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Theoretical interpretation of the second-order Ramanspectra of caesium halidesA.M. Karo, J.R. Hardy, I. Morrison

To cite this version:A.M. Karo, J.R. Hardy, I. Morrison. Theoretical interpretation of the second-order Ra-man spectra of caesium halides. Journal de Physique, 1965, 26 (11), pp.668-676.<10.1051/jphys:019650026011066800>. <jpa-00206333>

668.

THEORETICAL INTERPRETATIONOF THE SECOND-ORDER RAMAN SPECTRA OF CAESIUM HALIDES

By A. M. KARO (1), J. R. HARDY (2) and I. MORRISON (1),

Résumé, 2014 Les auteurs ont calculé les spectres Raman du second ordre de CsCl, CsBr et CsI.Pour les deux derniers, on peut comparer leurs résultats avec les spectres observés. On constateque les spectres prévus en admettant une polarisabilité et un chevauchement des ions sont en bienmeilleur accord avec l’expérience que ceux obtenus en traitant ces ions comme des charges rigidesponctuelles.Par la première méthode, l’accord général est remarquablement exact, même sans tenir compte

de l’influence du vecteur d’onde sur le tenseur de polarisabilité de Raman.La raison probable en est que les principaux pics dans le spectre théorique proviennent de la

superposition de points critiques dans les courbes de dispersion à deux phonons, qui se trouventsur les axes de symétrie. Les présents résultats semblent montrer que le spectre Raman observéfournit une image assez exacte de la densité d’états à deux phonons, et peut servir pour vérifieret préciser le calcul théorique de la matrice dynamique.

Il apparait également que des spectres à haute résolution, obtenus à basse température, révéle-raient beaucoup plus de détails, si les spectres calculés sont dignes de foi.

Abstract. - The results of calculation of the second-order Raman spectra of CsCl. CsBr, and CsIare presented. For the last two crystals the results can be compared with observed spectra. Itis found that the spectra derived by allowing for the polarizability and overlap deformation of theions are in much better agreement with the experimental results than those derived by treatingthe ions as rigid point charges. In the first case the overall agreement is remarkably close, eventhough no allowance has been made for the wave-vector dependence of the Raman polarizabilitytensor.The probable reason for this is that the main peaks in the theoretical spectra arise from super-

positions of critical points in the two-phonon dispersion curves which occur along symmetry axes.The present results tend to support the view that the observed Raman spectra provide a reasonablyclose mapping of the two-phonon density of states and can be used both to test and refine theore-tical calculation of the dynamical matrix.

Also, it appears that low-temperature, high-resolution spectra should reveal considerably moredetail if the computed spectra are reliable.

Lf JOURNAL DE PHYSIQUE TOME 26, NOVEMBRE 1965,

Introduction. - The alkali halide crystals areprobably the best examples of solids whose valenceelectrons can be adequately described by the" tight binding " approximation, since the pertur-bation of the free-ion functions is relatively smalland is due primarily to first-neighbour overlapeffects. It is thus possible, as was done byL6wdin [1] following Landshoff [2], to calculate thecohesive energy quantum mechanically from thezero-order (i.e. free ion) wave functions and obtainvery close agreement with the observed values.Thus, it would seem that the effects of configu-ration mixing are relatively small, whereas thecohesion of the inert gas solids, where the isolatedatoms have a closed-shell electronic configuration,is only possible because of the admixing of excitedstates by the crystal perturbation, resulting in theVan der Waals attraction.

In an earlier paper [3] we have presented theresults of a systematic treatment of the lattice

(1) Lawrence Radiation Laboratory, University of Cali-fornia, Livermore, California, U. S. A. (Sponsored by theU. S. Atomic Energy Commission).

(2) Theoretical Physics Division, A. E. R. E., Harwell,Didcot, Berkshire, England.

dynamics of most of the NaCI-structure alkalihalides. The aim of this work was to remove themain shortcomings of Kellermann’s original theoryby the inclusion of the effects of electronic distor-tion in the harmonic potential function in so far asthey alter the dipole-dipole component of this

potential.Thus, we retained his assumption of a central

repulsion, effective between first neighbours only, Ito describe the short-range component of the poterj-tial.As regards the electronic distortion, we reco-

gnised two components :(a) The field-induced polarization dipoles, given

by the product of the self-consistent crystal polari-zability of the particular ion being considered, andthe effective field at the centre of that ion, wherethe dipole is assumed to be situated.These self-consistent polarizabilities were taken

from the compilation of Tessman, Kahn and

Shockley [5] whose work also provides independentsupport for treating the electronic polarization inthis way because, not only were they able to finda clearly defined set of characteristic polariza-bilities, but also their most consistent set was,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011066800

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obtained by assuming that the effective polarizingfield was that acting at the ion centres in an arrayof point dipoles.

(b) The " deformation " polarization, which werestricted to the negative ions, and regarded as inde-pendent of the field-induced polarization. Itsexistence, as was first observed by Szigeti [6], is alogical consequence of the presence of the overlaprepulsion, which must distort the ions to someextent. In a polar vibration this distortion has aresultant odd multipole component and, in parti-cular, a dipole component. Since we consider onlyfirst-neighbour repulsion, it follows that this dipoleis determined by the configuration of the negativeion and its six first-neighbour positive ions. Thus,by a generalization of Szigeti’s [6] original work onuniformly polarized crystals, we could derive thedeformation dipole (D. D.) moment on any givenion in an arbitrarily distorted crystal.The results of these D. D. calculations for the

NaCI structures were very different from the rigidion (R. I.) results, both as regards calculated fre-quency spectra and frequency v. wave vector

(õ. v.q) dispersion curves. In particular, the D. D.( w . v . q) curves for N aI and KBr agree closely withthose obtained from the original " shell model "calculations [7, 8] which, in turn, agree with themeasured [8, 9] dispersion curves much better thanthe R. I. results.The most striking improvements are those which

occur near the zone boundary, particularly forlongitudinal modes propagating along [100] direc-tions. These are remarkable for two reasons :

first, they are predicted, not " forced " by thechoice of input parameters ; and second, the netchanges in the individual frequencies produced bycorrection for both types of electronic polarizationare relatively small differences between very muchlarger changes produced by the individual polari-zation mechanisms acting separately. It seems tous important to emphasize these two points, parti-.cularly the second, since they provide good evi-dence for asserting that the D. D. approximationis not too far from an exact description of theeffective potential function governing the nuclearmotion. Conversely, one can argue that since theeigenfrequencies for a given crystal are very sensi-tive to the dipolar part of this potential, while thebasic assumptions of the D. D. approximation, ifvalid for one alkali halide, should be reasonablyvalid for the whole sequence, it is of considerableimportance to verify that the validity of the D. D.theory is not confined to either one particularcompound or to only those alkali halides with theNaCI structure. In our earlier work [3] we wereable to verify the first of these points ; however,we have not previously investigated the caesiumhal des and that is the object of the present paper.Owing to the large neutron absorption cross-

section of caesium there seems very little prospectof the direct determination of dispersion curves,and one will have to resort to more indirect testsof the theory, the best of which, at present, seemsto be that provided by the second-order Ramanspectrum. For both NaCI and CsCI structuresthis is the lowest-order spectrum, since the first-order line is forbidden. We shall restrict our

attention to that part of the spectrum which arisesfrom double phonon creation, and we shall alsoassume that all the crystal normal modes are intheir ground state. Consequently, our results areonly directly comparable with observed spectrameasured at 0 OK.. In practice, even at 300 OK,these combination bands are generally distinctfrom any difference bands which may be present,and as a first step, we can compare our predictedspectra with the available experimental measu-rements made at room temperature. What weactually calculate is the " two-phonon density ofstates " p( 6)1 + W2) which gives the number ofpairs of phonons, of equal and opposite q vectorssuch that the combination frequency (6)1 + w2)lies between 6) and w +A w. Thus, the onlyselection rule we include is the conservation ofcrystal momentum. Obviously this cannot pro-vide a full theory of the Raman spectrum, sincethe strength of a given combination 1 + 2 mustalso depend on the matrix element M(12)2. Re-

cently Cowley [10] has used an extended shellmodel [11] to make a full calculation of the Ramanspectra of NaI and KBr. However, this theory isused to derive the Raman spectra from the measu-red [8, 9] dispersion curves, since these are fittedat the outset by adjusting the extra parametersintroduced by the extension of the shell model.Our problem is rather different, since what we

would like to know is whether it is possible to usethe Raman spectra as a means of discriminatingbetween various potential functions in the absenceof any other data.We have already investigated NaCI [12] by cal-

culating p( 6)1 + w2) for both R. I. and D. D.

potentials and found, somewhat surprisingly, thatthe D. D. curve reproduced the experimental spec-trum of Welsh et al. [13] remarkably well. uotonly were the main peaks in approximately thecorrect positions, but the general shape of the curvewas correct ; whereas, the R. I. density had 4completely different shape and bore no resemblanceto the observed spectrum.

This result indicates that one should certainlyinvestigate the caesium halides in the same way,complementing the calculation of p( Ù1 + (õ2) withthe calculation of combined dispersion curves,«õl + Cù2) v.q, along symmetry directions. Inthis way it is possible to assign the various featuresof the distribution function to critical points on thedispersion surfaces. Also, there is the practical

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point that, to reveal all the details of p(w, + (Ù2),one needs a very fine mesh of sample q vectors,and it seems best to push the calculations to thecomputational limit in this respect before intro-ducing selection rules. This difficulty is likely tobe most pronounced for CsBr and CsI, on which weshall concentrate our attention (there seem to beno measurements available for CsCI itself, and CsFhas the NaCI structure and does not concern ushere), since the ionic masses are comparable forCsBr and almost equal for CsI.

In this paper we shall confine ourselves to thepresentation of results for Raman spectra anddispersion curves ; the details of the calculationwill appear in a later paper which will contain a

systematic account of the whole of the work. It

is, however, worth mentioning here that the finaldynamical matrix has the same form as that foithe NaCI structure [14], but the various constituentmatrices are modified. Apart from the fact thatthe dipole-dipole coupling coefficients have had tobe rederived for the CsCI structure, the matrixwhich describes the short-range interactions is alsomodified, together with that which specifies thedeformation dipole.

Results. - These will be presented separatelyfor CsBr and CsI together with the appropriateexperimental data. Unfortunately, the charac-teristic temperatures of the phonons involved aresuch that the observed Stokes and anti-Stokes

components at 300 OK have nearly the same inten-sities because of the high degree of thermal exci-tation. Thus, although it is still possible to discri-minate between combination and difference bands,one must remember that the observed relativeintensities are not directly comparable with thetheoretical values, since at high temperature theobserved line strengths are

which implies a strong relative enhancement of thelow-frequency end of the spectrum.

Caesium bromide. - In figures 1 and 2 we showthe computed densities of states (R. C. D. S.) andcombined dispersion curves, first for the rigid ion(R. I.) model and second with both field-inducedand deformation polarization of the ions included.The repulsive interaction is confined to first neigh-bours, the T. K. S. [5] polarizabilities are used, andit is assumed that the deformation dip’ole expe-riences the field at the negative ion centre.

FIG. 1. - Two-phonon density of stades histogram anddispersion curves for CsBr computed for rigid ions (R. I.).Inset shows the corresponding smoothed curve with thepositions of the observed features (see fig. 2).

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Inset into figure 2 we also show the observed spec-trum [15] with the various features marked a, b,c, etc... The corresponding points on the theore-tical spectra are also indicated. The actualR. C. D. S. curves are smoothed through a histo-gram obtained from a sample of 64 000 q vectors

in the whole reduced zone, while the dispersioncurves are plotted using a sample of points fivetimes as dense along the symmetry directions.As in the case of NaCI [12] the two curves look

very different, and the D. D. results are in muchbetter overall agreement with the experimentaldata. The only really significant discrepancy isthe presence of the peak at 1.8 X 1013 s-1 in thecalculated spectrum, but it is possible that thisoccurs at too low a frequency because theTO (X) + fiA ( X) combination is misplaced,since the present model does not allow us to fit mo,the T. 0. frequency at 1. To remove this discre-

pancy we will have to introduce non-Coulomb

second-neighbour interactions and it should be

possible, assuming that these are central, to deter-mine the relative magnitudes of the + + and -----components, by fitting the observed values of Woand this tranverse-mode combination frequencyat X.

It is evident that we expect better measu-

FIG. 2. - Two-phonon density of states and dispersion curves computed for CsBr allowing for ionic deformation(D. D.1. , The insets show the smoothed theoretical distribution and the observed spectrum [15], and the combinationbands are labelled a, b, etc... Features marked with a query may be third-order lines.

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rements to reveal a great deal of extra structure ;certainly the present experimental data are notgood enough to rule out the presence of the weakerfeatures, particularly at the high-frequency end ofthe spectrum. The combined dispersion curves

are extremely complex and show a large numberof almost degenerate singularities, and one can seethat the various peaks do not generally occurexactly at symmetry-point combination frequen-cies. This is because the dispersion curves areoften very flat over the zone surface and thestrong features arise from superpositions ofextrema which, although close in frequency to asymmetry point combination, are nowhere near itin q space. This result seems particularly impor-tant since it supports the contention that it is thedensity of states which determines the spectrumrather than the matrix element, and it indicatesthat selection rules only valid at symmetry pointswill not greatly affect the Raman spectrum.

It would seem fair to regard the present resultsas encouraging and an indication that the D. D.theory is not too far from the truth. Furthermore,it appears that we can use the present results as abasis for including short-range, second-neighbourinteractions between both types of ion, and it isextremely useful to be able to determine theirrelative magnitudes as one expects them to be ofcomparable size.

This refinement is certainly well worth makingI

because there is certainly no guarantee that it willproduce a better fit to the whole Raman spectrum-or indeed as good a fit as the present D. D. curve.Should it prove successful in this respect, we willthen have strong evidence to support the assump-tion that the second-neighbour forces can be deri-ved from two-body central potentials.

This is important from the viewpoint of exten-Mng our theoretical understanding of the effectivepotential function, since we would like, if possible,some assurance that any shortcomings of the pre-sent theory are not primarily due to failure totreat second-neighbour interactions (e. g. Van derlvaal’s forces) in a more refined manner. If thiscan be established, then further refinement of thedipolar approximation will be justified.

Caesium iodide. - The computed R. C. D. S.and combined dispersions curves for this crystal areshown as figure 3 (R. I.) and figure 4 (D. D.).Once again we show the positions of the observedroom-temperature peaks on the theoretical spectra,

i jt

FIG. 3. - Computed two-phonon histogram and dispersion curves for CsI ; R. I. model.Inset shows smoothed spectrum with observed features marked.

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together with the actual experimental spectrum [16](inset in fig. 4) taken at 300 OK. In this case the

experimental data are somewhat better than the

results for CsBr [15], and we have also indicatedthe positions of certain " shoulders " in the obser-ved spectrum which appear to be more clearlydefined than some of the high-frequency peaks.

It is again evident that the (D. D.) R. C. D. S.curve is in much better agreement with the experi-mental results. Moreover, the fit to those featureswhich are reliably established is remarkably good,although the high frequency end of the spectrumprobably contains a significant contribution fromthe third-order Raman spectrum. If this is so it

strengthens the case for concentrating our atten-tion on the clearly defined shoulders at this end ofthe spectrum. These are almost certainly due tosecond-order processes owing to their relativelyhigh intensities, and it is gratifying to find thattheir presence is predicted by the (D. D.) R. C. D. Sspectrum.As in the CsBr spectrum, the theoretical

TO (X) + TA ( X) combination at 1. 5 X 1013 s-1occurs at too low a frequency, and the argumentsregarding the inclusion of second-neighbour forces

Fm, 4. - Computed two-phonon histogram and dispersion curves for CsI ; D. D. model. Insets show smoothedtheoretical distribution and the observed spectrum [16]. Features marked with a query are either doubtful orpossible third-order lines.

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also apply to Csl. Once again, the positions ofthe major peaks do not tally exactly with those ofsymmetry point combinations, and for the samereasons as were given for CsBr, each peak.arisingfrom a superposition of sharp critical points lyingalong symmetry directions.

It is interesting to observe that the predictedmultiplet structure of the two strongest peaks isconfirmed experimentally. It also seems clearthat an improved spectrum taken at low tempe-ratures should reveal considerably more structureif the present theoretical results are generallycorrect.

Discussion. - In figure 5 we show the predictedR. C. D. S. spectrum and combined dispersioncurves for CsCI (D. D. theory). Unf ortunately,there are no experimental data with which to com-pare these so that we have no test of the validityof our theory for a crystal where the positive ionis very much heavier than the negative ion. Thisstatement is also true for the NaCI-structure alkalihalides, although in our earlier paper [3] we wereable to account for the specific heat data for KF ;however, the present work indicates that compa-rision of theoretical and experimental Ramanspectra would provide a more stringent test. Forth s purpose RbCI would seem to be the best mate-rial, both theoretically and experimentally.The main reason for making these extensive

tests is to understand, as far as possible, the beha-viour of the " deformation dipole moments ".These are the least well-defined parameters in thetheory. First of all there is no real evidence thatthe overlap deformation is adequately representedby point dipoles, and secondly, if one accepts thepoint-dipole approximation, then there remains theproblem of where these are located. At present weare forced to put them at the ion centres, since weare only able to derive the fields at these points.As Cowley et al. [11] have pointed out, it is possiblethat an intermediate position would be better andthis can be tested, to some extent, by assumingthat the deformation polarization experiences theeffective field at the positive ion centre and it isproposed to do this.

Equally important is the question of whether ornot the deformation is confined to the negativeion. If this is not so, then the net deformationdipole is the resultant of two opposed dipoles.In the work on KCI [14] it was found that the

dispersion curves and frequency spectra were criti-cally sensitive to any division of this kind, so

much so that one could say that overlap defor-mation of the positive ion must be negligible, atleast for the K and Na halides.

FIG. 5. - Computed two-phonon spectrumand dispersion curves for CsCI (D. D.) model.

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Obviously, this assumption needs testing formore extreme cases such as CsCI.

‘

At present these calculations remain to be car-ried out, but in figure 6 we show three sets of

single-phonon dispersion curves computed for CsI.In this case the first two are standard R. I. andD. D. results, but the third set has been derivedby including only field-induced polarization.

FIG. 6. - Single-phonon dispersion curves for CsI along 4i, A and E for :(a) Rigid ions (R. I.).(b) Polarizable deformable ions (D. D.).(c) Ions polarizable by the local field, but not by overlap deformation (P. D.).

It will be observed that the neglect of defor-mation polarization has much less effect on the

stability of the CsCI lattice than on that of theNaCI structure, since the latter is nearly or comple-tely unstable if one omits the deformation polari-zation. This result tends to support the theore-tical explanation; given elsewhere [17], of the

instability under pressure of the NaCI phase of RbI,since this was based on the behaviour of the

dipolar forces in the NaCI phase.

Discussion

M. BURSTEIN. - It is appropriate at this time,after the presentation of the theoretical papers ofCowley and of Hardy and Karo to state thatprogress in the interpretation of infrared andRaman lattice vibration spectra has been jointlydependent on the contributions of both experi-mentalists and theorists. Without theoretical (or

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experimental) neutron phonon dispersion curves,it would not be possible for the experimentalists togive an adequate interpretation of the data. The

experimental data also provide the theorists witha means of establishing more precisely the modelsthey are using. This is in fact reminiscent of thework on semiconductors, where the joint effects ofexperimentalists and theorists made it possible toelucidate in some detail the energy band structureof semiconductors. The detailed energy band cal-culations of Herman, Philips and others were

guided by information provided by experimen-talists and vice versa.

M. COWLEY. - I would like to suggest thatcalculations of joint density of states are easier ifthe extrapolation of Gilat and Dolling is used.

M. HARDY. - One finds in the combined densityof states critical points not associated with combi-nations of single phonon critical points, but withcritical points in the two phonon dispersion curves(Ref. to Loudon) and also we have found in thesingle phonon density of states of NaCI, certainsharp maxima and minima associated with secondderivative singularities.However, one must beware of using matrix ele-

ments which are overconstrained, e.g. one whichsatisfies a point selection rule, but also vanishesthroughout the zone (1).

(1) A somewhat fuller account of the present work hasnow been prepared as a Lawrence Radiation LaboratoryReport (Ref. No U C R L-14398) showing, among otherthings. detailed critical point assignments. Copies of thisreport are available on request.

REFERENCES

[1] LÖWDIN (P. W.), Some Properties of Ionic Crystals,Uppsala, 1948.

[2] LANDSHOFF (R.), Phys. Rev., 1937, 52, 246.[3] KARO (A. M.) and HARDY (J. R.), Phys. Rev., 1963,

129, 2024. See also Phil. Mag., 1960, 5, 859.[4] KELLERMANN (E. W.), Phil. Trans. Roy. Soc., London,

1940, A 238, 513.[5] TESSMAN (J. R.), KAHN (A. H.) and SHOCKLEY (W.),

Phys. Rev., 1953, 92, 890.[6] SZIGETI (B.), Proc. Roy. Soc., London, 1950, A 204, 51.[7] COCHRAN (W.), Phil. Mag., 1959, 4, 1082.[8] WOODS (A. D. B.), COCHRAN (W.) and BROCKHOUSE

(B. N.), Phys. Rev., 1960, 119, 980.[9] WOODS (A. D. B.), BROCKHOUSE (B. N.), COWLEY

(R. A.) and COCHRAN (W.), Phys. Rev., 1963, 131,1025.

[10] COWLEY (R. A.), Proc. Phys. Soc., London, 1964, 84,281.

[11] COWLEY (R. A.), COCHRAN (W.), BROCKHOUSE (B. N.)and WOODS (A. D. B.), Phys. Rev., 1963, 131, 1030.

[12] KARO (A. M.) and HARDY (J. R.), To be published.[13] WELSH (H. L.), CRAWFORD (M. F.) and STAPLE (W. J.),

Nature, London, 1949, 164, 737.[14] HARDY (J. R.), Phil. Mag., 1962, 7, 315.[15] STEKHANOV (A. I.), KOROL’KOV (A. P.) and ELIASH-

BERG (M. B.), Soviet Physics, Solid State, 1962, 4,945; see also NARAYANAN (P. S.), Proc. Ind. Academyof Sciences, 1955, A 42, 303.

[16] KRISHNAMURTHY (N.) and KRISHNAN (R. S.), Ind. J.Pure and Appl. Phys., 1963, 1, 239.

[17] HARDY (J. R.) and KARO (A. M.), Proc. Int. Conf. onLattice Dynamics, Pergamon Press, 1964, 195.