SOME time ago we presented the results of a systematic investigation into the lattice dynamics of most of the NaC1-structure alkali ha1ides.l However, certain salts were omitted from this initial study, mainly because of the absence of the required input data, and among these was CsF. This was unfortunate since this compound is, in many respects, an extreme case, with the positive ion being both very much heavier and considerably more polarizable than the negative ion. Since that time, sufficient experimental data2 have become available for us to be able to treat this crystal, and we wish to report in this paper the results of our calculations.
Also we wish to report the results of an associated calculation of the second-order Raman spectrum of this crystal. We believe that these calculations will be of direct and immediate interest to experimental workers studying phonon-assisted electronic transitions at defect centers in CsF, defect vibrations in CsF, and the secondorder Raman spectrum of CsF. In the first place, apart from a much earlier calculation by one of us (A. M. K.): there is no other theoretical work on the lattice dynamics of CsF. In the second place, our present work has revealed that the extreme properties of CsF are reflected both in the calculated frequency spectrum and the calculated second-order Raman spectrum. Both of these have a very striking appearance which should be well worth investigation by any, or all, of the experimental techniaues mentioned above.
The calculations presented here were carried out within the framework of the deformation dipole (DD) model which is fully described in an earlier paper.4 However, we have recently made an investigation into the Debye-Waller factors of all the cesium halidesJ6 including CsF, and we have found it necessary, particularly for CsF, to include short-range interactions between second-neighbor negative ions to obtain good agreement with experiment (i.e., within the quoted experimental error). We have therefore included these in the present calculations. The two disposable force constants involved are fitted to the shear modulus C44 and the infrared dispersion
freauencv wo. (We have assumed that all the interionic potentials central are and that Our model therefore automatically satisfies the Cauchy relation C12=C44, a reasonable approximation for CsF.) The input parameters required for this modifications of the DD model are shown in Table I. The calculations were carried out for a uniform sample of 64 000 wave vectors q in the first Brillouin zone and the six normal mode eigenfrequencies determined for each point. Then, in order to obtain a smooth distribution function. instead of the conventional histogram method we represented each sample frequency by a Gaussian entered on the exact frequency.
The resultant distribution, shown in Fig. 1, is a superposition of all these Gaussians with their common halfwidth chosen to produce the maximum smoothing consistent with the preservation of genuine structure. Dispersion curves were also calculated along the principal symmetry rections, and the results are shown in Fig. 1. The principal Van Hove ~ingularitiesa,s~s ociated with symmetry points and axes, are labeled in accordance with the notation of Bouckaert et al.,' subscripts indicating the order of increasing frequency. As can be seen. the sDectrum has several extreme features, e.g., the two very sharp acoustic-mode peaks, the large gap between optic and acoustic branches, and the striking, alm~s t~rectan~ulpaera,k in the optic branch. These features, if genuine, should certainly manifest themselves in both the ~honon-assistedo ~t i c atlr ansition spectra of defects in CsF and in impurity-induced infrared lattice absorpti Recently we have also completed a calculation of the second-order Raman spectrum of NaF,g and have obtained very good agreement with experiment. We therefore carried out a similar calculation for CsF, the results of which are shown in Fig. 2. It is assumed that the temperature of easurement is 300°K and that we are dealing with radiation incident along a (100) crystal axis and scattered through 90" along a (010) axis. Under these circumstances, our assumptions regarding theRaman polarizability tensor predict that the scattered radiation will be completely polarized and therefore the calculated curves refer only to this component. Two separate choices of the polarizability tensor elements, which we believe represent two opposite extremes in the behavior of these quantities, were taken, and the predicted spectra for both variations (1 and 2) are shown in Fig. 2. (For the actual construction of the polarizability ensors, see Ref. 9.) gain we have used a Gaussian smoothing technique, and for clarity we have shown the difference spectra i.e., that part arising from processes in which one honon is created and another destroyed) separately rom the combination spectra, which involve either ouble-phonon creation or destruction. Both types of pectra have a very unusual appearance, since instead of aving relatively smooth structure they consist of iscrete, very sharp lines. In addition, unlike the results or NaF where the combination spectra for variations 1 nd 2 were very similar, the present data show very ifferent combination as well as difference spectra. Because off this, it seems to us that it would be very worthwhile to have an experimental test of our predicted spectra using a modern laser Raman spectrograph. If the spectrum is indeed strongly polarized for the (100) scattering geometry, then it should easily be possible to discriminate between the two theoretical variations. It should be stressed, however, that our results are contingent on the validity of our polarizability tensor model which assumes only first-neighbor interactions? This assumption, while apparently valid for NaF, may well be less trustworthy for CsF, where the dominant polarizability is that of the positive ion.
However, we would emphasize that reliable data on the relative influence of first- and more-distant-neighbor displacements on the Raman polarizability of a given ion in a given crystal, as one proceeds through a sequence of crystals, are of primary importance in understanding the second-order Raman effect. The only way such information can be obtained is by predicting the Raman spectra on the basis of a series of assumptions and testing prediction against experiment. Thus, our primary object in presenting these calculations of the Raman spectrum of CsF is to demonstrate the logical consequences of making the same assumptions about the nature of the crystal Raman polarizability tensor for CsF at the heavy end of the fluoride sequence as for NaF at the light end.
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