (4)
a i = 2a(fi jey fi, z - e zx e yj , + a 2 s z j, (5)
2 = a2 fe + a**) - e(e + e y ,) + 4 + e, 2 2 , (6)
3 = 2ea, (7)
a 4 = 2 z- (8)
riting eqs (3)-(8) we have assumed (without any loss of generality) that the wave
:>r k lies on the xz plane (i.e. /? = 0). The four (in general distinct) eigenvalues
n by the solution of (3) leads to the eigenvectors which can be easily determined
i (2). It may be noted here that (5) of Yeh [3], for the eigenvectors is not always
icable, which can be easily verified, for example, for A representing identity
sformation, or coordinate rotation by n/2 about, say x axis. In determining
sigenvectors corresponding to a specific root of (3) we have adopted the following
;edure. We substitute a particular solution for y in the matrix corresponding
J) and construct the adjoint matrix. Next we identify any nonzero element in
adjoint matrix. The corresponding minor is a basis minor which leads to the
ivectors. If all the elements of the adjoint matrix are zeros (i.e. the rank of the
ficient matrix equals one) we refer to the nonzero element of the coefficient matrix.
corresponding components of the eigenvector are zero. The other two com-
;nts are chosen in order to maintain the orthogonality of the ordinary and
lordinary components. Let the electric field eigenvectors be denoted by
= 1, 2, 3, 4). The corresponding magnetic field vectors can be expressed in terms of
; follows.
q ff = (-yp ffy ,yp-ap,ap ffy ). (9)
rms of the eigenvectors the transfer matrix for a particular layer of width d can be
essed as follows:
(10)
I-I1JC JIZJC *O.* ^H-AI
and P is the propagation matrix (4 x 4) having the form
P = diag(e~ iy
, e~ ', e
(12)
The dynamical matrix given by (11) is defined such that it becomes block diagonal
in the absence of mode coupling. Thus the amplitudes of the electric field vector
corresponding to y l and y 2 (Vs an d yj represent the wave of the same polarization.
1.0
0.8
0.6
0.4
0.2
0.0
(b)
irter
137(85
Figure 2. Transmission coefficient T y> , as a function of k Q d (in units of n) for (a)
N m = 4 and (b) N m = 5 for N p = 100, n = 2-3, n e = 2-208, n, = 7 = 2-3.
392
Pramana - J. Phys., Vol. 46, No. 6, June 1996
Arbitrary orientation of optic axis
'act in the absence of damping and evanescent waves, root of (3) are real and occur in
rs with opposite signs, representing the forward and backward propagating plane
ves. We choose yi,y 3 (y 2 ,74.) to represent the forward (backward) waves. The
iracteristic matrix M m for the layered medium with N layers can be written as
M (T) =
(13)
13) superscript) refers to thejfth layer with dielectric tensor 0) and width d (J} . In
.er to relate the amplitudes of the waves in the medium of incidence and in the final
iium one needs to know the corresponding dynamical matrices. The relation
ween the amplitudes can be expressed as follows:
I A,
\
(14)
:re A i+ , A^ and A t+ (B i+ , J3 t -_ , 5, + ) represent the amplitudes corresponding to the
dent, reflected and transmitted x polarized (_y-polarized) wave. We also assumed
t the wave is incident from the left of the structure. Superscript i(t) in the dynamical
:rices refers to the medium of incidence (transmission). With the help of (14) four
2s of transmission coefficients can be defined:
T
*~
T =
l xy
(15)
(16)
0.1'74 ' 0.476 ' 0.-/78 ' 0.^80 ' 0.-/82
T ~
yx
B s .
B.
(17)
(18)
The various reflection coefficients can be defined in an analogous manner.
3. Numerical results and discussion
In this section, we present the results of our numerical investigations of periodic and
quasiperiodic layered structures. A general Fortran code was developed along the lines
discussed in the previous section which can deal with any number and sequence of
anisotropic layers with arbitrary orientation of the optic axis and for arbitrary angle of
incidence. In all our calculations we have chosen the same anisotropic medium and
only varied the orientation of the optic axis from layer to layer. It is well-known that
a periodic variation in the orientation of optic axis along the plane of stratification can
lead to well defined band gaps. This has already been exploited to create very narrow
10.4
10.6
10.8
k d
11.0
11.2
11.4
1.0-
0.8-
0.6-
0.4-
0.2-
0.0-
11.4
10.4 10.6 10.8 11.0 11.2 11.4
Figure 4b.
Figure 4. Transmission coefficient T yy and T xy as a function of k d (in units of n) for
a sole filter for (a) N p 50 and (b) N p = 100. Other parameters are as follows:
n Q = 2-3, n e = 2-208, n. = n f = 2-3, e i = e 3 = n* and s 2 = n^.
band sole filters. However, to the best of our knowledge, the case where the orientation
of the optic axis is changed on the plane of incidence in a periodic fashion has not been
dealt with in sufficient detail. Thus, in dealing with periodic structures we consider
two cases, namely, (a) optic axis on the yz plane and (b) optic axis on xy plane. In
the context of quasiperiodic structures also, we deal with the above two cases though
our code can handle any other orientation. In all our calculations we have chosen
the anisotropic material to be uniaxial and we have considered normal incidence (i.e.
a = 0). In what follows we present the results separately for periodic and quasiperiodic
structures.
3.1 Periodic anisotropic layered media
(a) Optic axis on the yz plane: In this case we take 1 = 2 = n^ and 3 = n*. It is clear
that, with optic axis on the yz plane the ordinary waves will have x-polarization and
there is no polarization mixing (i.e. T xy = T yx = 0). So far as the ordinary waves are
v ManaiaKsnmi ei ai
concerned, the layered medium will act as uniform slab having the total width of the
structure and the ordinary refractive index n . Since there is no coupling between the
x and y polarized waves and the matrix M (T) has block diagonal structure, one could
have a 2 x 2 matrix formalism for this particular case. One may be tempted to think
that in the context of the extraordinary waves one can replace the anisotropic layers by
isotropic ones with 9 dependent refractive indices. This approach is not correct since
the eigenvectors as well as their projection on the plane of layers depend on 9. We
consider a layered medium consisting of N p stacks. Each stack is assumed to have N m
layers each with width d. In a y'th layer the Euler angles are given by
(19)
>, = 0..sin
(/-I).
AL '
(20)
Equation (19) corresponds to the situation when the optic axis is rotated through 2n
in each stack. Obviously, the period of the structure will depend on the value of N m .
1.0 -,
0.8-
>.0.6-
>,
" 0.4-
0.2-
0.0
1.0-
0.8-
0.4-
0.2-
0.0-
1.0-
0.6-
0.4-
0.2-
0.0-
n fib = 9
1.0
1.2
1.4
1.6
"fib =12
1.0 1.2
n fib= 15
(a)
1.8
2.0
.6 18
r ' i i i i i i i 1 1
1.0 1.2 1.4 1.6 1.8 2.0
k o n o d B
0.6-
1.46 1.48 1.50 1.52 1.54
1.490 1.495 1.500 1.505 1.510
Mod B
Figure 5b.
Figure 5. (a) Transmission coefficient T yy as a function of k n d B (in units of n) for
the Fibonacci sequence for w f . b = 9, 12 and 15 (top to bottom) and (b) expanded
version of (a). Other parameters are as follows: n = 2-3, n e = 2-208, n ! = n f = 2-3,
Equation (20) corresponds to the situation where is varied in a periodic fashion with
amplitude of modulation given by Q m . The purpose of this special case is to show that
the bandwidth of the gap can be controlled by varying the depth of modulation B m . The
results of (19) are shown in figure 2a (for N m = 4 and N p = 100) and in figure 2b (for
N m = 5 and N p - 100), where we have plotted T yy as a function of fe d. Other
parameters were chosen as follows: n = 2-3, n e = 2-208 (corresponding to LiNbO 3 at
A = 0-633 fim), n t = n f = 2-3. It is clear that for 7V m = 4 a period consists of effectively two
slabs and the gaps occur at [3],
k d(n
= mn, m integer.
(21)
With the increasing number of periods the gap becomes more well defined. The
location of the gap can be controlled by changing N m . The results for N m -5 and
different values of m , namely O m Q-2n and Q-4n. It is clear from figure 3 that we can
double the gap width by doubling the depth of modulation.
(b) Optic axis on xy plane: The main purpose of this subsection is to show the
applicability of our code to recover known results. We take e i = & 3 = n^ and 8 2 n^.
Note that we have taken the optic axis along the y axis (in the last section the optics axis
was along z axis). This is just to enable us to affect the rotations on the relevant planes in
a direct fashion, xy rotation is affected by varying cp from layer to layer and keeping
= \l/ = 0. The structure is assumed to consist of N p periods. Each period (labeled by
A and B) having two slabs (with equal width d and with (p a + 6 and cp h = 6), where
n/($N p ) such that the Bragg-solc condition is satisfied. The sole resonances occur
[3] at
k Q d(n n e ) = mn, m-integer
(22)
and one of these resonances are shown in figure 4a (N p = 50) and 4b (N p = 100). The
following parameters were chosen for calculations: n = 2-3, n e = 2-208, n { = n f = 2-3. We
have plotted T yy and T xy as functions of k Q d. The curves for T xx and T yx are analogous.
As expected for sole filters, at sole resonances the conversion ofx polarization to y and
1.0-
0.8-
0.6-
0.4-
0.2-
0.0-
(a)
10
15
20
25
1.0-
0.8-
0.6-
0.4-
0.2-
0.0-
~l >-
5
15
_, j-
20
1.0-
0.6-
0.4-
0.2-
0.0-
10
~20~
~25
k d
Figure 6a.
398
Pramana - J. Phys., Vol. 46, No. 6, June 1996
Figure 6b.
Figure 6. Transmission coefficient (a) T yy and (b) 7^ for n fib = 9, 12 and 15 (curve
from top to bottom) for the Fibonacci sequence. Other parameters are as follows
1 = e 3 = n and 2 = n e .
versa is maximized. Moreover, one can drastically narrow down the band widtl
with an increase in the number of periods (compare figures 4a and 4b).
3.2 Quasiperiodic anisotropic layered medium
We consider two types of anisotropic layers, namely A and B, with widths d A and d
respectively, and Euler angles A ,<p A ,A A and B ,<p B ,iA B , respectively. The Fibonacc
sequence and the Fibonacci numbers are generated using the following recursion schemf
C CCF- F4-F (Jl
li j+i~~' i ./-i^j' / j+i~ j-i^ j \
with S Q = A,S 1 = B and F = F^ = 1. As mentioned earlier such layered media cai
exhibit interesting features like self similarity and scaling and they have been probed t
explore weak localization in optics [8, 9]. Like in the periodic case, we consider the tw
specific situations and present the results separately.
(a) Optic axis on the yz plane: For this case we take e, = 2 = n and e 3 = n 2 e . 6 A = n/'.
(p A = ijs A = Q and 6 B Q,(p s = \l/ B = 0, n e d A = n d B . The numerical values were chose
12 and 15 are shown in figure 5a. A comparison of the plots on figure 5a reveals their
self similarity. In order to have a more direct proof of the same, we have selected specific
portions and expanded the horizontal axis and shown the same curves in figure 5b.
Note the different scales of the horizontal axis. The self similarity features are obvious
from figure 5b.
(b) Optic axis on the xy plane: We take e]i=s 3 = no and & 2 = nl, A = tJ/ A = Q,
(p A = - <5, B = \I/ B - 0, cp B = S. The results for T yy (T xy ) for n fib = 9, 12 and 15 are shown
in figure 6a (figure 6b). The parameters chosen for calculations were as follows:
n = 2-3, n e = 2-208, n { n f 2-3, b = 0-1. It can be easily seen from figures 6a and 6b
that the curves corresponding to n fib = 9 and 15 are self similar. Like in Kohmoto and
Sutherland 1987, for the chosen geometry and set of parameters the corresponding
dynamical map has a six-cycle and thus features repeat after six generations. It may also
be noted that the scale factor is very close to one. This is because we have chosen the
constituent slabs to be of the same material and a change in the orientation of the optic
axis in the xy plane by an angle 0-1 leads to a weakly quasiperiodic medium. In order to
prove this, one needs to calculate the invariants associated with the dynamical matrix
map, which can be treated as a measure of quasiperiodicity. Investigation are on to
study these features in more detail and they will be reported elsewhere.
4. Conclusions
In conclusion, we have developed a very general code to deal with anisotropic layered
media and applied the code to various periodic and quasiperiodic layered structures. In
particular, we have dealt with the case where the optic axis is rotated on the plane of
incidence and showed that the location and width of the gaps can be controlled simply
by changing the depth of modulation of the angle of rotation. We have also investigated
quasiperiodic layered media with optic axis on the plane of incidence or on the plane of
stratification. In both the cases we have demonstrated the self similarity and scaling in
the transmission coefficients. In this paper we restricted ourselves only to normal
incidence though our code can also handle cases of oblique incidence. It is well-known
that in case of oblique incidence one can excite the guided and surface modes which are
of great practical importance. Such studies are underway and will be reported
elsewhere.
Acknowledgements
The authors (SDG and VML) are grateful to the Department of Science and Technol-
ogy, Government of India, for supporting this work.
References
[1] D A Holmes and D L Feucht, J. Opt. Soc. Am. 56, 1763 (1966)
[2] D W Berreman, J. Opt. Soc. Am. B62, 502 (1972)
[3] P Yeh, J. Opt. Soc. Am. 69, 742 (1979)
[4] P Yeh, Optical waves in layered media (New York, Wiley, 1988)
[5] D S Bethune, J. Opt. Soc. Am. B6, 910 (1989)
[6] H L Ong and R B Meyer, J. Opt. Soc. Am. 73, 167 (1983)
[7] G Joly and N Isaret, J. Opt. (Paris) 17, 21 1 (1986)
[8] M Kohmoto, B Sutherland and K Iguchi, Phys. Rev. Lett. 58, 2436 (1987)
[9] S Dutta Gupta, Recent developments in quantum optics, edited by R Inguva (Plenum,
New York, 1993) p. 15
[10] S Dutta Gupta and D S Ray, Phys. Rev. B40, 10604 (1989)
[1 1] S Dutta Gupta and D S Ray, Phys. Rev. B41, 8047 (1990)
[12] G S Ranganath and Y Sah, Opt. Commun. 114, 18 (1995)
[13] H Goldstein, Classical mechanics (Narosa Publishing House, New Delhi, 1985) p. 146
PRAMANA Printed in India Vol. 46, No. 6,
journal of June 1996
physics pp. 403-410
Optimal barrier subdivision for Kramers' escape rate
MULUGETA BEKELE 1 -*, G ANANTHAKRISHNA 2 and N KUMAR 3 ' 1
*On leave from: Department of Physics, Addis Ababa University, P.O. Box 1176, Addis Ababa,
Ethiopia
Department of Physics and 2 Materials Research Centre, Indian Institute of Science, Bangalore
560 01 2, India
3 Raman Research Institute, C.V. Raman Avenue, Bangalore 560080, India
MS received 1 March 1996; revised 12 April 1996
Abstract. We examine the effect of subdividing the potential barrier along the reaction
coordinate on Kramers' escape rate for a model potential. Using the known supersymmetric
potential approach, we show the existence of an optimal number of subdivisions that maximizes
the rate.
Keywords. Kramers problem; activated processes; reaction rates.
PACS Nos 05-70; 31-70; 87-15
1. Introduction
The problem of surmounting a potential or, more generally, a free energy barrier is
a classical one that appears in all processes having thermally activated kinetics. This
problem was originally addressed by Kramers in the context of a bistable potential
energy curve [1]. He provided an approximate solution for the rate of escape over the
barrier in a high barrier low noise limit. In the commonly encountered high friction
limit, the bistable potential is usually parameterized in terms of the height of the barrier
at the potential maximum and the width of, or the distance under the barrier
connecting the initial and the final states (potential minima). Since Kramers' original
work, there has been a number of refinements as well as varied novel applications of his
solution, and a large volume of literature exists on this [2, 3].
There are, however, situations where the initial and the final states are separated by
a barrier which is so high that the estimated reaction rate is very small, and yet the
reaction actually turns out to proceed at a substantially higher rate. The enhancement
is attributed to the catalytic action, notably of an enzyme in a biochemical reaction that
forms a 'transition state' complex with the substrate giving a reduced barrier height [4].
We, however, envisage here an alternative scenario where the enzyme effectively
reduces the activation energy by subdividing the reaction path into a number of
discrete steps each requiring a much smaller barrier crossing. These subdivisions are
expected to correspond to the discrete conformational/configurational changes of the
macromolecules. proteins say. Besides looking for a physical consequence of the barrier
suoaivision on me reaction rate in me nign iricuon nmii. mis we nave aone ror
a W-s.haped model potential barrier whose subdivision can be well parameterized. Our
analysis of the problem is based on the supersymmetric potential technique [5-7].
2. The methodology
It is sufficient for our purpose to note that in the high friction limit, Kramers' escape
problem is one of solving the Smoluchowski equation (SE):
where P(x, t) is the probability density associated with the particle position,
U'(x) = dU/dx with U(x) being the 'double well'-potential, D the diffusion constant and
/J = (kT)~ l is the inverse temperature. With the ansatz
P(x,t) = (l)(x)Q- pU(x)/2 Q- it ) (2)
the SE is converted to an Euclidean Schrodinger equation for </>:
H + <K=E + <K (3)
with H + A + A being positive semi-definite, where E + = A/D and
This Hamiltonian H + corresponds to the motion of a particle in the potential
U"(x). (6)
For a high barrier, the escape rate is determined by the smallest nonzero eigenvalue,
A 1 = DE Ij. , of the SE where E + is the eigenvalue of the first excited state of eq. (3). O n the
other hand, this eigenstate is degenerate with the ground state of the 'supersymmet-
ric partner potential' V^ (x) given by
^U"(x) (7)
*
so that H_ (j>_ = E_ 0_ with H_ = AA + and _ = j. . The problem thus boils down
to finding the ground state eigenvalue of this 'partner' potential.
3. The model and its solution
3.1 The model potential and parameterization of subdivision
For simplicity we consider a symmetric W-potential. For a full characterization of this
potential we require two parameters, namely, the height 17 and the width under the
(a)
-LO x. 2
(c)
i
1
1
1
i
I
1
i
1
J
1
1
J
i
i
... j
Figure 1. (a) The model potential, (b) Plot of the subdivided model potential: U(x)
versus x, when N 3. Note the change in the slope of the left- and right-confining
walls from that of (a), (c) Plot of V_(x) versus x (not to scale), (d) A figure showing
a step of the subdivided potential found between the intervals x n and x a+ 1 .
potential 2L (see figure la). We now subdivide the barrier between the initial and final
states into a series of smaller connecting barriers of many steps (see figure Ib). In order
to examine the effect of barrier subdivision on the reaction rate systematically, it is
necessary to parameterize the subdivision in a physically, reasonable manner. Consider
the step located between x n and x n+l (figure Id). We choose U lt U 2 and the associated
widths a, b (shown in the figure) such that U i /a = U 2 /b. This choice simplifies the
calculation further. Note that x n+ 1 x n = a + b. If we have a total of N such equally
spaced steps from the top of the barrier on either side, then L = Na + (N l)b, while
U = NU l (N~l)U 2 .We introduce a parameter p defined as
p =
(N- 1)17 2 (N-l)b
NU t
Na
(8)
The aim is, given 17 and L , to find the escape rate for different values of barriei
subdivision consistent with the high barrier limit, i.e. for various values of N. Such
parameterization is physically reasonable as it not only keeps the barrier height and its
width fixed, it also keeps the area under the barrier approximately constant as N ii
which for the considered potential takes the form
where u l =j^U l . Note that the potential V_ (x) is a series of attractive and repulsive
delta-potentials superimposed over a constant potential (see figure Ic). Changing the
variable x to y = x/Na leads to a new Hamiltonian h. given by
d 2 + N
/ 3 _^H_ = -~ + ( w ?) 2 -2 U ? [%-)>)-% ->>-,)], (11)
oy n =-N
where a Q = Na t u = Nu lt a t = l/N, b l = p/(N - 1) and y M = w^ + ^ ). With this,
h _ ^? (r) = e_ </> (>'), where <;_ is a dimensionless quantity equal to o_
3.2 Solution
We use transfer matrix method to find the ground state energy. The ground state wave
function <j)_ is of the form /le~* y + Be ky peaked around the positions of the delta
potentials. Consider one period of the potential, say, the interval between y n and y n+l .
Assume the wave function of the form
r/>j(y) = A,^"^'-^ + B n Q k(y ~ y -\ (12)
for the interval y,,_ l + a l ^ y ^ y n and the wave function of the form
<M>') = C n e- fc() - J '- fll) + D B e* (J '- J ''- fl ' ) (13)
for the interval y n ^y^y n + a l with fc = [(?) 2 e_] 1/2 . By matching the wave
function at >, i.e. ^(y,,) = 2 (>')> an< ^ tne integrating eq. (11) around y n noting that
there is a negative delta-potential of strength 2w, i.e. $\(y n ) $'2(9 ) 2?^> t (y fl ) = 0,
we get
_
'U/
relating the two pairs of amplitudes (a = /fc). Next, matching the wave function
having amplitudes C n and D n with the wave function having amplitudes A n+i and
B n+ 1 (found in the interval y n + a l ^ y ^ y n+ 1 ) and integrating eq. (1 1 ) around y n + a l
noting that a positive delta-potential of strength 2u is located there, we get
c
The transfer matrix, T, relating the amplitudes y4 ?J + j , B M + , to the amplitudes A n and B n ,
406 Pramana - J. Phys., Vol. 46, No. 6, June 1996
will then be a product of the matrices T, and T 2 , i.e., T = T 2 T, . The amplitudes jus
before the end of the AMh step A v . B N are related to the amplitudes A Q , B at the top lei
side of the barrier by a product of W of these transfer matrices
TPA'
Symmetry of the potential about y = implies that the ground state wave function i
symmetric, i.e., ( y) = </>?.(y). Using matching and integration at and around th<
origin where there is a negative delta-potential with the symmetry property of the wavi
function relates A Q and J? ,
I + a
Since we are concerned with a bound state solution, B N = 0. Using this and eq. (18) ir
eq. (17) enables us to get
(l-a)(T%+(l+a)(T N ) 22 = 0, (19;
the lowest positive solution of which gives us the value of e_ when M O , p and N are
specified. The expressions for the matrix elements (T N ) 21 and (T N ) 22 are given in the
Appendix.
4. Results and discussion
Now we consider the solution of eq. (19). The result could be better appreciated if we
compare it with the corresponding escape rate for the original W-potential with nc
barrier subdivision (figure la). Applying the same technique as above, the equation
-corresponding to eq. (19) to be solved is
a 4-(l-a )e 2fc <> = 0, (20;
where a = w / fc o' /c = [^-e_] 1/2 with U = \PU . In this case, e_ = L 2 _. The
inverse of DE_ is the time required to go from one minimum to the other in figure(lal
and we define the corresponding escape rate as DE_ for the original 'W potential. Then
the ratio, f N , of the escape rate, DE_, over the potential with a certain barriei
subdivision to that of escape rate, DE_ , over the original W-potential is given by
We call this ratio, / N , as the enhancement factor. It may be worthwhile pointing ou
here that we have used the first passage time from one minimum to the other in th<
original potential as a scale factor. This is because the subdivided potential is rugged 01
the 'down hill part 1 as well, which could give rise to a considerably different transit timi
compared to the situation if only 'sliding down 1 on a smooth line were allowed.
There are only two parameters in our model, namely u , which is the total barrie
height and p, which essentially represents the steepness of the local barriers. We chos
u (so as to be in the high barrier limit) holding p fixed and explored the enhancemen
30
20
10
U -12.0
10 15 20 25
N
Figure 2. Plots of/ w versus N for three different values of u (i.e. 6-0, 9-0, 12-0) with
fixed p(= 0-8). Note that all the optimal values occur at N = 9.
40
30
20
10
u =9.0
X XX x ...
: X P-^'x
* p-.8
p-.4
5 10 15 20 25 30
N
Figure 3. Plots of/ N versus N for four different values of p (i.e. 04, 0-6, 0-8, 0-9) with
fixed M O (= 9-0).
factor, / N , for various values of barrier subdivision, N. Figure 2 shows plots of f N versus
N for three different choices of w with fixed p ( 0:8). For this case, the enhancement
factor at the optimal barrier subdivision, N op , increases as w is increased reaching
a value as high as 35 for u = 12-0, while N op remains constant (here 9) suggesting that
408
Pramana - J. Phvs., Vol. 46, No. 6, June 1996
values of p. This is due to the fact that for these values of p, the enhancement factor :
less than unity for low values of N and correspond to the situation where U 2 > U 1 .]
We have verified that these trends are general. Thus, there is an optimal value c
barrier subdivision, JV op , at which the escape rate takes a maximum value. Th
existence of N op may be readily understood by a reference to the potential K_ (x). Th
binding energy of this localized ground state of the individual negative energ
delta-functions and its lowering due to mutual overlap of the neighbouring boun<
states (banding effect) are oppositely affected by N.
It may be worthwhile to mention here that in addition to changing the terrain (sa;
steepness) of the intermediate barrier (connecting the initial and final states) by tin
subdivision, the outer barriers were also made to change their steepness accordingly fo
the sake of simplicity (see figures la and Ib). Because of this increased steepness, the;
become more confining than with their original slope and, thus, give rise to ai
overestimation of the enhancement factor. We have verified this by retaining tin
original slope of the outer barriers. However, the main features remain the same. On th<
other hand, due to the monotonic increase of the optimal enhancement factor with th<
barrier height, its value could be much larger than the ones considered here for barriei
heights that exist in chemical and physical processes.
We remark that this problem can be viewed, approximately, as that of finding the
mean first passage time of a biased random walk [8]. However, this would implj
assigning values to the forward and the backward transition rates for the individua
sub-barriers taken in isolation as input, i.e. assuming that the potentials to the left anc
right sides of each sub-barrier are totally confining, and then using these input values tc
calculate the global escape rate for the coupled sub-barriers. We have found that while
this gives an optimal barrier subdivision for the escape rate consistent with the present
result, the enhancement factor is considerably over-estimated by this random walk
approach. The present SUSY-based calculation goes beyond this uncontrolled ap-
proximation.
It would be interesting to examine and optimize the effect of an athermal (possibl)
colored) noise (the 'blow torch' of Landauer [9,10]) on one of the steps of oui
subdivided potential curve. This is under investigation.
In conclusion, we have shown that the Kramers' rate for the escape over a giver
potential barrier, in the high barrier high friction limit, can be substantially enhancec
by subdividing the barrier optimally. This might provide an alternative scenario foi
certain activated processes where the measured escape rate is substantially higher thar
that anticipated.
Appendix A
To find the elements of the matrix T N we decompose the transfer matrix T as a produc
of three matrices, i.e.
T-RAL (A3
Pramana - J. Phys., Vol. 46, No. 6, June 1996 40
matrices L and R are, respectively, made up of the left- and right-eigenvectors of T such
that LR = RL = I. With this decomposition,
TN RA^r (&,')}
KA JL ()
whose two elements of our interest, (T N ) 2} and (T N ) 22 are expressed as
* ' '"^ :N 1 (A3)
v 21 ~ Q
and
V1 ;22 ~ 2(2
/. are the eigenvalues of T given by
2 ll 22-^
with Ty as the matrix elements of T and Q = [(7^ , - T 22 ) 2 + 4T 12 T 2 , ] 1 /2 .
Acknowledgements
The authors would like to thank Dr A M Jayanavar for bringing ref. [7] to our
attention. One of the authors (MB) would like to thank The International Program in
Physical Sciences, Uppsala University, Sweden for financial support for this work. He
would also like to acknowledge Dr M P Joy for his technical advice on computation.
References
[1] H A Kramers, Physica 7, 284 (1940)
[2] P Hanggi, P Talkner and M Borkovee, Rev. Mod. Phys. 62, 251 (1990)
G Fleming and P Hanggi (eds), Activated barrier crossing: Applications in physics, chemistry
and bioloyy (World Scientific, Singapore. 1993)
[3] V 1 Mel'nikov, Phys. Rep. 209, 1 (1991)
[4] See example, Lubert Stryer, Biochemistry. 2nd edn. (W.H. Freeman and Co.. New York,
1980) Ch. 6
[5] M Bernstein and L S Brown, Phys. Rev. Lett. 52, 1933 (1984)
[6] H R Jauslin. J. Phys. A21, 2337 (1988)
[7] K Schonhammer, Z. Phys. B78, 63 (1990)
[8] I Gefen and Y Goldhirsch, Phvs. Rev. A35, 1317 (1987)
[9] R Landauer, J. Stat. Phys. 53, 233 (1988)
[10] N G van Kampen, IBM J. Res. Dev. 32, 107 (1988)
Ratios of B and D meson decay constants with heavy
quark symmetry
A K GIRI, L MAHARANA and R MOHANTA
Physics Department, Utkal University, Bhubaneswar 751 004, India
MS received 20 February 1996
Abstract. SU (3) flavor symmetry allows the decay constants f D and / Dj as well as //, and j' Rj t<
be equal. But due to 5(7(3) flavor symmetry breaking the ratios' f B Jf Bj and f D jf Di are deviatec
from unity. We have estimated these ratios in the heavy quark effective theory and obtainec
A/A = 0-93, f D Jf Di = 0-94 and the double ratio (A./A)/(A/A) = 0>99 -
Keywords. Decay constants; heavy quark effective theory.
PACS Nos 11-30; 13-20; 14-40
1. Introduction
In recent years, considerable progress has been made towards a QCD based and mode
independent description of hadrons containing a heavy quark, the so-calJed heavj
quark effective theory [1-4]. This progress has been achieved by assuming an infinite
mass limit for the heavy quark and in this limit, two new symmetries beyond those
usually associated with QCD arise. These two symmetries are the spin symmetry anc
the flavor symmetry. The effective theory of hadrons containing a single heavy quart
has been applied to various areas of phenomenology [5-11] and successful results have
been obtained for bottom quark and charm quark systems. The idea of the heavy quark
symmetry is considered to be effective only for heavy quarks whose masses m Q be
significantly larger than QCD scale A QCD . Thus the hadrons containing oquark and/oi
6-quark may provide laboratory to test the heavy quark effective theory. The pseudo-
scalar decay constant is one of the first physical quantities studied in the context o
heavy quark effective theory. The decay constants for D d and D s mesons are denoted b}
f D and f D , respectively and are equal in the chiral symmetry limit where the up, dowi
and strange quark masses go to zero and 5(7(3) flavor symmetry for the light quarks i;
an exact symmetry implying f D Jf Dj = 1 and analogous relations for B meson system
i.e. f B /f B = 1. However in nature, the quark masses m q ^0, hence the chiral SU(3
symmetry is broken, its effects on the B and D meson systems are observed by their mas;
differences [12]
m B m Bi ~ m D m Dj ~ 100 MeV. ( I
Due to the symmetry breaking effect the ratios f B Jf Bi and f D jf Dj deviate from unit)
These ratios play a significant role in the possibility of constraining the Cabibbo
Kobayashi-Maskawa matrix. This can be seen from the fact that within the standar
41
A K Giri et al
model, the mixing between B s and B s occurs with the parameter x s = (AM/F) Bj
given by [13]
Y L r m 2 (f 2 R )n I V* V I 2 F(m 2 /m 2 } f?1
s ~ 2 B,W\J B,B i )'l8,\ y is v ib\ r \. m t/ m w)- \ z ->
Equation (2) shows that the ratio x s /x d is independent of the top quark mass m t , the
experimental determination of the ratios implies the ratio | V ls /V td \ is known, once
(fs^ajfl^s) an d tsJ^Bj h ave Deen calculated or known. Several investigations have
been done for evaluation of the B and D meson decay constant ratios following various
approaches. A summary of earlier studies can be found in ref. [14]. Lattice calculation
indicates that the ratio can be calculated much more accurately than either f Bs or
f Bj due to cancellation of some systematic uncertainties and lattice calculation of
Bernard et al [15] yield f B Jf Bj ^ f D Jf Da ^ H. Using QCD sum rules Dominguez [16]
has shown that f B Jf B<i = 1-22 and f D Jf Di =1-21. Incorporating heavy quark and chiral
perturbation theory, Grinstein et al [17] obtained f B Jf Bt = 1- 14 and f D Jf Dj = H . To be
more specific the double ratio (f B Jf B )l(f D Jf D ) is very close to unity in all the above
calculations as small corrections to both numerator and denominator are cancelled. In
a recent letter, Oakes [18], using chiral symmetry and basic quantum mechanical
arguments, has shown the double ratio to be 1-004.
In this investigation we calculate the ratios f B /f B and f D /f D in heavy quark effective
theory and show that the double ratio is very close to unity. Since we have used the
basic assumptions of HQET, our results are independent of any model-dependent
parameters and depends only on the quark and hadron masses. Hence, the only
uncertainties in our calculations are due to the uncertainties present in the quark
mass terms.
2. Theory
A convenient framework for systematically analyzing the heavy particle systems is
provided by the so-called heavy quark effective theory developed by Georgi [3]. The
basic observation is that as the mass of the heavy quark W Q - oo, its velocity v becomes
a conserved quantity with respect to soft processes. Hence in HQET the effective heavy
quark field h$(x) is related to the original field Q(x) by
, (3)
and is constrained to satisfy the relation
#?(*)- *?(x). (4)
Before turning to the detailed calculations it is important to understand what HQET
tells us about the decay constants, since it provides the only results that follow directly
from QCD. At leading order in the effective theory the decay constant follow the scaling
In the effective theory the decay constants for the D d and D s mesons are defined in terrr
of hadronic matrix elements as
where q = (5, d) for D s and D d mesons respectively. The decay constants for B meson
are defined by equations analogous to (6). These decay constants are equal in th
SU(3) flavor symmetry limit. In reality the QCD hamiltonian contains a quarl
mass term J#*(x) = 'L i m i q i (x)q i (x), which breaks the symmetry. Therefore the axia
vector current is not exactly conserved and the divergence of the axial current cai
be given as
The current in the full theory can be expanded in terms of the operators in thi
effective theory as [19]
rQ = cmn, (8
where C(^) is the short distance coefficient which depends on the renormalization scal<
ILL and at leading order C(n) = 1. Using (8) the modified (7) in the effective theory is giver
as
dfy 5 h c ) = i(m e + m)qj 5 h< . (9
To obtain the ratio of the decay constants we have to evaluate the matrix elements o
the operator present in the above equation. Matrix elements of the operator on the l.h.s
of (9) can be most concisely computed employing a compact trace formalism. For the
relevant matrix elements of the operator one can write from ref. [18]
where F(^) is the scale dependent low energy parameter independent of m Q , denotes the
asymptotic value of the scaled decay constant given as jp(^) = <Jm D f D [20], ^ is the
scale at which the effective current is renormalized. A is a parameter characterizing the
properties of the light degrees of freedom defined as [19]
A = m JD m c , (11
and
denotes the spin wave function of the D q meson [20].
Using (11) and (12) we obtain from (10) that
Using (9) and (13) one can easily obtain the ratio of the decay constants for D meson a
f n m,
To evaluate the matrix elements consistent with Lorentz invariance and heavy quark
spin symmetry, we introduce the interpolating fields for the heavy mesons as
where q v is a light antiquark which combines with a heavy quark h t . of velocity u to form
the appropriate meson. The current given in the matrix element is the interpolating
current with the quantum numbers of the heavy meson, which can induce the
generation of a heavy meson out of vacuum. Hence we can immediately obtain
(17)
where M = (Q\q v q v \Qy is a 4 x 4 matrix [21] and we have used the heavy quark
propagator as <0|/iJ?/jj?|0> = (1 + ^)/2. Lorentz invariance implies that
M = A(v 2 )I + B(v 2 )?, (18)
where A(v 2 } and B(v 2 ) are functions of the scalar variable v 2 . Since v 2 1 the functions
4(1) = A and B(l) = B(say), (19)
are universal constants. Substituting the value of M from (18) in (17) we obtain
(20)
Using (20) we obtain from (14) and (1 5) the exact expressions for the ratios of the decay
constants as
' D, _ I'-'Uj I -"V, "V if c ' "'s \ /2J\
and
fa, l m Bj m B-^\fm h + m s
, \ n i- < 22 )
JB< ^ m B,\ m ^- m bJ\m h + m df
We take the current quark masses as m rf =10MeV, m s = 150MeV, m c = l-3GeV,
m b = 4-3 GeV and the masses of B and D mesons are M Dj = 1 869 MeV, M D = 1968-5 MeV,
M B = 5375 MeV and M Bj = 5279 MeV from ref. [12]. With these values we obtain the
ratios to be
^ = 0-93 (23)
and
'^ = 0-94. (24)
JD<
The double ratio is given as
U' B Jf B ,,W D J.f D } = v-99. (2:
3. Discussion
We have tried to calculate the ratio of B and D meson decay constants using heav
quark effective theory. Considering the SU(3) flavor symmetry breaking in the ligh
quark sector, we obtain the exact expressions for the ratios of the decay constants. Th
matrix element contained in the expression is calculated in HQET, consistent with th
Lorentz invariance and heavy quark spin symmetry. Thus we obtain the ratios in term
of quark and hadron masses and substitution of the masses yield the result
f B Jf Bj = '93 and f D Jf Dj = 0-94 and the double ratio (f B Jf B )/U' D jf D ) = 0-99. It has beei
argued by Grinstein [22] using both heavy quark and chiral symmetry that the doubli
ratio be equal to unity with sizable corrections for the light and heavy quark sector. Thi
double ratio, in our case is very close to unity with a correction factor of 1 %. Sino
a direct measurement of f B through the leptonic decay will be extremely challenging
because of the very small branching ratio and difficult signature, a measurement of f,
is much more feasible, so a precise measurement of f D Jf Dj will determine the value o
fajfas which is a factor in determining the relative strengths of B s B s and B d B t
mixing. These mixings give valuable information on the elements of Cabibbo-
Kobayashi-Maskawa matrix, i.e. from the measured value of both these mixings on*
can extract | VJV^ d \ from their ratio.
Acknowledgements
We are thankful to Profs B B Deo, A Das and N Barik for useful discussions. One of the
authors (RM) would like to thank CSIR, Government of India, for a fellowship.
References
[1] N Isgur and M B Wise. Phvs. Lett. B232, 1 13 (1989); 237, 527 (1990)
[2] E Eichlen and B Hill. Phys. Lett. B234, 51 1 (1990)
B Grinstein, Nncl. Phys. 239, 253 (1990)
[3] H Georgi, P/JV.V. Lett. B240, 447 (1990)
[4] A Falk, H Georgi, B Grinstein and M B Wise, Nucl. Phvs. B343, 1 (1990)
[5] M E Luke, Phvs. Lett. B252, 447 (1990)
[6] T Mannel, W Robert and Z Ryzak, Phvs. Lett. B254,. 274 (1990); B259, 359 (1991)
[7] J L Rosner, P/IV.V. Ret: D42, 3732 (1990)
[8] M Tanimoto, Phvs. Rev. D34, 1449 (1991)
[9] M Neubert, Phys. Rev. B264, 455 (1991)
[10] Z Hioki, T Hasuike, T Hattori, T Hayashi and S Wakaizumi, PA vs. Lett. B299, 1 15 (1993
[11] U Aglietti, Phys. Lett. B281, 341 (1992)
[12] Review of Particle Properties, Phys. Rev. D50, Part 1 (1994)
[13] P J Franzini, Phys. Rep. 173, 1 (1989)
[14] P J O'Donnell, Phys. Lett. B261, 136 (1991)
[15] C Bernard, J N Labrent and Amarjit Soni, in Proceedings of Lattice' 93, Amsterdam 1 99:
edited by P vanBall and J Smit, Nucl. Phys. (Proc. Suppl) B30, 465 (1993); Phys. Rev. D4<
2536(1994)
[16] C A Dominguez, in Proceedings of the Third Workshop on the Tau-Charm Factory
(Marbella, Spain, 1993)
[17] B Grinstein, E Jenkin, A Manohar, M J Savage and Mark B Wise, Nucl. Phys. B380, 369
(1992)
[18] R J Oakes, Phys. Rev. Lett. 73, 381 (1994)
[19] M Neubert, Phys. Rep. 245, 259-395 (1994)
[20] M Neubert, Phys. Rev. D46, 3914 (1992)
[21] M B Wise, CALT-68-1721, Lectures presented at the Lake Louise Winter Institute,
February 17-23, 1991
[22] B Grinstein, Phys. Rev. Lett. 71, 3067 (1993)
Diquark structure in heavy quark baryons in a geometric
model
LINA PARIA and AFSAR ABBAS
Institute of Physics, Bhubaneswar 751 005, India
e-mail: lina@iopb.ernet.in
afsar@iopb.ernet.in
MS received 24 January 1996; revised 1 May 1996
Abstract. Using a geometric model to study the structure of hadrons, baryons having one, two
and three heavy quarks have been studied here. The study reveals diquark structure in baryons
with one and two heavy quarks but not with three heavy identical quarks.
Keywords. Heavy quarks; diquark; baryons; geometric model; mass formula.
PACS Nos 14-20; 14-80; 12-70; 02-10; 12-90
The study of the heavy quark systems [1] is an important issue in particle physics. It has
even become more so in the recent years [2] as several existing and planned machines are
expected to produce a barrage of experimental information in the near future. The field
has received a boost from the so called heavy quark effective theory [3-5]. An interesting
property which arises in this theory is the existence of the diquark structure for two heavy
quarks (QQ) in a baryon consisting of QQq, (where q is for a light quark) [6, 7].
Now the study of diquark is almost as old as that of the quarks. Existence or
nonexistence of diquark structure in baryons with (i) qqq, (ii) qqQ, (iii) qQQ, (iv) QQQ
has been a problem of much interest [8, 9] and has recently been reviewed [10]. Most
studies have been done within the framework of potential models, bag models and
string models [10]. The cases (i), (ii) and (iii) have been well studied but not (iv). In our
study in addition to cases (i), (ii), (iii) we shall study case (iv) as well. However, the
information extracted on the diquark structures obtained by different people are often
in direct conflict with each other. Some consensus has been achieved but there is still
much confusion. The situation can be best summarized by quoting the statement of the
five authors of the review article [10] "... sometimes we do not agree among ourselves
about the nature of diquarks".
Our aim in this paper is to study this important dual problem of the heavy quarks
and of the diquark structure in baryon within a framework which is complementary to
the potential models and the bag model pictures. This is a geometric model made use of
recently [11,12] in the context of light baryons. In fact a diquark structure was
obtained therein [12]. This view complements and supports the view that the nucleon is
deformed in the ground state. This was obtained within the configuration mixed wave
function picture in a quark potential model [13].
quark
quark r 2 quark
(b)r,=o (0^-=^
c. it Z
=o
Figure 1. Geometrical arrangement of quarks separated by i\.r 2 in the general
case and three other cases as discussed in the text.
In the geometric model of hadrons, quarks are at different positions in a collective
picture [11,12]. All the excited states of hadrons are obtained by rotations and
vibrations of quarks in a collective mode. For simplicity, we will be considering only the
rotational excitations. The total wave function of baryons is given by
where iA c ,iA sf and ^ r stands for the wave function corresponding to color, spin-
flavor and the geometric degrees of freedom respectively. From the antisymmetriza-
tion of the color state, the product of the wave function i^ sf (g) \j/ t has to be totally symmetric.
To understand the geometric structure of baryons, we are considering the four
possible geometrical arrangements of quarks as in figure 1: General case (a): two quarks
are separated by a distance r 2 , while the third quark is at a distance r l from the center of
mass of the first two quarks. Case (b): r 2 = 0; i.e. two quarks are at the same place while
the third one is separated. Case (c): r l /r 2 = x /3/2: i.e. three quarks are sitting at the
vertices of an equilateral triangle. Case (d): r t = 0; i.e. all quarks are equidistant and
lying on a line. The point group structure for inertia tensor (G,), for equivalent particles
418
Pramana - .T. Phvs.. Vol. 46. No. fi. June 1QQ6
(G E ) and for distinct particles (G D ) are given below respectively.
General case (a):C 2y , C 2v ,S l
Case(b):C w ,C a . P ,C aor
Case(c):D 3A ,D 3 ,,S 1
):> x/) ,Z)^,C xt , (2)
As we are considering baryons with a heavy flavour (charm) the corresponding spin
flavor group symmetry is SU af (8). The group SU sf (8) is broken in the following chain of
subalgebras,
= SC7 f (3)<8>l7 c (l)<8)SC7 8 (2)
is SU&) / Y (l)<8> U e (l)SU,(2)
=> S0,(2)<8> C/ Y (l)(g) J7 c (l)<g)SO a (2). (3)
The decomposition of the relevant representations of 5f/ sf (8) into SU f (4)<S)SU s (2) are
120^ 4 20 2 20
168^ 2 20 4 20 2 20 2 4
. (4)
The quantum numbers of the baryons JV, A, A c , S c , E cc , Q wc under SU sf ($) group symmetry
are given in table 1. Figure 2 is the schematic diagram drawn on the basis of table 2 in [12].
This shows the variation of levels with a change in the shape of the baryons (as stated in the
cases ((b), (c), (d))) for L = and L = 1. Here Lis the orbital angular momentum and K is the
projection of the orbital angular momentum on the body fixed axis with L = K, K + 1,
K + 2, . . . . The notation used here is: K = + indicate L = 0, 2, 4, ... and K - - indicate
L = 1, 3, 5, .... This diagram is a slightly modified version of the one by Halse [12].
Note that the spectrum generating algebra chain in eq. (3) implies that the symmetry
is broken diagonally. (It is because of this, that the oquark is much heavier than the M-,
d-, and s-quarks and is not expected to affect our analysis.) This means that the energy
levels may depend on the eigenvalue of the Casimir operators of the group chain. Hence
the 5(7(8) mass formula is
M 2 = Ml + a
231
)- l
39 1
b\C 2 (SU(4)}-~- \+c
C 2 (SU(3)}-~ \+d\s(s
4- l)] + aL + /tfC, (5)
where C 2 is the eigenvalue of the Casimir operator for different representation, C is charm
quantum number, s is spin value, / is isospin, and a, b, c, d, e.f. a, j? are parameters. The term
nl ' and RK in the mass formula comes from the rotation of the svstem. The nnerntors
r 2 =0
r2
=
II I
Figure 2. Schematic variation of the levels [_g] K with the change in the shape of the
baryons. Note that X = + or - is explained in the text. (Warning: the symbols +/
are not superscript and are not symbols for parity: see text.) The vertical axis is labelled
by the energy (schematically) and the horizontal axis is labelled by the values of ^ and r 2
as explained in the text. The dashed lines labeled by L" represent negative parity levels
and the solid lines represent positive parity levels. The dotted line represent the state
[3]0 - for the baryon containing at least one heavy quark (i.e. not for N and A). The
arrows indicate the location of different baryons as per our assignment.
are defined in such a way that for the ground state of A c all the terms except Mj vanish.
The justification for using the linear term in L in the above formula arises from the fact
that our model is intrinsically related to the string-like bag model (as is evident from
figure 1). This has also been pointed out in [11].
We treat Kalman and Tran's result [14] on the heavy quark baryons as a good
representative sample of the theoretical studies in this area. In addition, these results are
1 TH 7_1
I \ /> ~ IS\"/
uncharmed baryons, classified according to isospin (/), strangeness (S)
and charm (C).
SU t (3) SU ( (4) SU fs (8) SU S (2)
Particle ISC (piP 2 ] (PiPaPa) [0] Spin
N
i (11)
(HO)
[3]
2>
[21]
1 1
[111]
1
2
A
| (30)
(300)
[3]
3
2
[21]
i
A c
001 (01)
(110)
[3]
1
2
[21]
1)1
[HI]
2
(01)
(001)
[21]
1
[HI]
2
E c
1 1 (20)
(300)
[3]
1
[21]
1
(20)
(HO)
[3]
1
[21]
i>!
[HI]
i
2
i 2 (10)
(300)
[3]
3
2
[21]
i
(10)
(110)
[3]
1
[21]
i!
[HI]
i
o
003 (00)
(300)
[3]
3
[21]
I
published in great detail making them suitable as a point of reference. We will treat these
numbers as experimental numbers. Note that we do not consider S C ,Q C ,Q M , etc. as the
corresponding data do not exist. We do not expect much changes in our basic conclusions
when the real experimental numbers become available. The available experimental data
are from [15].
For each baryons, we arrange the states in such a way that the energy will increase from
lower to higher excited states. Knowing the internal quantum numbers, spin-flavour
symmetry of each baryon considered is given in table 1. Knowing the Casimir operator
value of the relevant representations and using the mass formula we can specify each states
of these baryons in a particular [0] L" representation. Here n is the parity of the state and
[g] is the SU"(8) representation in the standard Young Diagram formalism giving the total
number of boxes in each row, e.g. [21] denote the two boxes in first row and one box in
second row. The notation (Pip 2 p$) in table 1 denote the difference in the number of boxes
between the rows, i.e. p t = ^ ~k ( + , where A,- is the number of boxes in the ith row of the
Young Diagram. Our approach for assignment of the states of baryons is that of Halse [12]
where the mass formula is used as a guide for these assignments.
c
W)
rt o
*-* _D
go
If
*^ o
*"* T3
C 2
o
"O (U
(U Ui
C co
'S "3
C T3
O P
o m
c o
fl G
'C rt
W
3 <i
^D r ~''
I I I I
m m
i ii i LI? i i i i
m m r: ' m '
i i i i P 4 . *~* i i ^*
o
i ii i
O i i i i i i i i
I 1 i I ^-H 11 *-H
rn CM CN (N (N
422
Pramana - J. Phys., Vol. 46, No. 6, June 1996
/ r e c?rT T rgy State ^^ iS * (2282) ' With L = 0> * = i * = 0, (1 1 0) representa-
tion of SI/ (4) and (0 1) representation of 517(3). Now using the Casimir operator value
ot each representation in the mass formula we get M 2 = Mj and the state f (2282) eoes
to the [0] L n representation as [3] + . ~
(ii) The next excited state J*(M) is f (2653), with L= l,s = iK = 0,(l 10) representa-
tion of 517(4) and (0 ^representation of SC7(3). This assigns the state in the [0] L*
representation as [3] 1 (which gives a reasonable fit) rather than the assignment
[21] 1 and so on. In this way the states of each baryons considered in table ! are
assigned and a global fit of the states are able to give the parameter values as-
M = 6-303, a =-0-317, 6-0-163, d= -0-152, a = 1-297, = 1-022, c = 9-5025
e = 1 8-7795, / = - 1 8-994. All parameters are in GeV 2 .
The low excited states of JV, A, A c , S c , H cc , Q ccc have been given the assignment shown in
table 2. The location of these particles as per our assignment is also indicated in figure 2.
We find that the nucleon has the structure falling in between cases (b) and (c)
(figure 2). The position of A is found to be slightly to the right of N (see figure 2). This is
in contrast to what Halse [12] had obtained for N and A. This is because in his fit he was
trying to include a single star state % (1550) which existed then (i.e. in the 1986 data set),
but does not exist anymore (i.e. in the 1994 data set [15]). Instead a new state (1750)
has risen whose presence makes the above difference. However, in agreement with
Halse [12], we obtain a diquark structure in A.
For A c and D c we see that the order of the representations is such that its geometric
structure tends to move towards the case (b) (figure 2). This indicates that for one heavy
quark baryon, the diquark structure exists; i.e. two light quarks (qq) can form a diquark
in (qq-Q) while restoring the C 2v symmetry. This is in agreement with the result of
Lichtenberg [9]. In the case of E cc baryons also, there is a diquark structure. So one can
say that in (QQ-q), the two heavy quarks come together to form a diquark. This view is
in agreement with others [6, 7, 8, 10]. We see that though the diquark structure exists in
both the baryons containing one and two heavy quarks, the nature of the diquark in
these two cases is different due to the C 2v symmetry considerations. Here we consider
only the low angular momentum states so that the effect on the diquark structure due to
higher angular momentum is not being looked into.
The three heavy quark baryon Q ccc shows the structure of case (c) (figure 2). This is
quite reasonable as all the three quarks are equivalent. This is a new interesting result,
since not much work has been done by others in the three heavy quark case. So the
conclusion of our model is that the diquark structure exists in one and two heavy quark
baryons but not in the three identical heavy quark baryons.
Acknowledgement
The authors would like to thank the referee for useful comments.
References
[1] W Kwong, J L Rosner and C Quigg, Annu. Rev. Nud. Part. Sci. 37, 325 (1987)
[2] J G Koerner and H W Siebert, Annu. Rev. Nud. Part. Sd. 41, 511 (1991)
[3] N Isgur and M B Wise, Phys. Rev. Lett. 66, 1130 (1991)
[4] B Grinstein, Annu. Rev. Nucl. Part. Sci. 42, 101 (1992)
[5] M Neubert, Phys. Rep. C245, 259 (1994)
[6] M J Savage and M B Wise, Phys. Lett. B248, 177 (1990)
[7] A F Falk, M Luke, M J Savage and M B Wise, Phys. Rev. D49, 555 (1994)
[8] S Fleck, B Silvestre-Brac and J M Richard, Phys. Rev. D38, 1519 (1988)
[9] D B Lichtenberg, J. Phys. G16, 1599 (1990)
[10] M Anselmino, E Predazzi, S Ekelin, S Fredriksson and D B Lichtenberg, Rev. Mod.
65,1199(1993)
[11] F lachello, Phys. Rev. Lett. 62, 2440 (1989)
[12] P Raise, Phys. Lett. B253, 9 (1991)
[13] A Abbas, J. Phys. G18, 89 (1992)
[14] C S Kalman and B Tran, Nuovo. Cimento. A102, 835 (1989)
[15] Review of particle properties, Phys. Rev. D50, 1173 (1994)
AMANA Printed in India Vol. 46, No. 5,
journal of June 1996
physics pp. 425-429
mi-empirical formulae for the A and neutron-hole
dilator frequency
Z RAHMAN KHAN and NASRA NEELOFER
)artment of Physics, Aligarh Muslim University, Aligarh 202002, India
received 24 November 1995
tract. We have obtained an expression for the oscillator frequency in inverse powers of the
lear mass number, by equating the spacing of the outermost levels of a square well, found to
learly constant to the oscillator spacing for which the spacings are also constant. The
nulae for the oscillator frequency obtained here are compared with similar formulae obtained
>ther authors. A reasonable qualitative agreement is found to exist between our formulae of
and ha> N and those given in the standard literature, obtained mainly from size consider-
ns. Our derivation is based only on the assumption that a particle-nucleus potential exists.
r reference to particle-hole states is made purely for a rough comparison of our parameters,
jrwise nothing hinges on that description.
words. Semi-empirical; oscillator frequency.
:SNo. 21-80
ntroduction
i nucleon-nucleus potential-that describes several average properties approximately,
xpected to have an interior region of constant density with a diffused tail. For
lium and heavy nuclei, the Woods-Saxon (W-S) potential offers a good representa-
i of the average nucleon-nucleus potential. However, the harmonic oscillator has
n widely employed as the nucleon-nucleus potential in many structure calculations.
; most important reason for this choice is that many calculations can be performed
lytically using oscillator wave functions. Further, leaving aside matters of detail, the
illator density is not too bad. These and other reasons may be regarded as
ification for the use of the oscillator.
,et us consider the use of the oscillator in the shell model calculations. In most simple
:ulations, the closed shells and the closed sub-shells are ignored and only nucleons
he one or two outer shells are taken into account. These nucleons are usually
Bribed by oscillator wave functions. For these outer nucleons the W-S wave
stion is not expected to be far too different from the oscillator wave function. When
12 the oscillator rather than a more realistic potential, one hopes that parameters of
realistic, it may not be too far off the mark either. We may, therefore, regard it as
a qualitative description. The success of our calculations [1] shows that the oscillator
roughly describes not only the outermost neutron-holes but also those in the inner orbits as
well as the A-particle in its various orbits. One may be surprised because for medium and
heavy nuclei, the nucleon-nucleus or A-nucleus potential is more like a W-S potential than
an oscillator, but a relevant fact is that for slightly large angular momentum /, the upper
levels of the square well (we hope this is also true of potentials like W-S) are nearly equally
spaced whereas all levels of the oscillator are equally spaced [6]. Thus, we can always
equate the spacing of the upper levels of a square well to the spacing of a certain oscillator.
This leads to a formula for the oscillator frequency in inverse powers of the mass number.
The situation of the inner orbits is not at all crucial for our present purpose. Our central
premise is that a particle-nucleus potential exists and it resembles a W-S potential which
may be roughly approximated by a square well. Any reference to the particle-hole picture of
hypernuclear excitations is made purely for a rough comparison of our parameters and no
more.
The dependence of oscillator spacing on the mass number has already been obtained by
many authors. A A-oscillator spacing formula derived by Lalazissis et al [2] is based on the
idea of approximating the A-nucleus potential in a simple model, to an oscillator-like
potential. Using virial theorem [7] for the oscillator potential, ftco A is then expressed in
terms of the expectation value of the kinetic energy to establish the dependence of fao A on A.
We may note that the A-nucleus potential is assumed to be oscillator-like to begin with.
The nucleon-oscillator frequency (hco N ) has been expressed in terms of A~ liZ and
A~ 1 in [3] by equating the r.m.s. radius of an oscillator to that of a W-S form. Recently,
the proton- and neutron-oscillator spacing formulae have been obtained by Lalazissis
and Panos [8] using the r.m.s. radius of the nucleons derived from the density of the
neutrons and protons separately [9].
The work of other authors [2, 3, 8] is based on size considerations. Our derivation is
based mainly on energy level separation considerations, namely that the spacing of the
first few levels of the square well are nearly equally spaced and so may be equated to the
spacing of an oscillator. We confine ourselves to the square well because, at present, we
are not able to carry out the required calculations for a W-S potential. The derivations
based on these ideas are presented in the next section. Results and discussion are
presented in 3. The last section is conclusions and summary.
2. Derivation of the formula
For K R, the spacing AE, for neighbouring levels of same t for a square well of
depth V Q , by nuclear particle, is given by [6]
426 Pramana - J. Phys., Vol. 46, No. 6, June 1996
Semi-empirical formulae
where K Q = \_2mV Q /h 2 ^ and R is the nuclear radius which is written as r Q A^ Here
we shall take R = r' A + A, used extensively in the standard literature [10-121 We
notethatthespacingA^isconstantforagivenpotentialasfortheharmonicoscilLr
The energy levels of the three-dimensional harmonic oscillator are given by
where the symbols have their usual meaning. It follows that neighbouring levels of the
same { arise from the change of n by unity. Thus, AJS, can be equated to 2fe,
.
K Q R
Substituting in the above, R given in terms of A and A, and neglecting higher powers of
A, we have
The dependences of co on A is of the same form as given for A in the standard literature
[2]. It is slightly different for the case of the nucleon [3]. However, we see in the next
section that in actual practice, the difference is rather small. The values of the
coefficients of A ~ 1/3 and A " 2/3 depend upon the particular choice of F and r' . As such,
too much emphasis cannot be placed on quantitative agreement especially as we are
using a square well.
3. Results and discussion
From low-energy scattering experiments, it is well-known [10] that V R" - constant,
where n lies between 2 and 3. It is also known [13] that the level at zero energy, i.e. a just
bound level, in a square well potential, remains unaltered if V R 2 - constant.
For a A-particle, taking K OA = 21-7 MeV and r = l-30fm, as given by Walecka [14]
for a square well potential, and using the simpler formula = r Q A i/2 , we find the
constant in the equation V R 2 = constant. Then, taking for F OA , the more reasonable
value of 30 MeV, the value of A-well depth in infinite nuclear matter [7], we get the new
value of r = 1-11 fm for the equivalent square well potential. The value of A, the
additional constant term in the expression of R, is subject to variations depending upon
the way the nuclear radius is defined or measured. We may choose A = 0-70 fm, a value
obtained from early optical model analyses [11], for a rough calculation of the
coefficients in the formula for fto> A . The value of r' Q is then obtained from least square
fitting of the radius R = r' Q A~ i/3 + A, with R = r Q A~ 1/3 , over a large range of mass
numbers, taking r' as the adjustable parameter and taking r = 1-1 1 fm as found above.
The value of r' is found to be 0-97 fm. With these choices, the coefficients of A~ 1/3 and
A ~ 2/3 are 37-05 MeV and 26-74 MeV, respectively. The energy spacings obtained here
miA ttrlfl-. tU/-vn> nii,a.-n ii-i POT Aiffai- 1-nr oK/-nt 1 A/TaV in tli i=> Irmr tn aoo nntnKpr rporinn TViic
In a similar manner, we obtain the coefficients of A 1/3 and A 2/3 applying expression
(2) for neutron oscillator spacing (#<%). On plausible grounds, taking K ON , the depth of the
neutron-nucleus square well potential to be roughly lOMeV more than the depth of the
A-nucleus square well potential, i.e. the neutron-nucleus depth to be about 40 MeV and
taking r and A to be same as obtained for A, the value of the coefficients of A~ 1/3 and
A~ 213 are 46-63 MeV and 33-65 MeV, respectively. We may take the same to apply for the
neutron-hole. Now, our formula cannot be directly compared with that given in [3],
obtained by equating the r.m.s. radius of an oscillator to a Fermi distribution
ha) N = 38-87 A' 1 ' 3 - 23-24 A~\ (3)
as the power of A in the second term is not same as given in (2). However, the level
spacings using the formula obtained here and that given in [3] do not differ much. This
kind of agreement may be considered sufficient for our purpose.
One may hope that better values of the coefficients would be achieved, in both the
cases of ftco A and hco N , for medium and heavy nuclei, if a W-S or some other similar
potential is employed and a better search of the potential parameters is carried out.
Here, we were interested in the qualitative dependence of the oscillator frequency on the
nuclear mass number. The discussion provides one more justification of use of the
harmonic oscillator.
4. Conclusions and summary
Here, we have banked mainly on energy considerations as the basis of obtaining the
formula for ftco A and h(o N . The other authors [2,3, 8] have banked on size consider-
ations. Our main drawback is use of the square well for the A as well as the nucleon or
the nucleon-hole. Our simple derivation of the formulae does not rest in any way on the
assumption of A-hypernuclear excitation arising from particle-hole states. That picture
has been referred to solely for a rough comparison of our parameters.
Due to difference in the nuclear potential depths of the A-, E- and S-particles, one
expects that the level spacings for these different spectra could be quite different. The
observed spectrum would, therefore, provide information on the nuclear potential
depths for the different particles.
Incidentally, the work here provides some additional justification for using oscillator
wave functions in many nuclear calculations.
Acknowledgement
The authors are grateful to Dr Mohammad Shoeb for some discussion.
References
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PRAMANA Printed in India
journal of VOLW, INC. o,
physics June 1996
pp. 431-449
Positron scattering from hydrocarbons
RITU RAIZADA and K L BALUJA
Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
MS received 29 January 1996
Abstract. The total cross sections for positron impact on hydrocarbons have been calculated
using the additivity rule in which the total cross section for a molecule is the sum of the total cross
section for the constituent atoms. The energy range considered is from a few eV to
several thousand eV. The total cross sections for positron impact on an atom are calculated by
employing a complex spherical potential which comprises of a static, polarization and an
absorption potential. We have good agreement with the experimental results for hydrocarbons
for positron energy ^ 100 eV. Our results also agree with the available calculations for CH 4 and
C 2 H 2 which employed full molecular wavefunctions beyond 100 eV. Our absorption cross
sections also agree with molecular wave-function calculations for C 2 H 2 and CH 4 beyond 100 eV.
We have shown the Bethe plots for e + ~C and e + -E scattering systems and Bethe parameters
have been extracted. We have fitted the cross section for positron impact on hydrocarbons in
the form ff l (C II HJ = nflr'' + mcE~'' in the energy range 300-5000eV where a = 195-0543,
b = 0-7986, c = 371-1757 and d = 1-1379 with in eV and o, in 10~ 16 cm 2 .
Keywords. Positron; hydrocarbons; static potential; polarization potential; absorption
potential; inelastic and total cross sections; Bethe plot; independent atom model; additivity
rule.
PACSNos 34-80; 34-90
1. Introduction
The total cross sections (including elastic plus energetically possible all inelastic
channels) for positron scattering from various molecules have recently been measured
for energy ranging from a few eV to several hundred eV in various laboratories. The
experimental and theoretical data have been summarized earlier [1-5]. The surge in
experimental activity is due to the availability of positron beams of good intensity and
the ease with which the positrons can be detected.
The comparison between positron and electron scattering data on a particular target
gives better insight into the scattering mechanism. The effect of polarization is more
important in the case of positron scattering because of the cancellation effects between
the repulsive static potential and the attractive polarization potential. Due to the lack
of symmetry of the molecule, there are additional degrees of freedom like rotation and
At Inw imnarf p.nerfrifis. the effect of oositronium formation must be taken
reduced to a single channel problem. In spite of this simplicity, the task is still difficult
due to the lack of availability of wavefunctions of complex molecules like hydrocarbons.
However, at high energies, an independent atom model [6] has often been used to
calculate the elastic cross sections. In this model, the constituent atoms are considered
as separate scattering systems and interference occurs between the electron waves
scattered by individual atoms. The interference term is dependent upon the geometry of
the molecule. The inelastic effects within the independent atom model can be incorpor-
ated by using the optical theorem [7]. This leads to great simplification in which the
total cross section for positron impact on a molecule can be expressed as an incoherent
sum over the total cross sections of the consitituent atoms. This additivity rule has
been used earlier for electron impact on various atoms and molecules [8-11]. The
validity of additivity rule has also been studied for electron-impact ionization cross
sections [12].
The present work deals with the calculation of total cross sections for positron
impact on hydrocarbons by employing the additivity rule. The total cross-sections for
carbon and hydrogen atoms can be conveniently calculated by an optical potential
method [13]. This method takes into account positron-atom interaction to all orders of
the perturbation. The total cross sections for positron impact on various hydrocarbons
have been experimentally measured [14-17].
2. Theory
The positron-atom scattering system is described by the Schrodinger equation
t ...,T K ,T), (1)
where the total wavefunction of the N target electrons (r N ) and the positron (r) is
expanded in terms of channel functions t ,
Hr l5 ...,r N ,r)= ^.(r !,..., r^(r), (2)
i=l
where M is the number of channels included. After expanding F { (r) in terms of
Legendre polynomials, the radial part for each partial wave satisfies the coupled
equation
(3)
J
where C/ -(r) is the coupling potential given by
Z _ " / _J_
^ fc = i \ *"fc ~
Z is the nuclear charge of the atom, H is the atomic hamiltonian and k { is the wave
number in the ith channel.
At intermediate and high positron impact energies, we replace S^l/y^-F^r) by
(1986)
Expt. Floeder et al(1985
--- TheoreticalBaluja and
Jain (1992)
This work
100 1000 5000
Positron Energy (eV )-*
Figure 1. Total cross-sections for positron impact on CH 4 .
annel problem is reduced to a single channel problem and we can replace eq. (3) by
(5)
ien all the channels are closed, K opt (r) is real which becomes complex when some
annels are open. The imaginary part of V opi (r) then accounts for absorption effects
e to loss of flux in the open channels. We express V opt (r) as [18]
lere F st (r) is the static potential, V pol (r) is the polarization potential and V abs (r) is the
sorption potential. All these potentials are dependent on atomic charge density. The
Pramana - J. Phys., Vol. 46, No. 6, June 1996
433
Ritu Raizada and K L Baluja
12
10
o 8
e + -C 2 H 2
J Expt. Sueoka and Mori gg)
Theoretical Baluja and
Jain(1992)
This work
10
100 1000 5000
Positron Energy (eV )-*
Figure 2. Total cross-sections for positron impact on C 2 H, .
static potential K st (r) is repulsive and is calculated at the Hartree-Fock level by
employing the independent-particle model [19] and is parameterized as
K.frt = -Q(rt. (7)
where
(8)
Expt Sueoka and
Mori (1986)
This work
100 1000
Positron Energy (eV)-
5000
Figure 3. Total cross-sections for positron impact on C 2 H 4 .
is calculated exactly and is given by [20]
1
(9)
! polarization potential V^r) is based on the correlation energy of a single positron
nhomogeneous electron gas [21]. This polarization is different from the correspond-
ctron case [22], because the positron distorts the electronic charge cloud differently
the corresponding electron case. Near the nucleus, the two polarizations differ,
as asymptotically they both behave as -(a/2r 4 ) where a is the static dipole
^ability of the atom. The correlation energy is calculated from the ground state
ation value of the hamiltonian which describes the fixed positron interacting with the
electrons. The positron polarization potential has been interpolated for the whole
region and is dependent only on the target density via a density parameter r s ,
(10)
o
r 12
u
_ 8
10 100 1000
Positron Energy ( eV ) -*
5000
Figure 4. Total cross-sections for positron impact on C, H,.
The expressions for different radial regions have already been given [18]. The density
p(r) for carbon is taken from the independent-atom model [19]
p(r) = ((Z - l)/4nr 2 d) [te/(l + hT) 2 ~] [ - 1 + 2te/(l
(11)
where T = e*- 1 and = r/d. For carbon, a = Il-878a 3 . The imaginary part V abs (r)
represents the total loss of flux into all accessible channels. We have employed the
semi-empirical form [23-26] of electron case V~ bs modified for positron case V+ bjt (r) as
prescribed earlier [18]. They are related as
(12)
This particular form has been shown to give good results for total cross sections for
positron impact on several molecules. The factor 2 accounts for the fact that exchange is
436
Pramana - J. Phys^ Vol. 46, No. 6, June 1996
24.
20
16
M 12
36
e - C 3 H
$ Expt. Floeder et al (1985
This work
10
100
Positron Energy (
1000
5000
Figure 5. Total cross-sections for positron impact on C 3 H,.
absent in positron scattering. It may be pointed out that a factor of 1/2 approximately
accounts for exchange effects in the evaluation of K a " bs (r). So to remove this factor,
a factor of 2 appears in V* bs (r). The factor l/^/kr approximately accounts for Ps
formation at lower energies. It is now a standard procedure to solve the radial
Schrodinger equation. We transform this equation into a set of first order coupled
differential equations for the real and imaginary parts of the complex phase functions.
These equations are solved by a variable phase approach. The S matrix and the elastic
and inelastic cross sections are evaluated by standard formulas [18]. The positron-
atom scattering total cross sections are obtained by summing the elastic and inelastic
components.
The number of partial waves included depend upon the energy of the incident
30
20
10
e+-C 3 H 8
Expt. Floeder ei al(1985)
This work
10 100
Positron Energy (eV )
1000
5000
Figure 6. Total cross-sections for positron impact on C 3 H 8 .
Born phase shifts are employed. The integration is carried up to a radius where all the
components of the interacting potentials are negligible. Convergence was tested by
taking various step sizes.
3. Results and discussion
In figure 1, we display the theoretical e + -CH 4 total cross section values (cr t ) in the
energy range 10-5000 eV along with the experimental points of Floeder et al and
Seuoka and Mori [15]. Floeder etal [14] measured o t values in a transmission
experiment utilizing positrons from a 22 Na source and a tungsten moderator. They
obtained o t values in the energy range 5-400 eV for various hydrocarbons like
methane, ethane, ethene, propane, propene, cyclopropane, n-butane, isobutane and
1-butene. Seuoka and Mori: [15] measured a t values for methane, ethane and ethylene in
the energy range 0-7-400 eV by using a retarded potential time of flight method. Their
values were not absolute but were normalized to e + -N 2 data [27]. We notice that our
peak in cr, occurs at 50 eV against experimental peak at 30 eV. Beyond 100 eV, we are in
good accord with both the experiments, The effect of rotational excitation is expected
to be small for this spherical hydride because the first non vanishing moment is
100 1000
Positron Energy (eV)-*-
5000
Figure 7. Total cross-sections for positron impact on C 4 H 8 .
octupole in nature. We also compare our theoretical values with that obtained by
Baluja and Jain [1] in which the optical potential was determined from the molecular
wave function at the Hartree-Fock level. The good agreement between the two
theoretical curves indicate that the role of bonding is not significant at higher energies
for this spherical system. There are other experimental data available [28,29] for
e + -CH 4 cross sections but we have not shown these because these experimental points
almost coincide with the experimental data shown in figure 1.
In figure 2, we display cr, values for e + -C 2 H 2 scattering system along with the
experimental points [17]. The values reproduce the hump at 40 eV. Excellent agree-
ment was obtained with the experiment beyond 80 eV. The values in the low energy
region are not in good agreement with the experimental results. This shows that the rule
of additivity is not reliable at low energies. Since C 2 H 2 is a linear molecule, the
rotational excitation may not be insignificant in the low energy region. We obtained
good agreement with other calculation [1] which employed molecular wavefunctions.
This once again reveals that the effect of bonding is not very significant. In figures 3 and
4, we have shown a t values for e + -C 2 H 4 and e + -C 2 H 6 system respectively along with
the experimental points [15]. For C 2 H 4 , we have excellent agreement with the
experiment beyond 80 eV. However, for C 2 H 6 we are in good accord with the
42
36
24
o 1
1
5 Expt. Floeder etal(1985)
This work
10 100 1000 5000-
Positron Energy (eV)--
Figure 8. Total cross-sections for positron impact on C 4 H 1 .
experiment only beyond 100 eV. For C 2 H 6 our points are higher than the experimental
points in the energy range 30-100 eV. Our additivity rule predicts a,(C 2 H 5 )>
<7,(C 2 H 4 ), but the experimental points reveal the trend c t (C 2 H fl )< ff,(C 2 H 4 ) for
positron impact energies below 50 eV. This is perhaps due to the fact that the electronic
orbits in ethane molecules are saturated in comparison to ethylene. A similar situation
prevails for C 3 H 6 , C 3 H 8 , C 4 H 8 , C 4 H 10 and C 6 H 6 .
These results are shown in figures 5-9 respectively along with experimental results
[14, 16]. For all these cases, we have good agreement with the experimental results
beyond 100 eV. Our extensive study on these hydrocarbons indicate that the rule of
additivity is able to predict good cross sections for energies beyond 100 eV. Our model
predicts the same cross sections for different isomers of a hydrocarbon. For example,
the total cross sections for rc-butane and isobutane are identical. In fact this is borne out
by the experimental results [14] which further lends support to our additivity model.
We have calculated the total cross sections for positron impact on hydrocarbons by
the following form
-d
(13)
10 100
Positron Energy (eV)
1000
5000
Figure 9. Total cross-sections for positron impact on C fi H 6 .
abs , for e + impact on
where a and b refer to the fit of total cross sections for carbon atom; and c and d refer to
the hydrogen atom. The values of these constants are a =195-0543, b = 0-7986,
c = 371-1757 and d = 1-1379. E is in eV and <r t is 10" ' 6 cm 2 . The fit is valid in the energy
range 300-5000 eV and is better than 5% in the energy range 300-5000 eV.
In figure 10, we have displayed our absorption cross sections, a
H 2 . We have compared our results with the calculations of optical model potential [1]
in which molecular wavefunctions for H 2 were used to generate the potentials. The
experimental results [30] are also shown. The measured points are a sum of the
ionization cross sections and the Ps formation. The information on other inelastic
channels like excitation and dissociation is lacking, so it is not taken into account. Our
low energy (below 50 eV) results are lower than the molecular wavefunction results due
to the use of different thresholds used. The calculations involving molecular wavefunc-
tions used 8-62 as the Ps threshold whereas we used ionization potentials for hydrogen
atom as the threshold value. The peaks in a abs differ by about lOeV for the two
calculations. At energies beyond 50 eV, the two curves seem to merge. Moreover, our
theoretical curve is in better agreement with the experimental results. This implies that
Pramana - J. Phys., Vol. 46, No. 6, June 1996
441
o Expt. Fromme et al(l988)
Theoretical
Baluja and Jain(1992)
This work
10 100 1000 5000
Positron Energy (eV ) -<~
Figure 10. Absorption cross-sections for positron impact on H,.
the ionization channel is the dominant one among all inelastic channels since Ps
formation cross sections is very small at energies above 80 eV. Both the theoretical
curves give an upper bound to the ionization cross sections because the ionization-
cross sections is one of the components of the absorption cross section.
Our o- abs cross sections for C 2 H, are shown in figure 11 along with the molecular
wavefunction calculations [1] employing a model optical potential approach using
4-61 eV as the threshold value. We used ionization potentials for H (13-605 eV) and
C (1 1-26 eV) as thresholds for the addivity rule. As seen from figure our result lies lower
than the molecular wave function calculation due to our high threshold value. At
energies above 100 eV the two curves are very close to each other. Our simple model
exhibits a peak at 50 eV which is 20 eV higher than the peak given by the other
AA1
Pramana- T Phvc Vnl
NTn
.Tuiu> IQQft
-- Theoretical
Baluja and Jain (1992)
100 1000
Positron Energy (eV )-*-
5000
Figure 1 1. Absorption cross-sections for positron impact on C 2 H 2 .
:ulation. There are no experimental results available for inelastic cross sections for
*2-
'he <r abs cross sections for the methane molecule are shown in figure 12 along with
other calculation employing molecular wavefunctions [1], which used 6-18 eV as
threshold value. Once again due to the difference in threshold values used in the two
>retical models, our peak lies higher by about 20 eV. Beyond 100 eV, the two curves
in good accord with each other. There is no experimental information available for
astic channels for this molecule. However, since the ground state of CH 4 is a singlet
e and all the excited singlet states are repulsive in nature, we expect absorption cross
ions to be very near to the ionization cross sections plus the dissociation cross
ions for these excited states. These cross sections are not available for positron
act on CH 4 . However, we have displayed the corresponding electron cross sections.
D Expt Ionization Winters
(1975)
A Expt lonizationAdamczyk
et at (1966)
+ Expt. Ionization plus
dissociation, Winters
(1975)
\ \ Theoretical
i 4. Baluja and Jam(1992)
This work
10 100 1000
Positron Energy ( eV )
5000
Figure 12. Absorption cross-sections for positron impact on CH 4 .
The measured ionization cross sections [31,32] and the ionization plus dissociation
cross sections [32] are shown. The absorption cross sections lie between ionization
cross sections and the ionization plus dissociation cross sections. It is interesting to
note that the absorption cross sections merge with the electron ionization cross section
results at energies above 400 eV. It is well-known that at high energies, the ionization
cross sections for electron and positron impact are nearly equal. This is due to the fact
that according to first order perturbation theory, the Born cross sections are indepen-
dent of the charge of projectile. Due to the cancellation effect of the repulsive static
potential for e + -CH 4 and the attractive polarization potential, the positron cross
2 ln(E/RJ
Figure 13. Bethe plot for positron impact on H.
sections plus the dissociation cross sections for the positron case would be lower than
the electron case and the expected results may lie on our absorption curve.
It is well-known from Bethe' s asymptotic theory of inelastic scattering that the total
inelastic cross sections can be parametrized in the form
(14)
(E/R)
where R is the Rydberg energy. The constant Mf ot is the total dipole matrix element
squared and is related to S( 1,0) [33]. The equation suggests that a plot between
ff^ bs (E/R)/4nal against \n(E/R) is a straight line which is known as Bethe plot. We have
shown these plots for positron impact on hydrogen and carbon in figures 13 and 14
respectively. The Bethe parameter for hydrogen are Mf ot = 0-846, C tot = 0-479 and for
carbon these are M 2 t = 2-59 and C, ot = 0-309. M t 2 ot is obtained by using the linear
portion of the Bethe plots. It is worth mentioning that these parameters have been
derived when the threshold is kept at the ionization limit, and we observe that the law of
additivity gives good results for positron impact on hydrocarbons. When the threshold
In (E/R)
Figure 14. Bethe plot for positron impact on C.
is kept at the energy of positronium formation, then for hydrogen we get Af f ot = 0-909,
C tot = 2-2567 and for carbon we get M t 2 ot = 545, C tot = 0-1708. Our value of M 2 1 for
hydrogen agrees with the exact value of M 2 ot =1-0 [33], which can be calculated, by
evaluating S(- l,fc) = (/c)~ 2 [l + (1 + /c 2 /4)~ 4 ] in the limit A:->0. In the closure ap-
proximation M, 2 ot = Z/A(Ryd), where A is the mean excitation energy of the atom with
Z electrons, A can be calculated with the knowledge of the polarizability a d and the
value of <> 2 > by the relation a d = (2/3) r 2 >/A). For carbon atom, a d = 1 1-878 aj, and
<r 2 > = 13-7372a 2 at the Hartree-Fock level and we get A = 20-98eV. This yields
M t 2 1 = 3-89. The total dipole element squared M t 2 ot can also be known from the
knowledge of oscillator strengths and the relevant energy levels and the value of M 2 on .
For carbon atom, we consider only the first three excited states ls 2 2s2p 33 jt) 03 P and
3 S which have their energy levels at 64089 cm" 1 , 75255 cm" 1 and 105799cm" 1 with
respect to the ground state ls 2 2s 2 2p 23 P e .
The values of oscillator strengths for these transitions from the ground state are
respectively 0-1, 0-1, 0-25. The total contribution of these three states is 0-5764 to the
value of M t 2 1 . To get the contribution M? n of ionization; we made Bethe plot using the
- with A = 6.8eV
with A=13.6eV
i i i i 1 1 1 1 1 i i i i 1 1 1 1 1 i i i i i 1 1
Figure
Positron Energy (eV)
15. Total cross-sections for positron impact on H.
ionization cross sections values [34]. We obtained M. 2 on = 3-96 giving M t 2 ol = 4-54. This
value is derived from electron data and is of comparable magnitude with our positron
value of M t 2 ol = 5-45. We emphasize that our optical model potential is capable of giving
good values of the absorption cross sections and the total cross sections for hydrogen
and carbon atoms provided the threshold is kept at the Ps formation threshold.
However, the rule of additivity works for hydrocarbons only if the threshold is kept at
the ionization limit. We also display our total cross section results for hydrogen and
carbon in figures 15 and 1 6 respectively. We compare our results for hydrogen with the
experimental results [35] and we notice good agreement beyond 80 eV.
In conclusion, we state that the rule of additivity works well for positron impact on
hydrocarbons when the total cross sections for the constituent atoms are calculated by
12
10
in 6
withA=4.46eV
withA=11.26eV
\
10
100 1000
Positron Energy (eV )-*
5000
Figure 16. Total cross-sections for positron impact on C.
employing an optical model potential and the absorption potential is calculated
keeping the threshold at the ionization limit.
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Nuclear Physics
Signature inversion in the K = 4" band in doubly-odd 1:>2 Eu and 156 Tb
nuclei: Role of the /i 9/2 proton orbital Alpana Gael and Ashok K Jain 51-66
Conservation of channel spin in transfer reactions
V S Mathur and Anjana Acharya 67-74
Semi-empirical formulae for the A and neutron-hole oscillator frequency
M Z Rahman Khan and Nasra Neelofer 425-429
Atomic and Molecular Physics
Depopulation of Na(8s) colliding with ground state He: Study of collision
dynamics A A Khan, K K Prasad, S K Verma,
V Kumar and A Kumar 373-380
453 Subject Index
.7 +
Multiconfiguration Hartree-Fock calculations in Cr 5 + , Mn 6 + and Fe
S N Tiwary, P Kumar and R P Roy 381-3*
Positron scattering from hydrocarbons Ritu Raizada and K L Baluja 431-4*
Lasers, Optics and Spectroscopy
Spatial and time resolved analysis of CN bands in the laser induced
plasma from graphite S S Harilal, Riju C Issac, C V Bindhu,
Geetha K Varier, VPN Nampoori andCPG Vallabhan
Self-similar solutions of laser produced blast waves K P J Reddy
A distributed feedback dye laser based on higher order Bragg scattering
S Sivaprakasam, Ch Saradhi Babu and Ranjit Singh
Nonlinear Schrodinger equation for -optical media with quintic non-
linearity G Mohanachandran, V C Kuriakose and K Babu Joseph
Plasma Physics
Scaling laws for plasma transport due to ^-driven turbulence
C B Dwivedi and M Bhattacharjee
Condensed Matter Physics
Perturbation theory of polar hard Gaussian overlap fluid mixtures
Sudhir K Gokhul and Suresh K Sinha
Structural study of aqueous solutions of tetrahydrofuran and acetone
mixtures using dielectric relaxation technique
A C Kumbharkhane, S N Helambe, M P Lokhande,
S Doraiswamy and S C Mehrotra
Composite Anderson-Newris model and density of states due to
chemisorption: Quasi-chemical approximation
jR Guleria, P K Ahluwalia and K C Sharma
Transient and thermally stimulated depolarization currents in pure and
iodine doped polyvinyl formal (PVF) films P K Khare
frequencies P J Singh and K S Sharma 259-27)
A model for the reflectivity spectra of TmTe F Nayak 271-27
Evidence for superconductivity in fluorinated La 2 CuO 4 at 35 K:
Microwave investigations
G M Phatak, K Gangadharan, R M Kadam,
M D Sastry and U RK Rao 277-28
Harmonic generation studies in laser ablated YBCO thin film grown on
<IOO> MgO Neeraj Khare, J R Buckley, R M Bowman,
G B Donaldson and C M Pegrum 283-28
A Compton profile study of tantalum B K Sharma, B L Ahuja,
Usha Mittal, S Perkkio, T Paakkari and S Manninen 289-29
Brief Report
Current algebra results for the B D systems
V Gupta and H S Mani 239-24
as Afsar
e Paria Lina
arya Anjana
e Malhur V S
iwalia P K
e Guleria R
jaBL
e Sharma B K
nthakrishna G
e Bekele Muiugeta
a Ch Saradhi
e Sivaprakasam S
417
67
99
289
403
299
of Gell-Mann-Okubo
223
q deformation
ass formula
ijaKL
e Raizada Ritu 43 1
2rjee N
ausal dissipative cosmology 213
ham Aroonkumar
e Banerjee N 213
sle Muiugeta
ptimai barrier subdivision for Kramers'
cape rate 403
ttacharjee M
e Dwivedi C B 229
Ihu C V
c Harilal S S 145
'as S N
e Bagchi B 223
idov G N
'aves with linear, quadratic and cubic
ordinate dependence of amplitude in
ystals 245
man R M
2 Khare Neeraj 283
dey J R
s Khare Neeraj . 283
idra B P
obile interstitial model and mobile electron
Ddel of mechano-induced luminescence in
loured alkali halide crystals 127
idhuri S
i the structure and multipole moments of
ialiy symmetric stationary metrics 17
idhury D K
atic and dynamic properties of heavy light
;sons in infinite mass limit 349
Das K C
see Chaudhuri S 17
Das Pratibha
see Choudhury D K '349
Donaldson G B
see Khare Neeraj 283
Doraiswamy S
see Kurnbharkhane AC 91
Dutta Gupta S
see Mahalakshmi V 389
Dwivedi C B
Scaling laws for plasma transport due to
ijj-driven turbulence 229
Gangadharan K
see Phatak G M 277
Giri A K
Effect of heavy quark symmetry on the mass
difference of B-system in minimal left right
symmetric model 41
Ratios of B and D meson decay constants
with heavy quark symmetry 41 1
Goel Alpana
Signature inversion in the K = 4~ band in
doubly-odd 152 Euand 156 Tb nuclei: Role of
the h 9/2 proton orbital 51
Gokhul Sudhir K
Perturbation theory of polar hard Gaussian
overlap fluid mixtures 75
Guleria R
Composite Anderson-Newns model and
density of states due to chemisorption:
Quasi-chemical approximation 99
Gupta V
Current algebra results for the B D systems
239
Harilal S S
Spatial and time resolved analysis of CN
bands in the laser induced plasma from
graphite 145
Helambe S N
see Kumbharkhane A C 91
Issac Riju C
see Harilal S S
Jain Ashok K
see Goel Alpana
Jose Jolly
see Mahalakshmi V
145
51
389
455
see Mohanachandran G 305
Joy M P
Painleve analysis and exact solutions of two
dimensional Korteweg-de Vries-Burgers
equation 1
Kadam R M
see Phatak G M 277
KarG
Objectification problem, CHSH inequalities
for a system of two spin- 1/2 particles 9
Kataria S K
see Sastry S V S 357
Khan A A
Depopulation of Na(8s) colliding with
ground state He: Study of collision dynamics
373
Khare Neeraj
Harmonic generation studies in laser
ablated YBCO thin film grown on <100>
MgO 283
Khare P K
Transient and thermally stimulated
depolarization currents in pure and iodine
doped polyvinyl formal (PVF) films 109
Kumar A
see Khan A A 373
Kumar N
see Bekele Mulugeta 403
Kumar P
seeTiwarySN 381
Kumar V
see Khan A A 373
Kumbharkhane A C
Structural study of aqueous solutions of
tetrahydrofuran and acetone mixtures using
dielectric relaxation technique 91
Kuriakose V C
see Mohanachandran G 305
Leach P G L
seePatelLK 331
Lokhande M P
see Kumbharkhane AC 91
Lopez-Bonilla Jose L
An identity for 4-spacetimes embedded into
E 5 219
Mahalakshmi V
Linear periodic and quasiperiodic anisotropic
layered media with arbitrary orientation of
optic axis A numerical study 389
Maharaj S D
seePatelLK 331
Maharana L
Self-interacting one-dimensional oscillators
203
Mani H S
see Gupta V 239
Manninen S
see Sharma B K 289
Mathur V S
Conservation of channel spin in transfer
reactions 67
Mehrotra S C
see Kumbharkhane AC 91
Mehta Mitaxi P
Time dependent canonical perturbation
theory III: Application to a system with
nonconstant unperturbed frequencies
323
Mittal Usha
see Sharma B K 289
Mohanachandran G
Nonlinear Schrodinger equation for optical
media with quintic nonlinearity 305
Mohanta R
seeGiriAK 41,411
Nampoori VPN
see Harilal S S 145
Nayak P
A model for the reflectivity spectra of TmTe
271
Neelofer Nasra
see Rahman Khan M Z 425
Nunez- Yepez H N
see Lopez-Bonilla Jose L 219
Ojha Bharti
see Chandra B P
127
Paakkari T
see Sharma B K 289
Paria Lina
Diquark structure in heavy quark baryons
in a geometric model 417
Patel L K
Cosmic strings in Bianchi II, VIII and IX
spacetimes: Integrable cases 331
see Vaidya PC 341
Pegrum C M
see Khare Neeraj 283
Perkkio S
see Sharma B K 289
Phatak G M
Evidence for superconductivity in fluorinated
La 2 CuO 4 at 35 KJ Microwave investigations
277
PrasaH K K
Rahman Khan M Z
Semi-empirical formulae for the A and neutron-
hole oscillator frequency 425
Raizada Ritu
Positron scattering from hydrocarbons 431
Rakhecha Veer Chand
Geometric phase a la Pancharatnam 315
RaoNN
Coupled scalar field equations for nonlinear
wave modulations in dispersive media 161
Rao U R K
see Phatak G M 277
Reddy K P J
Self-similar solutions of laser produced blast
waves 153
Roy RP
see Ti wary SN 381
RoyS
see Kar G 9
Sastry M D
see Phatak G M 277
Sastry S V S
Effective potentials and threshold anomaly
357
Sharma B K
A Compton profile study of tantalum 289
Sharma K C
see Guleria R 99
Sharma K S
see Singh P J 259
Shrivastava R G
see Chandra B P 127
Singh P J
Dielectric behaviour of ketone-amine binary
mixtures at microwave frequencies 259
Singh Ranjit
see Sivaprakasam S 299
Singh Seema
see Chandra B P - 127
Sinha Suresh K
see Gokhul K Sudhir 75
Sitaram B R
see Mehta Mitaxi P 323
Sivaprakasam S
A distributed feedback dye laser based on
higher order Bragg scattering 299
Tiwary S N
Multiconfiguration Hartree-Fock calcula-
tions in Cr 5+ , Mn 6+ and Fe 7+ 381
Vaidya P C
A spherically symmetric gravitational
collapse-field with radiation 341
Vallabhan C P G
see Harilal S S 145
Varier Geetha K
see Harilal S S 145
Verma S K
see Khan A A 373
Vishwamittar
see Mamta 203
Wagh Apoorva G
see Rakhecha Veer Chand
315
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