MESON PRODUCTION AS A SHOCK WAVE PROBLEM

NASA TECHNICAL TRANSLATION

NASA TT F-15,937

MESON PRODUCTION AS A SHOCK WAVE PROBLEM

W. Heisenberg

Translation of u Mesonenerzeugung Als
Stosswellenproblem", Zeitschrift fur

Physik, Vol. 133, 1952, pp. 65

79

'{WSA-TT-F- 15937) MESON PRODUCTION AS A
SHOCK UAVE PEOBLEH {Scientific Translation
iservice) 25 p HC $4.25 CSCL 20H

874-34191

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SEPTEMBER 19 74

STANDARD TITLE PACE

I. Report No.

NASA TT F-15,937

2. Government Accession No.

4. Title and Subtitle

Meson production as a shock wave
problem

7. Author(s)

■W. Heisenberg

9. Performing Organiiotion Name and Address

SCITRAN

iiox 5456

Sanf.a B «rh a r aj fiA S31QS

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National Aeronautics and Space Administration
Washington, D.C. 20546

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5. Report Date

October 19 74

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NASw-2483

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Translation

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15. Supplementary Notes

Translation, of "Mesonenerzeugung Als Stosswellenproblejn'
Zei'tscjirift fiir Physik, Vol. 133, 1952, pp. 6 5 . - 73- ■■:"." "'"

16. Abstract

The production of many mesons in the collision of two
nucleons is described as a shock wave process
represented by a nonlinear wave equation]. The
quantum- theoretical features of the process can then
be considered approximately according to the principle
of correspondence, as we are dealing with a "high
quantum number process". Statements on the energetic
and angular distributions of the various types of
mesons arise from the discussion of the solutions
of the nonlinear wave equation.

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11

MESON PRODUCTION AS A SHOCK WAVE PROBLEM
W. Heiseiiberg

The experimental information gained in recent years / ^ *
on the origin of the jz-j mesons makes it seem very probable
that many mesons are often produced at once in the collision
of two high-energy nucleons. It has for a long time been
established that a strong interaction of nucleons with mesons,
and particularly between mesons, can lead to such multi-
plication [4]. For a quantitative estimate, one can compare
the energy dissipation in the meson field with turbulence in
flow fields [5], or , as Fermi [3] has done, one can think of
a temperature equilibrium being attained at the moment of
collision, from which the energetic distribution of the
mesons can be calculated.

The following considerations, however, are intended
to take up the problem from the viewpoint which the author
presented in 1939 in relation to the Yukawa theory [4].
Meson production will be considered as a shock wave process,
described by a nonlinear wave equation, and it will be shown
that through such a treatment one can arrive at quantitative
results for the spectral and spatial distribution of the mesons.

I. Perceptual Description of the Shock Wave

In the following, meson production is always described in
the center-of-mass system. Transformation into the laboratory
system can be undertaken without difficulty as a supplement,
and has been done in earlier works; it need not be explained

Numbers in the margin indicate pagination in the original
foreign text.

here [5] .

a) In the center -of -mass system, both nucleons approach / 66
each other from opposite directions (Figure 1) until they
overlap in a certain region (shaded in Figure 1). The nucleons
are shown as flat discs. Because of the Lorenz contraction,
their thickness is less by the factor of J/-1-/S 2 ! (P = center
of mass velocity) than their diameter, which one can take to

be of the order of magnitude of the Compton wavelength of the

-13
i. e., on the order of 1.4 ■ 10 cm. At the moment

yt-rs cm X ■ ' r

Figure 1

of collision the velocity of the nucleons changes], so that in
their total region^ energy is transferred to the meson field.
In the first moment of the shock wave, then, the entire energy
of the meson field is concentrated in the thin flat layer
which was filled by both nucleons at the moment of collision.

b) If one ignores the interaction of the mesons,
it would expand after the first moment according to the wave
equation

■ : u<p-&tp = o \ (l)

(or according to a complex linear wave equation which contains
the different meson types).. The; spectral and angular distri-
bution of the meson wave would then no longer change in the
course of the wave expansion. They could, therefore, be
determined by a Fourier expansion of the wave at the first

moment. We find that the energy contained in the meson wave
between the frequencies k and k + dk (k Q corresponds to the
energy of a single meson) would be nearly independent of k Q
up to frequencies having their wavelengths of the order of
magnitude of the thickness of the layer in which the
collision occurs; i. e., on the order of --p • ( *\ i- s

the meson mass). For h>h>» z
rapidly as a function of k

yi-F

the intensity will decrease

d£=const-dk f or l k <^k „

(2)

Correspondingly, for the number of mesons in the interval

dk one obtains

dn — const -r-°- f or| • k <,k 0t

(3)

Figure 2 shows the course of <p] on the axis perpendicular to
the plane of emission (shortly after the act of emission).
It also shows de/dk \ and dnidk \ , under the assumption (1).

/ 67

Spectrum (3) corresponds to the well-known x-ray braking
spectrum of the electrons. Even if a considerable part of
the nucleon energy is transferred to the meson field, it
never leads to a large number;. of emitted mesons, because the
energy of an individual meson would average ~^o m \ .

Figure 2 a-c.

c) In reality, though, we cannot ignore the interaction
of the mesons. The wave expands according to a nonlinear
wave equation. Only in the limiting case of low intensity
does it transform approximately into the linear form. The

\9-

a—j:.'
a

Figure 3 a-c.

nonlinearity has the result, which we shall recalculate later,
that the singularity at the head of the wave is somewhat rounded
off. As a result, energy is transferred from the short to the

"I- -I
a " b

Figure 4 a-d.

long waves during the expansion process, and the spectral
distribution at the end of the expansion process falls off
more rapidly than if (1) were valid. Qualitatively,
one obtains the relations shown in Figure 3.

The spatial expansion is shown in Figure 4 a-d.

At the moment of impact the entire energy is concentrated
in the layer of the two nucleons (a). Then two shock fronts

move out to the right and left. The major portion of the energy] / 68

still resides in the two shock fronts, but there is also

wave excitation in the space between them, which contains the

rest of the energy" (b). Now the shock fronts proceed farther.

The excitation in their wake spreads over a wider space, and

that near the starting point becomes a new wave expansion.

The energy in the shock fronts has become smaller. It has

shifted into the remaining wave region [Jand, therefore, to

greater wavelengths (c)].J On continued advance, the excitation

at the center decreases. A true wave forms, propagating faster

in the direction of the shock fronts than perpendicular to it,

because waves of short wavelength have a higher propagation

velocity (group velocity). Only at, very slight intensity will

the excitation spread to all sides, even though with the

velocity of light. The energy in the shock fronts has by now \

become so small that here, too, the nonlinear it ies play no

important role. Continued progress is according to the

usual linear wave equation (d) .

In this perceptual description we have so far completely
ignored the quantum theoretical aspects of the problem. That
is a quite useful approximation, as it deals with the production
of many mesons; that it, with a process having high quantum
numbers. The work mentioned above [4] describes in detail how
to undertake the corresponding transformation into quantum
theory. Here it is sufficient to take the following qualitatively
from Figure 4d: A large part of the energy is radiated out
in all directions in the form of mesons having wavelength's
comparable with the diameter of the disc; i. e., with \fx\ .
In the direction perpendicular to the axis the momentum will
only rarely be able to be greater than -A because the Fourier \
coefficients of such waves become very small. But the momentum
in the direction of the axis can be greater because the shorter
wavelengths appear in the shock wave front proper. Therefore,

mesons with the energy k are generally emitted only in
an angular region of the order of magnitude */* \ about the
two primary directions. The heavier mesons are also emitted
principally only in the shock wave front.

II. Solution of the Shock Wave Equation

a) The expansion of the shock wave depends on the form
of the nonlinear wave equation based on the mesons. But it
can be shown that there is a limiting case for "strong"
interaction in which the spectral distribution of the mesons
can be stated independently of the particular form of the wave
equation.

If we consider first only the spectral distribution, and / 69
not the directional distribution, the solution of the non-
linear wave equation can be eased by some simplifications:
Consider the plane in which the emission occurs to be extended
to infinity, and the layer infinitely thin. Then, because of
the Lorentz invariance of the wave equation, ?l can depend
only upon s = 2 2 — * 2 \ . The partial differential equation thus
transforms into an ordinary differential equation, the
solution of which can be discussed more easily.

Two nonlinear wave theories will be considered as examples:

1. The equation discussed by Schiff [10] and Thirring [12]
in relation to the nuclear forces :

2. A wave equation which arises from the Lagrange function

following the pattern of the earlier work by Born [1]. Some
time ago, Born commented that nonlinear theories of this type
have singular solutions in a smaller degree than the linear
theories. That was used at the time for the self -energy of
electrons, but it also applies to meson production. Previous
studies on meson production have already been based on the
Lagrange function (5) [4],

On 1: For <p="p{ s )\ the first of these two equations trans-
forms into

4^(s4f)+^+^ = °" \ (4a)

For n = o\ one returns to the linear wave equation (1) and
the solution is then

<P

'«/o(*l$ I for S> °

(6a)
v = ° I for s<o,

Here a is a constant of integration; see also Figure 2. For
jj=j=o\ one can give an exponential series expansion with s = 0:

<p = a[l — (x 2 ~+ 7}a?)s + i [v? + 3?; a?) («« + v fl 2 ) s 2 - -\ ]

j f or 5 >0
-0 for s<0.

(6b)

We see immediately that (4) deals with a "weak" interaction

which changes nothing with respect to the discontinuity of the

wave function at the shock wave front. This is related to the

fact that the theory characterized by (4) is one of the group / 70

of renormalizable theories. The coupling parameter, y\ , has

the dimension of a pure number. It has already been established

in various ways that the renormalizable theories contain only

"weak" interactions which do not in general give rise to
multiple production of mesons.

0n| 2: The situation is different, however, for the wave
equation characterized by (5). For <p=<p{s)\ it reads:

(7)

If we assume that « = o\ (vanishing rest mass of the mesons),
then the solution can be written immediately:

for s^o

= o

(8)

In the general case («4=o)\ we can again state series
expansions. We set

<p = -j*f(0r -C = sk s

and obtain

/ff)^(i + -f + ^i^-'£«4—-) I for t<*| ]

=

[ft + &)

for ■ ?.» 1 ,
for C^o.

(9)

(10)

The constants y and d|are unambiguously determined by the
integration constant, a, but their values have not been
calculated.

One can see that here the nonlinear ity has extensively
changed the nature of the solution. The discontinuity of <p\
at s = has disappeared. Only <p'\ behaves discontinuous ly,
In the vicinity of s « 0, v] behaves like] }'*/.. ]■ '

8

If one expands <p{s) = <p(x, t)\ at a given time into a Fourier
integral according to the wave number, k, then, except for
constant factors, one obtains an expression of the form

for the coefficients tp(kj)\ for large values of k (£~£ 3>*}\. The

factor t 2 clearly arises because of the fact that during the

entire expansion process, energy flows continuously from the

head of the shock wave into the other parts of the wave, and,

therefore, into the lower frequencies. Actually the supply

of energy in the head of the shock wave is infinite here. This

is a necessary consequence of the assumption that the shock / 71

wave begins in an infinitely thin plane layer, because from

this assumption we concluded that the solution of <p(x, t)\ depends

only upon t*—x*\ > and is therefore invariant with respect to

the Lorentz transformation in x,t space. But a finite energy

momentum vector would indicate a direction in this space,

and thus could not be part of an invariant solution.

Actually, of course, the shock wave s£arts in a layer of
finite thickness ~ ^ • The energy-momentum vector is
finite and the rise of the Fourier amplitudes in (11) comes
to a stop after a certain time when the energy supply of the
wave front is exhausted. Then the Fourier coefficients for
large values of t fall off more strongly than as k '
as a function of k for A > A o» = y^ L Thus, one obtains for
the intensity distribution

"£- const £ I for ' x^S*..-^ \ (12)

and

—— = const -r-/

(13)

for the same region.

This is the form of the spectrum which was discussed
previously in relation to multiple generation [4, 5] and is
also presented in Figure 3.

The wave equation (7) taken from Born's theory [l]
represents a typical case of a "strong" interaction and leads
to multiple production of mesons. The coupling parameter
has the dimension of the fourth power of a distance.

b) Now it will be shown that the spectrum (12) and (13)
quite generally corresponds to the limiting case of strong
interaction, independent of the particular form of the
Lagrange function and independent of the special properties
of the particles involved.

We begin; with an arbitrary Lagrange function for a
scalar wave function <p\ and its first derivative fyisx^ .
Because of the Lorentz invariance, L can depend only on 9>\
and 2(?*J | * For ver Y small values of <p\ and *d<pjdx T \, L must

transform into the Lagrange function of the ordinary wave
equation (1). Now we inquire about the value of 2j{TxJ \ in
the vicinity of s = o(s>o)| . For s-+o\ , 2(101 can e it her become

infinitely large, take on a finite value, or approach zero.

Next, we can exclude the last of these three possibilities, / 72

because then the nonlinearity would play no part just at the

critical point, s = 0. But that is impossible because for

the usual wave equation (1), (1) 2(11") at tlie critical point\

is by no means zero, but infinite.

Of the two remaining possibilities, the second obviously
gives the smoother curve for y\ at the singular point. Thus,
it corresponds to the stronger interaction. Here, in the

10

vicinity of s = 0, we get

.2(J5)T— «ufr— !- (+0 H 4, " ) ' I u*>

from which

<p(s) ---'const J/S j / -I r \

so that behavior as in (7) and (10) follows.

c) But one can give a still more general proof for (12)
and (13), which also applies for arbitrary particles of high
spin value. It has already been mentioned under I la that in
the limiting case in which the shock wave begins in an
infinitely thin layer its total energy content must be infinite,
because the wave function is then invariant to rotations in
x,t space. Now the energy spectrum of the mesons falls off
more steeply the greater the energy dissipation due to the
interaction is. Thus, to the extent that the spectrum has
the form of a power law at \ all (and that could apply for most
of the simple wave equations), it cannot fall off more sharply
than in (12) and (13), because here the total energy still
diverges for '■Xo*,-+ 00 \ (namely, logarithmically). The spectrum
(12) and (13) therefore just corresponds to the limiting case ;
of strong interaction. Thus, as has already been said, the
Lagrange function (5) taken from Born's theory gives only a
special example of a theory with strong interaction. But
the spectrum, (12) and (13), remains correct also for very
much more complex Lagrange functions which contain various
types of mesons as a solution for the limiting case of small
interaction, if we deal with a theory with strong interaction.

11

III. Application to Meson Production

The multiple production of mesons will now be treated
quantitatively, with the assumption, of strong interaction.

a) One of the most important quantities for character-
ization of a meson shower is the average energy of the mesons
in the center-of-mass system. To a very crude approximation,
one can consider the spectrum (12, 13) as exactly valid between
Ao = «j\ (rest mass of the type of mesons concerned) and ^o—^m-\

Then we have

/ 73

Ao tn

(16)

and it follows that

b~ — * —

]g k o>»

.1 —

*i

for h 0M >x i .

(17)

For k 0lH £,r. t \ the type of mesons concerned would not occur at all.

In reality, the spectrum will have to contain the factor
kdk just because of the phase space volume, and will not
have the form of (12, 13) at all for small k. Furthermore,
it will not disappear completely for A >^om\> but only
diminish more strongly than in (12) and (13). One can try

de i = A i -

kdk n

*i i

"ami

(18)

as probably a somewhat better solution. Then we obtain

12

■• - Ai •' r -(l+2^-2«l/T+^),

x; 4

(19)

_ t + yT+^ lg i±Vl±^

k~-x± «

(20)

where we set xjk 0m = <x\\

Both approximations, (17) and (20), are plotted as functions
of lg(i/a)\ in Figure 5. The difference between the two curves
gives a measure for the inaccuracy of the entire estimate.

It appears from these calculations that in the limiting
case of strong interaction|,|the average meson energy increases
only logarithmically and that, therefore, the number of mesons
increases almost in proportion to the energy transferred into
the meson field in the center-of-mass system.

b) To be sure, the relations are complicated more by the / 74
occurrence of new types of mesons at higher energies. We can
assume that for sufficiently high values of M*^"^ the
relative proportion g^ of the meson species] ^is* independent of]
k , and depends only on the form of the shock wave equation.
In this region, then, the various species of mesons generally
occur in comparable frequency, but the g. need not be
simply proportional to the statistical weight of the species
concerned. We normalize

2ft.-* \ (21)

and set

A<-g t A. \ (22)

13

Figure 5

0,5

Eq.

. ( "L-

_ l

^-"—T"" ' ^

'Eq._C26)l

123*5678$

Figure 6

773]

Then, in the rough approximation of (16) and (17) we have

therefore?,!'

' i-~i K, \

x;

x;

=

for

. *Q»

for r-i^-Kt

(23)
(24)

(25)

For large A „,(\ w »k,-)\, therefore, the numbers in the various
groups of mesons behave as &M\ . As k decreases, the numb(
of heavy mesons decreases faster than that of the lighter. As

soon as k decreases below the value «; Lithe species of mesons

om ' ' c

in question disappears completely. Instead of (25), then,
in the approximation of Equations (18) to (20),\we would have

Si 4

^(1 + 24-2^+4

^-i + l/i+.iis— —Lj

(26)

14

The factor of &/x ( \ in (25) and (26) which is characteristic J-±=-
for the dependence of n. on k is shown graphically in Figure 6. ;
With the second approximation formula, there would still be
a small number of mesons of the type *,\ remaining even for
hm<'< i \ , as is also to be expected physically. \

c) If one wishes to make statements about the total
number of mesons emitted, one must also know the total energy,
e\, of the meson field in (25) and (26). For this quantity,
we can at first state only a maximum value: e\ can be no
greater than the kinetic energy of both nucleons in the
center of mass system before the collision.

Because in general only a fraction of this energy is
actually transmitted to the meson field, it is convenient to
introduce this fraction, y, as the "degree of inelasticity"
of the collision. Then we have (M = mass of the nucleons):

s = y-2M

(vr=F~ 1 )"l (27)

where

One would expect that for a central collision y would have a
value near 1, while only a small fraction of the kinetic
energy will be transferred to the meson field for a grazing
collision.

If we call the distance between the centers of the nucleons
at the moment of collision b, then we can consider the
overlap integral of the *■{ meson fields of the two nucleons
as a measure for the strength of the interaction. If one
simply sets y equal to this overlap integral as a very crude
estimate of the degree of inelasticity, one gets

Y = e-»* t \ (28) -

15

in which} x\ specifically signifies the mass_ of the , tt\ mesons.) ■'■'■
It follows from this that the effective cross section for a
value of v\ between y\ and Y^-^y\i.s

to-ixbdb^g-fisft). \ (29)

If one wishes to define a total effective cross section,
one must define a minimum value of v\ . For instance, if one
wishes to determine the total effective cross section for
multiple production, one must establish as the mimimum value
of v\ that which will produce at least two mesons.

^fc^V (30)

.¥m'ui '

(Here E refers to the lightest type of mesons, i. e. , to
the ^ mesons . )

From (30) it follows that / 76

a = ^is'y^ \ (31)

and

y = - fg2 y — - ymin -r Ymin lg ^min) \ ,^2)

!

It must be emphasized that the estimation of the frequency \
distribution of the values of y i n Equations (28) to (32) I
is independent of the preceding considerations on the |

expansion of the shock wave, and must be considered as less
reliable. So far there is not enough observational material
to determine the frequency distribution of y experimentally.

Table 1, following, gives the total effective cross
section, the expected values of y, «,,\ and n x \ (number of the
:r-\and w-j mesons, respectively), their average energy, and,

16

TABLE

It*

E

10

,0. i

10'

10*

BeV

a

o,is

0,49 I

0,85

+.3

10-" cm 2

V

0,34

0,19 :

0,13

0,09

n !t

3.6 ±0,7

4,2±0,S !

5,2±0,S

s,o±i

«X

—

0,9+0,2 j

2,0 + 0,4

3,4 ±0,6"

"On

0,25 + 0,04

0,36 + 0,04 i

o,so" ±o,os

0,67 + 0,06

BeV

^Oa

— "*

1,0±0,2 ;

1.4±0,1S

2,0±0,1S

EeV

l«* i

iO,7±i

2-z/i ± 4 i

+0,5 ±6

39 ±12

■*

—

4,7 + 1 |

15±6

38±6

Translator's note:j Commas in numbers represent decimal points

finally, the number of mesons in the limiting case of y = 1
as functions of the primary energy, E (in the laboratory system).
Other types of mesons such as Hand x\ mesons are not con- \
sidered. In addition, we arbitrarily set .g x = 2g :i \ , that is,

g^=i>S K ~t\t i n order to take into account the relatively
great frequency of the ^ mesons found according to the newer
measurements in Bristol. These numbers will have to be revised
later on the basis of more accurate measurements. We use 0.61
BeV for the mass of the « J i meson. In order to express the
inaccuracy of the theoretical estimate, we have taken the
average of values obtained from (16, 17) or (18) to (20)
(except for the first two columns) and have listed half the
difference as the error.

d) The angular distribution of the emitted mesons appears
from the perceptual considerations in I. Of course, the
details of the angular distribution still depend on the shock
wave equation. But, quite generally, the momentum of the
mesons perpendicular to the primary direction will only rarely
be able to exceed fhe value x\ to any extent. As a rule, mesons
with the energy k are emitted in an angular region of the
magnitude %jk \ about the axis. The distribution of the ^mesons

/ 77

17

is, therefore, always anisotropic, while in the center of
mass system the distribution of the slower k} mesons will be
to some extent isotropic.

IV. Comparison with Experience

So far, only a few meson showers have been observed with-
out gray or black tracks . Only for such showers can one
assume with some probability that they were from collisions
of only two nucleons, without involvement of a larger
atomic nucleus. But if one also includes showers with a few
(two to three) thick tracks to test the theory, the perturb-
ation of the shower by the atomic nucleus will generally be
small. But because a secondary scattering of the mesons
produced could have taken place at the atomic nucleus, the
determination of the primary energy from the angular distri-
bution and the evaluation of the angular distribution itself
become very unreliable.

TABLE"

"2p

£

30

40

f 40

90

130

iOOO

2000

j 30000 BrV

«l + «x
yemp

9
' 0,5t

IS
0,8

1 ^

\ 1,0

10

o,3S

IS
0,61

9

0,16

12
0,17

i 21

! o.i

* Translator's note: Commas in] numbers indicate decimal points

Observations of showers suitable for comparison with the
theory have been presented so far by Teucher [11], the working
group at Bristol [2], by Schein et al. [9], Pickup and
Voyvodic [8] and Hopper, Biswas and Derby [6]. If one tries
to estimate the primary energies from the angular distribution
(which is quite uncertain in some cases) according to the
reports in the publications, we obtain the meson numbers for
the eight observed showers in the second column of Table 2,
if we assume a ratio of 1:2 for neutral to charged mesons.
These numbers are already somewhat uncertain because of the

18

neutral mesons. If we assume that the last two columns in
Table 1 are correct, we obtain an empirical value of y f° r
each of these showers, which is shown in the third column of
Table 2.

We note first that the meson numbers are actually not
unambiguous functions of the primary energy. The y values
fluctuate strongly, as was to be expected. But they are on
the average somewhat greater than would have been conjectured
according to Table 1. This could be due to the fact that
small showers can be surveyed more easily than large ones;
but it could also mean that the estimate in Equation (28) is
still too rough *• Also, the empirical values of y in/
Table 2 are themselves still quite uncertain because, for / 78
instance, the proportion of a-\ mesons is not accurately known.
Perkins [7] also reports relatively high y values, but one
must await still more experimental material.

It has been possible to measure two showers (Teucher, [11] »

and Hopper, Biswas and Derby [6]) so accurately that the average

energy of the mesons in the center of mass system could be

reported. In the first case (40 BeV, some 25 mesons) the

observed average meson energy is 0.29 BeV, compared with 0.31

BeV according to Table 1. In the second case (1,000 BeV, some

9 mesons) there is some uncertainty because of the possibility

*( Comment added in proof. At the Copenhagen Conference in
June, 1952, LeCouteur mentioned that the expected value of
Y in heavy material (e. g., in the photographic emulsion)
must be considerably greater than in hydrogen (Table 1
refers to hydrogen) because "grazing" collisions can
occur only with nucleons at the edge of the atomic nucleus.
Powell has also reported on new experiments indicating
that the particles designated here as x\ mesons can be
separated into two groups with masses of 0.74 and 0.54
BeV, with quite different properties.

19

that some of the particles observed could have been *i mesons,
not taken into consideration by the authors. (According to
Table 1, one should expect some 3 ^mesons among 9 mesons.)
If we ignore that, the observed average ^ meson energy in the
center of mass system was 0.44 BeV, compared to 0.50 BeV
according to Table 1. Thus, these two measurements confirm the
relatively low meson energy of Table 1. ^ n t^e other hand,|
Perkins [7] reports the value of 1.5 BeV as the average energy
of mesons from a series of showers with a primary energy of
10 to 10 BeV. This is considerably higher. Here, though,
we must consider the uncertainty in the measurement of the
primary energy. Any error in the primary energy generally
increases the average meson energy, as this has the smallest
value just in the center of mass system.

On the frequency of the k j , mesons we have as yet only the
statement of the Bristol group that it is comparable with that
of the ^, mesons at high energies [7]. For the present,
this ratio cannot be determined from the theory. (in Table
1, we arbitrarily set gJi^'A .)

With respect to the angular distribution, it is observed
that the distribution in the center of mass system is rather
isotropic for showers of low energy, while distinct accumulations
appear about the primary direction and the opposite direction
in showers of high energy. This corresponds exactly to the
picture of Ic. In fact, mesons of high energy appear always
to be distributed anisotropically, and in particular, quite
generally, the «-\ mesons (Perkins [7]). The degree of the
anisotropy also corresponds to the theoretical estimate.

On the v whole, then, one has the impression that the
formulas derived in III under the assumption of "strong"

20

interaction satisfactorily represent experience; and that,
therefore, the interaction of the elementary particles at
high energy actually belongs in the group of the "strong"
interactions first studied by Born.

REFERENCES

1. Born, M. Proc. Roy. Soc. Lond.,Ser. A, Vol. 143, 1933,

p. 410.

Born, M. and L. Infeld. Proc. Roy. Soc. Lond.,Ser. A,

Vol. 144, 1934, p. 425; Vol. 147, 1934, p. 522; Vol. 150,
1935, p. 141.

2. Camerini, U. , P. H. Fowler, W. 0. Lock and H. Muirhead.

Phil. Mag. Vol. 41, No. 7, 1950, p. -413.

3. Fermi, E. Progr. theor. Phys. Vol. 5, 1950, p. 570.

Phys. Rev. Vol. 81, 1951, p. 683.

4. Heisenberg, W. Z. Physik, Vol. 113, 1939, p. 61.

5. Heisenberg, W. Z. , Physik, Vol. 126, 1949, p. 519.

6. Hopper, V. D. , S. Biswas and J. F. Derby. Phys. Rev. Vol. 84,

1951, p. 457.

7. Mr. Perkins has kindly provided us with the results of his

next work before publication.

8. Pickup, E. and L Voyvodic. Phys. Rev. Vol. 82, 1951, p. 265.

Vol. 84, 1951, p. 1190.

9. Schein, M. , J. J. Lord and J. Fainberg. Phys. Rev. Vol. 80,

1950, p. 970; Vol. 81, 1951, p. 313.

10. Schiff, L. J. Phys. Rev. Vol. 84, 1951, p. 1.

11. Teucher, M. Naturwiss. Vol. 37, 1950, p. 260; Vol. 39,

1952, p. 68.

12. Thirring, W. Z. Naturforsch. Vol. 7a, 1952, p. 63.

Translated for National Aeronautics and Space Administration under
contract No. NASw 2483, by SCITRAN, P. 0. Box 5456, Santa Barbara,
California, 93108.

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