GIFT OF
MOLECULAR DIFFRACTION OF LIGHT
MOLECULAR DIFFRACTION
OF
LIGHT
0. V. RAMAN, M.A., HON. D.Sc.,
PALIT PROFESSOR OF PHYSICS IN THE CALCUTTA UNIVERSITY
PUBLISHED BY THE
UNIVERSITY OF CALCUTTA
1923
Gift
PRINTED BY ATDLCHANDRA BHATTACHARYYA,
A I THR CALCUTTA UNIVERSITY PRESS, SENATE HOUSE, CALCUTTA.
To
SIR ASUTOSH MOOKERJEE
with
the author's warmest admiration and esteem.
.*
518434
PREFACE
The fundamental importance of the subject
of molecular diffraction came first to be recog-
nized through the theoretical work of the late
o
Lord Rayleigh on the blue light of the sky,
which he showed to be the result of the scatter-
ing of sunlight by the gases of the atmosphere.
It is proposed in this small volume to review
the present position of the subject and to dis-
cuss the general theory of the molecular scatter-
ing of light in all refractive media, including in
a comprehensive survey the case of gases,
vapours, liquids, crystals and amorphous solic^.
Experimental observations in support of the
theory are detailed, and reference is made to
various phenomena in which molecular diffrac-
tion plays a part. The case of moving media
is also briefly dealt with.
In the course of the work, references are
made to cases in which the classical wave-
theory apparently fails to explain the facts
relating to molecular scattering in a satisfactory
manner, and in the final chapter an attempt is
made to consider these cases in terms of the
conception introduced by Einstein in 1905 that
light is not continuous wave-motion but consists
of discrete quanta moving through space,
viii . PREFACE
In the writing up of this essay, I have
been greatly assisted by the researchers
working in my laboratory, references to whom
will be found in the pages below. To these
gentlemen, I wish to express my heartiest
thanks. I am specially indebted to Mr. K. R.
Raman athan M.A., Madras University Research
Scholar, for very valuable help in the
preparation of the volume and in the carrying
out of the experimental work. I am also under
great obligations to Sir Asutosh Mookerjee,
Vice-Chancellor of the University, for the
co-operation which made the publication of the
volume possible.
I desire also to record my indebtedness to Mr.
A. C. Ghatak, B.A., Superintendent of the
University Press and his staff for the quick and
efficient manner in which the volume has been
printed and got up.
j
CALCUTTA
C. V. RAMAN
llth February, 1922. '
CONTENTS
CHAPTER I
PAGE.
Fundamental Principles ...
CHAPTER II
Scattering of Light by Gases ... 14
CHAPTER III
Atmospheric Scattering and Twilight
Phenomena
CHAPTER IV
Molecular Scattering in Liquids
CHAPTER V
The Colour of the Sea and the Albedo
of the Earth ...
CHAPTER VI
Scattering of Light in Crystals ... 76
CHAPTER VII
Scattering of Light in Amorphous Solids 85
CONTENTS
CHAPTER VIII
PAGE.
The Doppler Effect in Molecular
Scattering ... ... ... 88
CHAPTER IX
Molecular Diffraction and the Quan-
tum Theory of Light ... 95
MOLECULAR DIFFRACTION
OF LIGHT
CHAPTER I
FUNDAMENTAL PRINCIPLES
1. The whole edifice of modern physics is
built up on the fundamental hypothesis of the
atomic or molecular constitution of matter. fti
common with other branches of the science,
physical optics has to concern itself intimately
with the attributes of these molecules or atoms
under different conditions and in different
states of aggregation and the manner in
which they determine the observed properties
of substances. The propagation of light
through refractive media is in a special
degree related to and determined by the
molecular structure of those media. The ques-
tion is, does any departure from perfect regular-
ity of the light-propagation arise from the
discontinuous structure of the medium ? The
% MOLECULAK DIFFRACTION OF LIGHT
answer to this question forms the subject of the
present essay. Under the description of the
Molecular Diffraction of Light, we may include
such deviations from simple wave-propagation
as can be attributed to the ultimate structure
of matter.
Rayleigh's Theory
2. The principles on which the problem of
molecular diffraction may be handled, at least in
the case of gaseous media, were first indicated
by the late Lord Eayleigh in one of his well-
known papers on the origin of the blue of the
sky.1 Reduced to its essentials, as has been
done by Schuster,2 the treatment is on the follow-
ing lines. The individual molecules in a gas
through which the primary waves of light pass
are regarded as secondary sources of radiation,
each molecule acting more or less as it would
in the absence of its neighbours. There is of
course a definite phase-relation between the
primary wave when it reaches a given molecule
and the secondary wave emitted from it. In the
direction of propagation of the primary waves,
the secondary radiations emitted by all the mole-
cules in a given layer are in identical phase, for,
the differences in the phase of the primary wave
1 Philosophical Magazine, XL VII. 1899, pp. 375-384, Scientific
Papers, Vol. IV, p. 397.
* Theory of Optics, 2nd edition, p. 325.
FUNDAMENTAL PRINCIPLES 8
when it reaches different molecules are exactly
compensated by the acceleration or retardation
due to the scattered waves having to traverse a
shorter or greater path, as the case may be. In
other directions, however, owing to the molecules
being distributed at random within the volume
of the gas, the phases of the scattered waves do
not stand in any invariable relation to each other,
and hence, in order to find the average expecta-
tion of intensity of the scattered light emerging
from within the gas, the intensities, not the
amplitudes, of the waves scattered by the indivi-
dual molecules should be added up. "In all ordi-
nary cases, there is very little difference of phase
between the primary wave reaching an individual
molecule and the secondary wave sent out by it
so far as the direction of original propagation of
the wave is concerned. When, however, tlte
effect of all the molecules contained in a stratum
parallel to the plane of the primary wave is in-
tegrated by the usual method of sub-division of
the stratum into Eresnel zones, and the result-
ant is combined with the primary wave, a change
of phase appears which may be identified as the
retardation associated with the passage of waves
through a refractive medium. A relation is
thus obtained between the scattering power of
the molecules, their number per unit volume
and the refractivity of the medium. Thus, tak-
ing the light vector in the primary waves to be
4 MOLECULAR DIFFRACTION OF LIGHT
represented by R0 cos (<»t — 1%) where # is measur-
ed from the position of the scattering molecule,
the vector in the scattered wave arising from it
may be written as
Al cos («>t — Ir). RQ sin ojr ............. (1).
when r is the distance from the molecule and 0
is the angle between the vibration at the origin
and the scattered ray. On carrying out the
calculations indicated, the relation obtained is
w A* - ^'O*-!)1 (^
1 i ~ NX* ............... (2)'
where N is the number of molecules in unit
volume, X is the wave-length of the light and /*. is
the refractive index of the gas.
3. The energy scattered by the molecules
in the interior of the gas must be derived from
the primary beam, and hence the intensity of
the latter must suffer an attenuation as it passes
through the medium. The coefficient of attenu-
ation may be readily evaluated by a simple
calculation of the total energy scattered by an
individual molecule and then multiplying this
by the number N of molecules per unit volume.
We thus obtain the intensity of the transmitted
light to be I=I0e-*c where * the coefficient
of attenuation is given by the relation
3)
3 NX*
This expression for the attenuation
coefficient may also be derived directly by
FUNDAMENTAL PRINCIPLES 5
a more accurate investigation which does not
neglect the small difference of phase between
the primary wave and the secondary waves
originating at a molecule. It is then found
that on compounding the effects of the primary
wave with those of the secondary waves arising
from a stratum of molecules, there appears in
addition to the alteration of phase of the
primary wave, also a small diminution in
its intensity which is exactly that expected
in view of the lateral scattering of part of
the energy.
Criticisms of the Theory
4. In connection with the treatment outlined
above, two distinct points come up for remar^:.
According to Lord Rayleigh's treatment, (/A— 1)
is proportional to the number of particles per
unit volume of the gas, in other words varies
directly as the density when the pressure is in-
creased. In other words, the theory leads to
Gladstone and Dale's law for the relation bet-
ween refractive index and the density. , It is
well-known however that this law is only an ap-
proximation, a more accurate relation between
the refractive index /*- and the density P being
the Lorentz-Mossoti formula
a2—!
a i o — constant, p (4).
6 MOLECULAR DIFFRACTION OF LIGHT
The derivation of this formula has been
discussed by Hayleigh1 and it is clear that to be
quite strict, the treatment of the problem of
molecular diffraction should be modified so that
it leads to (4) as the relation connecting the
density with the refractive index. It may be
mentioned however that in the case of gases at
ordinary pressures the error involved in overlook-
ing this point would not be appreciable.
5. A more important question is the justi-
fication for the view that the phases of the
waves laterally scattered by the individual
molecules are absolutely at random, so that
their energy-effects are additive. In the earlier
treatment given by Lord Rayleigh, this was by
no means made entirely clear, and on a super-
ficial view of the matter it might be questioned
(as indeed it has been by Sir Joseph Larmor)2
whether the phase relation of the scattered
waves arising from the molecules in any small
volume is indeed in reality subject to such large
and arbitrary variations that the energies due
to the individual molecules may be summed up
without any sensible error from their mutual
interference. Larmor points out that in the
1 Philosophical Magazine, Vol. XXXIV, pp. 481-502, 1892,
Scientific Papers, Vol. IV, p. 19.
2 Phil. Mag., Jan. 1919, p. 161. These comments of Larmor were
made with reference to a paper by Rayleigh in the Phil. Mag. for
Dec. 1918, dealing with the general problem of the light emitted
from a random distribution of luminous sources.
FUNDAMENTAL PRINCIPLES 7
case of a gas at atmospheric pressure, there are
106 molecules in a cubic wave-length so that the
scattered waves arising from adjacent molecules
differ in phase by only 10 ~2 of the period and
asks, in view of this closeness of packing of
the molecules whether, if they could be regarded
as fixed while the radiation was passing, they
ought not in conjunction to scatter far less
than they would do separately ? The difficulty
will perhaps appear even more acutely if we
consider a gas at fairly high pressure, say 64
atmospheres. The average difference of phase
for adjacent molecules is in this case only
2*5xlO~3 of a period, and the mean free path
of a molecule would be only about 0*02 X 10 ~5
cms., that is, less than ^0 of the wave-length of
sodium light. Could we in the circumstances
consider the phases of the scattered waves aris-
ing from individual molecules to be distributed
entirely at random ?
6. The difficulty referred to in the preceding
paragraph appears to have impressed Larmor so
greatly that in order to find a way of escape
from it, he has suggested a somewhat different
logical basis for the empirically observed correct-
ness of the result deduced from E/ayleigh's
theory. It seems best to quote Larmor's own
words : " The molecules of the atmosphere are
in thermal motion, with velocities in correlated
directions which are at ordinary temperatures
8 MOLECULAR DIFFRACTION OF LIGHT
of the order of 10 ~6 of that of the radia-
tion. The wave-length of the radiation scatter-
ed from them will thus vary within a range of
10~~6of itself. If the phases of the scattered
radiations are correlated at first, after traversing
10 6 wave-lengths or 50 cms. they will have
become fortuitous, and the energy effects thus
additive. This consideration, if justified would
find the source of Lord Rayleigh's principle in
the uncoordinated thermal motion of the mole-
cules."
Justification of the Principle of Random
Phase
7. With regard to the remarks by Larmor
quoted above, it may be pointed out that the
difficulty raised cannot be evaded in the
manner proposed by him. The suggestion
made is that the phases of the scattered
radiations emerging from the column of gas
may be correlated at first but after traversing
106 wave-lengths or 50 cms., they would
have become fortuitous and the energy-effects
thus additive. If this were correct, we should
find that the aggregate intensity of the scatter-
ed light should be small immediately after
emergence from the column of gas, that is
when it is observed within a distance of a
centimetre or two from the track of the primary
FUNDAMENTAL PRINCIPLES 9
beam, and should increase at a greater distance
from it. Such a result is obviously quite in-
admissible, besides being contrary to experience.
The fallacy lies in the assumption that the
change of wave-length (Doppler effect) has an
effect on the relative phases of the scattered
waves, whereas in reality it has none. To make
this clear, we may consider two neighbouring
molecules A and B. The scattered waves originat-
ing £rom them travel outwards with an identical
velocity which is quite unaffected by any move-
ments of these molecules. The phase-differ-
ence at any epoch therefore remains unaffected
as the waves move out, being exactly the same
as when the portions of the wave-train under
consideration left the molecules. In other
words, the phase-difference at each stage is e^-
actly the same as if the molecules had remain-
ed fixed from the instant of emission of the
scattered light. The scattering from any appre-
ciable volume of gas would thus remain unaffect-
ed if all the molecules were assumed suddenly
to be fixed in their instantaneous positions, and
the Doppler effect due to their movement exerts
no influence whatever on the observed results.
Larmor's suggestion therefore clearly fails.
8. What then is the justification of Ray-
leigh's principle ? The answer to this question
becomes plain when we consider the implications
contained in the propositions under discussion.
2
10 MOLECULAR DIFFRACTION OF LIGHT
In order that the phases of the scattered waves
arising from the individual molecules should be
entirely fortuitious, it is clearly necessary and
sufficient that the distribution of the molecules
in the space enclosed within the walls of the
containing vessel should be itself entirely
fortuitous. This again in its turn would be true,
if the probability that a given molecule is found
within a small specified volume is independent of
the presence of any other molecules, in ether
words if the probability that two or more given
molecules are found together within a specified
space is the product of the probabilities of each
of them separately being found within the space.
This will be true provided the total volume of
the molecules or rather of the spheres of in-
fluence within which their mutual action on
each other is sensible forms a sufficiently small
fraction of the total space occupied by the mole-
cules. This is precisely the condition necessary
that the relation between the pressure and
volume of a gas should be that given by Boyle's
law. In other words, we have a truly random
distribution of the molecules provided the
compressibility of the gas at the pressure under
consideration does not appreciably deviate from
that derived from Boyle's law. So long as this
is the case, Rayleigh's principle must be sub-
stantially valid, and neither the closeness of the
packing nor the smallness of the free path of
FUNDAMENTAL PRINCIPLES 11
the molecules in relation to the wave-length of
light can influence the result appreciably.
9. A precisely similar result is also arrived at
if we investigate the condition necessary that
the light scattered by an appreciable proportion
of the molecules in the given volume may be
extinguished by their mutual interference. It
is obvious immediately that if the molecules be
distributed uniformly throughout the containing
vessel, we may divide up the entire volume into
a large number of very small equal elements
each containing a few molecules, and take them
off in pairs situated at such distances from each
other that in any specified direction, the scatter-
ed waves from the components of each pair
differ in phase by -* and therefore cut each
other out by interference. In such a case,*^
is clear that there would be no scattered light
emerging from within the gas. (A few elements
of volume might be left over surplus and un-
compensated near the boundaries of the vessel.
These would give a surface-effect with which
we are not here concerned.) If however we
attempt to apply similar reasoning in the case
of an actual gas the argument breaks down.
The distribution of the molecules is no doubt
such that the density of the medium does not
vary by any appreciable fraction of itself when
we consider any appreciable volume, say one
cubic wave-length. But when the sub-division
12 MOLECULAR DIFFRACTION OF LIGHT
of the space is carried further, deviations from
the equality of the number of molecules present
in equal elements of volume become relatively
more important, until finally when we consider
volume-elements of molecular dimensions the
probability that a molecule will be found inside
such an element becomes small and in the case
of a gas obeying Boyle's law with accuracy,
vanishingly small. Thus if we take two volume
elements of molecular size at a distance exactly
A/2 apart, the expectation that they would both
simultaneously hold molecules whose effects
would mutually extinguish one another is
vanishingly small. Thus again we see that no
appreciable proportion of the energy scattered
by the individual molecules is taken off as the
result of interference.
10. The foregoing discussion makes two points
clear. The validity of the principle of random
phase depends on the conditions being such that
the compressibility of the medium is given
with sufficient accuracy by Boyle's law. Second-
ly, the ultimate justification of the principle
rests on the complete non-uniformity in the
spatial distribution of the molecules in so far
as very small volume elements are concerned.
As we shall see later on, it is precisely these
factors, namely, the compressibility of the
medium and the non-uniformity of the spatial
distribution of the molecules, which enter into
FUNDAMENTAL PRINCIPLES 13
the general theory of light-scattering developed
according to the principles laid down by Einstein
and Smoluchowski, and which, as has been pointed
out by these writers, in the case of gases obeying
Boyle's law leads to results substantially identical
with those obtained from Rayleigh's formula.
It is important therefore to notice that in res-
pect of gases at any rate, the special theory
developed by Rayleigh and the more general
theory of Einstein and Smoluchowski rest on
exactly the same logical bases and differ only
in the detailed mode of calculation of the in-
tensity of the light scattered.
CHAPTER II
SCATTERING or LIGHT BY GASES
11. In view of the very satisfactory explana-
tion by Lord Rayleigh and Schuster of the blue of
the sky and the observed degree of transparency
of the atmosphere on the basis of molecular
diffraction, it became obviously a question of
great importance to detect, and if possible, to
measure, the scattering of light by dust-free air
in the laboratory. The first successful attempt
in this direction was made by Cabannes.1 Later
work on the experimental side of the subject,
including scattering by other gases and vapours,
has been done by Prof. R. J. Strutt 2 (the present
Lord Rayleigh), by Cabannes3 himself, by Smolu-
chowski4 and by Gans5.
12. The methods adopted by these investi-
gators are essentially similar. The gas is contained
in a cross-tube dead-blacked inside. An intense
Cabannes — Comptes Rendus, CLX, p. 62, 1915.
R. J. Strutt— Proc. Roy. Soc., XCIV, p. 453, 1918.
Cabannes— Ann. de Physique, Tome XV, pp. 1-150.
Smoluchowski — Bulletin De la Academie Cracovie, p. 218, 1918.
R. Gans— Ann. der Physik, 10, 1921.
SCATTERING BY GASES 15
beam of light is sent along one of the tubes, and
the scattered light is observed in a perpendicular
direction. Owing to the extreme faintness of
the scattered light, the background has to be
perfectly black in order that the track of the
beam may be visible. The best arrangement to
secure this is that adopted by Strutt in his
later work. He used as a prolongation of the
observation tube a curved horn blown out of
green glass and covered outside with black
paint. The object of the glass horn is to reflect
any stray light that falls on its mouth
repeatedly towards the narrow end and thus
to absorb it. With such a background the track
of a beam of sunlight concentrated by a lens
in dust-free air is easily visible. Of course, the
gas under observation has to be carefully freed
from dust before introduction to the chamber
by slow filtering through a tube tightly packed
with cotton .wool, and in the case of gases at-
tacked by light, care has to be taken to exclude
rays having any chemical action.
Intensity and Polarisation of the
Scattered Light
18. According to Lord Rayleigh's calculation,
the intensity of the light scattered by one cubic
centimetre of a gas having symmetrical mole-
cules in a direction perpendicular to the incident
16 MOLECULAR DIFFRACTION OF LIGHT
beam should be proportional to (/*— I)2.1 The
experiments of Strutt led him to the conclusion
that this was so, within the limits of experi-
mental error. The following table gives his
results : —
Gas.
Scattered light.
Refractivity.
Air (assumed)
Hydrogen
Nitrous Oxide
1-00
0-230
3-40
1-no
0-229
3-12
Ether vapour
26-0
27-1
The careful experiments of Cabannes,1 showed
however, that although the law was true in its
main features, there were differences in the
value of the observed scattering from the calcu-
lated values too large to be explained as being
due to experimental error.
14. On the assumption of symmetrical mole-
cules, the light scattered in a direction perpendi-
cular to the incident beam should be completely
polarized with the electric vector perpendicular
to the plane containing the incident and scatter-
ed beams. Strutt examined the polarisation of
the scattered beam and obtained for the first
time the remarkable result that, in many gases,
the scattered light is only partially polarised.
1 Cabannes (loc. cit.) has calculated the scattering co-efficient
on the basis of the electromagnetic theory and obtains a value
2 Sir2
— (/i2-!)2. When (/x-1) is small, this reduces to f G*-l)2,
^?i A TtA
(See also Schuster Proc. Roy. Soc., XCV11I, p. 248.)
SCATTERING BY GASES 17
The experimental method adopted by Strutt
for the examination of polarisation was to place
a double image prism with its principal section
perpendicular to the incident beam in the path
of the scattered light and obtain an image of the
luminous track on a photographic plate. Two
images were in general obtained, a strong one
with the electric vector in the direction indicat-
ed by the ordinary theory and a weak one with
the electric vector in the perpendicular direc-
tion. The two images could be made of equal
intensity by inserting a nicol between the double
image prism and the camera and properly orient-
ing the nicol, and from the known angle between
the principal planes of the nicol and double image
prism, the ratio of the weak component to the
strong could be calculated.1
15. The imperfect polarisation of the light
scattered by gases has also been observed visually
and measured in experiments undertaken at the
author's suggestion by Mr. K. R. Ramanathan
at Calcutta. Eor this purpose, an apparatus was
used similar to that of Lord Rayleighand the gas
was illuminated by means of a concentrated beam
of sunlight, great care being taken to shield the
observer's eye from extraneous light. With air
at ordinary pressure, the intensity is not sufficient
1 la his earlier work, Strutt used a aeries of graded blackened
photographic plates in the path of the stronger component BO as to
get the intensities of the two components equal.
3
18 MOLECULAR DIFFRACTION OF LIGHT
to make more than a rough photometric estimate
feasible, but when we use carbon dioxide which
scatters nearly three times as much light as
air, fairly accurate measurements are possible by
visual observation. Such a comparison leads to
a value 10^ for the ratio of the weak to the
strong components as against 9*9^ obtained by
Cabannes and ll'T% obtained by Strutt. More
accurate measurements can be made visually with
the gases at higher pressure and an apparatus
is nearly ready for the purpose.
16. I give below for comparison the values of
the ratios of the weak component to the strong for
different gases obtained by Strutt and Cabannes.
The figures give the weak component as a
percentage of the strong component.
Gas. Strutt. Cabannes.
HB 3-83 Between 1 and 2
Na 4-06 „ 2-5 and 2-8
Air 5'0 „ 37 and 4-0
O2 9-4 „ 5'1 and 5'4
CO2 11-7 „ 9-5 and 9-9
Argon <0'5 <0'8
He <6.5
Strutt estimates the error of his results to be
not more than 6% . In view of the great care
that Cabannes also seems to have bestowed on
his work, it is remarkable that Strutt's results
should be systematically higher than those of
SCATTERING BY GASES 19
Cabannes.1 One reason that suggests itself for
this systematic difference is the difference in the
quality of the light employed by the two experi-
menters. Strutt used a carbon arc, while Caban-
nes used a mercury arc, the active radiations being
4358, 4046 and 3650 A.U., the rest of the radia-
tions being filtered out. Since both the experi-
menters used the photographic method, it is the
violet and ultraviolet that would have been most
effective. Considering the very great intensity
of the carbon arc in the region of 3000-4000 A.U.
it is possible that the effective wave-length in the
case of Strutt's experiments was smaller than in
those of Cabannes. The question of the influence
of wave-length on the ratio of the components
in the imperfect polarisation of the scattered
light is one of great importance, and is bemg
examined experimentally by Mr. Ramanathan at
the author's laboratory.
Explanation of Imperfect Polarisation.
17. The imperfect polarisation of the
scattered light has been explained on the basis of
a suggestion made tentatively in a much earlier
paper by the late Lord Rayleigh2 that the
molecules have three principal axes of sym-
metry and that they are oriented at random.
1 In his earlier work, Strutt got results which are in better
agreement with those of Cabannes.
' Phil. Mag. /Vol. XXXV, pp. 373-381, May 1918.
20 MOLECULAR DIFFRACTION OF LIGHT
His method consists in resolving the primary
vibrations along three mutually perpendicular
directions in the molecule and introducing
separate co-efficients of radiation for the diffe-
rent axes and integrating the effect due to
a large number of molecules in all possible
orientations. He obtains for the ratio of the
weak component to the strong in the scattered
radiation the value
+ Ca-AB-BC-CA
3(A2+B2+Ca)4-2(AB + BC + CA)
where A, B, C are three parameters character-
istic of the molecule and to some extent, depen-
dent on the frequency of the incident light.
Taking the imperfection of polarisation into
account, Cabannes has shown that the intensity
of the scattered light is not given by the formula
Since P differs for different gases, the inten-
sity of the scattered light would not be propor-
tional to the square of the ref ractivity, but to
SCATTERING BY GASES 21
18. The following table shows the nature
of the agreement between the observed1 and
calculated values according to Cabannes :—
Patio of Intensities of Scattered Light.
l
'
Observed (/*,—!)» / n» *+Pt
-
0-829 0-90 0823
Co« 331 2-53 312
Argon
Co« 2-62 2'35 2-65
Air
Co3
2-93 2-80 3-07
o,
0-255 0-276 0-255
O3
19. Sir J. J. Thomson2 has calculated the
ratio of the weak to the strong component in
the light scattered at different angles with
simple molecular models for the hydrogen
molecule and comes to the conclusion that, with
two positive charges at A and B and two
electrons rotating in a circle at the opposite ends
of a diameter in a plane bisecting AB at right
angles, the ratio of the minimum to the maximum
intensity of the components of the scattered
1 Cabannes, pp. 1-150, Ann. de Phys., 1920.
2 Phil. Mag, 393, XL, 1920.
22 MOLECULAR DIFFRACTION OF LIGHT
light would only be 0*4 per cent, while the
actual experimental value is nearly 4 per cent.
But with two electrons kept in equilibrium by
a modified inverse square law, a value for the
ratio nearly the same as the experimental
ratio is obtained. His calculations indicate
that although the polarisation is imperfect in
a direction perpendicular to the incident beam,
it may be perfect in a different direction.
Experimental work on the intensity of scattering
and polarisation in other than transverse direc-
tions might therefore prove of interest. Born l
and later, Born and Gerlach,2 have tried to
calculate the scattering on the basis of the
Bohr- Sommerf eld models of the molecules.
Their results also indicate a dependence of
the imperfection of polarisation on the frequency
of the incident light, the imperfection increasing
as the natural frequency of the molecule is
approached. The values which Born obtains
for the imperfection of polarisation do not
however agree with the experimental results.
The position appears to be, therefore, that models
based on the quantum theory have not yet
succeeded in solving the problem of molecular
scattering.
20. It is also pretty certain that Rayleigb's
law must break down when the frequency of
1 Ver. Deutsch. Phys. Gesellsch. 16, 1918.
2 Zeit. fur. Physik, 374, 1921.
SCATTERING BY GASES 23
the incident light is sufficiently increased. The
phenomenon of resonance radiation is sufficient
proof of the fact. The transition from ordinary
scattering to resonance would be very interest-
ing to study, although the subject is beset with
considerable experimental difficulties. It would
also be of interest to study by the scattering
absorbing gases like chlorine on either side
of the region of absorption.
CHAPTER III
ATMOSPHERIC SCATTERING AND TWILIGHT
PHENOMENA
21. Following upon the publication by the
late Lord E/ayleigh of his brilliant idea that the
scattering of light by the molecules of air
accounted in large measure both for the blue
light of the sky and the observed degree of
transparency of the atmosphere, the subject was
taken up by Lord Kelvin l and by Prof.
Schuster 2 and it was shown that the suggestion
was in quantitative agreement with the facts.
The subsequent development has been largely
a matter of detail and owes its interest to the
importance of the problem from the standpoint
of solar and terrestrial meteorology rather than
that of theoretical physics. Among the prin-
cipal contributions subsequent to the pioneer
investigations referred to above may be men-
tioned especially the work of Abbot and Eowle 8
and the theoretical researches of Prof. L. V.
1 Baltimore Lectures, 1904, pp. 301-322.
s Treatise on Optics, 2nd edition, p. 329.
* Annals of the Aetropbysical Observatory, Vol. II, and Astrophysical
Journal, 38, 1913.
ATMOSPHERIC SCATTERING 25
King1 in which an attempt is made to take
secondary scattering into account and to discuss
the disturbing effects produced by atmospheric
"dust." A large amount of detailed work, chiefly
of an observational kind on the character and
intensity of sky-radiation and on atmospheric
absorption has also been published. The main
result has been the confirmation of Rayleigh's
theory, but nothing essential has been added
to it except perhaps the recognition of the
importance of taking into account the selective
absorption in certain regions of the spectrum
exercised by the gases of the atmosphere and by
the water- vapour present in it.
22. The newer work of Cabannes and of the
present Lord Eayleigh in their laboratory ex%
periments on molecular scattering by gases and
the subsequent theoretical discussions of their re-
sults have however opened up novel issues. Two
new facts have emerged, namely, the imperfect
polarisation of the transversely diffracted light,
and the influence of this imperfect polarisation
on the intensity of the scattered light. A third
point is also suggested by theory that the
magnitude of the imperfect polarisation may
depend to an appreciable extent on the wave-
length of the incident light. It is natural to
ask the question, is there any evidence of
1 Philosophical Transactions of the Royal Society, A 212, 1913,
4
26 MOLECULAR DIFFRACTION OF LIGHT
these effects to be found in the observations on
sky-radiation? Then again, a perusal of the
literature shows that several interesting pro-
blems relating to molecular diffraction in the
atmosphere have not as yet been the subject
of mathematical treatment. Notable amongst
these is the explanation of twilight phenomena
regarding which very little theoretical work
has been done. It is proposed in this chapter
briefly to review the outstanding problems
relating to atmospheric scattering which are of
interest from the standpoint of theoretical
physics and to indicate the lines of advance.
The Polarisation of Skylight.
23. As mentioned above, the first novel issue
which is raised by the newer work is the extent
of polarisation of molecularly diffracted light.
As is well-known, the light of the sky observed
in a direction 90° remote from the sun is strongly
but not completely polarised, the degree of such
polarisation depending not only on the wave-
length of the light under consideration but also
to a large extent upon the altitude of the sun,
the meteorological condition of the atmosphere
and other factors. The defect of polarisation
under ordinary conditions is in fact so consider-
able that not more than a small fraction of it,
if at all, is that inherent in molecular diffraction.
ATMOSPHERIC SCATTERING 27
Much the larger part arises from disturbing
factors, such as dust, thin clouds or haze,
secondary scattering due to the self-illumination
of the atmosphere and light reflected from the
earth's surface. We may ask, is it at all possible
to eliminate these factors altogether or to disen-
tangle their effects and establish the imperfect
polarisation to molecular anisotropy by observa-
tions of skylight? At first sight this may seem
very difficult, but a little consideration will
show that the attempt is not quite so hopeless
as may be thought. As is well-known, dust
and haze are largely confined to the lower levels
of the atmosphere. This is beautifully illustrated
by the aeroplane photographs secured by Luckiesh1
which show a well-marked dust or haze horizon
lying at an altitude of about a mile above thtf
earth/ s surface. Mr. Evershed has mentioned
to the author in conversation that from the
observatory at Kodaikanal which is above the
dust-level, its rise and fall with the change of
seasons can be seen against the dark back ground
provided by a distant mountain. It is clear
therefore that by making the observations on a
high mountain on a bright clear day, it should
be possible practically to eliminate the effect
of dust and haze on the polarisation of sky-light.
The disturbing factors then left to be dealt with
1 Franktn Institute Journal, March 1919, p. 311.
28 MOLECULAR DIFFRACTION OF LIGHT
would be the secondary scattering and earthlight.
The influence of secondary scattering may be
reduced very considerably by making the obser-
vations at the extreme red end of the visible
spectrum. On a clear bright day, the sky as
seen at a mountain observatory through a deep
red glass appears almost perfectly black, but
there is ample illumination, if the observer's
eyes are carefully screened from extraneous
light, to allow the extent of polarisation to be
determined with the help of a double-image
prism and a nicol. The effect of earthshine
on the polarisation may be estimated by utilizing
the data obtained by Luckiesh1 on the albedo
of different types of landscape from aeroplane
observations. Under such conditions it should
evidently be possible to eliminate the disturbing
influences and to detect the residual effect due
to molecular anisotropy.
24. In order to make a test on these points,
the writer made the ascent of Mount Dodabetta
(8750 feet above sea level) in the Nilgiris on
the forenoon of the 4th December, 192] . The
sky was beautifully clear, free from cirrus
clouds and almost completely black as seen
through a red filter. The weaker component
of polarisation was found to have 13% of the
intensity of the stronger component. According
1 Frank. Inst. Journal, loc. cit.
ATMOSPHERIC SCATTERING 2&
to Luckiesh, the albedo of landscape covered
by grass or fields varies from 0'05 to 0*10,
and of landscape covered by woods from 0'03
to 0'05. That of barren land is greater, ranging
from 0*10 to 0*20. It was estimated that the
average albedo of the Nilgiris and the surround-
ing country could be taken as 0'08. As an
outside estimate therefore, earthshine when
the sun is 45° above the horizon would not
give rise to an imperfect polarisation exceeding
A% . L. V. King has calculated the imperfect
polarisation due to secondary scattering at the
level of Mount Wilson (5886 feet) and found it
to be 5^ at the red end of the spectrum. The level
of Mount Dodabetta is much higher (8750 feet)
and the disturbing factors are therefore less,
but some allowance must be made for** the
fact that the region of spectral transmission
of the filter used extends to slightly shorter
wave-lengths, and we therefore retain King's
figure of 5% as the effect due to secondary
scattering. A total of 9% out of the 13%
actually observed is thus accounted for, and the
remaining k% is ascribable to molecular
anisotropy. This is in good agreement with
the latest experimental results of Lord Eayleigh
obtained in the laboratory.
30 MOLECULAR DIFFRACTION OF LIGHT
Polarisation of Twilight.
25. Another very interesting way in which
the problem may also be dealt with is by observa-
tions on the polarisation of the sky immediately
after sunset. In this case, it is not necessary to
use any light-filters or to work at a mountain ob-
servatory, and the measurements may be made on
any clear evening at a low-level station. If the
polarisation of the light of the zenith sky in the
evening is determined from time to time, it will be
found that as the sun approaches the horizon and
sinks below it, there is a rapid improvement in the
completeness of polarisation, followed subsequent-
ly by a slow and steady deterioration with deepen-
ing twilight. Kimball l who observed the pheno-
menon suggests that the improvement of the
polarisation is due to the earth-illumination
being cut off when the sun sets. This explana-
tion does not appear to be adequate as it does
not account for the large magnitude of the effect
or the rapidity with which it occurs. Tor
instance, in some observations made at Calcutta
by the author and by Mr. K. E. Kamanathan, it
was found that 40 minutes before sunset the ratio
of the intensities of the components of polarisation
was 30 per cent., 20 minutes before sunset it
was 20 per cent., at sunset it was 14 per cent., 20
minutes later it was 15 per cent., and then gradu-
ally rose again to 30 per cent. In view of the
1 Mount Weather Observatory Bulletin, 1911.
ATMOSPHERIC SCATTERING 31
low albedo of landscape already quoted above,
we can hardly suppose that such effects could be
merely due to the cutting off of earthshine.
The greater part of the effect really arises in
another way. As the sun approaches the horizon,
the thickness of the atmosphere which his rays
have to traverse rapidly increases, and the actual
intensity of illumination of the first kilometer
or two of the atmosphere above the observer
becomes exceedingly small. At higher levels,
however, the weakening of the sun's rays is not
so great, and as we proceed upwards to the
layers of the atmosphere in which the barometric
pressure is considerably smaller than the sea-
level value, the intensity of the sun's rays rapidly
increases, until finally at a great height it reaches
practically its noon-day value. The effective Scat-
tering layers of the atmosphere are thus its high-
level dust-free portions. Thus immediately after
sunset, the effect of the low-lying dust and
of the earth-shine is automatically eliminated.
Further, the great diminution in the effective mass
of air and the increase in the effective wave-length
of the transmitted rays which illuminate it should
result in a considerable diminution of the effect of
secondary scattering. It should also be noticed
that the illuminating rays being horizontal, and
the extension of the earth's atmosphere being
chiefly horizontal, secondary scattering should
have a much smaller influence than when the sun
32 MOLECULAR DIFFRACTION OF LIGHT
is at a high altitude. This is easily seen on consi-
dering the directions of vibration in the incident
light, in the primarily scattered light which
reaches the observer, and in the scattered light
arriving from different directions which after
a second scattering also reaches the observer. In
fact, a careful consideration shows that if the
molecules of the atmosphere were spherically
symmetrical, the zenith sky immediately after
sunset should be almost completely polarised, the
defect of polarisation if any, not exceeding 5 or 6
per cent. Actually, however, a defect of about
W% is observed even on the clearest days, show-
ing that there is a residual effect of k% or 5%
arising from molecular anisotropy.
26. When the sun sinks very far below the
horizon, much the greater part of the atmos-
phere above the observer enters the region of
shadow and the influence of secondary scattering
on the polarisation again becomes prominent.
Some very curious effects may be observed, one
of which is that the region of strongest polari-
sation in the sky, instead of following the move-
ment of the sun, actually recedes from it.
The Problem of Secondary Scattering.
27. In attempting to extend the work describ-
ed in the preceding pages to different wave-lengths
in the spectrum and to put it on a very precise
quantitative basis, we naturally come up against
ATMOSPHERIC SCATTERING 33
the problem of evaluating the effect of secondary
scattering on the polarisation. This had been
attempted by Soret in order to explain the
existence of " neutral points " in the sky.1
More recent work is that of L. V. King already
quoted in which he has used the theory of
integral equations in order to find the result of
self -illumination of the atmosphere. In order
to apply his method to the determination of the
state of polarisation of sky-light, King had to
make two simplifying assumptions: firstly,
that the effect of the curvature of the earth may
be neglected : secondly, that the portion of the
scattered radiation due to self-illumination is
independent of the angle of polarisation of the
incident radiation. As regards the first assump*-
tion, it should be remarked that it is the curva-
ture of the earth that determines the horizontal
extension of the portion of the earth's atmos-
phere which contributes the primarily scattered
light which is again re-scattered by the part of
the sky under observation. Its neglect is thus
prima facie justifiable only if it can be shown
that the actual brightness of the sky in a hori-
zontal direction is the same as for an infinitely
extended atmosphere. As regards the second
assumption, we have only to remember the case
just discussed — that in which the sun's rays are
1 See Humphreys " Physics of the Air," Chapter on Optics of the
Air.
5
34 MOLECULAR DIFFRACTION OF LIGHT
nearly horizontal — to see that it may lead to
results which do not agree with facts. It would
seem therefore that there is a real need for a
discussion of secondary scattering in which the
curvature of the earth is taken into account and
the result is fully worked out without any
assumptions except perhaps the negligibility of
multiple-scattering of the third and higher
orders. If such calculations were made, it may
prove possible to establish the imperfect polari-
sation for different wave-lengths due to mole-
cular anisotropy by comparison with observa-
tions made at high-level stations. Perhaps the
use of a simpler mathematical method than that
adopted by Prof. King may render the problem
tractable.
The Influence of Atmospheric Dtist.
28. The curves showing the brightness of
the zenith sky as a function of the wave-length
obtained by the observations made at Washing-
ton and figured in Prof. King's paper show a
sudden kink amounting practically to a dis-
continuity at a wave-length of 0*61 /*.
A similar jump also occurs in the curves for
polarisation of the zenith sky. In the curves for
the Mount Wilson observations, undulations
also occur but at a shorter wave-length, about
0*45 /*. These effects are clearly due to the
influence of "dust," but precisely how they
ATMOSPHERIC SCATTERING 35
arise does not appear to have been fully ex-
plained. The suggestion may be ventured that
the effect is due to diffraction, the wave-length
at which the bend occurs being determined by
the average size of the dust-particles. In this
connection, some interesting observations made
by the author and by Mr. Bidhubhusan Ray
may be quoted.1 When suspensions of sulphur
are used containing particles comparable in size
with the wave-length, both the transmitted light
and the scattered light show oscillations of
intensity depending on the relation of size be-
tween the particles and the wave-lengths used,
and the polarisation of the scattered light also
shows striking fluctuations. It seems possible
that dust may give rise to somewhat similar
results in relation to atmospheric extinction,
scattering and polarisation. At a higher level
such as Mount Wilson, the average size of the
particles remaining floating in the atmosphere
would naturally be smaller and this would
explain the occurrence of the bends at smaller
wave-lengths in this case.
29. The foregoing suggestion is put forward
for what it is worth. Careful experimental
determinations of the average size of atmospheric
" dust " at different levels would be necessary
in order to establish its correctness,
1 Proc. Roy. Soc., Oct., 1921, p. 102, and Proc. Ind. Assoc. for the
Cultivation of Science, Vol. VII, Parts I and II, 1922.
36 MOLECULAR DIFFRACTION OF LIGHT
Twilight and Afterglow.
30. A very interesting application of the
theory of molecular diffraction is in the explana-
tion of the various phenomena attending twilight
or dawn, especially the manner in which the
total illumination due to twilight diminishes with,
the movement of the sun helow the horizon, the
distribution of brightness in the different parts
of the sky and its variation with the altitude of
the sun, and so on. The impression appears to
prevail that twilight phenomena are so complex
in their nature that no simple calculations con-
cerning them are possible. Thus for instance,
Prof. W. J. Humphreys in his book on the
Physics of the Air remarks, after giving an
account of the various effects observed — " The
foregoing descriptions which of course apply
equally to dawn are by no means universally
applicable. Indeed, the sky very commonly is
greenish instead of purple, probably when the
atmosphere is but moderately dust-laden.
Furthermore, the explanations are only quali-
tative. A rigid analysis, even if the distribution
of the atmosphere and its dust and moisture
content were known, — which they are not, nor
are they constant — would be at least difficult
and tedious." With reference to these remarks,
it may be pointed out, that twilight really arises
from the illumination of the higher levels of the
ATMOSPHERIC SCATTERING 37
atmosphere which may be regarded as dust-free,
at least under normal conditions. Further, as
we have seen in considering the explanation of
the polarisation of twilight, the transmission of
sunlight through the lower dusty levels is really
negligible under these conditions, and practi-
cally the whole of the observed effect arises
from light which has throughout its course
passed through the higher levels. Hence, we
are entitled to regard the problem as one of
practically simple molecular diffraction, and
the complications arising from secondary scat-
tering are far less important than might be
imagined. The possibility of giving a quanti-
tative theory of twilight is therefore much less
remote than has been suggested by various
writers on the subject.
31. Kimball and Thiessen1 have given data
based on photometric measurements of clear sky,
twilight and other natural illumination intensities
on a fully exposed horizontal surface. These
values are given in Table I.
TABLE I.
Relative Illumination Intensities. Intensity in
Surface of Illumination Horizontal. Foot candles.
Zenithal sun ... ... 9600'0
Twilight at sunset or sunrise ... 33*0
„ centre of sun 1° below horizon 30*0
1 Monthly Weather Review, 44, p. 614, 1916.
38 MOLECULAR DIFFRACTION OF LIGHT
TABLE I — continued.
Relative Illumination Intensities. Intensity in
Surface of Illumination Horizontal. Foot candles.
Twilight centre of sun 2° below horizon 15*0
„ 3° „ ... 7'4
„ „ 4° „ ... 3'1
5° „ ... 11
6° „ ... 0-40
(End of civil twilight)
7° ... 0-10
8° ... 0-04*
8°-40' ... 0-20
9° ... -015
10° ... '008
The above table shows that the brightness of
twilight changes rapidly when the sun is more
than about 4° below the horizon. The author
has attempted to explain the observations of
Kimball and Thiessen quoted in Table I quan-
titatively on the basis of molecular scattering.
The method adopted is to divide up the whole
atmosphere above the observer into a series of
horizontal layers, and to find the effective
mass of air in each layer illuminated by the
direct rays of the sun, secondary scattering
being neglected. In making the calculation,
allowance must be made for the diminution
of intensity of the sun's rays before they
reach the air-mass under consideration, and the
ATMOSPHERIC SCATTERING 39
cosine of the angle at which th^e diffracted rays
illumine the horizontal surface of the photometer
must also be included as a factor. Approximate
methods of numerical quadrature were used, and
it was found that the observations of Kimhall
and Thiessen were quite satisfactorily explained,
at least as regards the relative values of the
illumination for different altitudes of the sun
after sunset. But as regards the ratio of full
sunlight to the intensity of twilight a dis-
crepancy appears which has not up to the time
of writing of this volume been cleared up. It is
possible that the discrepancy is in some way due
to refraction of the sun's rays in passing horizon-
tally through the earth's atmosphere. But this
can only be settled by further investigation.
Sufficient work has been done, however, to show
that the problem of twilight at least in its essen-
tial features, is capable of being subjected to
numerical computation of intensities from theory
for detailed comparison with the observations.
CHAPTER IV
MOLECULAR SCATTERING IN LIQUIDS
32. As early as the year 1899, in his first
paper on the scattering of light in the atmos-
phere,1 the late Lord Eayleigh clearly emphasised
the principle that his theory of molecular
scattering is not applicable in the case of highly
condensed media such as dense vapours, liquids
and solids, for the simple reason that the molecules
in them possess only a greatly restricted freedom
of movement. The distribution of the molecules
cannot in the circumstances be regarded as a
simple random arrangement, and hence the
phases of the scattered waves arising from the
individual waves are not uncorrelated. The
total energy scattered by a volume of a liquid
or a solid cannot therefore by any means be
equated to the sum of the energies scattered by
the individual molecules in it. In the face of
this clearest possible declaration of principles,
some recent writers, notably Fowle,2 and Caban-
nes3 have put forward the obviously incorrect
1 Phil. Mag., Vol. XLVII, pp. 375.384 (1899). Scientific Papers,
Vol. 4, p. 397.
» Astrophysical Journal, Vol. 38, p. 392.
9 Annales De Physique, Tome XV, pp. 1-150.
SCATTERING IN LIQUIDS 4l
suggestion that Rayleigh's thdory is applicable
also in the case of liquids. How far such an
assumption must be from the truth can be realis-
ed easily in the light of the discussion of funda-
mental principles contained in our first chapter.
As we have seen, it is the degree of approxi-
mation of the compressibility of the medium to
that given by Boyle's law which is the measure
of the degree of applicability of the principle of
random phase on Rayleigh's theory. As is well-
known, the compressibility of a liquid or a solid is
usually only an extremely minute fraction of what
it would be if Boyle's law were applicable. This
itself is sufficient to show that we shall be
greatly in error if we attempted to extend the
principle of additivity of the energy effects of tke
individual molecules to the case of liquids. In
fact, Strutt has already found that liquid ether
scatters a great deal less light than the vapour
in proportion to the relative density of the two
media.1 We can easily see why this should be
so. Owing to the near approach of the mole-
cules to each other in the liquid state they
occupy a large proportion of the total volume
of the containing vessel. Hence the non-
uniformity in their spatial distribution is far less
striking than in the case of gases, and in conse-
quence there is a partial correlation of the phases
of the waves starting out from the individual
1 About |th according to Strutt ; Proc. Roy. Soc., Vol. 95, p. 175.
6
42 MOLECULAR DIFFRACTION OF LIGHT
molecules which entails as the result of inter-
ference, a great falling off in the total energy
scattered. A very interesting calculation1 which
was made by llayleigh of the energy scattered
by a cloud of particles having a restricted free-
dom of arrangement clearly illustrates this
principle.
The Einstein-Smoluchowski Theory.
33. The complexities of the problem of mole-
cular diffraction in liquids are so great that we
have evidently to proceed by statistical methods.
Fortunately, this has already been accomplished
in' great measure in the beautiful " theory of
fluctuations " developed by Einstein2 and Smolu-
chowski3 and used by the latter especially to
explain the peculiar opalescence exhibited by
fluids near the critical state. In this theory,
scattering is considered not as due to individual
particles but to small local variations of density
arising from the heat movements of the mole-
cules. These variations are quantitatively deter-
mined by Boltzmann's principle. Smoluchowski's
statistical thermodynamical reasoning gives for
the mean square of fluctuation of density in
volume V of density p« an expression, which
1 Phil, Mag., Dec. 1918, p. 449,
'- Ann. der. Phya. 33 (1910), p. 1275.
J Ann. der Phys. 25 (1908), p. 205. Also, Epstein, Ency. Math.
Wiss, Band V. 3, p. 520.
SCATTERING IN LIQUIDS 43
except in the immediate neighbourhood of the
critical point is equal to1
where R is the gas constant, Nt is the number
of molecules in a grammolecule, P» the compres-
sibility corresponding to density p» equal to
•_! <lp
v ' dp
At right angles to the incident light the inten-
sity of scattered light is given by the expression'2
(Ac) being the variation of the dielectric
constant.
Now as these individual local variations ^are
irregularly distributed, the phases of the various
scattered beams are also quite arbitrary, and
hence for calculating the total intensity of
scattered light we have merely to sum up the
above expression over the total volume 0. The
expression contains a factor
which may be evaluated by use of the Mosotti-
Lorentz law
= const, p
1 Boltzmaun, Wien. Ber. 63, p. 397. A. Einstein, Ann. d. Phy. 19,
p. 373.
2 Bayleigh— Phil. Mag. 1881, p. 81.
44 MOLECULAR DIFFRACTION OF LIGHT
Differentiating we obtain
fi1. ... (3)
Substituting in this the value given above for
fluctuation of density we obtain
0 v
(4.
N1 9
Thus the intensity of light scattered by a cubic
centimetre of fluid at right angles to the incident
rays is
18'
-18' ~N7~ ~A7~
In the case of gases
l
fi0 = — and /x*+2=3, nearly
and fji differs only slightly from unity. The
formula then reduces to Rayleigh's result.
Equation (5) may be applied with confidence to
find the intensity of light scattered in liquids,
for the work of Perrin and others on the Brown-
ian movement in liquids has furnished a strong
confirmation of Einstein's fundamental work on
the subject and has shown that the energy of
translation of molecules in a liquid is the same
as in the gaseous state of matter. The formula
SCATTERING IN LIQUIDS 45
thus expresses in a perfectly general manner
the scattering power of a fluid associated with
its ordinary refractivity taken together with
the non-uniformity of optical density result-
ing from molecular movements. It is a point
worthy of notice that according to the
formula the scattering power of liquids is pro-
portional to the absolute temperature, if we
leave out of account the changes which would
result from variation in compressibility and re-
fractive index with temperature. The constant
N! is a pure number independent of the parti-
cular state of molecular aggregation of the
substance or its density.
34. It must be remembered of course, that
the whole theory depends for its validity^ on
Maxwell's electro-magnetic equations for the
propagation of light, and the assumption of a
continuous interaction between the molecules and
the impinging light-waves.
Experimental Study.
35. To determine whether the absolute
scattering power of liquids for light is correctly
given by equation (5), some preliminary observa-
tions have been made by the writer and by
Mr. K. Seshagiri Eao working in his laboratory
at Calcutta. As is quite obvious, it is of the
highest importance to get very pure liquids.
40 MOLECULAE DIFFRACTION OF LIGHT
Water as is well known is difficult to get free
from motes. On examination the ordinary tap
water showed a very strong scattering when a
beam of light was sent through it. The track
was practically white and showed innumerable
motes floating about in the water. Repeated
filtration through several thicknesses of Swedish
filter paper made an improvement, the track
being now of a bluish colour, and a still better
result was obtained when an earthenware filter
was used. Suspended matter was however still
in evidence, and the track was also much
brighter when viewed nearly in the direction of
the source than when seen transversely or in the
opposite direction. A somewhat casual attempt
was then made to clear the water by adding
alkali and alum and thus throwing out a gela-
tinous precipitate of aluminium hydroxide.
This made a further improvement, but small
particles of the precipitate remained floating
about, apparently because the depth of the water
was insufficient and the appearance of the
track of the beam was not very prepossessing.
The next attempt was made with ordinary dis-
tilled water which had been prepared without
any special precautions and stored for some time
in the chemical laboratory. This gave imme-
diately a much smaller intensity of light-scatter-
ing than the tap water had done after several
attempts at filtration. For purpose of observation,
SCATTERING IN LIQUIDS 47
i
the distilled water was put into a stoppered
glass bottle with square sides and allowed to
stand. Test observations from day to day of the
scattered beam with a double image prism and
a set of Wratten colour filters showed a pro-
gressive improvement. After about a fortnight's
standing, the track of the light was hardly
conspicuous unless a dark background was pro-
vided for it to be viewed against, and the defect
of polarisation at the violet end of the spectrum
was much less striking than it was when the
observations were begun. Small motes were
still to be seen, particularly when viewed in the
direction of the source, but the track was of a
blue colour and it was judged that the greater
part of the observed luminosity was probably
due to the water itself. A sample of water
which had been distilled at the Calcutta Mint
and stored for 3 months also showed the blue
track very well. Allowing it to stand for some
time improved matters appreciably.
36. Eor a quantitative estimate, the bright-
ness of the beam in the water was compared
directly with that of its track in saturated ether
vapour. The latter was contained in a pear-
shaped bulb with a long neck which was covered
over with black paint and formed the " black
cave " against which the light scattered by the
vapour was observed. The bottle and the 'bulb
were set side by side and a parallel beam of
48 MOLECULAR DIFFRACTION OF LIGHT
light passed through both. An Abney rotating
sector was placed in front of the water bottle
and the opening of the sector varied till the
tracks appeared to be of equal intensity in both
vessels as judged visually. The opening of the
sector gives the ratio of intensities, a correction
being made for the loss of light by reflection
in the passage of the direct and scattered
pencils through the glass Avails. The deter-
minations made in this way were not anything
more than approximate estimates. The scatter-
ing of light in saturated ether vapour has been
measured by comparison with air by Rayleigh
and shown to be accurately proportional to the
square of its refractivity. Using this result,
the observation showed the scattering power of
the sample of water used was 175 times that
of dust-free air at N.T.P. From theory we find
taking for air
£=0-987xlO-c cm2 dynes'1 T=273°, /*= 1-000293 and
for water at 30°c, /2=43-5 10'1' cm3 dynes"1, T=303,
//,=!• 337
that volume for volume, water should scatter 140
times as strongly as air at N.T.P. This, though
not agreeing exactly with the observed value is
only slightly smaller and the difference may be
explained as due fco the effect of residual
suspended particles in the water used in the
experimental work.
SCATTERING IN LIQUIDS 49
i
37. More accurate measurements were made
a month later by Mr. K. Seshagiri Kao when the
sample of water had still further improved.
The method used was the comparison of inten-
sities by a double image prism and a nicol.
Sunlight was used as the source of light and
a long-focus lens was used to give an intense
and nearly parallel beam. The two bottles were
placed on either side of the focus and the track
viewed through two parallel slits. The double
image prism was placed so that the four images
seen were in a line with the planes of vibration
horizontal and vertical. The weaker image from
the water was adjusted so as to appear just in
contact with the stronger image from the
ether vapour. By the nicol, these two were
reduced to equality. The ratio was then given
by I1/T2=tan26» where 6 is the angle through
which the nicol is rotated from the zero position.
Measurements by this method gave the ratio of
scattering of water at 25° to air at N,T.P. as 158.
The ratio was still higher than that given by
theory. Possibly it might be due to the motes
not having been completely eliminated. But
it seems more probable that the higher ratio
may be due to the anisotropy of the molecules of
water which is evidenced by imperfectness of
polarisation. According to the theory of Eins-
tein and Smoluchowski, the light scattered in
a direction perpendicular to the incident rays
7
50 MOLECULAR DIFFRACTION OF LIGHT
should be completely polarised. As already
mentioned in a previous chapter, the work of
Cabannes and Strutt shows that most gases
depart from this ideal sphericity of molecules.
Cabannes has amplified Rayleigh's theory by
considering the anisotropy of the molecule and
shown that the expression for the intensity of
scattered light should be multiplied by
where p is the ratio of the weak component of
polarisation to the strong one. It is evident
from the formula that the larger the value of p,
i.e.., the greater the departure from sphericity, the
greater is the intensity of scattered light. It is
clear that a similar correction must also be
made in the case of liquids. The value of p
for the specimens of water used is 12% and for
air k% . When this correction is applied the
theoretical ratio comes out at 160 and is in
fair agreement with that actually observed.
Determination of the Avogadro Constant.
38. More accurate comparisons of intensities
by photographic methods are in progress. It is
also intended to measure the co-efficient of
scattering absolutely using liquids completely
freed from motes by repeated slow distillation
in a vacuum/ From a knowledge of the
SCATTERING IN LIQUIDS 51
*
absolute scattering power, the values of R, T, ft
and /* being known, it should be possible to
calculate the Avogadro constant Nl from experi-
ments on liquids in much the same way as
Cabannes has done with gases. The method by
which it is proposed to measure the co-efficient
of scattering is as follows. The intensity of
the scattered light will be of the order of 10~7
of that of the incident beam. To make com-
parisons we have to reduce the intensity of the
incident beam considerably and it is proposed
to effect it in the following manner. A very
short focus lens will be used to condense the
light. The light coming to its focus will diverge
very rapidly. The radius of the solar image
at the focus will be of the order of a millimetre
while at a distance of about 2 metres, owfng
to great divergence the intensity will have been
reduced in the ratio of about 10 ~4 or 10 ~5.
Eurther reduction will be made by a rotating
disc with a small radial slit at the edge. The
width of the slit will be varied till the light
incident on a fixed aperture placed behind it is
reduced to the same intensity as the image of
the track in the water, as determined by photo-
graphy. Incidentally the ^~4 law will also be
tested.
39. According to formula (5), the scattering
power is proportional to the absolute temperature
of the liquid apart from any variations in /* and ft
52 MOLECULAR DIFFRACTION OF LIGHT
due to the same cause. This effect should be
particularly noticeable in the case of liquids
such as ether whose compressibility increases
rapidly with temperature. In the case of water,
we should not expect much change as both the
compressibility and the refractive index dimi-
nish with rise of temperature. These points are
also under investigation.
Molecular Scattering and Transparency
of Liquids.
40, Since the energy of the light laterally
scattered is derived from the primary beam,
there must result . a certain attenuation in the
intensity of the latter in its passage through the
liquid, the magnitude of which may be readily
calculated from the co-efficient of scattering.
The multiplying factor necessary is —~- which
represents the result of integration over a sphere
completely enclosing an element of volume of
the scattering fluid. The transmitted light is
given by the formula I =I0 e~al where I is the
length of the path traversed through the
liquid and
As in the case of atmospheric scattering
we may expect that the co-efficient of attenua-
tion a will exactly indicate the observable
SCATTERING IN LIQUIDS 53
i
transparency of the medium in those parts of the
spectrum for which it does not exercise any
selective absorption. Erom the data already
given and the known values of R, T, and N15 a
may be readily determined for any value of the
wave-length.
41. Prom the observations of various experi-
menters1 it is known that water exercises a
selective absorption on the longer wave-length
side in the visible spectrum and also in the ultra-
violet region. Measurements of the co-efficient of
absorption in water have been made by various
investigators, but the values obtained by each
are hopelessly different from those of others.
Calculations on the basis of Evans's and Aschkin-
ass' values show that water is actually six to
ten times less transparent than it should be.** It
is not evident from their papers whether they
had taken care to get the water mote-free.
The most reliable measurements of any hitherto
made appear to be those of Count Aufsess. This
experimenter used double-distilled water and
convinced himself that it was free from suspend-
ed matter. It was found by him that the
selective absorption in the visual region ceased
for wave-lengths less than 558 /*//, . For the two
1 Evans : Proc. Roy. Soc. (1894), Vol. 57.
Aschkinass : Wied. Ann. 55 (1895).
Krenssler : Ann. der Phys. (1901), Band 6.
Aufsesa : Ann, der Phys,, Vol. 13, 1904 j also Kayser's Hand-
buch, Vol. 3, p. 392.
54 MOLECULAR DIFFRACTION OF LIGHT
ware lengths 522 w and 494 w* Aufsess gives
as the co-efficient of absorption 0'00002. For
these two wave-lengths the co-efficient of attenua-
tion p calculated from the formula given above
is respectively 0'000022 and 0'000029. The
agreement of observation and theory is signifi-
cant. It is desirable that farther accurate
measurements for different wave lengths for
carefully purified water up to the extreme
violet end of the spectrum were available so
that the increase of the co-efficient of attenua-
tion inversely as the fourth power of the wave
length could be tested. It would be interesting
to determine by careful experiment whether the
intensity of the light scattered by water follows
the fourth power law exactly. It would be
also interesting to investigate scattering at or
near an absorption band and to investigate what
becomes of the energy absorbed, whether it
appears as selective scattering or is merely stored
up in the liquid. If there is any selective scatter-
ing we might expect deviations from the fourth
power law in that region of the spectrum.
Observations of Polarisation.
42. Reference has already been made to the
imperfect polarisation of the light transversely
scattered by liquids. The detailed results
on this point will now be described. The
SCATTERING IN LIQUIDS 55
determinations of polarisation for water have been
made by visual methods. Sunlight was used as
the source of light and the track was viewed
through a small rectangular aperture. The
double image prism was so set that the two
images were in line and just touching each other,
and the directions of vibrations were horizon-
tal and vertical. The two images were brought
to equality by a nicol on either side of the zero
position. Half this angle gave the angle e through
which the nicol was rotated from the zero position
and the ratio of the two intensities was of course
given by tan20. Measurements were made in
different regions of the spectrum by using
Wratten colour screens.
The results are given below —
Red. Yellow. Green. Blue. Violet.
13-2 10-3 11-5 15-3 21*7
Inspection of the values shows an increase
of polarisation in the red and violet regions with
a minimum value at the yellow. It is well
known that water has an absorption band in
the red and another in the ultra-violet. The
experiments thus indicate that near the absorp-
tion bands the imperfectness of polarisation
increases. It will be noticed that the figures
show a rapid increase near the violet end of the
spectrum. This may be partly due to the small
particles still remaining suspended in the liquid
56 MOLECULAR DIFFRACTION OF LIGHT
whose influence will only be greatly evident in
the region of shorter wave-lengths. Bat that
the phenomenon is real, is shown by the fact that
at the red end where the influence of the particles
is small there is a slight perceptible increase of
polarisation. It would be interesting to make
observations at and near the absorption bands,
using carefully purified liquids. Another point
worthy of investigation would be to observe
what influence the temperature has on the
polarisation. It is well known that water in
liquid form exists as molecular aggregates and
that temperature has a great effect on them.
We might expect therefore a change of polari-
sation with temperature.
The Orientation of the Molecules.
43. It is important here to notice that im-
perfect polarisation of the transversely scattered
light is noticed not only when the primary beam
is unpolarised, but also when the latter is itself
completely polarised. To test this point a nicol
was placed so that the incident light passed
through it. It was found that when the plane
of polarisation was vertical or horizontal the
scattered light was a maximum or a minimum
respectively. In the latter case, that is, when
the scattered light was a minimum, it was viewed
through a second nicol and its intensity was
SCATTERING IN LIQUIDS
found to be independent of the plane of polari-
sation of the latter. It could however be ex-
tinguished by two nicols or a double image prism
and a nicol, thus showing that it was unpolaris-
ed light. A similar phenomenon is also observed
in the case of molecular scattering in gases and
is a consequence of the fact that the orientation
of the molecules in fluid media is arbitrary.
The observed intensity of scattering is the result-
ant effect of molecules in all possible positions
and orientations, so that the weaker component
of polarisation stands in no definite relation of
phase to the stronger component, in other words,
the scattered light consists partly of common
light. The intensity of the Tyndall cone as
observed by a nicol when the primary be%m
is uupolarised is given by the relation
J=C1+C2 cos2 8 where 8 defines the orientation
of the plane of polarisation of the observing
nicol.
Relative Scattering Power of Different
Liquids.
44. The observations of Strutt with liquid
ether, and of the present writer with Mr. Sesha-
giri Eao on water have clearly shown that the
absolute scattering power of liquids is much
smaller than that of the corresponding vapours
making allowance for the difference of density.
8
58 MOLECUIAR DIFFRACTION OF LIGHT
This diminution is clearly explained on the
Einstein-Smoluchowski formula as the result
of the extreme smallness of the compressibility
of the liquid which more than sets off the result
of the increased refractivity. Naturally, there-
fore, we should also expect the Einstein-Smolu-
chowski formula to give the relative scattering
power of different liquids correctly. The oppor-
tunity for testing this point is furnished by
some recent observations of W. H. Martin on
light-scattering by dust-free liquids.1 Martin
found a strong defect in the polarisation of the
light scattered by all the liquids observed by
him, the defect increasing with the light-
scattering power. The Cabannes factor
6_7 in the intensity is thus very important.
The necessary data for compressibility and re-
fractive index are not forthcoming for all the
liquids experimented upon by Martin. So far
as the available data permit, the results for
the liquids listed in column I of the table below
have been compiled and the relative scattering
powers shown in column II without applying
the Cabannes correction, and in column III after
applying the Cabannes correction. It will be
seen that the computed ratios in column III
and those given by Martin's observations shown
in column IV agree tolerably. It is to be noted
1 Journal of Physical Chemistry, Vol. 24, 1920, p. 478.
SCATTERING IN LIQUIDS
59
TABLE II
WATER is TAKEN AS THE STANDARD
I
II
III
IV
Liquid.
Calculated from
Einstein-Smolu-
chowski formula
without correction.
Calculated from
formula with Cab-
annes's correction.
Experimental
results of
Martin.
Water
I'OO
i-oo
. i-oo
Ether
4-53
4-78
3-94
Methyl Alcohol
2-03
2-04
2-67
Ethyl Alcohol
2-86
2-87
3*00
Benzene
6-38
19-5
15-17
Toluene
560
17'1
16-6
that the Cabannes correction cannot be applied
when P is more than 50^ , and this maximum
Talue must be used in the formula when the
observed defect of polarisation exceeds 50^ .
Transition from the Liquid to the Gaseous State.
45. As we have seen, the Einstein-Smolu-
chowski formula, when corrected for the effect
of molecular anisotropy gives results in fair
agreement with observations in non-fluorescent
liquids, and it also automatically reduces to the
Rayleigh formula in the case of gaseous media.
Further, the formula which was originally
developed in order to explain the observed
60 MOLECULAR DIFFRACTION OF LIGHT
enormous light-scattering power of gases at tem-
peratures slightly above the critical point has
been quantitatively confirmed for this region by
the very fine measurements of Keesom1 on the
opalescence of ethylene. In view of these strik-
ing successes of the formula, we may, primd
facie, feel confident that it would correctly re-
present the sequence of phenomena throughout
the entire range of transition between the liquid
and the gaseous states. But, surprisingly enough,
the law seems to break down in the case of
gases under high pressure. Strutt has shown
experimentally that the scattering by saturated
carbon dioxide at 21°C at a pressure estimated
at 60 atmospheres and a density 114*7 times the
density at atmospheric pressure is 102 times the
scattering at the latter pressure. This agrees
fairly satisfactorily with the Rayleigh formula.
But when we calculate the scattering accord-
ing to the Einstein-Smoluchowski formula,
the value of the ratio is given by
where pl and ftx refer to carbon dioxide under pressure and
fi and /A refer to the gas at ordinary pressure.
46. In the following calculation, the unit of
pressure is taken to be 1 atmosphere.
Now =1
A.
and/i-1 at 2PC>=4-50xlO~4x
1 Annalen der Pbysik, 1911, Band 35, p. 591.
SCATTERING IN LIQUIDS 61
We may put (/A* — l)=2(/x— 1)
and )Ma -f2 =3
The compressibility of the condensed vapour
may be obtained in either of two ways ; one, by
making use of the experimental isothermal of
CO 2 for 21°C and the other by calculation on the
assumption of a suitable equation of state. The
value of p obtained from Andrews' isothermal
curve 2 23*5° is TTT. On assuming Clausius' equa-
tion of state
(which is found to represent the isothermals of
CO2 at high pressures with great accuracy3), we
get for the co-efficient of compressibility
fl- -L^H=!!ZL6 a' 2a*Q-&) **
" v dp v ' T(v+c)« ' TO + c)3
Taking jo=60 atmos.
v= — — - of the volume at the atmospheric pressure at
Alt)
21°C
= -L . .??£ of the volume at 0° C.
1 1 5 *j / o
and the constants
a=2-092
6=0-000866
and c=0-00094»
We get /^jj-j
1 Kaye and Laby's Tables.
a Phil. Trans. Roy. Soc., p. 575, Part II, 1869.
* Jeans : Dynamical Theory of Gases,
62 MOLECULAR DIFFRACTION OF LIGHT
We may take the mean of these results ^-— as the com-
17*5
pressibility of the vapour at 21°C.
The value of /x1* is easily calculated from the data given
by Dr. Phillips.1
It comes out to be 1'099 and the value of
O
£i(fti*_l)2(/Ai«+2)2 to be 5-38xlO"~
Hence
5-38xlO-»
l)a(fi* +2)a 6-29 x 10-6
=855.
whereas the actual scattering observed by Strutt
was only 102. It seems very remarkable that a
law which holds good for such widely different
conditions as (1) a gas at ordinary pressures,
(2) in the immediate neighbourhood of the criti-
cal point and (3) for liquids, should not also hold
good for saturated vapours below the critical
temperature. The reason why the law appa-
rently fails is not clear. The question is one of
very great importance and its solution may be
expected to throw light on the mechanism of
scattering. What is urgently wanted is a care-
ful determination of the scattering co-efficient
over a wide range of pressures and temperatures,
from the state of vapour through the critical
point to the liquid. If it is indeed found that
Strutt's results are confirmed for the whole region
of temperatures and pressures below the critical
point, it might mean that the arrangement of
1 Phillips : Proc. Roy. Soc. A 97, p. 225,
SCATTERING IN LKdUIpS 63
the molecules in space is of far less importance
in determining the phase of the scattered waves
than is assumed in the treatments so far given,
and that the attempt to explain the molecular
scattering of light on the basis of the classical
theories of electromagnetic wave-propagation
and the continuous interaction between light
and the electrons is really a failure. We may
then be forced to adopt explanations based on a
discontinuous type of action, exactly as in the
theories of photo-electricity, ionization, and
so on.
47. A related question is the imperfect polari-
sation of the scattered light. In all the cases
investigated by the authors and by Martin, the
scattered light from the vapour is found to ^be
more perfectly polarised than that from 'the
liquid. Why this should be so is not clear.
There are no observations available regarding
the polarisation of the light scattered by vapours
under pressure. The changes in the polarisation
of the scattered light in the transition from the
gaseous to the liquid state should be investigated
side by side with its intensity.
48. The discussion given here has perhaps
raised more difficulties than it has solved. But
this only demonstrates the importance of the
subject and the need for an extended study
of the phenomena both from an experimental
and a theoretical standpoint.
CHAPTER V
THE COLOUR OP THE SEA AND THE ALBEDO OF
THE EARTH
49. To an observer situated on the moon or on
one of the planets, the most noticeable feature
on the surface of our globe would no doubt be
the large areas covered by oceanic water. The
sunlit face of the earth would appear to shine by
the light diffused back into space from the land
and water-covered areas. The character and
intensity of the radiation thus sent back would
depend on various factors : firstly, sunlight
diffused back by the gases of the atmosphere
over the whole surface of the earth: ; secondly,
the sunlight incident on the oceans and returned
partly after reflexion at the surface of the water,
and partly after diffusion within its body ; thirdly
the light reflected back from cloud-covered areas
and the lower dusty levels of the atmosphere ; and
fourthly, the light scattered by the land-masses.
When we consider the fact that nearly three-
quarters of the surface of the globe is covered
by oceanic water, we begin to realise that the
molecular scattering of light in liquids may
possess an astronomical significance, in fact con-
tribute in an important degree to the observed
THE COLOUR OF THE SEA1 65
albedo of the earth. The " earthshine" on the
moon for instance may owe not a little to the
light diffused out from the oceanic water as the
result of molecular diffraction.
50. In intimate relation with the problem
of the albedo of water stands the question of the
colour of the sea. A detailed discussion of the
subject is appearing in a separate paper,1 and it is
sufficent here to deal with the matter only so far
as it illustrates the theoretical principles of our
subject.
Colour and Polarisation of the Light
Scattered in the Sea.
51. The method of observation used by the
writer is sufficiently described in a prelimmai^
communication that appeared in Nature? : — As
Tyndall and others have remarked, the reflec-
tion of sky light at the surface of the water is
an embarrassing feature in making observations
of the colour of the sea. Its influence may how-
ever be eliminated in the following simple way.
Light reflected at the polarising angle from the
surface of a liquid may be quenched by observa-
tion through a suitably oriented Nicol. Hence
by observing a tolerably smooth patch of water
through a Nicol at the polarising angle, the
surface-reflection may be got rid of. The Nicol
1 Proc. Roy. Soc. 1922.
1 Nature, November 17, 1921. p. 367.
66 MOLECULAR DIFFRACTION OF LIGHT
may be mounted at the eye-end of a card-board
tube so that it can be conveniently held at the
proper angle with the surface of the water and
rotated about its axis so as to get the correct posi-
tion for extinction of the reflected light. During
a recent voyage, the writer made some observa-
tions by this method in the deeper waters of the
Mediterranean and the Red Seas and found that
the colour of the sea so far from being extin-
guished when the sky-reflection is cut off, is seen
with wonderfully improved vividness and with
saturated hues. Even when the water is ruffled
or when it is viewed more obliquely than at the
polarising angle, the Nicol helps to weaken the
sky-reflection. Further, as is well-known, the
light of the sky is itself strongly polarised, and
this fact may, in favourable circumstances be
used to practically eliminate sky-reflection
from the whole surface of the sea. For this
purpose, the time most suitable is when the
sun has reached its maximum altitude and
.the observer should stand with his back towards
the sun and view the surface of the sea through
a Nicol. The part of the sky facing the observer
has then its maximum polarisation, especially
the low-lying parts, and the amount of polarisa-
tion is further enhanced when the light is re-
flected from the water at various angles of in-
cidence. By turning the Nicol about its axis,
the best position for extinction should be found
THE COLOUR OF THE SEA 67
and the whole surface of the sea will then be
found to glow with a vivid blue light emerging
from inside the water. Part of, this improve-
ment is also due to the fact that the Nicol in great
measure cuts off the atmospheric haze which
covers the more distant parts of the sea.
52. The obvious way of testing the light
from the sea for polarisation, that is, viewing it
through a Nicol and turning the latter about its
axis, is interfered with by the fact that the in-
tensity of the reflected light also varies at the
same time and obscures the variation in the in-
tensity of the light diffused from inside the
water. Even thus however, it is possible to
observe the polarisation of the scattered light,
the surface of the water appearing less j^lue
when seen through the Nicol in one position
than when viewed directly. Much the better
way of detecting the polarisation of the diffused
light, however, is to hold the Nicol at the proper
angle for extinguishing the surface-reflection
from the water and vary the azimuth of observa-
tion relatively to the direction of the sun's rays
entering the liquid. Striking changes in the
colour and intensity of the light diffused by the
water will then be noticed. The best time for
making this observation is when the altitude of
the sun is moderately large but not too great.
Obviously, if the sun's rays are too nearly verti-
cal, varying the azimuth of observation can make
68 MOLECULAR DIFFRACTION OF LIGHT
no difference. But when the sun's rays inside
the water proceed at an angle to the surface, the
variation of the azimuth of observation alters the
relation between the direction of the primary
beam and the scattered rays under test. When
the observer has his back to the sun, he looks
down practically along the track of the rays in-
side the water and the scattered light reaching
his eye is unpolarised inside the water and is not
extinguished in any position of the Nicol. The
colour of the scattered light is then seen as a
vivid but comparatively lighter blue. As the
azimuth of the plane of observation is swung
round, the intensity of the scattered light dimi-
nishes and its colour changes to a deeper blue,
until finally when the observer nearly faces the
sun,1 the intensity of the scattered light is very
small and it appears of a dark indigo colour.
If the polarisation of the scattered light were
complete and the direction of observation exactly
transverse to that of the primary beams inside
the water, the Nicol would have completely
quenched the light. This is however not actually
the case, evidently because we have to deal not
only with the scattering of the sun's direct rays
inside the water, but also with multiply-scattered
1 He cannot of course exactly face the sun as the reflection of the
sun's rays from the surface of the water would then interfere with
the observations. It is advantageous to choose a time when the
altitude of the sun is such that these reflections are also quenched by
the observing Nicol.
THE COLOUR OF THE SEA 69
light and also with the blue light of the
sky which enters the water and is then re-
scattered within it. It is evident that these
contributions to the luminosity of the water
would diminish the perfectness of the polarisa-
tion l and would give a much darker blue
than the primarily scattered rays.
53. The relatively deep colour of the secon-
darily scattered rays mentioned in the preceding
paragraph is also prettily illustrated by observing
the water on the shadowed side of the ship where
the sun's rays do not strike it directly. Such
water shows a much darker and deeper colour
than the contiguous parts exposed directly to
the sun's rays. A similar explanation maybe
given of the deepening of the colour of the.«ea
as the sun goes down. The lower the altitude
of the sun, the more important is the contribu-
tion of sky-light re-scattered within the water
to the observed luminous effect. The blue
colour of the sea as observed with the aid
of a Nicol when the sky is completely
overcast by clouds also appears of a distinctly
deeper tint than sunlit water. It is probable
that this may, at least in part, be due to
the importance of multiple scattering in
such cases.
1 Much in the same way as the polarisation of sky-light even at 90°
from the sun is incomplete. The imperfectness of the polarisation of
the molecularly-scattered light (due to asymmetry of the molecules or
other cause) also contributes to this result.
70 MOLECULAR DIFFRACTION OF LIGHT
54, The difference between the colour of the
parts of a wave sloping towards and away from
the observer is a very interesting feature.
When the surface of the sea is viewed through
a Nicol, the degree of contrast varies enormous-
ly as the Nicol is rotated about its axis. The
precise effect, of course, depends upon the rela-
tive intensity, colour and polarisation of the
light reflected from the surface of . the water at
different angles and of the light emerging from
inside the water. Broadly speaking, the pheno-
menon observed is that in one position of the
Nicol the sea appears almost flat and undisturb-
ed and in another position ruflled and full
of ripples. The visibility of the horizon which
depends on the contrast between sea and sky
also varies, in some cases very greatly, as the
Nicol is rotated.
The Albedo of Deep Wetter.
55. The phenomena described above make it
perfectly clear that the light molecularly diffused
from within the water is the principal factor
to be taken into account and that the colour of
the deep sea is not due to reflected sky-light as
has sometimes been suggested. That the reflec-
tion of skylight is at all noticeable arises from
the fact that the observer on the deck of a ship
views by far the greater part of the surface of
the sea at a very oblique angle, The position
THE COLOUR OF THE SEA ' 71
would be entirely different in the case of an
observer at a great height above the surface of
the water, e. g. , when flying in an aeroplane.
Since the reflecting power of water at normal
incidence is quite small (only 2% ), the lumino-
sity of the sea to such an observer would be
almost entirely determined by the diffusion of
light within the water.
56. That such diffusion must, in the case of
the deeper oceanic waters at any rate, be due to
molecular scattering and not to any suspended
matter may be inferred from the known great
transparency and freedom from turbidity of
such waters. It is extremely unlikely that,
under normal conditions at any rate, any
colloidal matter would remain for long in
suspension in salt water. Further, it should
be remarked that if sea-water did contain
any " motes " in suspension, they would not
appreciably influence the observed results.
For, " motes " scatter light in an unsymmetrical
manner, that is far more in directions approxi-
mating to that of the primary rays, and very
little in the opposite direction which, to an
observer above the surface of the water, is the
direction that really matters.
57. A simple calculation may be easily made
of the albedo of oceanic water. Since in round
numbers, water diffuses light 150 times as
strongly as an equal volume of air, a layer of
72 MOLECULAR DIFFRACTION OF LIGHT
the liquid 50 meters deep would scatter ap-
proximately as much light as 7J kilometers
of homogeneous atmosphere, in other words,
it should appear nearly as bright as the
zenith sky. This calculation however omits to
take into account two important factors, the
diminution in the intensity of sunlight before it
reaches the level of the water and its further
attenuation in the passage through the liquid
and also the loss in intensity of the scattered
light before it re-emerges from the depths. It
is the two last factors just mentioned which
together with the magnitude of the scattering
itself ultimately determine the total observed
luminosity of an ocean of liquid of very great
depth. Neglecting the effect of self-illumina-
tion within the liquid and also the contribution
which is made by diffuse sky-light which enters
the water and is then subsequently re-scattered
within the liquid — both of which may, in
certain circumstances, rise to importance — the
observable luminosity of a very deep layer
of liquid may be readily calculated. Eor
simplicity, we shall consider a case in which the
altitude of the sun is sufficiently great to
enable its rays within the water to be treated
as approximately vertical in direction, and the
intensity of the light scattered will also be
assumed to be observed in an approximately
vertical direction, e.g., by an observer in an
THE COLOUR OF THE SEA 73
aeroplane flying at some height above the
water. The coefficient of scattering in such a
case will be twice as great as when the scatter-
ing is observed laterally. Denoting it by
2B/^4 and the coefficient of absorption of light
in water by y, the total observed luminosity is
given by the integral
ZB f
X0
7
dx
x being the depth of any layer. For a suffi-
ciently great depth this reduces to BM*. For
the case of pure water, the values of y are
taken from the determinations of Count Auf-
sess for wave-lengths up to 522 /*/*, and for
shorter wave-lengths we may take them* to
be the same as the value of coefficient of at-
tenuation <* given by theory. The value of B
is in round numbers 140 times the coefficient of
lateral scattering by dust-free air. From these
data and making an allowance for the diminu-
tion of the solar intensity in transmission
through the atmosphere as on an average
day, the total luminosity of deep water for
different wave-lengths is expressed in Table
III in terms of the kilometers of dust-free
air at atmospheric pressure which would by
lateral scattering of full sunlight give an
equal effect.
10
MOLECULAR DIFFRACTION OF LIGHT
TABLE III
ALBEDO OF DEEP WATER
A in n/j.
658
622
602
590
579
558
522
494
450
410
14
Equivalent
kilometers
04
05
06
1-3
2-4
28
45
36
22
of dust-free
air.
58. If we take the scattering by 8 kilometers
of dust-free air as the standard and compare
with it the figures shown in Table III, it is seen
that in the light returned by the water, prac-
tically all the red is cut out, the orange and
yellow are quite feeble, but the green is greatly
enhanced, and also the blue, indigo and violet
but to a considerably less extent. The standard
of comparison, — (scattering by dust-free air)
being itself of a blue colour, it is clear that the
cutting out of the red and the enfeeblement of
the orange and yellow would result in the
colour of the light scattered by the water
being a* highly saturated blue. The enfeeble-
ment of the orange and yellow would however
considerably diminish the visual intensity which
at a rough estimate would probably not exceed
two or three times that of the zenith sky.
59. It will be understood from the figures
given in Table III, that the blue colour of the
light scattered by the water arises primarily
from the operation of the Rayleigh *~4 law, the
THE COLOUR OF THE SEA 75
absorption of the red and yellow regions of the
spectrum in the water resulting merely in the
colour being more saturated than it would
otherwise be. If the figures entered in the
columns of Table III had represented ratios of
comparison with white light, the presence and
predominance of the green would result in the
perceived colour being a greenish-blue and not
a deep blue colour. In other words, the blue
colour of the scattered light is really due to
diffraction, the selective absorption of the water
only helping to make it a fuller hue.
60. In connection with the foregoing
calculations, it should be remarked that certain
disturbing factors may arise. If owing to the
presence of organic or other dissolved matter^ in
the sea with a marked absorption in the green-
blue region of the spectrum, the transparency of
the water in this region be greatly diminished,
the albedo of the deep water may show a
great falling off. This is a possibility that
should not be overlooked, and how far it does
actually arise can only be determined by actual
observation. But the considerations set out
above make it clear that the light molecularly
scattered in the oceanic waters must play an im-
portant part in determining the total fraction of
the sunlight incident on the earth's surface that
is diffused back into space. A fuller discussion
of the matter would obviously be of great interest.
CHAPTER VI
SCATTERING OF LIGHT IN CRYSTALS
Introduction.
61. The well-known influence of temperature
(" Debye-effect ") on the intensity of X-ray re-
flection as illustrated, for instance, in the expe-
riments of Sir W. H. Bragg1 on rock salt indicates
that the atoms in the space lattice forming a
crystal are not absolutely fixed but oscillate to
some extent about a mean position ; the magni-
tude of this effect differs widely for different
crystals depending on the value of the " character-
ristic temperature " for the substance. Larmor2
has suggested that this thermal movement of
the atoms in the crystal should have an impor-
tant consequence, namely that when a pencil
of ordinary light traverses a transparent crystal, a
certain portion of the incident energy should
appear as scattered light. Such an effect, if
observable, would furnish us with direct visual
evidence of the reality of thermal oscillations
in solids. No theoretical calculation of the mag-
nitude of the expected effect has however
1 Phil. Mag. Vol 27, 1914, page 891.
• " " " 37, 1919 page 163.
SCATTERING IN CRYSTALS 77
appeared so far. Prof. E. J. Strutt1 (now Lord
Rayleigh) who experimented on the subject of
the scattering of light in solids found that the
track of a beam of light passing through a
block of transparent quartz could be detected
by photography and estimated that clear quartz
scatters light 8 times as strongly as dust-free
air. The effect was however ascribed by him to
inclusions which he assumed were present in
the quartz and not to the crystal itself. It
occurred to the present author that observations
with crystals such as rock-salt which show a
marked Debye-effect would be of interest and
that such crystals may be expected to show a
strong scattering of ordinary light capable of
direct visual observation. This expectation is
shown to be justified by experiment, anct it is
found that even in the case of quartz in which
owing to its high characteristic temperature
the effect is weaker, direct visual observation
of the scattering is possible.
Theory.
62. A theoretical discussion shows that the
observed effects are of the expected order of
magnitude and are thus really due to the thermal
agitation of the atoms in the crystal and not to
the presence of inclusions in the crystal. The
1 Proc. Boy. Soc. Vol. 95, 1919, page 479.
78 MOLECULAR DIFFRACTION OF LIGHT
principles on which we must proceed become
clear when we consider the hypothetical case of a
crystal in which the atoms occupy fixed positions
on a space-lattice, thermal movements being
assumed to be non-existent. The size of a cell
in the lattice being small compared with the
wave-length of the incident light, the crystal
may for practical purposes be regarded as a
continuous homogeneous medium of uniform opti-
cal density and can accordingly scatter no light.
As thermal movement disturbs the uniformity
of the medium and introduces local fluctuations
of optical density, the medium is no longer
homogeneous but shows irregular variations of
refractive index, which though small, nevertheless
in the aggregate, result in an appreciable scat-
tering of the light traversing the medium. The
intensity of this scattering can be calculated if
the average magnitude of fluctuation of optical
density is known.
63. It has already been pointed out in the
chapters on scattering in gases and liquids that
precisely the same considerations result in the
Einstein-Smoluschowski formula for the scatter-
ing power, namely,
where P is the compressibility, /* the refractive
index of the substance, A is the wave-length of
SCATTERING IN CRYSTALS 79
incident light and E, T, ^ are the constants of
the kinetic theory.
64. The success of Debye's theory in ex-
plaining the influence of temperature on X-ray
reflection by crystals suggests that the Einstein-
Smolu-chowski theory (which is based equally
with Debye's theory on the principles of statistical
mechanics) should enable the scattering power
of crystalline solids for ordinary light to be
determined. An important reservation is how-
ever necessary owing to the known failure of
the law of equipartition of energy in the case of
substances with a high characteristic tempera-
ture such as diamond. The formula for the
scattering power deduced on the assumption
that the translatory kinetic energy of the* in-
dividual atoms in the space-lattice is the same
as that of the freely moving molecules in gases
and liquids would obviously give us a result
much in excess of the actual values.
65. The scattering power being directly pro-
portional to the thermal energy, it is clear that
in order to obtain the correct result, we should
dimmish the value given by the formula in
the ratio which the actual heat-content of the
solid at the temperature of observation bears
to the heat-content determined on the principle
of equipartition of energy. A calculation made
on this basis and from the known compressibi-
lities and refractive indices gives a scattering
80 MOLECULAR DIFFRACTION OF LIGHT
power for quartz about 10 times and for rock-
salt about 40 times that of air at N.T.P.
Visual observations of scattering in crystals.
66. In view of the fact that the scattering of
light in dust-free air is easily visible, it is clear
that the observation of the scattering of much
greater magnitude in crystals indicated by the
theory should be a simple matter provided the
conditions necessary for success are attended to.
Sunlight is evidently the best source of light to
use in carrying out the experiment. A beam of
it being admitted into a darkened room through
an aperture and then focussed by a lens, the
crystal is placed at the narrowest point of
the cone of rays. In examining valuable
material, it is a good plan to use a filter to cut
out the heat rays to avoid possible damage to
the crystal. It is not at all necessary to use a
large block of crystal. In fact quite a modest-
sized piece of good quality will do, but it is of
the highest importance that all the faces of the
crystal should be scrupulously clean and highly
polished so that they do not scatter light. The
most suitable shape for the block is a cube or
a rectangular parallelepiped held with one pair
of faces quite square to the incident beam of
light, the track of the cone of light inside the
crystal being observed through another pair of
SCATTERING IN CRYSTALS 81
faces. A natural cleavage block of transparent
rock-salt thus seems very suitable for the
observations. If a crystal, say of quartz, is of
irregular shape or has oblique faces, a good
plan of getting rid of stray light is to immerse
the block in a square glass trough containing
clean distilled water. A dark background
should be provided against which the track of
the light passing through the crystal should be
viewed. Working in this way the scattering
of light in clear colourless quartz is very readily
observed visually. The Tyndall cone is quite
uniform and of a beautiful blue colour closely
matching that of the track of a concentrated beam
of sunlight in saturated ether vapour, and of about
a third of its intensity so far as can be judged
visually. The latter furnishes a convenient
standard of intensity, and the observed result is
thus of the order expected on theoretical grounds.
Accurate measurements by a photographic
method are at present being made in the author's
laboratory by Prof. Lalji Srivastava.
67. By a similar method, light-scattering in
rock-salt and in block ice can be very readily
observed, the track being of a blue colour. In
Iceland spar, the track is of a reddish tinge due
apparently to a feeble fluorescence. This may
be quenched by a suitable filter.
11
82 MOLECULAR DIFFRACTION OF LIGHT
Polarisation of the Scattered Light.
68. In making observations on the polarisa-
tion of the light scattered in crystals, account has
to he taken of the doubly-refractive or optically
active property of the material. In the case of
quartz, the difficulty may he avoided hy sending
the beam of light in a direction transverse to the
optic axis, and observing in a direction
transverse to the axis as well as to the track of
the primary beam. Using this method, it is
found that the light scattered transversely in
quartz is not completely polarised, the track
-being quite clearly visible through a nicol. The
cases of other crystals have not yet been
thoroughly examined.
69. There is a noteworthy feature in which
the light-scattering in crystals arising from the
thermal movements of the atoms stands on a
somewhat different footing from the case of
light-scattering in liquids or gases. It has
already been remarked in dealing with fluid
media that the transversely-scattered light con-
sists in part of common or unpolarised light even
when the primary beam itself is completely
polarised to begin with, and that this effect
arises from the arbitrariness of the orientation
of the molecules in such media. In crystals on
the other hand, according to the current ideas,
the positions and orientations of the atoms are
SCATTERING IN CRYSTALS 83
more or less definitely fixed, subject only to
small oscillations about the mean positions.
If this be the case, we should expect that
if the primary beam in the crystal is itself
polarised, the transversely scattered light should
also be polarised, though not necessarily
in the same way as in the case of spherically
symmetrical atoms. Observations have been
made by the writer to test this point. In order
more readily to detect the residual intensity
of the track of the beam in the crystal, the
method of " flicker " was used. The track was
caused to vibrate slowly up and down in the
crystal so that its existence or non-existence
could be detected. It was found that the track
of the beam could almost completely be quenched
by observation through a nicol when the primary
beam was itself polarised. But if the incident
light was unpolarised, it always remained quite
clearly visible in any position of the observing
nicol. The matter however remains to be
further tested by photographic methods.
Possible Influence of Temperature.
70. As in the case of the Debye-effect, we
should expect the light-scattering power of the
crystal to be enhanced by rise of temperature.
Some preliminary observations made with rock-
salt seem to indicate that there is such an effect.
84 MOLECULAR DIFFRACTION OF LIGHT
The technique of experimentation on light-
scattering with crystals placed in enclosures
capable of beiDg heated up or lowered in tem-
perature without damage to the surface of the
crystal requires however to be further developed.
CHAPTER VII
SCATTERING OF LIGHT IN AMORPHOUS SOLIDS
71. The methods of examination by the use
of X-rays introduced by Laue and by Professors
Sir W. H. Bragg and W. L. Bragg have thrown
much light on the problem of the structure of
crystalline solids, but our information regarding
the structure of amorphous solids like glass is
still scanty. "What information we do possess,
we owe to the recent work of Debye and
Scherrer by the X-ray powder method. They
find that most solids hitherto classified as amor-
phous are really composed of a large number
of minute crystals. Dehydrated colloidal silica
and stannic acid show the presence of such
crystalline aggregates in an otherwise amorphous
medium. Optical glass alone, of all the solids
investigated, does not show any crystalline
inclusions. Its diffraction photograph is exactly
the same as that of a liquid.
72. The essential difference, then, between
a crystal and an amorphous solid is that, in a
crystal, the atoms are similarly oriented and
arranged in a perfectly regular manner, whereas,
in an amorphous solid, there is no regularity of
arrangement of the molecules and there may
even be local fluctuations of density as in a
86 MOLECULAR DIFFRACTION OF LIGHT
liquid; only, these local fluctuations do not
alter rapidly with time as in the case of liquids,
but remain quasi-permanent for very long
periods of time. Why a mixture of complex
silicates like glass develops the phenomenon
of rigidity to such a high degree in a non-
crystalline condition, awaits explanation.
73. If, then, glass is an undercooled liquid,
we should expect the scattering power of glass for
ordinary light to approximate to that of a liquid
rather than to that of a crystal. Lord Rayleigh
in his paper on " Scattering by Solid substances,"
mentions that a specimen of Chance's Optical
Glass showed a scattering about 300 times that
of dust-free air. He was, however, inclined to
attribute the scattering to inclusions and
explained the observed imperfectness of the
polarisation of the scattered light as due to the
large size of the included particles. In view of
the fact that the closest scrutiny under a power-
ful microscope even with dark-ground illumina-
tion, fails to indicate the presence of any
visible inclusions, and in view of Debye and
Scherrer's X-ray analysis of optical glass, it
seems more reasonable to assume that the
scattering is really molecular. Its magnitude
is much larger than in the case of clear crystals
and agrees with what might be expected on the
basis of a non-uniform distribution of molecules
such as would have existed in the liquid state
SCATTERING IN AMORPHOUS SOLIDS 87
at the temperature of solidifaction of the
material. Lack of data regarding the com-
pressibility of melted glass at high temperatures
makes it impossible to make a quantitative
calculation of the scattering co- efficient on the
basis of the Einstein-Smoluchowski equation.
Observations made in Calcutta on a specimen
of optical glass show a scattering power nearly
four times that of pure water at ordinary
temperatures. The track of a beam of sunlight
is sky-blue in colour and is nearly, but not
completely, polarised when viewed in a transverse
direction. It does not show any fluorescence.
(Many specimens of common glass exhibit a
green, yellow or pink fluorescence when a beam
of sunlight is sent through them ; such fluore-
scence can be easily detected by examining the
scattered light through a double image prism, when
the two images would show different colours.)
74. Quantitative studies of the intensity and
polarisation of the light scattered by well-
annealed glasses of known composition at
different temperatures would yield results of
value regarding the molecular structure of
glasses and of amorphous bodies in general.
Experiments on the scattering of light in fused
quartz of optical quality would also be of
special interest in view of the recent observation
of Kayleigh that this material exhibits a feeble
double-refraction.
CHAPTER VIII
THE DOPPLER EFFECT IN MOLECULAR
SCATTERING
75. In the discussion of fundamental princi-
ples contained in our first chapter, we have
already had occasion to refer to the Doppler effect
arising from the uncoordinated movements of the
molecules and found that it has no influence on
the proportion of energy laterally scattered. We
may now briefly consider the question whether
it has any effect on the refractivity of the
medium. The light scattered by a stationary
molecule has the same wave-length in all direc-
tions as the incident radiation ; and if we leave
out of account the question of polarisation, there
is no direction specially favoured as regards in-
tensity as well. But in the case of a moving
molecule, the wave-length of the scattered light
is smaller in the direction of motion than in the
opposite direction or intermediate directions.
Since the molecule receives the incident radia-
tion with an altered frequency, its motion must,
according to the Hayleigh law of scattering,
alter the intensity of the scattering, the latter
being increased when the molecule moves against
the advancing waves and decreased when it
THE DOPPLER EFFECT 89
moves with the advancing waves. The velocity
of the scattered waves is however independent
of the movements of the molecules, and hence
the phase-relation between the advancing primary
and secondary waves remains unaffected. The
coherence of the primary and the scattered
waves in the direction of propagation of the
former on which the refractivity of the medium
depends continues therefore to subsist. Any
alteration in the scattering power of a molecule
must produce a corresponding alteration in its
contribution to the refractivity of the medium.
If we assume that the movements of the mole-
cules occur in random directions, the increased
scattering and refractivity due to the molecules
moving up towards the incident light is com-
pletely set off by the decreased scattering and
refractivity due to the molecules moving in the
opposite direction, and hence the refractivity
of the medium considered as a whole remains
unaffected. If however all the molecules have
a common direction of movement relative to
the advancing primary waves, the case is entirely
different. If the molecules move against the
direction of propagation of the primary waves,
the scattering by all of them is increased and
hence also the refractivity of the medium. If
the molecules move with the waves, the scatter-
ing is diminished and therefore also the refrac-
tivity. In other words, the velocity of light
90 MOLECULAR DIFFRACTION OF LIGHT
through the medium is increased or decreased
by a certain proportion of the common velo-
city of its ultimate particles. This is exactly
Eresnel's principle of the convection of light in
a moving medium, and in a paper appearing
in the Philosophical Magazine, Dr. Nihal Karan
Sethi and the present writer have shown that
the convection of light (Fizeau effect) in moving
gases can be explained in this way, and we
obtain (at least in the case of gases where the
molecules can be regarded as independent centres
of secondary radiation) a convection co- efficient
agreeing with the values given by Eresnel's
well-known expression and by the Theory of
Relativity. The extension of the same argument
to the case of liquids and solids will probably
not present insuperable difficulties.
Experimental Observations of Doppler Effect.
76. As is well-known, the Doppler effect in
the light reflected from a system of moving mirrors
was demonstrated experimentally by Belopolsky
and later by Prince Galitzin, and Stark's work
on the Kanalstrahlen has also established the
effect in the light emitted by electrically lumi-
nescent moving molecules. Recently Eabry and
Buisson1 have greatly simplified the laboratory
demonstration of the Doppler effect by using a
1 Journal de Physique, Tome 9, 1920, pp. 234-239.
THE DOPPLER EFFECT 91
rapidly- revolving paper disk, the edge of which
is illuminated by a mercury lamp and observed
through an etalon. It appears to the author that
it would be interesting and quite practicable to
make an experimental study of the Doppler effect
in light scattered by moving molecules. The
experimental arrangements most suitable would
probably be very similar to those adopted in
Eabry and Buisson's experiments. A flat re-
volving steel vessel containing compressed car-
bon dioxide or some suitable liquid may be
provided with glass windows through which
monochromatic light is admitted into it, the
scattered light being observed laterally. By
photographing the scattered light through an
etalon and reversing the direction of rotation,
the alteration of wave-length should be capable
of observation. Simpler still would be to ex-
periment with the light internally scattered
within a rapidly revolving disk of glass. It
would also be interesting to find in such cases
whether there is any difference in the behaviour
of molecularly scattered light and of fluorescent
radiation.
77. The widening of the lines in the spect-
rum of a luminous gas due to the Doppler effect
arising from the thermal movements of the
molecules in it has been discussed by several
writers, notably by the late Lord Rayleigh, and
has been established by laboratory experiments.
92 MOLECULAR DIFFRACTION OF LIGHT
It would appear worth while to examine experi-
mentally the similar effect which may he
expected to arise in the light scattered by a gas
at high temperature. Light from a source at
low temperature may he passed through a com-
pressed gas or a liquid at a high temperature
and the width of the lines in the spectrum of
the scattered light determined by photographing
it through an etalon or echelon spectroscope.
The magnitude of the effect that may be expected
has been discussed theoretically at the sugges-
tion of the author in a paper by Mr. Panchanan
Das.1 The astrophysical importance of the
Doppler effect in molecular scattering in such
cases as for instance, the light of the sun's corona
is fairly obvious, and has already been emphasised
by Fabry.2
PlancWs Law and Molecular Scattering.
78. The Doppler effect in molecular diffrac-
tion is also of theoretical importance from another
standpoint. Consider a space bounded by com-
pletely reflecting walls and containing enclosed
within it radiant energy corresponding to some
known temperature distributed amongst the
different wave-lengths according to Planck's law
of radiation. We may assume further that the
1 Bulletin of the Calcutta Mathematical Society, 1921, pp. 6-10.
8 journal De Physique. Tome 7, 1919, pp. 89-102.
THE DOPPLER EFFECT 93
enclosed space contains a few molecules of a gas
at the same temperature, and for simplicity also
assume that the molecules do not either absorb
or emit light but merely scatter the radiations
incident on them in accordance with the Rayleigh
law of scattering. Owing to the movement of
the molecules, the scattered energy will not
always have the same wave-length as the incident
waves, and hence the postulated conditions pro-
vide a mechanism for the interchange of energy
between different wave-lengths. If, further, we
assume that the molecules scatter the waves
incident on them continuously, the mechanism
provided for the interchange of energy would
operate according to the classical laws of electro-
dynamics, and the final distribution of energy
in the enclosure would not be that given by
Planck's law but would necessarily be that
consistent with the principle of the equipartition
of energy1 viz. —
f(X) d\ = STT RT X-* d\
In other words, the distribution of energy in the
enclosure which was postulated in the first
instance would be altered, and the thermodynamic
equilibrium of the system would be upset. As
the system was assumed to be initially at the
s£me temperature throughout, such a conclusion
is primd facie inacceptable, and we must therefore
draw the inference either that the Rayleigh law
1 Of. Jeans ; Report on Quantum Theory, § 10.
94 MOLECULAR DIFFRACTION OF LIGHT
of scattering is not valid or that the mole-
cules do not scatter the radiations incident
on them continuously. Since the Kayleigh
law of scattering is supported by experiment,
at least over a considerable range of wave-
lengths, it seems more reasonable to accept
the latter conclusion, and to infer that molecular
scattering of light cannot take place in a conti-
nuous manner as contemplated by the classical
electrodynamics. It seems to be difficult, how-
ever, to reconcile this with the hypothesis that
light is propagated through space in the form of
continuous waves, and we are apparently forced
to consider the idea that light itself may consist
of highly concentrated bundles or quanta of
energy travelling through space. This will be
further discussed in the following chapter.
CHAPTER IX
MOLECULAR DIFFRACTION AND THE QUANTUM
THEORY OF LIGHT
79. In the year 1905, Einstein l put forward
the hypothesis that the energy of a beam of light is
not distributed continuously in space but that it
consists of a finite number of localised indivisible
energy-bundles or " quanta " capable of being
absorbed or emitted only as wholes. The theory
had some notable successes to its credit, especially
the prediction of the photo-electric equation and
the explanation of the phenomena of ionisation
of gases by X-rays. Nevertheless it has been
felt that very serious difficulties stand in the
way of its acceptance. Maxwell's electro-mag-
netic theory conceives the energy of light as
distributed in a continuous manner through space
and offers a satisfactory explanation of whole
groups of phenomena, the mere existence of some
of which, especially those classed under the
heading of interference and diffraction, seems
very difficult to reconcile with the hypothesis of
light-quanta. The tendency has therefore been
to regard the propagation of light in space as
determined by Maxwell's equations, but that
1 Atmalen der Physik, p. 132, 17, 1905.
93 MOLECULAR DIFFRACTION OF LIGHT
these equations for some reason or other fail
when we have to' deal with the emission or ab-
sorption of energy from atoms or molecules. The
discontinuity is thus conceived to be limited to
the act of emission or the act of absorption or of
both. Historically, the quantum hypothesis had
its origin in the derivation of Planck's radiation
formula, and an assumption that the disconti-
nuity occurs only in emission is apparently
sufficient for that limited purpose. Hence, though
Planck's hypothesis of quantum emission, rein-
forced as it has been by the success of Bohr's
theory of line-spectra, has passed into general
acceptance, Einstein's idea of light-quanta has
apparently been regarded as unnecessarily revo-
lutionary in character. This feeling has perhaps
been strengthened by the considerable degree of
success which has attended the use of the " cor-
respondence-principle " recently introduced by
Bohr in which an attempt is made to effect a
reconciliation, limited though it be, between
Maxwell's theory and the quantum theory of
emission of light.
80. If, however, we view the matter from a
purely philosophic standpoint, Einstein's original
conception of the discontinuous nature of light it-
self has much to recommend it. It fits in with the
assumed discontinuous character of the emission
and absorption of energy as part of a consistent
and homogeneous theory, whereas the idea that
QUANTUM THEORY 97
emission and absorption are discontinuous while
the propagation of light itself is continuous
belongs to the class which Poincare has described
as " hybrid hypotheses." Such hybrid hypo-
theses may temporarily serve as useful planks to
bridge gaps in existing knowledge, but there is
little doubt that they must ultimately make way
for a more consistent system of thought. His-
torically, Maxwell's theory is the embodiment of
the belief of nineteenth-century physicists in the
validity of Newtonian dynamics as applied to
physical phenomena in their ultimate analysis,
and especially as applied to phenomena occurring
in the medium which was postulated as pervad-
ing all space. The belief in the validity of
Newtonian dynamics as applied to the ultimate
particles of matter has however received a rude
shock from the success of the quantum theory
as applied to the theory of specific heats, and
there seems no particular reason why we should
necessarily cling to Newtonian dynamics in con-
structing the mathematical frame-work of field-
equations which form the kernel of Maxwell's
theory. Rather, to be consistent, it is necessary
that the field-equations should be modified
so as to introduce the concept of the quan-
tum of action. In other words, the electrical
and magnetic circuits should be conceived
not as continuously distributed in the field but
as discrete units each representing a quantum
13
98 MOLECULAR DIFFRACTION OF LIGHT
of action, and possessing an independent
existence, somewhat in the manner of vortex-
rings in a perfect fluid. Interference and
diffraction phenomena may then be conceived
of as arising from the approach or separation,
i.e., crinkling of the mean "lines of flow" of
energy in the field.
81 . Bohr's theory has made the idea familiar
that the emission or absorption of light from the
atom or the expulsion of an electron involves
something in the nature of a catastrophic change
in the atom itself. If, therefore, we wish to
look for some experimental support for Einstein's
conception that light itself consists of quantum
units, we must consider those optical phenomena
in which obviously no such catastrophic change
in the atoms or molecules is involved. The
molecular diffraction or scattering of light is
obviously such a phenomenon, which stands in
the most intimate relationship with the general
theory of the propagation, reflexion, refraction
and dispersion of light. If we found that the
phenomena of molecular scattering of light are
completely and satisfactorily explained .on the
basis of the classical electromagnetic theory,
.the case against Einstein's conception would be
enormously strengthened. If, on the other hand,
we find that the classical theory based on the
idea of continuous wave-propagation breaks down
and fails to explain the observed facts, we should
QUANTUM THEORY 99,
naturally feel called upon to revise our ideas
regarding the nature of light itself.
82. In view of the foregoing remarks, the
fact already mentioned in a previous chapter that
the scattering power of compressed carbon
dioxide gas as determined by the present Lord
Rayleigh is far smaller than that which is indi-
cated by the Einstein-Smoluchowski formula
appears highly significant. The theoretical for-
mula expresses the scattering power of the
medium in terms of its compressibility and
refractive index, and is based on the conceptions
of the kinetic theory of matter and of Maxwell's
electromagnetic theory of light. It expresses
the scattering power of a gas at ordinary pres-
sures correctly, and also the scattering power
of liquids with tolerable accuracy. But it fails
altogether to express the scattering power of
compressed carbon dioxide gas under the con-
ditions of Lord Rayleigh's experiments, that
is, when it is in the form of a saturated vapour
below the critical temperature. There are three
possible alternatives in explanation of this failure ;
firstly that the derivation of the formula is not
valid for some reason or another in the parti-
cular conditions of Lord Rayleigh's experiment :
secondly that the conceptions of the kinetic
theory are invalid under those conditions : thirdly
that the continuous wave-theory of light does
not represent facts.
100 MOLECULAR DIFFRACTION OF LIGHT
83. In respect of the alternative explanations
referred to in the preceding paragraph, it may
be pointed out that the experimentally observed
result is precisely what might be expected ac-
cording to the conception that light consists
of discrete quanta moving through space. If
we imagine a stream of such quanta passing
through a highly compressed gas, scattering
of light would result when a quantum en-
counters a molecule and suffers a large-angle
deviation in its path. Such encounters would
occur according to the laws of chance ; in other
words, the molecules should be regarded not as
scattering light continuously but only occasional-
ly, and at any instant, only a small proportion
of the molecules distributed at random through
the gas are in action. Hence the total number of
quanta scattered in any appreciable interval of
time would be simply proportional to the number
of molecules per unit volume, and would be
practically independent of the actual manner in
which they are distributed in the space, so long as
a quantum is regarded as impinging on only one
molecule at a time and not on two or more
simultaneously. In other words, the principle
of additivity of the energies scattered by the
individual molecules would be applicable even
in the case of a highly compressed gas for which
Boyle's law does not apply. This is the result
actually obtained, whereas on the continuous
QUANTUM THEORY 101
wave-theory in which all the molecules are
conceived of as scattering light all the time,
the resultant effect would depend on their distri-
bution in space, and in the case of a highly
compressible gas would not be determined by
the additive principle. In fact, the observations
of Lord Rayleigh were regarded by him as
supporting the principle of additivity of the
energy-effects of individual molecules, and
this principle, as we have seen, cannot be
reconciled with the results of the classical wave-
theory under the conditions of the experiments.
84. Though, primd facie, the phenomena of
molecular scattering in highly compressed gases
seem thus to support Einstein's conception of
light-quanta, the cautious reader would naturally
wish to make sure that the two alternative expla-
nations of the result suggested above must be
excluded. So far as can be judged on the avail-
able evidence, neither of the two alternatives
seems very probable. In order, however, to ex-
clude them definitely, two series of experiments
have been undertaken in the author's laboratory
at Calcutta. In the first series of experiments
which is being carried out by Mr. K. H. Rama-
nathan, an attempt is being made to confirm
Rayleigh's result for the scattering by compressed
carbon dioxide and extend it to the case of
itnsatiirated vapours and also to gases at
temperatures considerably above the critical
102 MOLECULAR DIFFRACTION OF LIGHT
temperature. It is hoped to find the scattering
power of various gases and vapours besides
carbon dioxide over a wide range of pressures
and temperatures. If the experiments support
E/ayleigh's result, the experimental basis for
inferring the failure of the Einstein-Smolu-
chowski formula would be greatly strengthened.
In the second series of experiments which
has been undertaken by Mr. J. C. Kames-
wararao, an attempt is being made to study
the Brownian movement quantitatively in gases
and vapours under high pressures, in order to
find whether the energy of molecular movement
indicated by the kinetic theory agrees substan-
tially with that found in experiment. The results
of the two sets of experiments may well enable a
final judgment to be arrived at legarding the
validity of Einstein's conception of the propaga-
tion of light in quanta.
85. The belief in the correctness of the prin-
ciples of the wave theory is to a large extent
based on the quantitative agreement between the
co-efficients of reflexion and refraction indi-
cated by Fresnel's formulae and those found
in experiment. Already certain failures of
Fresnel's formulae are known, as for instance
the existence of reflexion at the boundary
between two media having equal refractive
index,1 and it seems important to make a
1 Rayleigh, Scientific Papers, Vol. V.
QUANTUM THEORY 103
careful re-investigation of the co-efficients of
reflexion and refraction in various cases, e.g.,
at the boundary between a liquid and its
vapour slightly below the critical temperature,
in order to find whether the quantitative agree-
ment between the results of the classical wave-
theory and the facts is really so brilliant as is
generally believed.
86. The phenomena presented by the
scattering of the X-rays and especially the
well-known failure to obtain any refraction of
X-rays will no doubt have to be re-discussed
in the light of foregoing remarks and the results
of the optical experiments.
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