7388 D. LIU, H. M. LU, J. R. HARDY, AND F. G. ULLMAN -44 11. EXPERIMENTAL PROCEDURES Raman-scattering measurements were made with a Spectra Physics 2016 argon-ion laser. The 514.5-nm line was used throughout this experiment. The laser power incident on the sample was about 80 mW. The sample was mounted in an exchange-gas-coupled liquid-nitrogen cryostat with computer-controlled temperature to within 0.1 K, which enabled Raman measurements to be made from 78 K to room temperature. A Spex 1401 75-cmfocal-length, f /6.8, double monochromator, combined with a thermoelectrically cooled photomultiplier, was used to collect and detect the scattered light. The slit widths of the monochromator used give a resolution of about 2.0 cm-'. The photon-counting time was about 10 '.s at each frequency, and the scanning was in steps of 0.5 cmThe crystals used were all grown from aqueous solu- tion by evaporation at constant temperature or with slow cooling. Rb2S0, and Cs2S04 powders used for crystal growing were obtained from Aldrich Chemicals Company with purities of 99.9%. K 2 S 0 4powders were obtained from Fisher Scientific Company with a purity of 99%. The crystals were polished into rectangular prisms with the surfaces perpendicular to the crystal axes. Buehler microcloth and 0.05 p m A1,O1 slurry were used for polishing. The directions of the crystal axes were identified with two polarizers. The orientations of the crystals were determined by comparing their Raman spectra with known results. The samples usually had dimensions of several millimeters in each direction. 111. EXPERIMENTAL AND CALCULATION RESULTS The measured Raman frequencies of the three alkalimetal sulfates are listed in Tables 1-111, together with the calculated values. The Raman spectra of the three crystals at liquid-nitrogen temperature in the external mode region are plotted in Figs. 2-4. According to the factor group analysis,5 the number of external modes should be 7Ag, 7B,,, 5B2,, and 5B3,. All the A, external modes were identified for all three crystals, as were the B,, and B3, external modes. There should be the same number of B1, external modes as A, external modes; however, not all could be identified. All of the internal modes of the three crystals were identified. The details of the calculation method to treat molecular ionic crystals were given in Refs. 7 and 8. The basic concept is to give separate consideration to intramolecu- PTABLE I-.-Expe-ri-ment-al- a-n-d cal-culated Raman fr--equenc-ies of K2S0, (in--cm-'1. 298 K A, 78 K Calc. 298 K B1, 78 K Calc. 298 K 107 134 159 B2, 78 K 78 107 141 147 165 Calc. 69 105 121 161 199 298 K 74 96 105 142 B3, 78 K 74 94 105 144 148 Calc. 34 73 120 154 180 . .-44 RAMAN SCATTERING A N D LATTICE-DYNAMICAL . 7389 lar and intermolecular interactions. The starting point is to perform electronic calculations for the whole molecular ion SO:- in the present case, to obtain realistic electron charge distributions and effective ion charges that contain the effect of electronic covalency. These distributions are used to describe intermolecular interactions in the same spirit as the original Gordon-Kim electron-gas model. These calculations also provide a harmonic expansion of the total energy of the molecular ion, which is used to describe the intramolecular interactions. The electronic calculations were performed using the software package G A U S S I A N ~ ~ .The electron charge densities for the alkali-metal ions K + and ~ b a+re from Ref. 10, and that for CS+ are calculated with the software of Liber- +man, Cromer, and ~ a b e r . " The effective ion charges used for the alkali-metal ion M + was 1.O, and those for the oxygen and sulfur ions were obtained from the electronic calculation of the SO:- group using the Mulliken population analysis method. The effective ion charge found for the oxygen ion was -0.9700 and for the sulfur ion 1.8801. First, the crystal under consideration was statically relaxed by minimizing both the total forces acting on each ion and the stresses to get a stable equilibrium structure. Then the vibrational frequencies of the crystal were calculated for the relaxed structure. The calculated structure parameters for the three alkali-metal sulfate crystals are listed in Table IV. The calculated Raman frequencies are listed in Tables 1-111 with the experimental values. IV. DISCUSSION The three alkali-metal sulfates K2S04, Rb2S04, and Cs,S04 have the same negative ion radical and have the same crystal structure. The only differences among the three crystals result from the properties of the positive metal ions. Different alkali-metal ions have different masses and different electronic structures. These factors, especially the electronic structure, have major influences on the lattice-dynarnical properties of the crystals. Since all three alkali-metal ions have closed-shell electronic structures, the major differences in the electronic structure of the three alkali-metal ions can be well described by ionic radius, electronegativity, and the polarizability of the inner-shell electrons. The values of the masses, ionic radii, electronegativities, and the ionic polarizabilities are listed in Table V. It can be seen that the differences in ionic radii are larger than the differences in 298 K 46 69 81 115 122 128 TABLE 11. Experimental and calculated Raman frequencies of Rb2S04(in cm-I). A, 78 K 47 70 76 81 118 128 136 Calc. 48 87 95 98 136 148 172 298 K 71 92 111 B ,1 78 K 71 75 92 115 147 Calc. 80 104 115 130 141 166 176 298 K 65 74 108 123 B2, 78 K 64 74 108 130 137 Calc. 64 83 99 150 155 298 K 63 78 116 124 B3g 78 K 63 76 106 122 129 Calc. 48 75 90 135 141 D. LIU, H. M. LU, J. R. HARDY, AND F. G. ULLMAN 298 K TABLE 111. Experimental and calculated Raman frequencies of Cs2S04(in cm-'). A, 78 K Calc. 298 K B 1, 78 K Calc. Calc. Calc. 0 50 100 150 200 Wave Number (cm-I) FIG. 2. Raman spectra of K2S04at 78 K. 0 50 100 150 200 Wave Number ( c m - l ) FIG. 3. Raman spectra of Rb2S04at 78 K. ?!! RAMAN SCATTERING AND LATTICE-DYNAMICAL . . . 7391 0 50 100 150 200 Wave Number (cm-I) FIG. 4. Raman spectra of Cs2S0,at 78 K. electronegativities, which are relatively quite small. The differences in ionic polarizabilities are large too. However, we expect that ionic polarizabilities contribute to the lattice-dynamical properties only through higher-order terms, although they have a major influence on the Raman-scattering intensities. The properties change gradually from K + to CS'. However, this does not necessarily mean that the vibrational spectra of the three crystals follow the same pattern and change gradually from K2S04 to Cs2S0,, because the vibrational spectra are not simply related to the properties of the alkalimetal ions. Changing from K + to ~ b c+an result in such large changes that some features of the two vibrational spectra no longer correspond. This can be seen from Figs. 2-4. However, some similarities in the spectra and a consistent variation from K2SOdto Cs2S04in both the crystal structures and vibrational frequencies do exit. For example, correspondence between the lowest A, vibrational modes and correspondences between internal modes of the three crystals can be found. The reason for the latter is that the influence of the crystal environment on the vibrational properties of the ~ 0 gro~up is~rela-tively small. A. Crystal structure It can be seen from Table IV that the lattice constants increase from K2S04to Cs2S04. The fractional coordinates of ions in the crystal unit cells are about the same for all three sulfates. There are some minor variations, and most of them follow a particular order. The alkali-metal-ion-oxygen-ion Gordon-Kim shortrange potentials for the three sulfates are plotted in Fig. 5. In the interion distance range of interest, i.e., around 6 a.u., the difference between the potentials of K2S04and Rb2S04is smaller than that between Rb2S04and Cs2S04. This is consistent with the values of the ionic radii of the three alkali-metal ions in that the difference between the first two is smaller than that between the last two. In the calculation the static relaxation for all three crystals converges rapidly. The calculated crystal structures are very close to the experimental ones, as seen in Table IV. The calculated lattice constants are smaller than the experimental values for all three crystals, but the errors are within a few percent. The calculated ion fractional coordinates in the crystal unit cells are very close to the experimental values. These results indicate that GordonKim potentials describe well the structural properties of these ionic crystals. The starting point of the static relaxation process was the experimental crystal structure, except for Rb2S04. When the experimental crystal structure was used, the relaxed structure gave several negative frequencies. When the experimental structure of Cs2S04 Parameter TABLE IV. Lattice parameters of Pnam alkali-metal sulfates (in a.u.). -- KzS04 Ex~t. Calc. Rb2S04 E x ~. t Calc. Cs2S04 Expt. Calc. D. LIU, H. M. LU, J. R. HARDY, AND F. G . ULLMAN -44 TABLE V. Ionic properties of K', Rb+,and Csi. Property K + Rb+ Mass Electronegativity Ionic radius A Polarizability 39.10 0.9 1 1.33 0.84 85.47 0.89 1.48 1.41 132.91 0.86 1.69 2.42 was used instead, the negative frequencies disappeared. This may indicate that the potential surfaces of the alkali-metal sulfates are very flat so that the relaxation process may easily converge on saddle points on the surface. Consequently, the calculation results for Rb2S04 reported here were obtained using the experimental structure of Cs2S04as the starting point for the relaxation process. B. Raman spectra As mentioned earlier, generally no obviously similar pattern can be found for the external-mode region among the Raman spectra of the three sulfates, especially in the spectral intensities. Since all the modes are identified for the Ag, B2g, and Bjg symmetry, it is possible to make comparisons for these modes. In general, the Raman frequencies decrease from K2SO4 to Cs2S04, and the changes are much larger for the external modes than for the internal modes. The internal modes of the three sulfates correlate well, and the changes follow a simple pattern. It can also be seen that the relative order of the Raman frequency values of different symmetries may change from one crystal to another. The decrease of vibrational frequencies from K2S04to Cs2S04is expected. It can be explained by the values of the ionic radii and the masses of the alkali-metal ions. For larger ionic radius, the separations between ions are larger and the Coulomb interactions between ions are smaller. Also, since the vibrational frequency of a harmonic oscillator is proportional to the reciprocal of the square root of the mass, larger values of masses of the ions should result in smaller vibrational frequencies. The intensities of the external modes of the Raman spectra increase from K2S04 to Cs2S04. For example, the lowest Ag mode has an intensity of 600 counts for K2S04, 10000 for Rb2S04,and 22000 for Cs2S04. Apparently, this is related to the larger polarizabilities of the inner-shell electron distributions of ~ b a+nd CS' ions. It can be seen from Tables 1-111 that the calculated Ra- man frequencies, especially those of the external modes, are in general agreement with the experimental value- s- . The agreement seems better for rubidium sulfate and cesium sulfate than for potassium sulfate. The calculated frequencies are higher than the experimental ones for K2S04 for the higher external-mode frequencies. This may be because the Gordon-Kim potentials for the K + ion might be a little too hard. The calculated internal frequencies for all three crystals are higher than the experimental values. The principal cause is that zero-point motion for these high frequencies is sufficient to ensure that the effective curvature of the potential for the oxygen motion is significantly smaller than that calculated by GAUSSIAN86. Common to the Ag Raman spectra for all three alkalimetal sulfates is a feature at about 900 cm-', shown in Fig. 6. It is strongest in the spectrum of cesium sulfate. This feature has a sharp rise and fall with a broad, relatively structureless, nearly flat top, differing from usual first-order Raman peaks. Its spectral shape and temperature independence suggest that it is not due to higherorder Raman scattering either. It is too strong to be attributed to an impurity, because of the relatively high purity of the growth materials. This feature has not been reported for other sulfate crystals and seems peculiar to the 8-K2S04structure sulfates. At this time we have not identified the origin of this feature. Isotope peaks of the totally symmetric internal mode 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 Distance (o.u.) FIG. 5. Gordon-Kim potentials between alkali-metal ions and the oxygen ion. 850 900 950 1000 1050 Wave N u m b e r ( c m - I ) FIG. 6 . Isotope peaks of the A, internal modes. -44 RAMAN SCATTERING AND LATTICE-DYNAMICAL . . . 7393 be identified for all three alkali-metal sulfates, as shown in Fig. 6. This agrees with the results of Montero, Schmolz, and Haussuhl. They gave a full explanation of these peaks, attributing them to the isotopes 0 1 7 and 0'' odcupying three nonequivalent positions in the sulfate ion. The weaker peak closer to the A , peak is attributed to 017, and the stronger one is attributed to 018.. The triplet structure of the 0 1 8 peak can be identified. It can be noted that the separation between the isotope modes becomes smaller from potassium sulfate to cesium sulfate. This indicates that the interaction between the sulfate ion and the alkali-metal ions becomes smaller for cesium sulfate. V. SUMMARY The Raman spectra of K2S04, Rb,S04, and Cs2S04 were measured at both room and liquid-nitrogen temper- atures. It was found that there is no straightforward correspondence among the external modes of the Raman spectra of the three sulfate crystals. A particular feature is found at about 900 cm-' in the A , Raman spectra of all three crystals. Its origin remains unexplained. The crystal structure and the vibrational frequencies of the three sulfate crystals were calculated using a recently extended Gordon-Kim method for the short-range potentials. The calculated crystal structures agree well with the experimental ones. The calculated Raman frequencies are in general agreement with experiment. ACKNOWLEDGMENT This work was supported by the Army Research Office. 'M. Gaultier and G. Pannetier, Bull. Soc. Chim. Fr. 105 (1968). 2K. Gesi, Y. Tominaga, and H. Urabe, Ferroelect. Lett. 44, 71 (1982). 3 ~ D.ebeau, Rev. Phys. Appl. 7,49 (1972). 4P. Venkateswarlu and H. Broida, Proc. Indian Acad. Sci. A 47, 230 (1971). 5S.Montero, R. Schmolz, and S. Haussuhl, J. Raman Spectrosc. 2, 101 (1974). 6 ~Scr.occo, Phys. Status Solidi B 91, K21 (1979). 'H. M. Lu and J. R. Hardy, Phys. Rev. Lett. 64, 661 (1990). *D.Liu, H. M. Lu, F. G. Ullman, and J. R. Hardy, Phys. Rev. B 43, 6202 (1991). 9 ~J. .Frisch et al., G A U S S I A N ~(~Carnegie-Mellon Quantum Chemistry Publishing Unit, Pittsburgh, PA, 1984). 1°E. Clementi and C. Roetti, At. Data Nucl. Data Tables 12, 177 (1974). "D. A. Liberman, D. T. Cromer, and J. T. Waber, Comput. Phys. Commun. 2, 107 (1971).