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Raman scattering from crystals of the diamond structureR.A. Cowley

To cite this version:R.A. Cowley. Raman scattering from crystals of the diamond structure. Journal de Physique, 1965,26 (11), pp.659-667. �10.1051/jphys:019650026011065900�. �jpa-00206332�

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RAMAN SCATTERING FROM CRYSTALS OF THE DIAMOND STRUCTURE

By R. A. COWLEY,Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada.

Résumé. 2014 On discute la théorie de la diffusion Raman de la lumière par des cristaux du typediamant, et on effectue les calculs détaillés pour la diffusion par le germanium, le silicium et lediamant. L’anharmonicité entre les modes normaux de vibration produit un élargissement de laraie à un phonon. La forme de cette raie dépend du comportement détaillé de la durée de vie et dela fréquence du mode optique de grande longueur d’onde. L’anharmonicité amène aussi un cou-plage entre les diffusions à un et à deux phonons, qui tend, en partie, à écarter le centre de la raieà un phonon de la fréquence du mode optique.

Les modèles en coquille adaptés aux courbes de dispersion mesurées servent au calcul des modesnormaux de vibration. Les éléments des matrices à un et à deux phonons de la diffusion Ramansont évalués au moyen de la méthode du modèle en coquille, appliquée à ces modèles particuliers,et les spectres sont calculés, en tenant compte des effets de l’anharmonicité de la diffusion, par unphonon. La comparaison des résultats avec les données expérimentales connues montre que lesdétails du spectre dépendent considérablement de la polarisation de la lumière et de l’orientationdu cristal.

Abstract. 2014 The theory of the Raman scattering of light from crystals of the diamond struc-ture is discussed and detailed calculations made of the scattering by germanium, silicon and dia-mond. The anharmonicity between the normal modes of vibrations gives rise to a broadeningof the one-phonon peak. The shape of this peak then depends upon the detailed behaviour of thelifetime and frequency of the long wavelength optic mode. The anharmonicity also introducesa coupling between the one and two-phonon scattering, part of which tends to shift the center ofthe one phonon Raman peak away from the frequency of the optic mode. Shell models whichwere fitted to the measured dispersion curves are used to calculate the frequencies of the normalmodes of vibration. The one and two-phonon matrix elements for the Raman scattering areevaluated by means of the shell model formalism for these models and the spectra calculatedincluding the effects of anharmonicity on the one phonon scattering. The results which arecompared with the available experimental results show that the details of the spectra dependconsiderably on the polarization of the light and on the orientation of the crystal.

LE JOURNAL DE PHYSIQUE TOME 26, NOVEMBRE 1965,

1. Introduction. -- The theory of the Rarnanscattering of light by the lattice vibrations of acrystal shows that it depends on the way in whichthe lattice vibrations modify the electronic polari-zability, PaB, of the crystal as described by Bornand Huang [1], and by Loudon [2].The intensity of the scattered light depends upon

the polarizations of the light and upon the crystalorientation. If the electric vector of the incident

light is defined by E, and the polarization of theelectric vector of the scattered light by n, theintensity of scattering per unit solid angle is [1]

where

The frequency of the incident light is wo, v isthe initial state of the crystal of energy ev, v’ thefinal state of energy e,,, and Q the change infrequency of the light. The expression (1.2) ismost readily evaluated by the use of the tech-

niques of many body theory (for example Abri-kosov, Gorkov and Dyzaloshinski [3]). TheFourier transform of the thermodynamic time orde-red Green’s functions G(P :yPa8’ Q) are easilyevaluated by the use of these techniques, and

The Raman scattering is now obtained by expan-ding the polarizability of the crystal as a powerseries and evaluating the Green’s functions bymeans of diagrams [4]. In each diagram the pho-nons associated with the PgT operator enter on theleft, while those associated with Pys leave on theright. In between they interact with one anotherthrough the anharmonic interactions in the crystal.The rules for calculating the contributions of thesediagrams have been given by Maradudin andFein [5] and by Cowley [6].The Raman spectra of the crystal can then be

obtained from the I,,,py8, appropriate for the parti-cular conditions of the experiment. The spectraare dependent upon the state of polarization ofboth the incident and scattered light, and also on

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011065900

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the crystal orientation. A table of the appro-priate combinations of the 1,,pys for some typicalexperiments is given in [4].Two difficulties arise in the detailed calculation

of the Raman spectra. Firstly the frequencies andeigenvectors of the normal modes of vibration areusually unknown. In this paper this difl’lculty isovercome by the use of models whose parameterswere fitted to give good agreement with measu-rements of the dispersion curves, as made by ine-lastic neutron scattering techniques. The modelsare described in detail by Dolling [7] and by Dollingand Cowley [8].The second difficulty arises because the influence

of the lattice vibrations on the polarizability of thecrystal is unknown. These calculations are per-formed by assuming that the interaction is of a

very simple form between nearest neighbour ions,as described in the next section. The detailedform of the Raman spectra of the diamond struc-ture materials depends on the behaviour of theone-phonon contribution. This is largely in-fluenced by the anharmonic interactions of the longwavelength optic mode of vibration with the othernormal modes. These interactions have been cal-culated by means of the parameters of the anhar-monicity determined from the experimental ther-mal expansion of the crystal [8].

In section 3 the detailed expressions for ,theRaman spectra are given, including the effects ofanharmonicity on the long wavelength optic mode.The calculation of the spectra and the results aredescribed in section 4.Measurements of the Raman spectra are fre-

quently used to deduce critical point frequencies(Loudon [2] and Bilz, Geick and Renk [9]). Someof the difliculties in deciding and interpretingthese critical point frequencies are described insection 5. The results are summarized and dis-cussed in section 6.

2. The polarizability. -- The electronic polari-zability of a crystal may be expanded as a powerseries in the phonon coordinates [1] :

where the A(q j ) are the sum of the phonon creationand destruction operators. The one-phonon con-tribution arises from the long wavelength opticmode of vibration in the diamond structure, thesymmetry of which shows that P«T(oj) is non-zeroonly if ce and p are unequal [2]. The coefficientsof the two-phonon term, Paa(q j j’), are more compli-cated and depend both on the wave vector q andthe indices j and j’ of the normal modes. Birman

[10], Johnson and Loudon [11] and Kleinman [12]have given the selection rules for these coefficientswhich result from the symmetry of the normalmodes.

In these calculations the coeflicients are deter-mined by assuming a form for the interactionbetween the ions giving rise to the polarizability.The two-phonon contribution to the polarizabilitymay be written as

where u(lx) is the displacement of the xth ion inthe Zth unit cell. The number of arbitrary coef-ficients needed to describe the interactions in equa-tion (2.2) is restricted by the symmetry of thecrystal and by making physical approximationsabout the form of the interaction. In these calcu-lations the simplest form of one parameter nearestneighbour interaction has been used in which

where X is a constant and R is the vector distanceof the ion (l’x’) from the ion (lx) and the inter-action is restricted to nearest neighbours. The

expressions for (l’x’) = (lx) were deduced by theuse of translational invariance. The coefficients ofthe two-phonon term in equation (2.1) are then

given byv,

The eigenvectors of the normal modes are writ-ten e(x, qj) and the frequencies m(qj) while X’ isanother constant.

This development of detailed expressions for thetwo-phonon contribution to equation (2.1) may becompared with those deduced earlier [4] by use ofthe shell model formalism. Equation (9) of [4]is very similar to equation (2.4) if the terms depen-ding on the displacements of the shells are neglec-ted. The one parameter interaction we have usedhere is equivalent to assuming that the interactionin equation (10) of [4] is a central force in which thefourth derivative terms are by far the largest.The parameters for the calculation of the Ramanspectra of the alkali halides were deduced earlierby assuming that all the short range interactionacts through the shells. In the case of thediamond structure this approximation necessarilygives rise to no Raman scattering, when transla-tional invariance is applied to the interaction.The parameter of’ the two-phonon Raman scat-tering in the diamond structure could not there-

661

fore readily be deduced from other measurements.The contributions included by the expressions

(2.3) and (2.4) are therefore very similar to thoseof equations (9) and (10) of [4]. However theshell model formalism shows that there is an addi-tional contribution to the two-phonon scatteringfrom two optic modes of long wavelength. Such ascattering is observed experimentally in dia-mond [13] and it is necessary to add in this extratwo-phonon term to explain the magnitude of thisscattering.

3. The Raman spectra. - The Raman spectracan be obtained from the expressions given in Iand the expansion for the polarizability (2.1).The detailed shape of the one-phonon spectra isdetermined by the anharmonic interactions invol-ving the optic mode of vibration. The extra con-tribution to the Hamiltonian from the anharmonicinteractions is written as

The anharmonic interactions alter the self-ener-gies of the normal modes. The three lowest order

diagrams are shown in figure 1, and their effect on

FIG. 1. - The three lowest order diagrams which contri-bute to the self-energy of the normal modes. Dia-gram (a) arises from the thermal expansion while theother diagrams arise from the interactions between thenormal modes.

the optic mode of vibration is to make its fre-

quency and lifetime dependent upon both thetemperature and the applied frequency (Q) :

The detailed expressions for the contributions toA(oj, Q) and r(oj, Q), shown in figure 1, are [5],

Diagram (a), figure 1, and the first term of thisequation arise from tlie thermal expansion of thecrystal, while the second term is from diagram (b).The function R(Q) is

The occupation number of the normal mode isn(qj) and its inverse lifetime

where

The diagrams which contribute to the Ramanspectra are shown in figure 2. The one-phononcontribution to Ixyxy is

FIG. 2. - The lowest order diagrams which contribute tothe Raman spectra of the diamond structure materials,

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Diagrams (b) also contribute only to IxYXY, and involves both one and two-phonon processes :

Diagrams (c) and (d) are the two-phonon contri-butions to 7Arjcy and I xx xx respectively ; dia-

gram (d) gives

2 n { exp (ph 0) - 11-1

The Raman spectra can now be obtained indetail by the use of these equations and those forthe coefficients in the expansion of the polariza-bility (2.1).

4. Calculations and results. - The one-phononscattering cross-section depends in detail on theanharmonic interaction of the long wavelength opticmode with the other normal modes. The calculationof the shift, A(0/, Q), and inverse lifetime, r(o/, Q),have been performed using the eigenvectors andfrequencies of harmonic models, obtained by fit-ting the parameters of the models to give excel-lent agreement with the dispersion curves as mea-sured by inelastic neutron scattering. The modelsare described in detail by Dolling [7] and byDolling and Cowley [8].

FIG. 3. - The shift in frequency, 3(oj, Q), and the inverselifetime, r(oj, Q), as a function of the applied frequency,í1, for the optic mode of germanium at 10 OK and 300 OK.

The anharmonic interactions V (I F . q )1 /1 12were deduced by assuming they were two bodycentral forces between nearest neighbours, andfinding the parameters which give best agreementwith the experimentally measured thermal expan-sion as described by Dolling and Cowley [8]. Thecalculation of the third contribution, diagram (c) offigure 1, to A(oj, Q) and to h(oj, Q) were thenperformed numerically by sampling q over 4 000wave vectors in the full Brillouin zone. Theresults for germanium at 10 oK and 300 OK areshown in figure 3 as a function of the applied fre-quency. The contributions to the shift from dia-grams (a) and (b) of figure 1 were not evaluatedbecause they do not alter with applied frequency,and furthermore in calculations for the alkalihalides they almost exactly cancel [6]. It is of finterest that both the shift and inverse lifetime arequite stiongly dependent upon the applied fre-

quency, Q. In any particular experiment theinverse lifetime is normally taken as given by thewidth of the one-phonon peak. This is roughlyapproximated by h(o j, 6J(oj)) and the shift in fre-quency by A(o/, 6J(oj)). These are listed in thetable at various temperatures for germanium, sili-con and diamond.

TABLE

THE SHIFT IN FREQUENCY A(oj, co(0/))AND THE INVERSE LIFETIME r(oj, M(0/))

OF THE OPTIC MODES, FOR DIAGRAM (c) OF FIGURE 1.UNITS 1012 c. p. s.

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Once the-form of,&(Oi, Q) and r(oj, Q) have beencalculated the shape of the one-phonon peak is

given by equation (3.7). The results for germa-nium are shown in figure 4, and for silicon in

figure 5 by the curves A and AA. Curve AA showsthe fine structure resulting from the frequencydependence of A(o/B Q) and r(oj, Q) while Ashows the principal peak. Since the numericalvalue of Pxp(o/) is unknown the intensity scalelisarbitrary.

FIG. 4. - The contributions to the Raman spectra of ger-manium at 300 °K. Curves AA and A show the one-phonon part. Curve AA is on 100 times larger scalethan A. B arises from the joint one-and two-phonontype of process, C is the two-phonon Ix yx y contribution,and D the Ixxxx contribution. - E results from the twophonon contribution from 2r phonons. The relativeintensity of the curves A, C, E is arbitrary as explainedin the text.

The two-phonon contributions have been evalua-ted by means of equation (3.9) and the expressionfor the matrix elements given in section 2. Theresults differ for the contribution to Ixyxy, curve C,and to Ixxxx, curve D, showing that the observedtwo-phonon intensity will depend on the polari-zation of the light and the orientation of the

crystal. The contribution involving both one andtwo-phonon processes, equation (3.8), contributesto Ix yxy alone and is shown in figure 4 and 5 ascurve B. Since the magnitude of the two-phononrelative to the one-phonon matrix element isunknown the curves C and D may be scaled by

FIG. 5. - The contributions to the Raman spectra of sili-con at 300 °K. The contributions are labelled simi-larly to these of figure 4 except the ratios of the scalesfor AA to A is 1/200. The critical point frequenciesfor r, X, W are also shown. Those with a horizontalbar arise from difference bands, and those dotted occuronly for Ixxxx.

any constant K 2 while curves B are scaled by K.A further two-phonon contribution identical forboth I xYxY and Ixxxx, arises from the term invol-ving two long wavelength optic modes and itsadditional contribution is shown by the curves Eof figures 4 and 5. The relative intensity of thiscontribution is not simply related to that of theother contributions.The detailed structure of the scattering near the

one-phonon peak is of considerable interest.Figure 6 shows the detailed shape of the one-

phonon contribution and it is clearly not sym-metric because of the frequency dependence ofA(oj, Q) and r(oj, Q). Figure 6 also shows thecontribution of the last part of expression (3.8)which is asymmetric about the one-phonon peakfrequency. This contribution will act so as to shiftthe observed peak away from the peak in the one-phonon contribution. In the cases calculated hereit appears that the size of the shift is very small,but nevertheless the detailed study of the shapesof one-phonon lines might well be rewarding.

It is also possibly of interest to point out thatthe size of the asymmetric term relative to the one-phonon term depends on the coefficient of the

664

expansion (2.1). Since the coefficients of this

expansion differ for neutron and X-ray scatteringand for infrared absorption the frequencies deter-mined using these different techniques may differ,because the relative size of the asymmetric termwill be different.

FIG. 6. - The detailed behaviour of the one-phonon con-tribution, A, and of the asymmetric part of B near theoptic mode frequency in germanium at 300 °K.

Diamond is unfortunately the only one of thematerials discussed here for which the Raman

FIG. 7. - The Raman spectra of diamond as measured byKrishnan [13], dotted line, and as obtained by thepresent calculations, solid line.

spectra has been measured. Figure 7 shows asketch of the experimental results of Krishnan [13]compared with the spectra as calculated by thesame methods as those described for germaniumand silicon. Since Krishnan does not state theorientation of the specimen, the appropriate com-bination of the IapY$ is unknown, the calcula-tions shown in figure. 7 are for Ixxxx + 3I g pg pwith a value of the parameter X chosen to giveapproximately a similar shape to that of the experi-mental results. It should be emphasized that thecurve shown in figure 7 is obtained by addingparts from all the contributions A-E shown in

figures 4 and 5, and far worse agreement wasobtained if any of them were neglected.The results show that the largest peak is dis-

placed in the calculations by about 2 (1012 c. p. s.)from the largest peak in the experiment. This

may possibly be due to errors in the harmonicmodels however, because the accuracy of the mea-surements of the dispersion curves on which theyare based is about 3 % [14].

In view of the uncertainties in the dispersioncurves, of the crystal orientation in the 1,Ramanmeasurements, and in the scale factor K, theagreement between the calculations and the expe-riment is probably as good as may be expected.The Raman spectrum of gallium phosphide has

been measured recently using a laser source, byHobden and Russell [15]. It is of interest to com-

pare the general shape of this spectrum with thepresent calculations. The spectrum of galliumphosphide shows two one-phonon peaks instead ofthe one for germanium and silicon. However thefine structure on the low frequency side of the one-phonon peaks is very similar in shape to that ofcurves AA in this region. Furthermore there are

peaks near the one-phonon peaks which mightresult from contribution B, and at higher fre-

quencies there are peaks which appear as thoughthey probably arise from contributions C and D.These features seem to suggest that a combinationof all these different contributions could give excel-lent agreement with the Raman spectra of siliconand germanium. ,

5. Critical points. - The fine structure of theRaman spectra of crystals is frequently inter-

preted in terms of the frequencies of the normalmodes of vibration at certain special points withinthe Brillouin zone. Van Hove [16] showed thatthe one-phonon density of states plotted againstfrequency exhibits a discontinuity of slope, whenthe frequency is equal to that of a point where thegradient of the dispersion curve is zero. These

points are known as critical points. This work hasbeen extended by Johnson and Loudon [11] to givethe shapes of the critical points in the two-phonondensity of states function.

665

The interpretation of the measured Ramanspectra by Bilz, Geick and Renk [9], Burstein,Johnson and Loudon [17] and Hobden and Russell[15] for example, is based on identifying the peaksand shoulders of the distributions with the fre-

quencies of the critical points. The calculationsperformed in this paper are not sufficiently detailedto show the discontinuities in slope, even thoughthey are comparable in resolution to the experi-mental measurements. In figure 5 the frequenciesof the critical points for r, X, L are shown forcomparison with the calculated curves for silicon.Although some of the peaks do occur at the samefrequencies as the critical points, many of themoccur slightly to one side of the peaks in the finestructure, or even are not readily correlated withthe structure. Krishnan [13] gives a list of fre-quencies deduced from his measurements of which10 are greater than a frequency of 75 (1012 c. p. s.),whereas the model based on the experimental dis-persion curves only gives the 21, critical point inthis frequency range. In view of these conside-rations it is not surprising that the dispersioncurves predicted by Bilz et al. [9] and others fordiamond, on the basis of critical points in theinfrared and Raman spectra, show several markeddiscrepancies with those subsequently measured byWarren et al. [14]. Similar difficulties occur in theanalysis of critical points in the two-phonon infra-red spectra of crystals as described in detail byDolling and Cowley [8].

Features of the measured spectra can clearlyonly be identified as critical points if the resolutionof the experiments is sufficient to be able to dis-tinguish the discontinuities in slope, to avoid thenecessity of having to rely on interpretating grossfeatures of the spectra as arising from criticalpoints. This necessitates performing very highresolution experiments. Useful information maythen be obtained about the frequencies of the cri-tical points if the type of critical point is deter-mined [11], and with the aid of the selection rules[8], [12]. Even under ideal conditions however,it seems unlikely that assignments can be madeunambiguously in the absence of measurements ofthe dispersion curves.

Since very high experimental resolution isneeded to establish the existence of critical points,it is necessary to examine the effects of anharmo-nicity on the critical points. Diagram (a) of

figure 8 shows the ordinary two-phonon processwhich gives rise to the critical points in the har-monic approximation. In an anharmonic crystalboth of the lines shown in diagram (a) can haveself-energy insertions. A typical diagram of thistype is (b). The contributions from these diagramsmay be calculated by replacing the harmonicGreen’s functions with the dressed Green’s

functions in the expressions for diagram (a).The result is proportional to

where P(qj, (0) is the spectral function of the mode(qi) ;

-

In general the expression (5.1) is complicated,However if we assume the modes have long life-times, the expression (5.1) is large when

and the peak has a width given approximately by

These results show that the width of a two-

phonon addition process is the sum of the lifetimesof the two independent normal modes. Since thecritical point behaviour depends on the frequenciesbeing well defined, it would clearly be impossibleto observe them once the lifetimes of the normalmodes became significantly large.

FIG. 8. - Two-phonon diagrams showingthe different ways anharmonicity can enter.

666

Moreover not all the diagrams can be decom-posed into two phonon diagrams with self-energyinsertions however. Diagrams of a more complextype are shown in figure 8 (c) and (d). The contri-bution from diagram (c) is proportional to

lS(qii j2’ Q) -R(P/i j2, Q) + R(qii j2, Q) S(Pil j2 0)], (5.2)

where S and R are the functions defined by equa-tions (3.4) and (3.6).The critical point behaviour arises because of

the S functions in (5.2), whereas the R functionsbehave smoothly. If the critical points are welldefined this type of diagram will not alter the fre-quency or the sharpness of the critical point com-pared with diagram (a). Similar arguments can beused to show that diagram (d) will also leave thebehaviour of the critical points unaltered. Dia-gram (e) shows another more complicated diagramincluding both self-energy parts and couplingbetween the modes.

It is of interest to compare the frequencies ofthe critical points with those obtained from one-phonon measurements. As remarked earlier all

one-phonon measurements have an asymmetriccontribution associated with them which tends toshift the frequency by amounts which vary accor-ding to the type of measurement. The two-

phonon critical points do not however have anyasymmetric contribution associated with them.Nevertheless this does not enable the size of theasymmetric contribution to the one-phonon resultsto be determined, since the asymmetric term is

only considerable if the inverse lifetime of thenormal modes become large. In this case the cri-tical point is not well defined and would bedifficult to identify.

6. Summary and conclusions. - The Ramanspectra of germanium, silicon and diamond havebeen discussed. The models which describe thefrequencies and eigenvectors of the normal modesgive good agreement with the measured dispersioncurves [8]. The deficiencies of the calculationsprobably lie in the treatment of the polarizabilityin terms of only a simple form of nearest neighbourinteraction.The detailed shape of the one-phonon peak has

been calculated including the frequency and tem-perature dependent lifetime of the optic mode, andalso the asymmetric term arising from the processinvolving both a one and two-phonon interaction.This latter term shifts the frequency of the opticmode slightly, while both effects make the peakasymmetric.The fine structure arises from all of the different

contributions : one-phonon, two-phonon and jointprocesses. The available experimental results [13,

15] suggest that all these contributions are neededto explain the measurements. The results alsoshow that the fine structure depends on the polari-zation of the light and on the orientation of thespecimen.The frequencies of the critical points of the

models are not given by the peaks in the finestructure. Critical points can only be satisfac-torily determined under conditions of very goodexperimental resolution. Even under these condi-tions and with the aid of the selection rules, theauthor believes that their interpretation can onlybe made satisfactorily under extremely favourablecircumstances, unless measurements of the dis-persion curves have already been made by othermethods.

It is hoped that these calculations will stimulatework on the Raman spectra of germanium andsilicon and particularly on single crystal specimensnow that laser sources are available. In parti-cular it would be of interest to study the detailedshape of the one-phonon peak under high reso-lution. The fine structure appears to depend onthe polarization of the light and on the orientationof the crystal. These experiments would undoub-tedly show that the simple nearest neighbour typeof interaction used here to describe the polari-zability is inadequate, and lead to a better under-standing of the processes involved in Raman scat-tering.

"

Acknowledgements. - The author is indebted toDr. G. Dolling for his assistance and encoura-gement in this work. In particular he providedthe shell models which gave the frequencies andeigenvectors used in these calculations.

Discussion

M. THEIMER. - Do you have some quantummechanical estimates concerning the relativemagnitude of nearest neighbours and second nea-rest neighbours effects on the polarizability of anatom ? -

M. COWLEY. - The calculation of the matrixelements is very difficult. Second neighbour inter-action is very important for the dielectric suscepti-bility, so it may be important for Raman scattering.

M. MARADUDIN. - In principle, every term inthe expansion of the electronic polarisability inpowers of the atomic displacements contributes tothe intensity of the first order Raman spectrum.Have you compared your theoretical results for

diamond with those of Miss Smith ?What is the origin of the contribution to the

Raman spectrum which you have labeled E in yourcurves ?

667

M. COWLEY. - The contribution E arises from adouble single phonon term in the shell model for-malism (Proc. Phys. Soc., 84, 281).

M. BURSTEIN. - In the case of the diamond and

perhaps also in the zinc-blende structure, it wouldseem to me that one should expect that the matrixelements for Raman scattering would have a dis-tinct orientation dependence, since the electron

density exhibits a localization along the (111) direc-tions which the present calculations do not takeinto account. In the case of NaCI type crystals,it appears, at least from the analysis of the 2ndorder spectra, that the phonons at the X point makea major contribution. One might expect, in thezinc-blende and diamond structure, that the (longi-tudinal) phonons with atom displacements along(111) would make the major contribution. This

would, for example include the longitudinal modeat the L point.

It is of interest to note that there is an inte-resting analogy between the second order infraredspectra of diamond type crystals and the secondorder Raman spectra of rocksalt type crystals. Inthe case of diamond type crystals which exhibit nolinear moment, one must involve the second ordermoment mechanism for infrared absorption. Themechanical anharmonicity can not lead to absorp-tion. In the case of rocksalt type crystals, onemust involve the second order terms in the polari-zability, since the first order scattering does notoccur. For this structure, the anharmonic cou-pling makes no contribution.

M. COWLEY. - The processes involved are

exactly analogous to those needed for infrared

spectra.

REFERENCES

[1] BORN (M.) and HUANG (K.), Dynamical Theory ofCrystal Lattices, Oxford University Press, 1954.

[2] LOUDON (R.), Adv. in Physics, 1964, 13, 423.[3] ABRIKOSOV (A. A.), GORKOV (L. P.) and DYZALO-

SHINSKI (I. E.), Methods of Quantum Field Theoryin Statistical Physics, Prentice Hall Inc., 1963.

[4] COWLEY (R. A.), Proc. Phys. Soc., 1964, 84, 281.[5] MARADUDIN (A. A.) and FEIN (A. E.), Phys. Rev.,

1962, 128, 2589.[6] COWLEY (R. A.), Adv. in Physics, 1963, 12, 421.[7] DOLLING (G.), Slow Neutron Scattering from Solids

and Liquids, Vol. II, I. A. E. A., 1963.[8] DOLLING (G.) and COWLEY (R. A.), To be published.[9] BILZ (H.), GEICK (R.) and RENK (K. F.), Lattice

Dynamics, p. 355, Pergamon, 1965.

[10] BIRMAN (J. L.), Phys. Rev., 1962, 127, 1093 ; Phys.Rev., 1963, 131, 1489.

[11] JOHNSON (F. A.) and LOUDON (R.), Proc. Roy. Soc.,1964, A 281, 274.

[12] KLEINMAN (L.), Solid State Communications, 1965, 3,47.

[13] KRISHNAN (R. S.), Proc. Ind. Acad. Sc.,1947 A26, 399.[14] WARREN (J. L.), WENZEL (R. G.) and YARNELL (J. L.),

Inelastic Scattering of Neutrons, Vol. I, I. A. E. A.,Bombay,1964.

[15] HOBDEN (M. V.) and RUSSELL (J. P.), Physics Letters,1964, 13, 39.

[16] VAN HOVE (L.), Phys. Rev., 1953, 89, 1189.[17] BURSTEIN (E.), JOHNSON (F. A.) and LOUDON (R.),

Phys. Rev., 1965, 159, A 1239.