Collective Molecular Motions in Liquids from Depolarized Light Scattering SEARBY AND PIERRE BY GEOFFREY PIERREBEZOT SIXOU Laboratoire de Physique de la Matiere Condens6e,t Parc Valrose 06100-Nice France Received 6th August 1976 The basic hypotheses used by some of the different theories (viscoelastic and microscopic) that have been proposed to explain the lineshapes observed in quasi-elastic depolarized light scattering are summarized and the results predicted by these theories compared and contrasted in different situations. These theoretical results are in turn compared with the experimental results obtained from different physical systems liquids with low viscosity composed of relatively small and simple molecules; liquids composed of more cumbersome but rigid molecules ; supercooled liquids ; liquid crystals in the isotropic phase.The coupling parameters obtained from VH and HH spectra ix. the coupling between molecular orientation and transverse or longitudinal waves in the liquid are assessed whereby it is shown that the transverse (shear) waves can be strongly propagative at low temperatures in a supercooled liquid but that as the viscosity is decreased these waves become pro- gressively softer and for low viscosities they have a purely diffusive character. Although most theories are adequate to explain the diffusive shear waves observed for low viscosities a theory using generalised hydrodynamics is necessary to explain the results observed in supercooled liquids. It has been known for over fifty years now that the spectral density of light scat- tered inelastically from a fluid reproduces the spectral density of polarizability fluc- tuations in the fluid and thus contains information on molecular movements.Light scattered with the same polarization as the incident light contains informa- tion about isotropic fluctuations in the fluid whereas depolarized scattering contains information about anisotropic fluctuations arising from molecular reorientations. The intensity of the depolarized spectrum generally weak is proportional to the optical anisotropy of the molecule. For simple liquids the depolarized spectrum is frequently Lorentzian with an inverse half-width which is equal to a correlation time for molecu- lar orientation.2 Leontovich3 first predicted that the depolarized line may contain a central dip corresponding to propagative but highly overdamped shear (transverse) waves in the liquid.This effect was first confirmed by Fabelinskii and co-w~rkers.~ Since then there has been a considerable amount of work in this field both theoretical and experimental. DISCUSSION OF DIFFERENT THEORETICAL APPROACHES The visco-elastic theory of Rytov5 assumed a priori the existence of a frequency- dependent shear modulus which couples the stress and strain tensors. Fluctuations in the strain tensor produce light scattering via a photo-elastic coefficient. The depolarized spectrum has a width l/z where z is the common relaxation time for the stress and strain tensors. The lineshape is non-Lorentzian because of modulation of the strain tensor produced by the damped shear (and longitudinal) waves.More iLaboratoire associ6 au C.N.R.S. no. 190. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS recently a number of schools have developed a rather different approach which we shall call microscopic. It is not possible to list all the authors who have contributed to this work but amongst the most frequently cited we may mention the schools of Keyes and Kivelson,”s Pecora,9~10Gershon and Oppenheim,’l-13 Ailawadi and Although outwardly these theories appear quite different there is a great deal of similarity in the type of approach. Briefly one chooses one (or more) “pri-mary ” variable(s) which is directly coupled to the local dielectric constant generally an orientation or polarizability tensor this (these) primary variable(s) is allowed to couple to one (or more) secondary variable(s) generally a momentum density tensor via transport equations.A secondary variable by definition does not couple directly to the local dielectric constant. Except where discussed explicitly we shall be con- cerned in the rest of this paper with the << two variable ”theories in which one primary variable couples to one secondary variable. The various treatments mentioned above differ in their choice and exact definition of the primary and secondary variables and also in the mathematical tools used. However with the exception of the results of Ailawadi Berne and Forster,15 the above treatments eventually predict the same form for the shape of the VH scattering spectrum.The physical interpretation of the para- meters in the various theories is of course different. The theory of Ailawadi Berne and Forster is somewhat different in that it contains two primary variables angular momentum and stress. Their published lineshape is fundamentally different from that of the other theories but does not reproduce the experimental spectra . It is important to note that these treatments were formulated for liquids composed of spherical top molecules. Nevertheless the application to molecules of a more general shape seems to be verified experimentally for liquids well above their freezing point.10.’6,17 At this point we mention an approach used by De GenneP in 1971 to calculate the spectrum of depolarized light scattered by a liquid crystal in the isotropic phase in the VH geometry.It does not seem to have been widely recognized that De Gennes’ theory predicts the same shape for the spectrum of the scattered light (with the exception of a Lorentzian term which he explicitly neglects) as the above mentioned “two variable ” theories. In fact Fleury and have remarked that the iso- tropic phase of a liquid crystal should not be very different from other liquids com- posed of very anisotropic molecules but fail to bring out the correspondence of De Gennes’ expression with that of the other theories. De Gennes’ approach is simple and physical and we feel it is worthwhile recalling the main steps in his argument. His primary variable is a traceless symmetric second rank tensor Qap describing the local orientational order in the liquid i.e.the tensor orientational order parameter. The secondary variable is the hydrodynamic velocity Y from which he constructs the shear rate tensor cap = apYp + abVa. He then writes down two coupled linear equations linking the fluxes Qap and cap with their conjugate forces. The force con- jugate to the shear rate tensor is the stress tensor and the force conjugate to the rate of change of the order parameter is obtained by expanding the free energy as a func- tion of powers of Qab and then differentiating with respect to Qa~. It is now sufficient to add Newton’s acceleration equation to obtain a closed system for the equations of motion. The flow birefringence effect and depolarized scattered light spectrum can be calculated directly.Moreoever De Gennes’ theory is valid not only for tempera- tures well above the nematic-isotropic transition of a liquid crystal but also close to this transition temperature where it is possible to include the effects of the critical divergence of certain parameters. One should note however that De Gennes has neglected the relaxation of the GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU elements of Qa~ which contribute to the scattered spectrum but which do not couple to the hydrodynamic modes. These elements decay with a single relaxation time and add an undisplaced Lorentzian to the spectrum. DISCUSSION OF VH SCATTERING The two-variable of theories Pecora,l0 Gershon and Oppenheim13 lead to a VH lineshape which may be written in the following form The intensity factor B is related to correlation functions of two elements of a sym- metric second rank tensor (defined below) the TIand I'2 are inverse correlation times of elements ofthe same tensor.The other parameters are defined below. The authors argue that they expect B = 1 and Tl= r2,in the limit of small k for Pecora and for reasons of symmetry according to Gershon and Oppenheim. The expression for the VH spectrum then reduces to the expression given by Keyes and Kivelson6 x (1 -R) + CO~/~~) I' cc sin2(e/2) 1 + co2/r2 (l/r)~0~~(9/2)((k~q~/p~I'~) 1/r + (k2qlpr-co2/r2)2 + (co/Q2 x (1 + (1 -R)k2y/pr)2' (2) Here 8 is the scattering angle; k is the scattering wave vector; q is the normal shear viscosity; p is the density; r (= 117) is an inverse correlation time.For Keyes and Kivelson Gershon and Oppenheim z is the correlation time for the anisotropic part of the polarizability density for De Gennes it is the correlation time of the orientational order parameter for Ailawadi and Berne (in their equivalent expression for I:) it is the correlation time of the angular momentum density for Pecora it is the correlation time of a second rank tensor introduced in a general fashion and identified first with the stress tensorg and later with the orientation density tensor.1° R is a dimensionless parameter expressing the coupling between the hydrodynamic modes and the anisotropy of the liquid. In the VH scattering geometry only the transverse hydrodynamic modes (shear waves) are able to couple to the primary vari- able.The results for the HH scattering geometry will be discussed in a later section. The expression for I has been written here in such a way that it can be seen that the shape of the spectrum is a function of two and only two dimensionless parameters R and k2q/pT. The first is a coupling parameter and the second is the ratio of two characteristic times. The first time l/T has already been discussed the second time p/k2qis the characteristic time for a viscous disturbance in the liquid. The normal- ized variable is co/F. We shall not repeat the results of Rytov's theory here but simply remark that if we put R 31 in eqn (2) then we find the same spectral shape as pre- dicted by Rytov.' However experimental evidence suggests that R N" 0.4.10*17921 Moreover in Rytov's theory z is identified with the relaxation time of the shear strain tensor whereas more recent arguments associate z with a correlation time for molecular orientation,* We now wish to discuss in detail the predictions of the microscopic theories under different physical conditions and compare these results with experimental evidence.COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS To do this we shall discuss the lineshape as a function of k2q/pI'. We divide the range of k2q/pI'into four rather arbitrary but characteristic regions (1) k2ullPI' < 1 (2) k2V/Pr< 1 (3) k2rllPI'> 1 (4) k2q/pT 9 1' The spectral shapes calculated using expression (2) are shown in fig.1 for these four values of k2q/pI' and at a scattering angle of z/2. For R -+0 or k2q/pI' -+0 the spectrum becomes a simple Lorentzian of half-width r. REGION k2q/pI'< 1 This situation corresponds to that of liquids composed of small simple molecules which have very short reorientation times (large r)and low viscosity. Such molecules are interesting because they can be very good approximations to a symmetric top as required by theory. Moreoever it should be possible to perform numerical model calculations for them. However from an experimental point of view the fine struc- ture arising from coupling to the hydrodynamic modes is almost impossible to resolve. As an example we show the depolarized spectrum of cyanopropyne CH,-C-C-C-N (a "rigid rod " molecule) in fig.2. The calculated value of k2q/pTat this temperature is 3 x The central dip as expected from fig. 1 is not observed. This is FIG.1.-VH lineshapes according to the microscopic theories [eqn (2)] for four characteristic values of kzq/pI'. The first three columns are normalised to constant F. probably because the instrumental resolution is 1.5 GHz. whereas the expected width of the dip is only 1 GHz. At present work is in progress to improve instrumental resolution and verify the existence of the dip. REGION k2q/pI'< 1 This zone corresponds to a large number of " normal "liquids and also " viscous " liquids far above their freezing point. It is the region which has received the most attention.The agreement between theory and experiment is always very good despite the fact that few of the molecules studied can really be approximated to symmetric tops. As an example we have chosen tolane DC-C Fig. 3 showsa comparison between an experimental spectrum and a computer fit (convoluted with GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU 1 I I I -50 0 +50 frequency/GHz FIG.2.-VH spectrum of cyanopropyne at 16 "C. k2q/pr= 3 x -12.5 0 *12.5 frequency/GHz FIG.3.-VH spectrum of tolane at 83 "Ctogether with a best fit using expression (2). The fitted curve has been displaced in order to be visible. k2q/pr= 0.099 R = 0.39. the instrumental function). The fitted curve has been displaced in order to be visible. Fig. 4 shows the evolution of the lineshape as a function of temperature.Note that the width of the dip [ ~c(k~qI'/p)~ see ref. (7)] is nearly independent of temperature. This is not evident from fig. 1 since the plotted curves have been normalized to con- stant r. The values of the coupling parameter R obtained from best fits are shown in fig. 5. The full circles are two-parameter fits (R,q) the crosses are one parameter fits where is given its static measured value. Contrary to the findings of Enright and Stoicheff for salo1,22 there is no significant difference between the two results. The reason for this is probably that tolane is a rigid reasonably symmetric molecule whereas salol is certainly not. The values of R are closely grouped around 0.4. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS I 1 I -12 0 +12 f requencyl GHz FIG.4.42) VH and (6)HH spectra of tolane showing the evolution with temperature.0.t -& ** +t +.+ + ++ .+ t 0.3 R 0.2 0.1 I 1 I I I I 1 1 0.0 FIG.5.-The coupling parameter R for tolane as a function of temperature 0 two parameter fits (R q) + one parameter fits (R). GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU Fig. 6 shows the values of R for a number of liquids studied in our laboratory as a reduced function of k2q/pI'. Although the scatter for some liquids is rather large the grouping of the different results around R E 0.4 is quite striking. This same value for R has been observed by many author^^^*^^*^^*^^^^ but it is not yet obvious from present day treatments why molecules as diverse as CS2 and ethyl benzoate should have very similar values for the coupling parameter.Much smaller values of R have been reported by Champion and Jacksonu for a number of n-alkanes. These mole- cules however are quite flexible which may well have an effect on the coupling and moreoever the values of R were deduced less directly from measurements of flow birefringence. ~~~ ,~ * 0.51 I 1 1 1 1 1 1 -$#no1 ll,...T ~ 0 0 10 -2 10 -1 1 EQ FIG.6.-Values of R measured in our laboratory as a reduced function of k2q/pr. pyridine; 0 acetophonone; 0ethyl benzoate; b tolane. REGIONk2q/pI'> 1 This region corresponds typically to the case of viscous liquids just above their freezing point. The width of the depolarized line is now roughly equal to the width of the dip that was observed at higher temperatures.The dip has disappeared but the lineshape is very non-Lorentzian. From a mathematical viewpoint the shear disturb- ances in the liquid are propagative in the sense that they are less than critically ~lamped.l~*~~ A typical example is shown for benzyl alcohol in fig. 7. The I and IE spectra at 12 "C are quite different the I spectrum is Lorentzian. Different authors have fitted expression (2) to experimental spectra in the range 30> k2q/pI'> 1 with varying degrees of success. Rouch et aZ.24claim good agreement in quinoline with k2qlpI' = 20. However Enright and Stegeman,22 Bezot et ~2Z.l~ find increasingly poor fits as k2q/pI' increases.Tsay and KivelsonZ5 have recalculated Keyes and Kivelson's expression for I retaining terms to higher order in k. They findz6 Expression (3) reduces to expression (2) in the limit k2q/pI' -g 1. Tsay and Kivelson state that expression (3) provides a good fit to the experimental spectra of super-cooled triphenyl phosphite up to at least k2q/pI'= 50. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS +12°C -28" c A-38" c -1'2 +i2 I I I -12 0 +12 frequencyIGHz FIG.7.-(a) VH and (b)HH spectra of benzyl alcohol showing the transition from k2q/pr> 1 to k2q/pr& 1. The VH spectra at -28 "Cand at -38 "Chave been amplified to show the appearance of the propagating shear waves. REGION k2y/pI' 1 This region corresponds typically to supercooled liquids approaching the glass transition temperature.Expression (2) predicts for R 1 a triplet spectrum and for R < 1 a central Lorentzian of width I' superposed on a wider Lorentizian of width r/(l-R),see fig. 1. In this latter case the shear waves have lost their propagative character and have become diffusive once more. In either case the integrated inten- sity of the central narrow Lorentzian is equal to the intensity of the rest of the spectrum. The modification proposed by Tsay and Kivels~n~~ [eqn (3)] does not substantially change the general features of the predicted spectrum. The experimental spectra however are quite different.17s22*27*28 At sufficiently low temperatures a very weak depolarized doublet appears on the wings of the central line indicating the presence of well defined propagating shear waves.Fig. 8 shows that it is not possible to fit an experimental spectrum using expression (2). Fig. 7and 9 show the way in which the depolarized doublet detaches itself from the wings of the centre line for supercooled benzyl alcohol and supercooled triphenyl phosphite. The corresponding I2 spectra show no sign of this structure. Tsay and Kivels~n~~ say that they have investigated the spectra of triphenyl phosphite down to -20 "Cbut do not report the appearance of the doublet. This structure is however very weak about 1% of the total de- polarized intensity and it is quite possible that it would pass unnoticed. This weak doublet was first observed in supercooled salol by Fabelinskii et aZ.,4and has been studied in more detail by Vaucamps et aZ.,27Enright and Stoicheff,22 and by Bezot Searby and S~XOU.~~.~~ These latter authors have studied the spectra of ethyl ben- GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU zoate and of benzyl benzoate.They show that the intensity of the depolarized doub-let decreases as the temperature is lowered and the corresponding transverse sound velocity increases. The large decrease in the intensity of the doublet is the main cause of the dis-agreement with expression (2). Both the intensity and the velocity tend to limiting values at very low temperatures as would be expected if a relaxation process were at work. They also show that in the case of benzyl benzoate the transverse sound velocity obtained from their spectra is in good agreement with ultrasonic measurements made by Barlow and Ergin~av.3~ Y I ii; ..... 2: . .I.I - .I i: . .I * :* s i; % c .-ln c al .c. .c I I -12.5 0 + !.5 frequency /GHz FIG.8.-Attempts to fit expression (2) to the VH spectrum of benzyl benzoate at -60 "C,k2q/pI' = 5.5 x lo1*(value obtained by extrapolation). * R = 1.0; -R = 0.9999 97; --R= 0.999995. The experimental and fitted curves have been normalised to the same intensity for the central line. Vaucamps et aZ.,27interpret the spectra of supercooled salol in terms of a modified viscoelastic theory in which there are two relaxation processes contributing to the frequency dependence of the shear viscosity and the shear modulus.They find that this phenomenological approach gives an accurate description of the experimental spectra. Enright and Stoicheff22use expression (l) the more general expression given by Andersen and Pecora and again find good agreement provided TI# T2and B # 1 in the supercooled region. The parameter B is related to the relative intensity of the de-polarized doublet. The fact that Vaucamps et al. have to assume the existence of two relaxation pro-cesses or that Enright and Stoicheff need to let elements of the polarizability tensor be strongly temperature (frequency) dependent suggestthat a third (at least) variable is necessary in the microscopic theories in order to completely describe the behaviour of the liquid i.e.one of the so-called "fast " variables can no longer be averaged out COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS tto"c -9.4"c d&-1 7.5"c -f2 0 +112 f requency/GHz FIG.9.4~) VH and (b)HH spectra from supercooled triphenyl phosphite showing the appearance of propagating shear waves as the temperature is lowered. The width of the central component is dominated by the instrumental width as is always the case for supercooled liquids. of the equations of motion. These difficulties almost certainly arise as the transla- tional motions of the molecules slow down into the hydrodynamic region and we suggest that a pertinent third variable would be one describing the translational correlations of the molecules. Keyes Kivelson and McTague 29 have calculated the spectra arising when an unspecified "fast " variable couples to the orientation tensor (a "slow " variable) and show that this coupling could account for the relative in- tensity of the depolarized broad background frequently observed.However they do not calculate the effect of coupling the hydrodynamic modes into the system. Andersen and Pecorag have formulated a three-variable theory in which the variables are a velocity tensor the symmetric part of the stress tensor and a molecular orienta- tion tensor. Again this theory was proposed to account for the broad background of the depolarized spectra and apparently no attempt has been made to compare the results with the spectra of supercooled liquids. More recently Quentrec30 has con- structed a new set of hydrodynamic equations for a dense fluid using a more general- ized version of De Gennes' approach.18 He introduces two microscopic variables into his equations of motion.The first is the tensor used by De Gennes and which de- scribes the local orientational order the second is an analogous tensor describing the local translational order of the molecules. However at present the calculations have not been applied to depolarized light scattering. LIQUID CRYSTALS Liquid crystals are particularly interesting because of the strong orientational correlations that may exist between molecules even in the isotropic phase.31 Stinson GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU Litster and Clark32 have examined the small angle scattering spectra of a liquid crystal MBBA.They do not show the observed I spectra but report good agreement between their experimental data and De Gennes' theory. It is curious to note that the values that they deduce for the coupling parameter R (G2p2/vq in De Gennes' theory) range from 0.2 at high temperatures to 0.4 close to the transition temperature. These values are lower than those observed for " normal "liquids (Region k2q/pI' < 1 above). Fig. 10 shows the I and I2 spectra of another liquid crystal 4-cyano pentyl bi- A FIG.10.-(a) VH and (6) HH spectra of a liquid crystal 4-cyan0 pentyl biphenyl in the isotropic phase 140 "Cabove the transition point. The flat top to the VH spectrum is indicative of an un- resolved " dip ".phenyl. The experimental conditions are different from those of Stinson in that the temperature is here far above the transition point. The flat top to the I spectrum suggests the presence of a shear wave dip but at the time of writing instrumental resolution was not sufficient to allow accurate lineshape analysis. It is interesting to remark that in the isotropic phase of a liquid crystal the re- orientation time z is frequently very much longer than for a simple liquid whereas the viscosity has the normal value of a few centipoise. This means that in such a case it is possible to obtain values of k2q/pI'& 1 under low viscosity conditions contrary to the case of supercooled liquids. In other words the translational correlations are always negligible compared with the orientational correlations and the simple two-variable theories should be adequate to explain the depolarized spectra even for k2q/pI' & 1.One should thus observe that in the low temperature limit the shear waves become increasingly diffusive in accordance with eqn (2). DISCUSSION OF HH SCATTERING The HH scattering geometry has received much less attention than has VH scat-tering. In the HH geometry the transverse velocity gradients cannot contribute to the scattering for reasons of symmetry; however coupling with the longitudinal gradients is now permitted. At scattering angles other than n/2 there is of course a further contribution from the normal isotropic Brillouin scattering. Keyes and Kivelson,6 Gershon and Oppenheim13 have calculated the shape of the 1; spectra using their microscopic theories.However in this geometry it is not COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS possible to express their results in an equivalent form. Keyes and Kivelson6 obtain the following expression where E = k2q/pI'; mB is the frequency of the longitudinal sound waves with wave vector k (Brillouin shift) ; S is the half width of the Brillouin lines; r has the same definition as before and R is a coupling constant between the longitudinal sound waves and the primary variable. We have assumed 0 = n/2. Using equivalences analogous to those used in re-writing Gershon's expressions for the VH spectra we may re-write his expres~ion'~ for Ig 1; cc [cos2(8/2)+ &sin2(8/2)I2 S S + m)2 + s2+ (mB -m)2 + s2 The intensity factors a1and a2are related to time independent correlation functions of elements of the polarizability tensor.The spectral shapes predicted by these expres- sions are shown in fig. 11 as a function of mB/r. We have set al = a2 in expression FIG.11.-HH lineshapes calculated from the expressions given by Keyes and Kivelson (K-K) expression (4) and by Gershon and Oppenheim (G-0) expression (5). The arrows in the last line show the position of the fine structure for K-K. Note that R runs horizontally and not vertically as in fig. 1. (5). The common features of these spectra are an intense central Lorentzian of half- width I'plus fine structure at the Brillouin frequencies. The calculated intensity of the fine structure tends towards zero as wB/r -+0 and also as mB/r -+ CQ.The strongest effect is expected when mB/r E 1. In the limit mB/F< 1the two expressions become equivalent and predict a dip in the spectrum at the Brillouin frequencies. The upper curve in fig. 12 shows that this dip is observed experimentally. In the opposite limit ws/r > 1 both expressions predict a structure whose intensity diminishes rapidly with GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU -50 0 +50 frequency /GHz FIG.12.-HH spectra of pyridine. The fitted spectrum (smooth line) calculated using expression (4) with only R as a variable parameter has been displaced in order to be visible. The small triangle at the centre of the line represents the contribution from elastic parasitic scattering.increasing wB/T. However Keyes and Kivelson predict a hump at the Brillouin frequencies whereas Gershon and Oppenheim predict a dip. Experimental spectra indeed show structure in this case but since the features are commonly very weak one understands why the IE structure has not often been analysed. Fig. 13shows that for ethyl benzoate the Zg fine structure at large wB/I',is certainly a hump not a dip in agreement with Keyes and Kivelson. Enright and Stoicheff 21 have analysed the Zg and Z$ spectra of CS2using the expressions given by Keyes and Kivelson. They find that the Zg spectra are well described by expression (4) over the temperature range 1 I I -8 0 +0 frequency / GHz FIG.13.-HH spectrum of ethyl benzoate at -102 "Cshowing the fine structurearising from coup- ling to the longitudinal sound waves.The position of the Brillouin lines in the VV spectrum is indicated by the arrows. There is an overlap because the Brillouin shift is greater than the free spectral range of the instrument. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS investigated. They also find that within experimental error the values of R obtained from the I and I; spectra are equal with an average value of R z 0.37. Searby Bezot and Sixou20 have undertaken a similar analysis for the case of pyridine. The IF fine structure in this case is more intense than for CS2and they also find good agree- ment between experiment and expression (4). The lower curve in fig. 12 shows a spectrum obtained with oB/T= 1.08 together with the corresponding best fit.The fitted curve has been displaced horizontally in order to be visible. However unlike Enright and Stoicheff Searby et al. do not find agreement between the values of R obtained from the I; and Ig spectra but find RVH= 0.4 and RHHz 0.65. The reason for this difference is not satisfactorily explained at the time of writing. CONCLUSION In conclusion we see that according to the value of k2q/pr(and therefore to a certain degree according to the type of liquid considered) the models proposed explain the experimental depolarized spectra with varying degrees of success. In the case of ordinary organic liquids (k2q/prz 1) the two-variable microscopic theories work very well perhaps deceptively well since the observed spectra do not seemat allsensitive to the detailed molecular structure of the liquid.The molecules may be symmetric tops or not rigid or partly flexible; even the coupling parameter R seems to have a near-universal value of around 0.4 in this viscosity region. The spectra of supercooled liquids (k2q/pI’9 1) are still difficult to explain quanti- tatively and probably need to be studied in the light of a more generalized theory. In the special case of liquid crystals in the isotropic phase the non-existence of a propa- gating doublet at high k2q/pI’ needs to be confirmed by a more systematic study. For liquids composed of small simple molecules and having a very low viscosity it is necessary to verify that the non-observation of the fine structure is due to lack of instrumental resolution and not to more fundamental reasons such as the impossi- bility of separating “fast ” and ‘‘slow ” variables.Lastly there remain a number of questions concerning the exact significance of the parameters RyHand RHH(=RvH?) and the calculation of their values as a function of molecular shapes and interactions. l R. D. Mountain Rev. Mod. Phys. 1966,38,205. R. Pecora J. Chem. Phys. 1968,49,1036. M. A. Leontovich Izvest. Akad. Nauk S.S.R. Ser. Fiz. 1941 5 148 (J. Phys. USSR,1941 4,499). V. S. Starunov E. V. Tiganov and I. L. Fabelinskii Zhur Eksp. Teor. Fiz. Pis’ma Red. 1967 5 317 (JETP Letters 1967 5,260). ’S. M. Rytov Zhur eksp teor. Fiz. 1957,33 514 (Sov. Phys. JETP 1958 6,401). T. Keyes and D.Kivelson J. Chem. Phys. 1971,54,1786. ’T. Keyes and D. Kivelson J. Chem. Phys. 1972,56,1876. T. 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