THEORIES OF DEPOLARIZED LIGHT SCATTERING

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THEORIES OF DEPOLARIZED LIGHT SCATTERING

N. Ailawadi, B. Berne

To cite this version:

N. Ailawadi, B. Berne. THEORIES OF DEPOLARIZED LIGHT SCATTERING. Journal de Physique

Colloques, 1972, 33 (C1), pp.C1-221-C1-229. �10.1051/jphyscol:1972138�. �jpa-00214928�

JOURNAL DE PHYSIQUE

Colloque Cl,.suppliment au no 2-3, Tome 33, Fkvrier-Mars 1972, page C1-221

THEORIES OF DEPOLARIZED LIGHT SCATTERING (*) N. AILAWADI and B. J. BERNE

(**)

Columbia University, New York, N. Y., 10027, U. S. A.

Rksumk.

-

On passe en revue et on compare les theories existantes des spectres de la lumikre depolarisee diffusee par les liquides non associes. On presente une thCorie complkte qui dans diverses limites se rCduit

h.

toutes les theories qui existent dans la litterature.

Abstract.

-

Existing theories for the splittings of the depolarized light scattering spectra in non-associated liquids are compared and reviewed.

A

comprehensive theory is presented, which reduces in various limits to a11 of the theories that exist in the literature.

Introduction. - Several theories [I]-[7] have been offered to explain the recently observed splittings in the depolarized light scattering spectrum of certain non- associated molecular liquids 181, [9]. The observed spectrum consists of a diffuse band (sharp band that may be split) sitting on top of a broad background.

Unfortunately, none of the theories that have appeared in the literature can account for the intensity ratio of the sharp and the broad bands [lo].

In the present paper we point out (a) some draw- backs of the existing theories, and (b) show that it is possible, in principle, to construct a unified theory of the observed depolarized spectrum. We shall illuminate this by considering, specifically, the I,, spectrum.

11. Theories of depolarized light scattering. - It is generally well known [ l l ] that the light scattering spectrum is determined by the spectrum of dielectric fluctuations. Thus, any theory of light scattering naturally divides into two parts

:

(a) The determination of those fluctuations in the system which directly give rise to dielectric fluctuations.

These we call

⁽⁽

direct

⁾⁾

fluctuations.

(b) The computation of the spectrum of the

(<

direct

⁾⁾

fluctuations. This involves deriving and

solving

^<(

equations of motion

>>

for the direct variables.

The equations of motion often are such that the direct fluctuations are coupled to other fluctuations which for the sake of simplicity we call indirect

^)>

fluctuations.

Previous calculations of the Brillouin spectra in liquids furnish an illustration of this [12]. In these theories the density fluctuation is a

^<(

direct

⁾⁾

fluctua- tion whereas the longitudinal current and the energy density are

⁽⁽

indirect

D.

The equations of motion that couple the direct and indirect variables are the equa- tions of linear hydrodynamics.

The depolarized spectrum is in principle much more

(*)

This work was supported

by a

grant from the

N. S.

F.

GP.

(**)

Alfred Sloan Foundation Fellow.

complicated than the isotropic Brillouin spectrum- The theories that have appeared in the literature are summarized in Table I. In this table

p, g,, ^E

are respec- tively, the number, momentum, and energy densities,

S j

is the the molecular rotational angular momentum.

density,

z$',

z$),

7;;)

are respectively the scalar, anti- symmetric and symmetric traceless parts of the micro- scopic stress tensor, uc' are the scalar, and symmetric parts of the distortion field (uij

⁼

Viuj where uj i s the displacement field).

a i j

and t i j are the symmetric parts of the polarizability density and orientationali fluctuations. From the table we see that the theories of A. B. F. I [5] and A. P. I [6] involve precisely the same sets of direct and indirect variables and, more- over, give precisely the same results. The theories o f Rytov [l] and Volterra [2], are viscoelastic theories in the classical sense since they involve displacement and distortion fields. Since these concepts only have.

direct meaning in connection with crystals and amor- phous solids, it is difficult to fit them into t h e conceptual framework of the other theories. Never- theless, if we identify the time rate of change of' the displacement field ;, with the velocity field V i

in the other theories, it immediately follows that A. B. F. I [5] and A. P. I [6] are equivalent to Rytov's, earlier theory [I], and that A. P. I1 [6] is equivalent to]

Volterra's theory 121. As a matter of fact, A. B. F. I [5], was formulated with this in mind.

In the theories of Ben-Reuven and Gershon [3],.

Keyes and Kivelson [4] and A. P. I1 [6], it is assumed that orientational fluctuations of the molecules cause the anisotropic dielectric fluctuations. From this point of view

cc Cij.

In fact, A. P. I [6] is derived with any second rank symmetric traceless tensor which couples to hydrody- namic variables (be it d2) or 5")). Thus, by choosing

z$',

as they do, they arrive at the same results a s A. B. F. I [5] or Rytov's [I]. If they had worked out the theory by taking c(') as the direct variable, they

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972138

N. AILAWADI AND B. 1. BERNE

Direct and indirect j?uctuations in dzyerent theories of the depolarized I,, spectrum observed in light scattering spectroscopy of molecular liquids

Direct Indirect

Theory fluctuations fluctuations Comments

-

- -

-

A. B. F. I {

7 8 )

> { g i > A viscoelastic theory (derivation of

Rytov's Theory)

A. B. F. I1

(")

{ zij

^{( 1 )}

, zij

⁽²⁾

) { Si, Sj > ^Uses

⁽⁽

Newtonian form

>)

of the stress tensor

Rytov Volterra

{ a$' } { g i I No account is taken of the antisymmetric stress

,

tensor

{

7:;'

> ^{{ g i I} Presence of antisymmetric stress tensor is reco- gnized but is not considered either as a direct or indirect fluctuation

{ u p } {

7;;)

> ^{z i j} is the stress tensor corresponding to the distor- sion field uij

Ben-Reuven and { a$;) { g i ) The indirect fluctuation is not explicitly stated

Gershon except that it is hydrodyn~amic

C. Y. { zi;' > ^{{ g ,}

^&?'

⁾ Reverses Volterra's results would have obtained the KK theory [4]. Were we to

assume further, following Leontovich [13], that the anisotropic dielectric fluctuations are determined entirely by the rigid molecule polarizability, aij, and furthermore that

then all the theories except A. 3. F. I1 [5] would be equivalent as far as the diffuse line of the observed depolarized spectrum is concerned. A. P. I1 [6], Volterra P] and Chung and Yip [7] went further and tried to account for the existance of a broad back-

(*) We wish to add a correction to our previous paper (ref. 5):

In section IV, we used the experimental value of

to obtain figures 3 and 4 rather than the value quoted there.

By using this value, the relationship SZ; = vqzlz is not satisfied.

As noted in footnote 17 of ref. 5, this discrepency also appears in Stegman's fit to Rytov theory. We would also like to point out that the use of the relation SL; = vqzlz in the second term of eq. (20a), yields

which gives a spike rather than a dip in the diffuse central line.

ground as well, by coupling z i j to I i j dynamically.

However, it has been pointed out by Keyes and Kivelson [lo] that while it is por;sible to get the width of the broad background, these theories do noy give the experimentally observed value of the relative intensity of the broad background with respect to the diffuse central line.

We would end up in a similar situation were we to couple the polarizability tensor a which is a cc direct

⁾⁾

fluctuation with the ((indirect

>>

fluctuations such as the momentum density

g

and the instrinsic angular momentum density S (along with the number density and energy density). Thus, it is clear that the observed broad background in the depolarized spectrum cannot be obtained by coupling the dielectric constant to only one direct variable. We present here a two coupling constant theory which could in principle correctly give the observed integrated intensity ratio of the broad background and the diffuse line.

111. Unified theory of dep~olarized spectrum. -

It should be noted that the only theory that includes an

antisymmetric tensor in either the direct or indirect set

of fluctuations is A. B. F. I1 [5]. In A. P. [6] and

K. K. [4] the existance of antisymmetric stress tensor

is allowed but is not used in this way. This is remar-

THEORIES OF DEPOLARIZED LIGHT SCATTERING Cl-223

kable since in fluids consisting of anisotropic molecules

which interact through non-central forces, the microscopic stress tensor must contain an antisymme- tric part, zg', as well as symmetric parts z!:', and, re'. The

⁽⁽

Newtonian form

^D

of these tensors is

where qv, q,,

q,

are the bulk, shear and rotational viscosities, and there V and o are the linear and angu- lar velocity fields respectively. The antisymmetric stress tensor arises because of intermolecular torques and is, in fact, the torque density.

The second thing that should be noted is that if the stress tensor has an antisymmetric part, it is necessary to consider the angular momentum density, Sj(r, t)

;

otherwise the conservation laws are not obeyed.

Although this last statement is correct without res- triction, it is easy to see from the

^ct

Newtonian

^)>

stress tensors together with the momentum conserva- tion law,

that the hydrodynamic equations are incomplete unless the angular momentum is considered. Unfor- tunately, only in A. B. F. I1 [5] is this fact realized and exploited. Nevertheless, even that theory is incom- plete since it presents a purely

((

hydrodynamic theory

D

in which the

⁽⁽

Newtonian form

^)>

of the stress tensor is used. This procedure gives rise to only the diffuse line. In what follows we show that when we take into account the microscopic relaxation of the stress tensor to its

⁽⁽

Newtonian form

D,

we can account for both the sharp and broac features of the depolarized Rayleigh spectrum.

First we note that since there are spontaneous fluctuations which give rise to antisymmetric and symmetric stress fields there is no reason to omit antisymmetric fluctuations in the dielectric tensor.

In fact, we write the off-diagonal elements of the dielectric fluctuations, dqj, as

6eij(r, t)

=

XI z&?)(r,

t)

+ X2 zl:)(r, t) . (2) This was the form adopted in A. B. F. I1 [5] whereas in the other theories XI was taken as zero. That S E , ~ couples to the stress tensor follows from the fact that we expect molecules to align in the presence of a stress field.

Some arguments have been raised, recently, which would require XI, in the static limit, to be exactly- zero [15]. These arguments are not clear to us since eq. (2) is consistent with physically known principles

;

and leads to no dissipation of energy under either uniform translations or uniform rotations. This can be checked immediately from the

⁽⁽

Newtonian form

n

of the stress tensor given in eq. (1).

In this paper, we do not assume local equilibrium values of the symmetric and antisymmetric stress tensor

;

but instead use the molecular expressions for these tensors. Also, we consider the set

as the complete set of variables, where the symbols have usual meaning. This is the simplest set of variables since they correspond to the conserved properties, mass, momentum, angular momentum, and energy.

Equations of motion for these variables can be derived using Zwanzig-Mori projection operator techniques [16], [17]. These equations are particularly simple if

:

(a) the Markov approximation is made on the memory functions and (b) spin diffusion [18] is neglected. These equations are presented in Appen- dix A.

Given the equations of motion it is straightforward to calculate the depolarized light scattering spectrum.

In particular, we compute the spectrum ZvH(q,

Q)

for 90° scattering which is

Combining eq. (2) and (3), and solving the equations of motion in the Appendix A for the pertinent stress tensor correlation functions yields the spectrum.

The spectrum is particularly simple if we neglect cross correlations between

z#)

and z!;) i. e. if we assume that < z!;'*(q) z$?)(q, t) >

⁼

0. Then

where

C1 = I XI l 2 < I z;:)(q) l 2 >

^;

Note that if the antisymmetric part of the stress tensor, T:;)(~, t) were not included as a direct variable, the term with C , in the spectrum would be zero and eq. (4) would reduce to the Rytov theory [I] or equi- valently A. B. F. I1 [5] of A. P. I [6]. Assuming, as in the Rytov theory [I], that the second term in eq. (4) describes the diffuse (sharp) Rayleigh line (which for some liquids is split), the first term in eq. (4) would then describe the broad Rayleigh line if

and

C1-224 N. AILAWADI AND B. J. BERNE

Condition

(a)

is a condition on the integrated intensity

ratio of the diffuse and broad bands,

and condition (b) is a condition on the relative ratio of relaxation of z e ) and z$).

While in general we do not know the relative magnitudes of the frequencies QT(q) and QA(q), it is reasonable to assume

[19]

that Q,(q) and QA(q) are the same order of magnitude. With this assumption X2/X, - ^43. Note that when QA(q) is comparable to QT(q) there may also be a splitting of even the broad background.

Let us recall that eq. (4) was determined on the assumption that z;;' and

7:;)

do not directly couple.

This is not rigorously correct. This coupling will lead to an additional contribution to the I,,(q, Q) spec- trum of the form

f(x;x2+X;X1)x

To the lowest order in q this term has a very compli- cated structure which is given explicitly in Appendix B.

This interference term is of the order qZ (to the lowest order in q) smaller than the contribution reported in eq. (4).

Since we are concerned with light scattering expe- riments where q - ^lo5 cm-l, we surmise that these interference terms will be small compared to the terms already included in eq. (4). Needless to say, light scattering experiments are done at finite q so that we cannot be certain about the relative smallness of the interference terms. It should be noted, however, that if these terms are retained, the equations of motion in the Appendix A, and consequently eq. (4), must be corrected since these equations are derived by omitt- ing terms of the same order in q as

Note that apart from a scalar factor, Cy represents the anisotropic fluctuations in polarizabillty of a system composed of rigid molecules. This set of dynamical variables includes all the subsets of variables consider- ed so far by different authors [I]-[7]. Following, A. P. 11

161,

we assume that initially

< z$'*(q) i:;'(q) >

^{= 0}

.

A similar relation is assumed for

< 4;'*(q) c:;)(q) >

Further, the time correlation function

< z!;)*(q) cg)(q,

t)

>

is at least of the order q2 whereas

< z!?'*(q) c:q'(q,

t )

>

is only of the order

qO.

This follows from requirement of isotropy in q. If we make the same approximations as before in deriving eq. (A. 2), we can write down the equations of motion given in Appendix C.

We now calculate the I,,(q, Q) spectrum from eq. (2) and (3) and retain terms to the lowest order in q. The part of the spectrum

< z$,l)*(q) .r$i)(q, Q) >

and

< z;:'*(q) z;;'(q, Q) >

is the same as before.

However,

< ^z;:'*(q)

^zg)(9,

^Q) >

and

< z;:)*(q) ~;:)(q, Q) >

are

< zg)*(q) zg)(q, Q) >

⁼

< ^z:;'*(~)

^.ti;'(q,

^{t )} > . and

< ~ g ) * ( ~ ) zg)(q, Q) >

⁼

2 F:e x We think that we are on safe grounds in ignoring the

terms which arise because of eq. (5). As a matter of < I zg'(q) l 2 > (s2+sr3) fact, all the theories mentioned so far have omitted

_{S3 +}

S2(r2 + r 3 ) + S ( r 2 r3 -rZ3 r32

+

such terms. It would be very difficult to derive a

consistent set of equations of motion to the order

required by inclusion of these interference terms. The poles of the correlation function (8) are, in the limit Nevertheless, were these terms inclused, they wouId

contribute to the splitting of both the diffuse line and (r2 + r3)2

^%'

(r2

j"3

-

^r23

r32), the broad background.

^-

S1 =

-

^rc =

-

r Z r 3

-

r23r32

r2 + r3

1V. Theory involving the orientational fluctuations.

-

In this section we consider, in addition to the set (A. I),

_{s 2 =}

- r s = -

the anisotropic orientational fluctuations C i j as well. 1

THEORIES OF DEPOLARIZED LIGHT SCATTERING C1-225

The correlation function (9) has a third pole certain circumstances interference effects (cross terms might be important). This theory is quite different

S3 =

- a; from other theories that have appeared in the Ijtera-

p2 ⁺

^{r 3}

^-

^{r2 r3}

^-

^{rz3 r3z}

ture [I]-[7] and can account in principle for the inten-

r 2

+

^{r 3}

sity ratio of the sharp line to the broad background.

which is the cause of the splitting in the diffuse line.

In this limit r, is the width of the central diffuse line and I', would be the width of what in Volterra 121, A. P. I1 [GI and Chung and Yip theories correspond to the broad background. If r2 e r 3 , i. e. when the shear stress relaxation is slower than the relaxation due to reorientations,

The width of the central line is then given by r2. ^This

case corresponds to Volterra [2] and A. P. I1 [GI

theories. In the reverse case corresponding to C. Y.

theory [7] r3 < r2 the reorientation relaxation is slower than the shear stress relaxation and accordingly, the central line has a width given by the reorientation relaxation. The integrated intensity ratio of the central line to the broad background is still given by eq. (6).

V. Conclusion.

-

In this paper we presented a viscoelastic theory which is and extension of the Rytov theory [I], by taking into account the presence of the antisymmetric part of the microscopic stress tensor.

This theory gives rise to a central split line and to a broad background. Moreover it suggests that in

Our previous work on the antisymmetric fluctua- tions [5] was based on the local equilibrium form of the stress tensor and could only account for the exis- tance of the sharp line. The magnitude of the rotational viscosity

q,

required for such a fit is about four-orders of magnitude smaller than the shear viscosity y,.

While an experimental value of y, is unknown, esti- mates of McCoy, Sandler and Dahler [20] for dilute gas of rough spheres give a value for rotational viscosi- ty of the same magnitude as shear viscosity. It is not quite clear if the same estimate for

q,

holds even for dense systems

;

the situation can be quite different for dense molecular fluids. It is, therefore uncertain if our previous work could account for the sharp line.

There is no such problem in our present work.

In section IV, we have presented a general scheme in which the ideas of all the previous work on this subject is incorporated. While, it is too early to tell physically what mechanisms are responsible for the observed depolarized spectra, it is fairly clear that two different coupling coefficients are needed in the dielectric fluc- tuations to account for the observations

[21].

Acknowledgements.

-

We have benefited from numerous conversations with Professor Daniel Kivel- son and Mr. Thomas Keyes.

Appendix A. - We restrict ourselves to the consideration of I,,(q,

SZ)

spectrum only

;

therefore only the reduced set of variables

(1) (2)

A

=

(gi, Sj, zij , zij )

^Y

(A. 1)

is considered. We use Zwanzig-Mori projection operator techniques to derive equations of motion for these quantities. Using the approximations given in the text, namely.

i) Markov approximation on the memory functions, and ii) neglecting spin diffusion,

the relevant equations of motion for the geometry usedin our previous paper are

;ii d gY(% t)

=

iq[+)(q, 1)

-t-

zg)(q, t)] (A. 2a)

- d Sx(q, t)

⁼

- 2 z;:'(q,

t )

dt (A. 2b)

d q,

t, =

- rl(q)z$i)(qy

t,

- q2 2(q) z$)(qy

t,

(A. 2e)

Cl-226 N. AILAWADI AND

B.

J. BERNE

where

and

(A. 30)

Note that eq. (A. 3) define p,(q) and pA(q) where p, is the usual high-frequency shear modulus at a wave vector q. ~l,(q), for want of a previous name, we call the high frequency rotational modulus. We have explicitly written in eq. (A.2) the dependence on q to the lowest order. Also

rl(q)

=

[< lYzlz:j'(q) l 2 > < I z:)(q) l 2 > - I < z;?*(q) 2;2)(q) > 1'1-I

^x:

- < ;$)*(q) e"(l-P)L(~ - P) ;/i)(q) > < z/j)*(q) z:)(q) >] (A. 4a)

- < ^{;;:)*(,) eit(l -P)L(l}

^-

^{P) ;;:Vq)} > < T;:)*(~) z;:)(~) >] ^{( A .}

^4c)

q 2 rZ1(q)

=l[<

I ~;:)(q) l2 > < I ~;:)(q) l2 > - I < 41)*(q) ~;:)(q) >

^{l2]-I x}

- < ;g)*(q) ewl-P)L(l - p)

;(2)(

q

)

> < zif)*(4) 2$:)(41) >] . ^{(A. 4d)}

In the low q limit

(A. 5a)

(A. 5b)

'(l)*(q)

^eit(l -P)L

- < zyz < I z;:'(q) (1 - l2 P) ;;:'(q) > < I z$)(q) > < z$:'*(q) z/S'(q) l 2 > ---I > ^{(A. 5c)}

THEORIES OF DEPOLARIZED LIGHT SCATTERING Cl-227

Appendix B. - We now use equations ( A . 2) and (2) and (3) to calculate the depolarized ZvH(q, 8 ) spectrum. First we assume no coupling between T $ ; ) ( ~ , t ) and z!f'(q, t).

From eq. (2) and (3) in the text, the IvH(q, 8 ) spectrum is

iv,(q, Q )

=

3 l

X I

l 2 [< z;:'*(q) z;:)(q, 0 ) > + < z;:'*(q) z;;)(q,Q) >] +

+ 3 1 X 2 I 2 [< z;;)*(q) z;f)(q, 8) > + < zg)*(q) T E ) ( ~ , S2) >] . ( B . l ) Neglecting terms involving q2 r 1 2 ( q ) in eq. ( A . 2 ) [which amounts to just disregarding the coupling between ze'(q) and z:)(q)], we can calculate the spectrum in eq. (B. 1)

which is just eq. (4) in the text.

Next we do not disregard the coupling between z;;'(q, t ) and z$?)(q, t ) but calculate the spectrum Z,,(q, Q ) only to the lowest order in q.

In this case, the spectrum Iv,(q, Q ) is not obtaihable from only eq. ( B . 1) but has, in addition, the cross terms given by eq. (5). Thus

IVH(99 Q)

=

t I X l l 2 [< z;:)*(q) ~ $ 3 9 , Q ) > + < z;:)*(q) z;:)(q,

0 )

>] +

+ : I

^x2

l2 [< zg)*(q) z%'(q, a) > + < 42'*(q) z31'(q, 52) >]

+ 3(x?

^x2

+ XI Xz) [< ~kk'*(q) zg'(q, Q ) + < zbi'*(q) zgl(q, 8) >] . ( B . 3) To the lowest order in q, the first two terms in eq. (B.3) are given by eq. (B.2). The interference (or cross terms) term has a very complicated structure,

The interference terms (B .4) and (B .5) are at least of the order q2 smaller than the terms in eq. (B .2).

Appendix C. - Equations of motion when the orientational JEuctuations are included.

-

These equations are

N. AILAWADI AND B. J. BERNE

The quantities r l ( q ) , r 2 ( q ) , r I 2 ( q ) and r 2 , ( q ) are defined

by eq

( A . 3 ) and ( A . 4 ) . The other quantities are

In the limit

q

^-t

0,

the off-diagonal elements of the friction matrix

are

related.

( C . 10)

( C . 11)

The

yx

components of

d 2 ) , d 2 )

and 5(2)) d o not couple to g or

S

and the equations of imotion for these involve only the friction coefficients

rij.

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Phys., 1971, 54, 1049.

[4] KEYES (T.) and KIVELSON (D.), J. Chem. Phys., 1971, 54, 1786 ; KEYES (T.) and KIVELSON (D.), ⁽⁽ LOW- frequency Depolarized VH-scattering from liquids Composed of Anisotropic Molecules )),

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[5] AILAWADI ( N . K.), BERNE (B. J.) and FORSTER @.), Phys. Rev., 1971, A 3,1472.

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[ I l l LANDAU (L. D.) and LIW:HITZ (E. M.), Electrodyna- mics of Continuous Mzdia (Pergamon Press, New York, 1960).

THEORIES OF DEPOLARIZED LIGHT SCATTERING C-229 See, for example, MOUNTAIN (R. D.), Chemical

Rubber Col, Critical Reviews, Solid State Scien- ces 1970,1,5.

LEONTOVICH (M. A.), J. Phys. (U. S. S. R.), 1941, 4, 499.

In this context, the word cc Newtonian ⁾⁾ implies local-equilibrium value. For more information, the rader is referred to : DE GROOT (S. R.) and MAZUR

(P.),

in Non-Equilibvium Thevmodynamics (North-Holland, Amsterdam, 1962).

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MORI (H.), Prog. Theoretical Phys. (Kyoto), 1965, 33. 423.

bring spin into and out of a volume element.

This has been analyzed by AILAWADI (N. K.), BERNE (B. J.) and FORSTER (D.), Phys. Rev., 1971, A 3, 1462.

[19] This seems to be the case for diatomic liquids where molecular dynamics calculations have been done.

For example in liquid CO 5 2 ~ and QA appear to be of the

same

order

of magnitude.

[20] McCoy (B. J.), SANDLER (S. I.) and DAHLER (J. S.), J. Chem. Phys., 1966, 45, 3485.

[21] A different theory has been proposed by KEYES (T.), KIVELSON (D.) and MCTAGUE (J. P.), ((Theory of k-Independent Depolarized Rayleigh Wing ^)), submitted to J. Chem. Phys. This theory arrives at the same result that slow and fast direct variables 1181 Spin &ffusion is not rotational relaxation. It has to couple with two different coupling constants to

do with translational diffusion processes which the dielectric fluctuations.