LOW FREQUENCY DEPOLARIZED VH-SCATTERING FROM LIQUIDS COMPOSED OF ANISOTROPIC MOLECULES

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LOW FREQUENCY DEPOLARIZED

VH-SCATTERING FROM LIQUIDS COMPOSED OF ANISOTROPIC MOLECULES

T. Keyes, D. Kivelson

To cite this version:

T. Keyes, D. Kivelson. LOW FREQUENCY DEPOLARIZED VH-SCATTERING FROM LIQUIDS COMPOSED OF ANISOTROPIC MOLECULES. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-231-C1-236. �10.1051/jphyscol:1972139�. �jpa-00214929�

JOURNAL DE PHYSIQUE Colloque C1, supplbment au no 2-3, Tome 33, Fivrier-Mars 1972, page C1-23 1

LOW FREQUENCY DEPOLARIZED VH-SCATTERING FROM LIQUIDS COMPOSED

OF ANISOTROPIC MOLECULES (")

T. KEYES (**) and D. KIVELSON

Department of Chemistry, University of California, Los Angeles, California, U. S. A.

Contribution Number 2853

RksumB. - On presente une analyse des diverses theories qui decrivent la partie etroite des ailes de la raie Rayleigh dans les spectres VH de la lumiere d6polarisee diffusee par des liquides composes de molecules de polarisabilite anisotrope. On Btudie A la fois la composante fine qui ne depend pas de k, de forme lorentzienne, aussi bien que le doublet ⁽⁽ transverse ⁾⁾ assez Btroit, qui depend de k.

On fait des comparaisons critiques avec les donnees experimentales en w e de tester la validite des diverses theories. On conclut que les fluctuations du tenseur dielectrique sont directement couplks a la densite de reorientation mol6culaire ; la reorientation moltculaire elle-m&me rend compte de la raie fine et son couplage i la densite de moment cinktique donne naissance au ⁽⁽ doublet ⁾⁾ transverse.

Abstract. - An analysis is presented of the various theories which describe the sharp Rayleigh wing line in the VH spectra of depolarized light scattered off liquids composed of molecules with anisotropic polarizabilities. Both the sharp, k-independent, Lorentzian component as well as the closely spaced, k-dependent ((shear n doublet, are studied. Critical comparisons are made with experimental data in order to check the validity of the various theories. It is concluded thal the fluctuations in dielectric tensor are coupled directly to the molecular orientation density ; the mole- cular reorientation itself accounts for the sharp line and its coupling to the momentum density gives rise to the ⁽⁽ shear ⁾⁾ doublet.

I. Introduction. - In recent months there has been considerable interest in the spectra of depolarized light off liquids composed of anisotropic molecules. scattered In numerous cases, the VH-spectrum consists of a broad background with a half-width of more than 5 cm-I, a sharp line with a half-width of about 0.2 cm-I, and a dip in the center which gives rise to a closely spaced k-dependent doublet (about 0.03 cm-l at a scattering angle of 90°).

2 n 8 k = - sin -

n

2

where

L

is the wave length of the incident beam and 8 is the angle of scatter. We shall concentrate on the low frequency spectrum, i. e. the sharp line with the central (< shear doublet ^)). The ⁽⁽ shear doublet ⁾⁾ was first observed by Starunov et al. [I], and has been studied extensively by Stegeman and Stoicheff 121.

A variety of theories have been proposed to explain the data and we shall analyze these theories in this article.

The theories are all valid at low k and can be summa-

(*) Supported in part by a grant from the National Science Foundation (GB-24552).

(**) Eastman Kodak Fellow.

rized by the following expression for the spectral density, IvH(o) :

The first term gives rise to a Lorentzian line with half-width, T, which is independent of k, i. e. the sharp line ; wo is a characteristic frequency which is proportional to k ;

R

is a dimensionless, k-independent parameter which determines the depth of the central dip ; C is a k-independent intensity factor. The second term is a negative Lorentzian of half-width, o i / T 4

r

^;

the subtraction of this second Lorentzian from the first results in a dip at the center of the sharp line which gives rise to a doublet with doublet separation, wsp,

Since wo is linear in k, the negative Lorentzian becomes sharper (and the doublet separation smaller) a t low k.

At w = 0, the ratio of the peak heights of the negative Lorentzian to that of the positive Lorentzian is

N = I (O) R cos 2 - 8

Ip(0) 2 '

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972139

C1-232 T. KEYES AND D. KIVELSON In all the theories except those of references [3]-[5],

the fluctuations in dielectric tensor are assumed to be proportional to a single dynamical variable which we call a ⁽⁽ primary variable ⁾⁾ [3]. In references [3]-[5], multiple primary variables are considered. The primary variable is then coupled to one or more ⁽⁽ secondary ⁾⁾ variables via linear transport equations, and the Fourier transform of the primary variable autocorrela- tion function is calculated. The choice of primary and secondary variables, the method of deriving the trans- port equations, and the values assigned to the transport coefficients vary widely from theory to theory as do the significance and values of o,, R and T .

In all the theories, the slowest secondary variable is the curl of the momentum density. If this variable were uncoupled from all others, its fluctuations for small k would be given by the Navier-Stokes equation and the correlation time would be (k2 y/p)-l, where p is the mass density and y the coefficient of shear visco- sity. In the theories outlined below we have associated the coefficient of shear viscosity with the slowest normal shear mode of the system for small, k, i. e.

where r & A R is the correlation time associated with the total stress tensor. This equation defines

r,,,,,.

Thus, from eq. (5) and (6) we obtain

and the o, 's differ in the various theories through the choice of

r.

If eq. (5) and (3) are combined, we can establish that

2 0; P R =

I [--

4ryk cos 012

where all the quantities on the right hand side are directly measurable. The value of

R

is a sensitive test of the various theories.

In the appendix, we relate the notationin the original articles with that used here.

11. Theories. - Rytov [6], developing a theory proposed by Leontovich [7], irssumed that the relevant fluctuations in the dielectric tensor are proportional to the anisotropic strain which relaxes with a characte- ristic time, rkHEAR, associatecl with the stres stensor, a.

As is the case in all theories where

r

⁼

rSHEAR,

^the

frequency, a,, is that of a propagating shear wave, Thus in all cases the width of the negative Lorentzian 7

is k2 y/p. The coefficient of shear viscosity is related to the high frequency shear modulus, p,, by the Maxwell expression,

o, (Rytov) = k

,/?.

See eq. (7). The ratio R in this theory is unity. The results are summarized in the table.

TABLE

SigniJicant quantities in dzflerent theories (I)

Primary Secondary

Theory Variable Variable

r

- - - -. -

Rytov Leontovich

Q"

(

Q" > ep ^SHEAR

^{1 P a}

Volterra

Keyes and Kivelson Andersen and

Pecora (') Ben Reuven and

Gershon

!!c Same as KK

rl Chung and Yip

Berne et al. ⁽⁴⁾

Experiment Debye Law aT/y .04 < R

2

^0.5

SHEAR too big ffT-

(I) See appendix and text for definitions.

(') Special two-variable theory with symmetric stress tensor as primary variable.

(3) ys M y and so R ^M 1.

(4) In eq. (2), the second term has a positive sign.

( 5 ) In reference 8, R is incorrectly set equal to y,/ (q

+

yc).

LOW FREQUENCY DEPOLARIZED VH-SCATTERING '21-233

Keyes and Kivelson (KK) 181, Andersen and Pecora (AP) [9] and Berne et al. [5] have developed two variable theories based on the Kubo, Felderhoff and Oppenheim, Zwanzig-Mori formulations of transport theory [lo]. These are microscopic theories. KK chose the second rank molecular orientation density tensor, Qq, as their primary variable and the curl of the momentum density, QP, as their secondary variable ; thus in their theory T-' is the characteristic time, r o l , for molecular reorientation ; w, is not the shear mode frequency (see eq. (7) and set

r

⁼ To). R is (y,/y) in the K K theory. y, is that part of the shear viscosity which is related to the orientational part of the stress tensor 1141 and y is the total shear viscosity ; there- fore ; R

<

1.

Andersen and Pecora (AP) use the same secondary variable as KK, but, after deriving the results for a general primary variable with a second rank tensorial character, they associate the primary variable with the symmetric stress tensor, oS, and

r-'

with a relaxation time, I , associated with this tensor. In the AP theory two different relaxation times, Ti' and T i ' , for the symmetric and antisymmetric parts of the stress tensor, respectively, are introduced, and

~t =

&rgl + ^&ril

⁼

v S +

^{~ t * ,} (lo) where and ,u: are the contributions to the shear modulus of the symmetric and antisymmetric stress tensors, respectively. The resulting expressions are summarized in the table. In order for the AP theory to be valid,

r, >> r,,

and provided p$ is not much greater than

&,

it follows that yS w y, and that R is about one.

In our analysis of the AP theory we have neglected cross-correlations between the symmetric and anti- symmetric stress, and we have also neglected the T i 2 term. We have only discussed the specific case discussed by Andersen and Pecora, that in which the primary variable is the symmetric stress tensor. The actual numerical work carried out by AP is not consistent with this specific formulation, but leads to R < 1, a result in agreement with experiment and with the general AP theory. This latter theory can, of course, be used to reproduce any two-variable theory ; if the primary variable is molecular orientation, it yields the K K theory and if it is the total stress tensor, o, it yields the Rytov theory.

Ben-Reuven and Gershon [ l l ] (BG) have formula- ted a theory based upon Mori's [lo] continued fraction representation of time dependent autocorrelation functions. They assume that the primary variable is the molecular orientation density, Qq. They then couple the orientation to a ((shear sound mode ^)>

which decays with a frequency, y. While this procedure does not correspond formally to the choice of a specific secondary variable, it is essentially equivalent to the choice of the momentum density, QP. The BG theory is a microscopic theory, and attempts to eva- luate all parameters from first principles. The results

are tabulated in the table. Although the BG theory does not specify y, it associates it with the damping frequency of the sound mode obtained from the Brillouin line ; as explained in the appendix, we have associated their y with k2 ylp. The BG coupling parameter, B,, is equal to k2 y, r / p . Ben-Reuven and Gershon propose a simple physical model in which B, = 2 @k2 k, ~/mg('), where Q, is a molecular geometric factor and g(" is the ratio of the time independent pair orientation correlation function t o that of a single particle orientation correlation func- tion ; both Q, and g ( 2 ) are or order unity.

Volterra [12] developed a three variable hydrody- namic theory to account for the central doublet, the sharp line and the broad background line. Although he did not assign microscopic interpretations to his variables, and he obtained his transport equations by means of hydrodynamic arguments, the ordering parameter, which he introduces as a primary variable, can be associated with the molecular orientation density, Qq, discussed above, and his two secondary variables can be associated with the momentum density QP, and the stress tensor, o. Volterra associates a time, r&LAR, with the relaxation of the stress tensor and a time, TB1, with the relaxation of the ordering para- meter. He makes the assumption that

which is equivalent to saying that the sharp line arises principally from the stress tensor relaxation while the broad line can be attributed mainly t o the reorienta- tional relaxation. As a consequence, even though Volterra uses the reorientation variable, Qq, as his primary variable, the low frequency part of his VH spectrum has the characteristic form expected for a theory with the stress tensor, o, as a primary variable.

In his coupled linear transport equations, Volterra does not introduce any direct coupling between the molecular orientation density and the momentum density, an interaction which is central to the KK theory. Volterra's results on the sharp line are summa- rized in the table, where D is a dimensionless parameter which can be expressed in terms of quantities intro- duced above,

and the decay time, T - ' , associated with the sharp line is

In measuring the depth of the dip it must be recalled that the spectrum predicted by eq. (1) is superim- posed upon a broad background.

Another three variable theory was developed by Chung and Yip (CY) [13]. The theory is, in many ways, similar to that of Volterra ; however, the stress tensor, o, is the primary variable in this theory, and the order- ing parameter, Qq, and the momentum density,

C1-234 T. KEYES A N D D . KIVELSON QP, are secondary variables ; also, in contrast to

Volterra's theory,

so that the ordering parameter is largely responsible for the sharp line and the stress tensor for the broad one. The results are given in the table where

Both Andersen and Pecora [9] and Keyes [14] have developed a three variable formalism by means of the Mori theory ; unlike their approach in the two-variable theory, in this theory, Andersen and Pecora have assumed that the primary variable is the orientation variable, Qp. In the limit that the orientation parameter is strongly coupled to the stress tensor, they show that their three variable theory reduces to their two variable theory with the stress tensor as primary variable.

Berne et al. [5] in a number of articles have deve- loped a somewhat different theory. They introduce two primary variables, the symmetric,

2,

and antisym- metric, oA, stress tensors. The fluctuations of the dielectric tensor, ^{6 8 ,} are directly dependent upon both these variables,

where X2 is a coupling parameter and rc is a dimension- less parameter. There are no cross-correlations between these components of the stress tensor. As secondary variables they introduce the intrinsic spin density, Q(", (angular momentum about the molecular center of masses), and the momentum density, QP. The intrinsic spin couples only to the antisymmetric stress tensor. In the first of these articles, Berne et al. assume that the stress has its Newtonian form and can relax only as the intrinsic spin density and linear momentum densities relax, whereas in the second article they allow for rapid relaxation of the stress tensor itself to its Newtonian form. In the first formulation the k-independent terms, which give rise to the predic- ted sharp line, are related to fluctuations in the intrinsic spin, which decay with a characteristic time, TJ- l ,

where

p is the mass density, m the particle mass, I the moment of inertia of the particle, and ^yA the ⁽⁽ antisymmetric viscosity ⁾⁾ defined in eq. (10). The results of Berne e f al.

can be written in the form of eq. (2), but the second term is positive rather than negative. This gives rise to a central spike rather than a dip, unless the requirement of eq. (5) is dropped. The theories of Berne et al.

differ from those discussed above because there are two primary variables. These authors have also inde- pendently developed a stress tensor theory similar to that of Andersen and Pecora and based upon a

two-variable Mori formalism, which reduces to the Rytov theory in appropriate limits.

Finally, we mention the theory of Keyes, Kivelson and McTague (KKM) [3] in which the molecular orientation density, Qq, and :x more rapidly varying quantity such as the stress tcmsor, Q('""), are both primary variables which also act as secondary variables since they are coupled via the transport equations.

The cross-correlations between Qq and Q('""' are very important, and it appears that the cross-correlation term can contribute a large component to the sharp line. The contribution of the cross-correlation to the sharp line is most likely nega.tive, which reduces the apparent intensity of the sharp line. If we now add, to the two variables above, the momentum density, QP, as a secondary variable, we expect both the autocorre- lation of the molecular orientation density and that of the cross-correlation between the molecular orien- tation and the rapid primary variable to have doublet features. Although the details depend very much upon the exact nature of the rapid primary variable, Q(fa"", and the strength of its coupling to the dielectric tensor, we expect the shape, if not the absolute intensity, of the sharp line together with its shear doublet to be similar to that predicted by tlie BG- and KK theories.

111. Discussion and comparison with experiment. - We believe that the sharp line arises principally from molecular reorientation and that

in contrast to the assumptions in the theories of Vol- terra, Rytov, Leontovich artd the symmetric stress tensor two-variable AP theory. In support of this statement we note that the Debye theory, which has has been eminently succesc;ful, predicts that [15]

where k, is the Boltzmann constant and < < a ⁾⁾ the molecular radius of an equivalent spherical molecule, and that for molecules with ¹² = 2.5

A

and y =

poise,

re

^M 10 GHz at room temperature, a width quite compatible with those found experimentally.

Furthermore, it has been estsrblished that [2]

independent of both scattering angle and temperature ; this tends to confirm the above theory and the asso- ciation of

r

with

re.

Experiments also indicate that

rsHEAR

^{2 10'} GHz for norrnal liquids [16]. Finally, it is the molecular polarizab~lity which directly deter- mines the dielectric tensor, and so only in cases of very strong coupling between the stresses and molecular orientation can we substitute: the stress tensor for the

~nolecular orientation density as the primary variable.

In the KK and BG theories, this limit corresponds to

LOW FREQUENCY DEPOLARIZED VH-SCATTERING C1-235 qc/q = 1, i. e., the stress tensor possesses only an of the cc dip depth >> to the sharp line ⁽⁽ peak height w orientational part. The theories of Rytov, Yip and is [2]

Chung, and Berne et al., and the symmetric stress

tensor two-variable AP theory, all use the stress tensor

W)

^{< cos2 -}

e

as the primary variable. I,(()) 2 ^' (22)

Note that in all the theories, the doublet splittings, o,,, have the same k-dependence, and this dependence has been verified by Stegeman and Stoicheff [2].

The behavior of the ratio, R, is a sensitive test for the various theories. For a number of liquids studied by Stegeman [2],

and

However, all the theories other than the BG, KK and general AP theories yield R x 1, independent of temperature. In the K K and BG theories, (qc/q) = R, and thus the values of (yc/q) are given by eq. (19).

Although formal expressions [8] can be obtained for (qc/q), it is difficult to see whether or not the values of qc/q required to fit the experimental data are reaso- nable ; however, it is possible in both the KK and BG theories to make estimates of the temperature depen- dence of qc/q. In the KK theory [8] the value of (vc/q) is

where u is an appropriate term in the anisotropic intermolecular potential ; thus q,/q is proportional to T-', in agreement with experiment. In the BG theory, the parameter, B,, which we have associated with k2 qc/pro, has been evaluated inamodel calculation [I I]

and found to be proportional to T ; the combination of this result with the Debye law, eq. (17), implies that q,/q, and hence R, should be independent of temperature, which does not seem to agree with the experimental results.

Experimentally, it is found that the doublet splitting frequency, a,,, is independent of temperature [2]. The KK theory predicts such a result ; this can be verified by combining the expression for o,, in the table with the facts that p is almost independent of temperature, q r , is proportional to T, and (qC/q)% is proportional to ^{T - l .} If q, in the BG theory is evaluated by means of eq. (21), this theory also yields o,, independent of T ; but if the BG model calculation [I I], described in the preceding paragraph, is used to evaluate qc, then theory yields ^o,, proportional to T%. In the CY theory, w,, is proportional to ~ ~

+

A)-%. The temperature ( 1 dependence of a,, in the other theories depends pri- marily upon the temperature dependence of ^p, ; if p, is independent of temperature then they predict that o,, is either independent of or only slightly dependent upon temperature.

Experimentally it is found that at w = 0, the ratio

A comparison of eq. (4) with the values given in the table indicates that those theories for which R = 1 cannot explain this phenomenon ; the GB, KK and generalized AP theories can account for this cr reduc- tion ⁾⁾ in dip. (In the symmetric stress tensor two- variable AP theory, qS w q and R ^w 1)

The theories discussed above are valid in the limit (o,,/r)

<

1, i. e. where the doublet splitting is small compared to the k-independent sharp line width.

The ratio (w,,/T) increases with decreasing temperature e. g. as T P 3 in the KK theory. Thus at low temperature (o,,/T)

>

1 ; this is born out by the data of Stege- man [2] on quinoline at - 210, and the various theories do indeed seem to break down there since no doublet is observed. A modified theory valid for (o,,/T) 2 1 has been formulated [17].

It is interesting to note that only in the KK and GB theories can the intensity factor C in eq. (1) be calcu- lated since it is readily related to the molecular aniso- tropic polarizabilities. However, the KKM3 theory discussed above casts doubt on these calculations of the factor C.

IV. Conclusions. - We believe that the KK, GB (not including the model calculation for Bk) and gene- ralized two-variable AP theories (not the one in which the primary variable is the stress tensor) can best explain the observed sharp line spectra with VH- shear doublets. The AP three-variable theory could also explain these observations but would not account properly for the broad line as explained elsewhere 131,

141.

V. HH-doublets. - The HH-depolarized scattered light spectrum at a scattering angle of 900 exhibits a weak Brillouin doublet which has been explained by theories analogous to those above for VH scattering. In particular we would like to comment on the KK theory [8]. The primary variable is the molecular orientation density ; Qq ; the secondary variables are the number density, the divergence of the momentum density, and the energy density. Propagating sound waves are predicted on the basis of the resulting four coupled linear transport equations. For quinoline, Stegeman [2] observes c< dispersive ^)> shaped Brillouin lines in HH-scattering at 900 ; since in this case,

r,

^m kc, where ^c is the velocity of sound, eq. (63) of reference [8] predicts just such a shape (cf. Fig. 2 and 3b of Ref. [8]).

In analogy with eq. (5) for shear modes, we associate the longitudinal sound absorption or Brillouin line width in the limit k -, 0 with the decay frequency, k2(+) (4 q/3

+

qs)/p, where q, is the bulk viscosity.

Thus in eq. (63) and (66) of Reference 8, q, (which

16

C 1-236 T. KEYES AND D. KIVELSON

determines the Brillouin line width) should be replaced bv

Similarly, as for VH scattering, (q

+

y,) in eq. (48) of Reference [8] should be replaced by q, but the factor q, remains unchanged.

VI. Acknowledgements. - We wish to thank Pro- fessor B. P. Stoicheff and Dr. G. I. A. Stegeman for sending us copies of unpublished material that cons- titutes most of the experimental work in the field.

We would also like to thank the National Science Foundation for its continued financial support.

Appendix A. - The following gives a brief sum- mary of the correspondence of notations used. (The notation is that of Ref. 8.) In the AP theory we have made the identifications

Volterra defines a T,, which gives the width of his sharp line. His r 2 , however, is the quantity which obeys our defining eq. (6) (eq. (31) of Volterra) ; thus,

Note that .Volterra's wSp is our oo. Since we have defined q by eq.

(9,

we note that (y

+

q,) in reference [8] should be redefined simply as y. Ben-Reuven and Gershon have used a parameter, y, which is the decay constant of the sound mode as reflected by the Brillouin line width ; for illustrative purposes we have assumed that y is the shear mode decay frequency and should be associated with k2 y/p. The symmetric viscosity qS used by Berne et al. is associated with the slow decay mode, and hence with q.

We can relate the total shear modulus p, to the symmetric

01:)

and antisymmetric (P:) moduli ;

I A

r;,

^{= (A. 1)} If this is combined with eq. (LO),

S Since for the specialized two-variable AP theory, t~;, = - Pw

-

(A .4)

r

⁼

rs

-g r,, it follows that

Chung and Yip define a T,, which obeys our defining equation for

rsHEAR

(eq. (6)), so this symbol need not be redefined. We have made the identification,

( A . 10)

5 ) and the A P expression for l~w,,/k2 cos 9 in the table

Go = r c

r,

may be rewritten as for Chung and Yip's orientational modulus, Go. In

Volterra's theory,

rl

=

r e .

References STARVNOV (V. S.), TIGANOV (E. V.) and FABE-

LINSKI (I. L.), Zh. Experim. Teor. Fiz. Pis. Red., 1967,5,317 ; Sov. Phys. JETP Lett., 1967,5,260.

STEGEMAN (G. I. A.) and STOICHEFF (B. P.), Phys. Rev.

Letters, 1968, 21, 202.

STEGEMAN (G. I. A.), Doctoral Thesis, University of Toronto, 1969.

KEYES (T.), KIVELSON (D.) and MCTAGUE (J. P.), J. Chem. Phys., 1971, 55, 4096.

KEYES (T.) and KIVELSON (D.), J. Chem. Phys (In press).

AILAWADI (N. K.), BERNE (B. J.) and FORSTER (D.), Phys. Rev., 1971, A 3, 1472 ;

A I L A ~ A D I (N. K.) and BERNE (B. J.), Preprint.

RYTOV (M. S.), Zh. Eksp. Teor. Fiz., 1957,33- 166,5 14, 671 ; Sov. Phys. JETP, 1958, 6, 130, 401, 513.

LEONTOVICH (M. A.), IZV. Akad. Nauk. S. S. S. R., Ser. Fiz., 1941,5,148 ; Bull. Acad. Sci. U. S . S . R., Phys., 1941, Ser. 4, 499.

[8] KEYES (T.) and KIVELSON (D.), J. Chem. Phys., 1971, 54, 1786; Erratum ibid., 1971, 55, 986.

[9] ANDERSEN (H. C . ) and PE:CORA (R.), J. Chem. Phys., 1971, 54, 2584.

[lo] MORI (H.), Progr. Theoret. Phys. (Kyoto), 1965, 33;

423 ; 1965, 34, 399.

FELDERHOF (B. U.) and OI'PENHEIM (I.), Physica, 1965, 31. 1441.

K u ~ o (R.), J. Phys. Soc. Japa,z, 1957,12,570.

ZWANZIG (R.), Phys. Rev., 1966, 144, 170.

[l 11 BEN-REUVEN .(A.)- and ^GERSHON (M. D.), J. Chem.

Phvs.. 1971. 54, 1049.

[12] VOLTERRA (V.), phY>. R e v , 1969,180,156.

[13] CHUNG (C. H.) and YIP(%), Phys. Rev., 1971, A4,928.

[14] KEYES (T.), Doctoral Thesis, UCLA, ^1971.

[15] BLOEMBERGEN (N.), PURCELL (E.) and POUND (R.), Phys. Rev., 1948,73, 679.

[16] PINNOW, CANDAU and LITOVITZ, J. Chem. Phys., 1968, 49, 354.

[17] KEYES (T.) and KIVELSO~ (D.), J. Chem. Phys., 1972 (In press.).