Selection rules for second order infrared and raman processes. I. Caesium chloride structure and interpretation of the second order Raman spectra of CsBr and Csi S. Ganesan, E. Burstein, A.M. Karo, J.R. Hardy To cite this version: S. Ganesan, E. Burstein, A.M. Karo, J.R. Hardy. Selection rules for second order infrared and raman processes. I. Caesium chloride structure and interpretation of the second order Raman spectra of CsBr and Csi. Journal de Physique, 1965, 26 (11), pp.639644. <10.1051/jphys:019650026011063900>. <jpa00206328> HAL Id: jpa00206328 https://hal.archivesouvertes.fr/jpa00206328 Submitted on 1 Jan 1965 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
The t,e JOURNAL DE PHYSIQUE TOME 26, NOVEMBRE 1965, 639. SELECTION RULES FOR SECOND ORDER INFRARED AND RAMAN PROCESSES. I. CAESIUM CHLORIDE STRUCTURE AND INTERPRETATION OF THE SECOND ORDER RAMAN SPECTRA OF CsBr AND CsI. By S. GANESAN (1), E. BURSTEIN (2), Department of Physics and Laboratory for Research on the Structure of Matter, University of Pennsylvania, U. S. A. and A. M. KARO, Lawrence Radiation Lab., Chemistry Division, Livermore, California, U. S. A. and J. R. HARDY, Atomic Energy Research Establishment, Harwell, Berkshire, England. Résumé. 2014 On donne les règles de sélection relatives à deux phonons pour les processus Raman et infrarouges dans les cristaux possédant la structure du chlorure de césium. Les règles sont tout à fait analogues à celles relatives à la structure du sel gemme, favorables aux processus de diffusion Raman du second ordre, et défavorables pour ceux du même ordre concernant l infrarouge. On montre, à l aide des courbes de dispersion théoriques, que les spectres Raman du second ordre de CsBr et CsI peuvent être interprétés par des paires de phonons aux points de symétrie de la zone de Brillouin. On trouve que chaque maximum observé est dû à plusieurs paires de phonons aux différents points de symétrie. Les multiples contributions aux maximums observés, ainsi que la grande polarisabilité des ions césium et halogènes, expliquent l intensité relativement forte du spectre. La limite du spectre d absorption infrarouge de CsBr s explique par l existence de paires de phonons aux points de symétrie A et 03A3. Abstract. 2014 Two phonon selection rules for Raman and infrared processes are given for crystals having the caesium chloride structure. The selection rules are quite similar to that of the rocksalt structure, favourable for second order Raman scattering processes and quite severe for the second order infrared processes. With the help of the theoretical dispersion curves it is shown that the second order Raman spectra of CsBr and CsI can be accounted for in terms of phonon pairs at symmetry points in the s. c. Brillouin zone. Each of the observed peaks is found to arise from several phonon pairs at the different symmetry points. The many contributions to the observed peaks together with the high polarizability of the caesium and halogen ions account for the relatively high intensity of the spectrum. The limited structure in the infrared absorption spectrum of CsBr can be accounted for in terms of phonon pairs at A and 03A3 symmetry points. 1. Introduction. second order (two phonon) Raman spectra of the alkali halides exhibit a wealth of structure. The corresponding second order infrared absorption spectra exhibits relatively little structure. In the case of the alkali halides having the rocksalt structure this striking difference in the character of the two types of second order spectra is primarily due to the fact that the selection rules for second order Raman processes are relatively lenient whereas the selection rules for second order infrared processes are fairly restricted. Using the selection rules for second order Raman processes in the rocksalt structure, Burstein, Johnson, and Loudon [1] were able to interpret the observed structure in the Raman spectrum of Nad, KBr, and NaI in terms of phonon pairs at X, L, and A symmetry points of the (f. c. c.) Brillouin zone corresponding to (1) Supported by the Advanced Research Projects Agency. (s) Supported in part by the U. S. Army Research Office, Durham. maxima in the combined density of states curves, In the case of NaCI the theoretical curves of Hardy and Karo [2] were used as a guide in making the assignments. The analysis indicated that phonons at (or near) the X symmetry point played a major role in the scattering, and it was, in fact, possible to establish a consistent set of frequencies for the phonon at the X point from the Raman data. In the case of KBr and NaI, the observed structure in the Raman spectra could be directly correlated with the neutron scattering phonon dispersion data of Woods, Brockhouse, Cowley, and Cochran [3] for these two materials. The second order Raman spectra of CsBr and CsI which have the caesium chloride structure also exhibit considerable structure and an analysis of the selection rules for the second order infrared for the caesium absorption and Raman spectra chloride structure by Ganesan [4] shows that the situation is similar to that for the rocksalt structure, i.e., the selection rules for the second order Raman spectrum are quite favourable, where as Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011063900
The Brillouin 640 those for the second order infrared absorption are relatively restricted. In the present paper we show that the peaks in the Raman spectra of CsBr and CsI can be accounted for by combinations of phonon frequencies at almost all the symmetry points in the zone. The analysis of the structure in the Raman spectra was carried out using the theoretical phonon dispersion curves of Karo and Hardy [5] as a guide for the assignment of phonon frequencies. 2. Symmetry of the phonons and selection rules. caesium chloride structure belongs to the space group 01. It is a set of two interpenetrating simple cubic lattices, with two atoms per unit primitive cell. The Brillouin zone is again simple cubic and it is shown in figure 1 with all the symmetry points indicated. We determine structure indicates that both the optical and acoustical phonons have even parity. This result differs from that given by Burstein, Johnson, and Loudon [1] who assign odd parity for the optical phonons and even parity for the acoustical phonons. The selection rules for the two phonon Raman and infrared processes are worked out in the usual way [7], [8]. These are given in Tables 2 and 3. We find that all combinations are allowed for second order Raman processes. For second order infrared processes overtones are strictly forbidden because of the center of symmetry [9], and no combinations are allowed at the r, X, M, and R points. These points exhibit the complementary nature of the selection rules for the two types of processes. TABLE 1 SYMMETRY POINTS AND PHONON SPECIES IN CAESIUM CHLORIDE STRUCTURE at FIG. 1. zone for caesium chloride structures with the symmetry points. the symmetries of the various phonons at these points in the zone, by placing displacement vectors on each atom and finding their transformation properties. At r, the centre of the zone, both the acoustic and optical phonons transform like r 15. We use here the notation of Bouckaert, Smoluchowski, and Wigner [6]. Along the symmetry directions [100], [111], and [110] the assignment of phonon symmetries making use of the compatibility relations is straightforward. At the zone boundaries, however, there are two assignments possible for each phonon, each assignment having a definite parity. Since each atom is a centre of symmetry, we study the displacement, x, of similar particles in adjacent cells to establish the parity of the phonons. The phase of the motion varies from cell to cell as q. x t where q is the wave ( 1) vector, 1 is the cell index and K is the particle index. We find that at X and R the phonons have even parity and at M the phonons have odd parity. This is further checked by looking at the compatibility relations between the various assignments along various directions. The phonon symmetries are given in Table 1. A similar analysis of the parity of the phonons at the L point for the NaCI Parity well defined only at r, X, M and R. TABLE 2 TWO PHONON INFRARED PROCESSES IN CAESIUM CHLORIDE TYPE STRUCTURES (3) These exhibit complementarity in the selection rules between Raman and infrared processes.
Frequency Frequency 641 TABLE 3 TWO PHONON RAMAN ACTIVE PROCESSES IN CAESIUM CHLORIDE TYPE STRUCTURES,, (4) These exhibit complementarity in the selection rules between Raman and infrared processes. The selection rules also give some information about the nature of polarization of the scattered radiation. As pointed out by Kleinman [10], the of the scattered radiation concept of polarization is not meaningful unless the direction of incidence and direction of observation are specified along with the polarization of the incident light. In the absence of such information for the second order we have not Raman spectra of Cs Br and Cs I, given the reduction of the Kronecker products for the phonon combinations. In the case of simple cubic lattices, Rosenstock [11] has shown that most of the critical points lie at the corners, along the edges and on the faces of the cube. Phillips [12], in his critical point analysis, shows that for this case r, R, M, and X must be ordinary critical points in all three modes. r and R are threefold degenerate ; M and X are twofold degenerate and lie in planes of symmetry. Further his analysis yields those points for which some components of the gradient vanish and these occur at the faces of the cube. The unique feature about the caesium chloride structure is that, except for the 1Y, A, and Xl points, all the other points occur on the same face of the zone. Analyzing the critical points required by symmetry one can get a symmetry set and from this using Morsels topological thorem (the existence of some critical points necessitates the existence of others) one can get a minimal set which is the smallest set satisfying these relations and containing the symmetry set. Phillips analysis gives the following results. For short range forces the symmetry set contains all the critical points. Introduction of second neighbour forces and hence long range forces, produces more critical points, however, the minimal set still contains all the additional critical points introduced by these forces. From his Tables we see that critical points appear at r, R, and X, and M for first neighbour forces and additional critical points appear at A, S, and E, etc..., when second neighbour forces are included. This can be seen in the Karo, Hardy phonon dispersion curves FIG. 2a. Frequency versus wave vector curves along [100] for CsBr and CsI. (After A. M. Karo and J. R. Hardy, to be published.) FIG. 2b. versus wave vector curves along [111] for CsBr and CsI. (After A. M. Karo and J. R. Hardy, to be published.) FIG. 2c. versus wave vector curves along [110] for CsBr and CsI. (After A. M. Karo and J. R. Hardy, to be published.) for CsBr and CsI given in figures 2(a, b, c). Accordingly for a proper analysis of the observed peaks
Microphotometer The 642 in the second order spectra, points in the zone. we consider all the TABLE 4 3. Interpretation of the second order spectra. 3.1. CAESIUM BROMIDE. first measurement of the second order Raman spectrum was carried out by Narayanan [13] who reported 5 peaks. A later measurement by Stekhanov and Koral kov [14] showed 9 peaks. The peaks in the CsBr spectrum (and also in the CsI spectrum) are relatively intense in contrast to the other alkali halides. Stekhanov and Koral kov attribute this to the high polarizability of both the caesium and halogen ions. Another feature of interest is the fact that the spectrum of CsBr (and also of CsI) is completely depolarized which is also in contrast to the spectra of NaCI type crystals which are appreciably polarized. The spectrum is shown in figure 3. SECOND ORDER RAMAN SPECTRUM IN CsBr FIG. 3. record of the Raman spectrum of CsBr. (A. I. Stekhanov and A. P. Korol kov [14].) Since the selection rules for the two phonon Raman scattering are very lenient, as in the case of NaCI type lattices, many phonon pairs are active in the second order spectra. From the theoretical dispersion curves of Karo and Hardy [5] shown in figure 2, we see that the spread of the spectrum is small and that critical points occur also at A and E. As critical points also occur at S, E, and T (not shown in figures 2a, b, and c) we also consider phonon combinations at these points. The branches at T, S and Z are called arbitrarily LO, y0i, TOa, LAi, TA1 and T A z in the order of decreasing energy.
Infrared Microphotometer 643 Taking the corresponding frequencies from the dispersion curves, we show in Table 4 that we can The inte that many combinations of An be of some reasonably account for all the lines. resting feature is phonon pairs contribute to a given peak. experimental polarization study may help but again due to the possibility of so many combinations, it may not resolve the ambiguity in the assignment. The infrared spectrum of CsBr was measured by Geick [14] in 1961, and is given in figure 4. It shows considerably less structure than the FIG. 5. record of the Raman spectrum of CsI. (R. S. Krishnan and N. Krishnamurthy [16].) FIG. 4. absorption spectrum of CsBr. (After R. Geick [15].) Raman spectrum as is to be expected from the rigour of the selection rules. We find that we can account for all the peaks in the spectrum in terms of phonon pairs at the A and 2 points. The assignments are given in Table 5. TABLE 5 INFRARED SPECTRUM IN CsBr whole spectrum is much less than in the case for CsBr. In common with CsBr the lines are found to be sharp and intense. From the dispersion curves for CsI shown in figure 2, we see that the spread of frequencies is much less than that for CsBr. The small spread in frequency is due to the nearly equal masses of the caesium and iodine atoms. It is most striking at the R point where the optical and the acoustical phonon branches are nearly degenerate. As in the case of CsBr we can account for the peaks in this spectra in terms of phonon pairs arising from the various symmetry points in the Brillouin zone. The phonon pair assignment for the observed peaks are given in Table 6. TABLE 6 SECOND ORDER RAMAN SPECTRUM IN CSI (15) Quite broad probably extra structure. 3.2. CAESIUM IODIDE. The second order Raman spectrum of CsI was measured recently by Krishnan and Krishnamurthy [15] (fig. 5). They observed 12 lines very closely spaced, extending from 19 cm1 to 181 em1. The extent of the
The 644 Conclusion. TABLE 6 (continued) selection rules for the CsCI structures are quite similar to those for the NaCI structures and it is possible with the help of the theoretical dispersion curves of Karo and Hardy to account for the observed structure in the Raman spectra of CsBr and CsI in terms of phonon pairs at various symmetry points in the s. c. Brillouin zone (s). Each of the observed peaks is found to arise from several phonon pairs at the symmetry points. This may be attributed, on the one hand to the high symmetry of the CsC] structure and the near degeneracy of the phonon (1) we can also expect phonon pair contribution to both Raman and infrared spectra from branches having equal or opposi le slopes particularly A and Z points which are also associated with peaks in the combined density of states. These are not included in our analysis, because of the uncertainty in the wave vector at which the slopes are equal or opposite and the corresponding large tainty in the phonon pair suni uncer or difference frequency. branches at the zone boundary, and on the other hand to the fact that all the symmetry points are located on the same face of the zone and are possible critical points. The small frequency spread of the second order Raman spectra of CsBr and CsI and the many contributions to each of the observed peaks, together with the higher deformability of the caesium and halogen ions account for the relatively high intensity of the observed Raman spectra. In view of the many phonon pair assignments for each of the observed peaks, it is not possible to use the Raman spectra to derive phonon frequencies at any of the symmetry points (7). Since the selection rules allow all phonon pair combinations and overtones at the symmetry points, one can expect the combined density of states derived from the theoretical dispersion curves to provide a reasonably close representation of the observed spectra (8). Among other things this approach provides the possibility of obtaining a verification of the theoretical phonon dispersion curves and under optimum circumstances it may provide information about the second order matrix elements. The relatively small number of second order absorption bands observed in the room temperature infrared spectra of CsBr is a consequence of Our analysis the very restricted selection rules. indicates that the major contribution to the bands comes from phonon pairs at the A and Xl symmetry points. The absorption bands are quite broad and may actually be made up of a superposition of several bands. Absorption measurements at low temperature may possibly yield additional structure. If so, this would allow us to make more definite phonon pair assignments for the observed peaks. ( ) An examination of the phonon pair symmetry representation indicates that polarization measurements will only have limited applicability in establishing unique phonon pair assignments for the observed peaks. (8) A detailed treatment of the combined density of states and the Raman spectra of alkali halides is given by Hardy and Karo in another paper published in this proceedings (J. Physique, 1965, 26, 000)., REFERENCES [1] BURSTEIN (E.), JOHNSON (F. A.) and LOUDON (R.), Phys. Rev., 1965, 139, A 1240. [2] KARO (A. M.) and HARDY (J. R.), Phys., Rev., 1963, 129, 2024. [3] WOODS (A. D. B.), BROCKHOUSE (B. N.), COWLEY (R. A.) and COCHRAN (W.), Phys. Rev., 1963, 131, 1025. [4] GANESAN (S.), Bull. Amer. Phys. Soc., 1965, 10, 389. [5] KARO (A. M.) and HARDY (J. R.), To be published. [6] BOUCKAERT (L. P.), SMOLUCHOWSKI (R.) and WIGNER (E.), Phys. Rev.. 1936, 50, 58. [7] BIRMAN (J L.), Phys. Rev., 1963, 131, 1489. [8] LOUDON (R.) and JOHNSON (F. A.), Proc. Roy. Soc., London, 1964, A 281, 274. [9] LOUDON (R.), Phys. Rev., 1965, 137, 1784. [10] KLEINMAN (L.), Solid State Comm., 1965, 3, 41. [11] ROSENSTOCK (H. B.), Phys. Rev., 1955, 97, 290. [12] PHILLIPS (J. C.), Phys. Rev., 1956, 104, 1263. [13] NARAYANAN (P. S.), Proc. Ind. Acad. Sc., 1955, A 42, 303. [14] STEKHANOV (A. I.) and KOROL KOV (A. P.), Soviet Phys., Solid State, 1963, 4, 2311. [15] GEICK (R.), Z. Physik, 1961, 163, 499. [16] KRISHNAN (R. S.) and KRISHNAMURTHY (N.), Ind. Jour. Pure and Appl. Phys., 1963, 1, 239.