Incommensurate instability and lattice dynamics of potassium selenate within a semiempirical rigid-ion model

PHYSICAL REVIE% BVOLUME 42, NUMBER 13 1NOVEMBER 1990 Incommensurate instability and lattice dynamics of potassium selenate within asemiempirical rigid-ion model I.Etxebarria and J.M. Perez-Mato Departamento de Fssica de la Materia Condensada, Facultad de Ciencias, Universidad del Pais Vasco, Apartado Postal 644, 48080 Bilbao, Spai n A. Criado Departamento de Fi'sica de la Materia Condensada, Facultad de Ciencias Fi'sicas, Universidad de Sevilla, Apartado Postal 1065, 41080Sevilla, Spain (Received 23 May 1990) The lattice dynamics of potassium selenate is analyzed using arigid-ion model with the selenate groups reduced to rigid bodies. The interatomic forces have been adjusted only using static structural data. The number of adjustable parameters varies from two to five. Such asimple model is already sufficient to reproduce semiquantitatively the phonon dynamics of the real system. In particular, the model exhibits the lattice instability leading to the existence of an incommensurate phase. The characteristics of the resulting soft mode agree with those observed experimentally. The calculated eigenvector, in excellent agreement with the experimental one, is rather unsensitive to the details of the interactions. This explains the strong similarities of the incommensurate modulations in most A2BX4 compounds. On the other hand, the form of the soft-phonon branch strongly depends on the force model. It is sufficient to fit the model to the static structure observed at 145 Kinstead of the one at room temperature, to provoke aconspicuous softening of the branch. The branch minimum is specially sensitive to some potassium-oxygen interactions. The relative size of the cations plays an essential role in the origin of the incommensurate instability. For comparison the results of asimilar analysis for Cs2Se04 are presented. In this case, the unstable or "soft" character of the lowest X2 branch disappears. I. INTRODUCTION Normal phase (Pnam ):INC phase X2, k=a(T)a =lockin phase k=(m /n )a* This general scheme is reproduced in all compounds, but some particular features can vary considerably. For instance, T, is 130 Kin KzSe04, but 553 Kin KzZnC14. The temperature range of the INC phase changes from 15 Kin Cs2CdBr4 to 155 Kin R12ZnBr4. The compounds of Zn do not exhibit aproper soft-phonon Aconsiderable number of materials of the type A2BX4 (K2Se04, Rb2ZnC14, Rb2ZnBr4, Rb2CoC14, Rb2Co8r4, K2ZnC14, etc.)exhibit acommon sequence of phase transitions as the temperature is varied. At acertain temperature range they are isomorphous to P-K2SO4, having space group Pnam and apseudohexagonal axis along the adirection. Below some temperature T;, this structure is unstable and the compounds transform into an incommensurately modulated (INC) phase with its wave vector along the aaxis (in most cases close to the value —, 'a'). The symmetry of the primary distorting mode (order parameter) is X2. At lower temperatures, asecond phase transition into acommensurate phase takes place and the wave vector locks into acommensurate value (for a comprehensive review see Refs. 1and 2). Thus, ascheme of the cornrnon transition sequence is branch, while in K2Se04 the existence of aXz soft mode was clearly observed. 'In the Zn compounds, the tetrahedral groups BX4 are disordered in the normal phase with two equivalent positions related by amirror plane. In contrast, x-ray structure determinations of the normal phase of K2Se04 have been successful using a nondisordered model. In this last case, however, large thermal ellipsoids indicate strong librations of the ionic groups and recently some spectroscopic anomalies in this compound have been interpreted in terms of adisordered model. ' Despite these differences, it is reasonable to assume that the normal-INC instability observed in all these compounds has essentially acommon origin. Acomparison of the eigenvectors describing the modulation or distorting mode in their INC phase supports this viewpoint. This eigenvector is shown in Table Ifor several A2BX4 materials. The atomic modulations corresponding to atoms belonging to the BX4 tetrahedral groups have been adjusted to arigid-body model and in the table only the fitted Fourier amplitudes of the syrnrnetry-allowed rotations and displacements of these groups are listed. A strong similarity among the different distortions is clearly observed (see also Ref. 8for acomparison considering all atoms independently). Only in the case of KzSe04, a significative difference in the relative weight of the amplitudes of the BX4 rotations along the aand baxes with respect to the other compounds is detected. The INC modulation is therefore not very sensitive to the details of 8482 1990 The American Physical Society 42 INCOMMENSURATE INSTABILITY AND LA j. IICE DYNAMICS. ..8483 TABLE I. Polarization vector of the modes describing the INC modulation of some A&BX4 compounds. Findicates the degree of fitting to rigid-body displacements. For each ion or ionic group in the asymmetric unit, the amplitude of the displacements (relative units X10 )and/or rotations (deg) together with the corresponding phase (/2m) are given. A(a) A(P) BX4 T. T. T. R„ Ry K2Se04 69/0. 102 92/0.321 58/0.825 2.27/0.036 3.68/0.643 92% RbqZnC14 69/0. 101 80/0.231 64/0. 771 5.30/0.024 2.55/0.654 89% Rb2ZnBr4 143/0. 110 147/0.238 118/0.778 9.66/0.025 4.27/0.654 92% KqZnC14 377/0.097 430/0.287 262/0. 828 14.49/0.013 9.71/0.626 73% Cs~CdBr4 83/0.067 84/0.078 43/0. 695 6.67/0.016 1.13/0.695 84% the interatomic interactions and seems to originate essentially in some of the common structural features of all these materials. This type of conclusion is consistent with some recent arguments, which introduce the concept of "latent symmetry" (a certain imperfect symmetry of the structure) as the ultimate cause of the appearance ofthe INC modulation. However, aclear understanding of the microscopic mechanism of the INC instability and its connection with the static structural properties of these structures is still missing. Apart from the phenomenological treatments in the frame of Landau theory, only afew works have attempted to explain the INC instability from amicroscopic viewpoint. Iizumi et al. suggest that the instability in K2Se04 is the result of acompetition between the first-, second-, and third-neighbor interlayer effective interactions in aone-dimensional local-mode layer model along the xdirection, with the effective cell corresponding to half the aparameter. This model implies the consideration that the local mode has stronger effective interac- 0 tions with atoms at distances of about 10 Athan with atoms at distances smaller than 3.8A. This strong assumption has been no obstacle for the model to be widely accepted and the argument has been reproduced in many subsequent works. More sophisticated models in the frame of mean-field approximation reduce the interactions to first- and second-neighbor layers. 'Other studies propose the anharmonic polarizability of the Xiona as the origin of the structural instability (at least in the case of KzSe04)."Finally, some recently reported investigations on the stability of these materials, including the introduction of the mentioned concept of latent symmetry, have been achieved through the analysis of the harmonic lattice dynamics of rigid-ion models with interatomic central forces. ' ' These models seem to be sufficient to reproduce the essential features of the energetic and vibrational properties of these materials. In particular, the lattice dynamics of potassium selenate has been studied from atheoretical point of view by Haque and Hardy. 'Using arigid-ion model with aprioristic values for the ionic charges, these authors derived the theoretical phonon branches along the Xline. The necessary parameters for the repulsive pairwise potentials were determined analytically using the set of equations resulting from the equilibrium condition of the structure observed at room temperature. For the pairs oxygen-oxygen and oxygen-selenium no functional form was assumed and only the necessary first and second derivatives at the distances corresponding to the actual bonds were determined. The resulting positive sign for the first derivative of the repulsive force between some of the atoms pairs was, however, incoherent with its nominal repulsive character. For the pairs oxygen-potassium aBorn-Mayer potential was considered with some ad hoc modifications for its second derivatives. The internal degrees of freedom of the selenate groups were included in the analysis. The obtained higher-frequency external modes deviated systematically to higher values than those observed experimentally. The model, however, was able to reproduce the existence of the soft-phonon branch by varying the second derivative of the repulsive force for a specific pair of oxygen atoms, and the calculated softmode eigenvector agreed quantitatively with the experimental data. ''The other simulations of the lattice dynamics of A2BX4 compounds have been performed using "ab initio" potentials. ' In the present paper anew analysis of the lattice dynamics of K2Se04 and its relation with the lattice INC instability is reported using asemiempirical rigid-ion model. We center our study on the compound K2Se04, as it is one of the best known from the experimental point of view, but the main aim is to clarify the lattice-dynamical origin of the structural instability leading to the INC phase in the whole A2BX4 family. One of the questions to elucidate refers to the abovementioned concept of latent symmetry. This concept has been introduced in lattice-dynamics studies that simultaneously stress the necessity of ab initio specific interatornic potential for reproducing the INC instability. However, if the INC instability can be traced back to rather general pseudosymmetric features of the structure of the normal phase, it would be more reasonable to expect that the lattice dynamics of any model of interatomic forces compatible with the observed structure should exhibit *'pathological" dynamical characteristics leading to the mentioned structural instability and the observed INC modulation. To check this point, the rigid-ion model used in the present study only considers semiempirical interatomic potentials. In addition, we expect that the use of afitted semiempirical model can lead to more rea1- istic results than first-principles models, as the first ones can absorb at least partially crystal-field effects. In the same line of simplifying the model without losing the essential structural and dynamical features of the 8484 I.ETXEBARRIA, J.M. PEREZ-NATO, AND A. CRIADO 42 material, asecond important characteristic of the present analysis is that the BX4 groups are considered rigid units. This contrasts with previous vibrational studies that have included the interactions between the atoms inside the tetrahedral groups. ''The internal interaction B-X has been even considered of some relevance in the transition mechanism in the case of Rb2ZnC14. As can be seen in Table I, the tetrahedral groups BX~ are approximately rigid in the normal-incommensurate transition in all the compounds. Therefore, their internal degrees of freedom are expected to be essentially inert and irrelevant in the transition. Only the external vibrations should play a major role in the lattice-dynamical model of the INC instability. By this means, the number of parameters of the interatomic force model to be fitted is reduced to two or five depending on the model used (see next section), which compares favorably with the 14 parameters for the static equilibrium and more than 20 additional parameters fitted using experimental vibrational data, which are considered in Ref. 12. The model parameters for the interatomic forces to be used in the dynamical analysis were optimized using the condition that the resulting atomic equilibrium configuration should be agood approximation of the observed structure (see Table II). As atomic equilibrium configuration, we mean the atomic positions and the unit-cell parameters, which minimize the potential energy with respect to any structural distortion compatible with the fixed orthorhombic Pnam symmetry. Consequently, for this configuration the derivatives of the total potential energy with respect to any structural variable, which is homogeneous in each cell and compatible with the Pnam symmetry, is zero. This requirement is necessary but obviously not sufficient to ensure that the structure is stable with respect to modulated distortions or symmetrybreaking modes in general. It is, however, enough to determine the values of the energetic parameters describing the model of interatomic forces. Once the model is chosen under the above conditions, the external phonon dispersion branches in the harmonic approximation have been calculated for the different models. Eventual instabilities of the structure with respect to symmetry-breaking modes will be accompanied by the existence of imaginary frequencies in one or more dispersion branches. We analyze thoroughly the eventual presence of these unstable modes as they can be considered the main factor leading to athermal structural instability of the Pnam structure. The eigenvector of the most unstable lattice mode can then be compared with the observed soft mode in the real system. By varying specific interatomic interactions we also investigate the essential factors that are responsible for the INC instability and the degree of sensitivity of the eigenvector of the modulation to the interaction model. For comparison, we present the results of asimilar analysis for Cs2Se04, which demonstrate the crucial role played by the effective volume of the Acations on the origin of the INC instability. Obviously, the present analysis, although microscopic, adopts amechanical picture of the material. Thermal effects are only considered from aphenomenological point of view, in the sense that we assume that thermal renormalization of the mode frequencies can be strong enough to stabilize at high enough temperatures the set of modes that are unstable in amechanical picture. The model of adjustable interatomic forces is partially adapted to the thermal effects, since it is the structure at a finite temperature that is taken as the observed structure in the fitting process. In this sense, it can be considered a set of thermal effective forces. II. INTERATOMIC FORCES We expect that the presence of the INC instability is essentially independent of the details of the interatomic forces. Therefore, the model is strongly simplified. In particular, only Coulomb rigid-ion and Born-Mayer repulsive forces are introduced. In contrast with previous studies, the Se04 groups are taken as rigid units in the room-temperature configuration. The interatomic potentials V(r, )are taken as V, (r,i)+V„, (r, ), with V, (r,j}=Q, Q, ir,i and V„,~(r,i)=A;, exp( B,Jrj ).— TABLE II. Experimental structure of potassium selenate at room temperature and at 145 K, and cesium selenate at room temper- 0 ature. The lattice constants are given in A, and the atomic positions in relative units (X10 ). The data are taken from Refs. 5, 17, and 23, respectively. K2Se04 (293 K) 7.661 10.466 6.003 K,Se04 (145 K) 7.597 10.375 5.975 Cs,Se04 (293 K) 8.377 11.276 6.434 Se K(a) K(P) O(1) 0(2) 0(3) 2242 1705 — 57 2931 3024 126 4200 843 7095 3471 5644 4251 2500 2500 2500 271 2500 2500 2228 1670 — 35 2918 3083 102 4195 823 7076 3440 5638 4274 2500 2500 2500 274 2500 2500 Se Cs(n) Cs(P) 0(1) 0(2) 0(3) 2348 1768 — 105 3032 3022 392 4188 900 7061 3525 5546 4178 2500 2500 2500 418 2500 2500 42 INCOMMENSURATE INSTABILITY AND LA IIICE DYNAMICS. . . 8485 As the Se04 tetrahedra are considered rigid, their internal interatomic interactions are not included. The repulsive forces between the selenium atoms and atoms external to the tetrahedra can also be reasonably neglected. Obviously the oxygen-oxygen interaction is considered only between atoms belonging to different Se04 groups. The repulsive interaction K-K was included in preliminary calculations, but its influence on the results was very weak, as they involve comparatively long interatomic distances. Consequently, energetic contributions of the K-K repulsive forces were neglected in further calculations. The potassium ions were taken to be fully ionized, QK =+le. Thus, the charge of the oxygen ion was related to the one of the selenium ions by the neutrality condition Qs, +4Qo =— 2e. As repulsive parameters Ao o and 8~~, we took those proposed in Ref. 15 (Aoo=55X10 kcal/mol, 8o=3.96 A'). The value of Bzowas fixed at 3.76 A, obtained from the values of BzK and Bo& and the approximate empirical rule 8;J.=(8;;+B~J)/2. In this relation, the value for Bx& 0 was taken to be 3.56 A'. This value results from considering the values of BK „and B„„given in Ref. 16 and the empirical rule mentioned above. It is well known that the parameters 8; are strongly correlated with the A; values, so that similar potentials can be described by different sets of both parameters. Therefore the values given apriori to the 8,¶meters are not adeterminant restriction in the present study. The remaining energetic parameters (Qs„A~o)were adjusted, as explained in the preceding section, using the condition that the resulting atomic equilibrium configuration should be agood approximation of the observed structure either at room temperature, or at 145 K just above T, (Ref. 17) (see Table II). We used for this process aprogram based on aSIMPLEX algorithm' that searched the energetic parameters minimizing the structural differences between the corresponding "equilibrium configuration" and the observed structure. The "equilibrium configuration" was calculated at each step of the SIMPLEX process by "relaxing" aconvenient initial structure (in most cases the experimental one at room temperature). The program wMIN working in mode 1 (Newton-Raphson energy minimization mode)' was used for this purpose. The numerical factor that was in fact minimized and quantitatively measured the difference between the observed and relaxed structure is defined by R=(I/X) g[(b,a, /a, )100] i=1,3 +gQ[bx, ;a, (10)]2 (3) where ha, and hx, are, respectively, the deviations of the lattice constants and of the atomic fractional positions from the observed values, while a,.are the experimental lattice constants in A. The sum in jextends to all the atoms in the asymmetric unit, and the sum in iis restricted to those components that are not fixed by symmetry requirements. Nis the number of terms in the sum. According to its definition, this parameter is equal to one in the hypothetical case that all unit-cell parameters deviate 1% and all the atomic coordinates differ in 0.1Afrom the experimental observation. This order of magnitude for the discrepancies between the simulated structural model and the real observation is normally considered acceptable in this type of study. In the present study, the factor Rattained values of the order of 0.1-0.3. It can be noted that these values are 1order of magnitude smaller than the one (R =1.42) corresponding to the relaxed structure obtained for Rb2ZnC14 in Ref. 9 using an ab-initio force model. The significance of the quantitative value of Rwas investigated by determining its value for the structural difference between two relaxed structures corresponding to force models differing only in the charge values an amount b,Qs, =O.le (EQo=0.025e). Typically Rtook values between 0.04 and 0.09. The best fit was finally obtained for Qs, =+l.30e (Qo= — 0.825e) and Ax a=97.7X10 Kcal/mol with R=0,159. The relaxed structure is presented in Table III. In the following, we shall call the resulting model of interatomic forces, model 1. The whole set of model pa- TABLE III. Relaxed structure of potassium selenate for model 1, model 2(293 K), and model 2(145 K), and of cesium selenate for 0 model 1. The lattice constants are given in A, and the atomic positions in relative units (X10 ). K2Se04 model 1 a7.662 b10.426 c6.074 K2Se04 model 2(293 K) 7.611 10.413 6.070 K&Se04 model 2(145 K) 7.556 10.335 6.028 Cs2Se04 model 1 8.352 11. 219 6.496 Se K(a) K(P) 0(1) 0(2) 0(3) 2333 4181 1644 856 — 76 6963 3006 3440 3156 5620 207 4261 2500 2500 2500 297 2500 2500 2319 4190 1657 887 — 95 7023 3030 3467 3071 5652 189 4214 2500 2500 2291 2500 1629 — 68 296 2997 25QO 3127 152 4191 2500 879 2500 7022 2500 3440 293 5648 2500 4252 2500 Se Cs(a) Cs(P) 0(1) 0(2) 0(3) 2363 4179 2500 1684 895 2500 — 111 6971 2500 3036 3506 437 3065 5537 2500 401 4191 2500 8486 I.ETXEBARRIA, J.M. PEREZ-MATO, AND A. CRIADO o TABLE IV. Parameter ofthe different force models considered (Ao p=55X10' Kcal/mol, Boo=3.96 Afor all models). R Qs.(e) Qo(e) BK.-O (A ) ~K(a)-o(1-2) (10' Kcal/mol) ~K(a)-O(3) ~K(/3)-O(1-2) ~K(f3)-0(3) Model 1 (K2Se04) 0.159 1.30 — 0.825 3.76 97.7 Model 2 (K,Se04-293 K) 0.132 1.34 — 0.835 3.76 114.7 102.6 83.3 118.5 Model 2 (K,Se0 -145 K) 0.105 1~34 — 0.835 3.76 106.6 97.4 75.4 100.0 Bcs-0 Model 1 (Cs2Se04) 0.109 1.44 — 0.86 3.50 122.5 rameters is listed in Table IV. Achange of the selenium charge of O. le did not change significantly the quality of the adjustment. Fixing Qs, to +1.6e is, however, enough to double the Rvalue. Any attempt to fix the charge of the selenium and oxygens to higher values failed. The wMIN program in mode 1was unable in these cases to relax properly from the experimental structure into an equilibrium configuration, indicating that the latter does not exist or differs considerably from the experimental one. This result contrasts with the analysis of the same compound in Ref. 12, where the charge of the selenium ion was fixed to +2e (Qo =— le). On the other hand, a recent molecular dynamics study of the related compound LiKSO& considered an oxygen charge even smaller than those obtained in the present study. 21 The results of the lattice-dynamics analysis of force model 1, described in the next section, led us subsequently to increase the complexity of the force model by assuming specific repulsive interactions for the pairs K-0 depending on the crystallographic type of potassium or oxygen considered. There are two types of inequivalent potassium atoms in the structure that occupy very different positions from atopological point of view (see Fig. 1). The K(P) occupy acavity of approximate hexagonal symmetry along the aaxis and is surrounded by six tetrahedra. Its separation from the oxygen atoms O(1) and O(2), on the face of the tetrahedra perpendicular to the aaxis varies from 2.74 to 2.82 A, while the shortest distance to an oxygen on the other vertex, O(3), is already 2.97 A. In contrast, the K(a) are situated approximately on some of the vertices of the hexagonal cavities (see Fig. 1), such that their shortest contact corresponds to an oxyoo oo oo oo oo K(P) I I K(a) oo oo oo oo oo FIG. 1. Projection along the aaxis of the structure of K2Se04 at room temperature showing the two kinds of inequivalent potassium atoms. 42 INCOMMENSURATE INSTABILITY AND LATTICE DYNAMICS ~ ~ ~8487 gen O(3) at 2.62 A, while the next ones are already at distances varying from 3.00 to 3.26 Aand correspond to oxygens of the type O(1} or O(2). It is well known that effective repulsive potentials can vary considerably when different ranges of atomic distances are considered, so that they are described by different Born-Mayer potentials. Also, ab-initio interactions like Gordon-Kirn potentials are given by different analytical expressions for different distance ranges. "It is therefore reasonable to assume aset of four different Born-Mayer K-0 potentials, if we take into account the very different interatomic distances relevant in each case. Accordingly, we split the initial unique K-0 Born-Mayer potential into four different potentials depending on the atoms considered: K(a)-O(1-2), K(a)-O(3), K(P)-O(1-2), and K(P)-O(3). The number of adjustable parameters in this generalized model increases to five: the effective charge of selenium and the four parameters Ax o(the parameter BK pis kept the same for the four different pairs and has the value given above). An optimal set of these parameters was determined by means of afitting process analogous to the preceding one. The process converged to a value of the Se charge of +1.6e (Qo= — 0.9e) with a fitting factor R=0.109. However lower values such as +1.3e did not deteriorate the adjustment significantly and Rincreased only up to R=0.136. The subsequent normal-mode analysis indicated that with Se charges as high as +1.6e, the system was unstable along the Xline. Therefore the Se charge was fixed to alower value (+1.34e )and only the parameters AKowere optimized. The resulting model, henceforth referred to as model 2 (293 K), is listed in Table IV. In Table III, the corresponding Pnam relaxed structure is also presented. As in the previous case, when higher values than 1.6e were considered no proper Pnam relaxed structure close to the experimental one could be found. The resulting configurational energy of the equilibrium structure [— 416.39 Kcal/mol for model 1, — 416.34 Kcal/mol for model 2(293 K)] is close to the experimental cohesion energy (— 413.76 Kcal/mol). Athird model [model 2(145 K) in Table IV] was considered using as observed equilibrium configuration the structure determined at 145 K(Ref. 17) (see Table II) and maintaining Qs, =+1.34e. Essentially, as aconsequence of the thermal contraction, adecrease of the effective K- 0repulsive interactions with respect to the roomtemperature model is obtained. Finally, for Cs~Se04, asimilar process was followed. A simple model with asingle Cs-0 interaction analogous to model 1for KzSe04 was fitted following the same procedure. As observed structure we considered the recently determined structure at room temperature (see Table II). The O-O interaction was kept the same as in potassium selenate, while the value of Bc,pwas fixed to 3.50 A'. The results are also given in Tables III and IV. The final optimized values of Qs, and Ac, owere +1.44e and 122.5X10 Kcal/mol, respectively, with R=0.109. Thus, the repulsive interaction A-0 increases considerably with respect to potassium selenate, in accordance with the corresponding increase of the unit-cell volume and the effective atomic size of the Aatoms. III. LATTICE DYNAMICS A. Introduction Using the same notation for the irreducible representations as in Ref. 3, the symmetry decomposition at the center of the Brillouin zone of the external normal modes 1s g ]gag 3g+ 5 A„+5B &u +7Bz„+7B3u Along the Xline (q =au '}the decomposition becomes 14K]+10K~+ 10K +14K and at the point X(q=—, 'a' )is 14X)+10Xq . The notation for the irreducible representations at the center of the Brillouin coincides with the one of Refs. 6 and 7. The number of branches Xz and X3 is the same, and the same happens for X, and X4. The compatibility relations are such that each pair of them are degenerate at the point X. Therefore, in an extended zone scheme each pair of branches Xz-X3, or X,-X4, reduces to asingle branch, and the total number of branches reduces to 24. In the following, we will use this extended zone scheme for the graphical representation of the phonon branches. Iizumi et al. determined by means of inelastic neutron scattering the existence of asoft-mode X~ branch connected with aX3 acoustic branch in KzSe04. The instability of this Xz branch at apoint close to 0.31a* takes place at atemperature consistent with the known transition temperature T, =130K. Several Raman and infrared spectroscopic measurements of KzSe04 have been reported in the last few years. ''The number of internal modes detected by this means in the normal phase differs from those corresponding to the usual structural model. As apossible cause, it has been suggested that the selenate tetrahedra have some kind of weak disorder, which cannot be detected by x-ray methods. 'On the other hand, the external Raman and infrared-active modes (Ag, 8&g, Bzs, 83g and 8&„,Bz„,83„,respectively) have been identified and measured without much problem in the frame of the symmetry analysis above. 'The experimental data reported in Ref. 6will be used for comparison with the present simulation. Asignificant fact can be deduced from these studies: apparently, the soft Xz branch becomes an optically inactive A„mode at the center of the Brillouin zone. This can be derived indirectly from the knowledge that aXz branch is only compatible with A„ or 83g symmetries at the zone center, and the lowest 8, measured Raman frequency is at the range of 45 cm while from the neutron data of Iizurni et al, the end of the soft Xz branch has values lower than 25 cm B. Phonon branches calculation The dispersion curves were calculated under the harmonic approximation using ageneral program, which al- 8488 I.ETXEBARRIA, J.M. PEREZ-MATO, AND A. CRIADO 42 D Rl CY Q ghJ of the experimental soft-mode eigenvector in Table I. The agreement between the phases is especially significan, but also the di8'erent atomic amplitudes follow in general terms the experimental pattern. Only a significant deviation to larger values of the amplitude of the selenate rotation around the xaxis and atoo small selenate ztranslation can be clearly observed. In contrast with the experimental situation, the X2 mode at the zone center has B3g symmetry. An increase of the Se charge leads, apart from the changes in the equilibrium structure, to astronger instability of the structure, with the soft-mode imaginary branch taking larger absolute values. REDUCED WAVE VECTOR 0.82. Model 2(293 EC) FIG. 2. Calculated dispersion curves plotted in the extended zone scheme corresponding to the model 1of K&Se04 (solid line) and Cs2Se04 (dashed line). For imaginary values of the frequency, — ~co~ is represented. lows us to consider rigid molecular units and therefore can restrict the calculations to the external modes. The external Born-von Karman formalism, in which the force constants are obtained in terms of molecular rotations and translations, is used. The dynamical matrix is built up using the method of Pawley where, using a pairwise potential model, force constants are calculated analytically as afunction of rotational and translational coordinates. "Self-terms" or force constants relating two displacements of the same molecular unit are obtained from the crystal translational and rotational invariance conditions. Short-range force constants were calculated for molecular pairs inside acutoff radius distance (8 A). For Coulombic interactions however, given their long-range character, the Ewald convergence method was used. C. Results 1. Model 1 This model is unstable for vibrations along the Xline, namely, an unstable dispersion branch of X2-X3 symmetry appears as shown in Fig. 2. No other branch along this line takes imaginary values. Aclear minimum with a rather large depth (5.32 meV) can be observed at apoint q=0.28a'. The mode eigenvector at this minimum listed in Table Valready reproduces most of the characteristics The system is stable with respect to all normal modes. Again the lowest nonacoustic branch along the Xline has X2 symmetry, but in this case the frequencies have real values. This branch (see Fig. 3) is rather flat and presents aweak minimum at q=0.22a'. The mode eigenvector at this point agrees with the experimental one much better than for model 1(see Tables Iand V). The previous discrepancies observed in the amplitudes of A„and 1, of the selenate group are corrected, but the deviation of the 'r, amplitude of the K(a) atom suffers aslight increase. The phases for the different atoms seem to be weakly dependent on the details of the interactions. They scarcely change between the two models and have an excellent agreement with the experimental values in Table I. The branch zone center has A„symmetry. However, for larger values of the Se charge (an increase of 0.02e is sufficient) the mode changes to 8& symmetry. The reason for this extreme sensitivity of the symmetry of the zone-center mode to the details of the interatomic forces is caused by the proximity of another Xz branch around the zone center. The two branches may "interact" and can easily interchange the symmetry of their zone-center modes. The frequency ofthe lowest B3& mode is specially sensitive to changes of the ionic charges and strongly decreases for larger charges. At Qs, =+1.36e the interchange of the extremes of the two X2 branches takes place. In addition, this lowering of the B3g mode causes, by asimilar effect, the destabilization (negative values for C~) of the acoustic branches along the yand zdirections corresponding to the shear e„which has the same symmetry. Further increases of the Se charge leads to lower frequencies all along the lowest X2 branch and strengthens the presence of aminimum at values close to TABLE V. Eigenvectors of the mode corresponding to the minimum in the soft branch of K&Se04 for (a) model 1, (b) model 2(293 K), (c) model 2(145 K), (d) modified model 2(145 K) changing the Az oparameter for the interaction K(a)-O(1-2). (e) Eigenvector ofthe mode X2 at 0.31a*for Cs&Se04. The data are given in the same manner as in Table I. K(a) K(P) Se04 T. T. T. R„ Ry (a) 61.2/0. 110 72.8/0. 318 38.2/0. 811 4.06/0.012 3.06/0.610 (b) 79.6/0. 135 80.3/0.320 68.0/0.816 2.23/ — 0.003 3.58/0.621 (c) 75.2/0. 126 83.3/0.319 62.6/0. 820 2.30/0.002 3.66/0.617 72.1/0.119 80.8/0.320 56.5/0. 830 2.70/0.008 3.63/0.613 (e) 64.6/0. 118 81.2/0. 319 45.3/0.824 0.96/ — 0.025 2.46/0. 614 42 INCOMMENSURATE INSTABILITY AND LA IlICE DYNAMICS. ..8489 6 VCV a bl CY O00.20.40.60.8 —, 'a'. For values of Qs, around +1.6e, which was the optimum value resulting from structural considerations (see Sec. II), the Xz branch is completely unstable, with a similar form to the one of Fig. 2for model 1. This was the reason to fix the value of Qs, in this model to 1.34e. Thus, the condition of complete stability of the system practically forced the A„symmetry at the zone center of the branch, as observed experimentally. The elastic constants and Raman frequencies calculated for model 2(293 K) are listed in Tables VI and VII, respectively, where experimental values are also given. Agraphical comparison can be seen in Fig. 4. In general, the agreement is within 5— 20%, which is excellent considering that the force model has only been adjusted using static structural data. However, aB, mode (the lowest one) strongly deviates from the experimental value. This strong deviation can be due to asymmetry interchange effect similar to the one commented above between the lowest modes of A„and B3g symmetry. In this case, both B,gand Bz„modes are compatible with the branches of X4 symmetry. In the simulation, the lowest X4 optic branch ends at the zone center with aBz„ mode (see Fig. 5). This symmetry could be interchanged with the B]g symmetry of the mode at the end ofthe next X4 branch ifthe force model is varied. For the sake of comparison with previous studies, and also as areference for eventual subsequent experimental REDUCED i%AVE VECTOR FIG. 3. Calculated dispersion curves of K&Se04 plotted in the extended zone scheme for model 2(293 K) (solid line), model 2(145 K) (dashed line) and model 2(145 K) decreasing the interaction of the K(a)-O(1-2) pairs (point line). The experimental points at 145 K(Ref. 3) are also shown. studies, the whole set of external phonon branches with this model is shown in Fig. 5. The density ofstates corresponding to the external modes is also shown in Fig. 4. The highest external frequencies have values that are significantly smaller than those obtained in Ref. 12, with abetter agreement with the experimental data. The global agreement of the mode frequencies can be confirmed by comparing the resulting specific-heat curve to the experimental results. This is done in Fig. 6, where the effect of the internal frequencies of the selenate groups have been included by adding to the density of states of Fig. 4four Einstein-type contributions, centered at the approximate experimental values of the selenate internal frequencies. The experimental specific-heat curve has been taken from Ref. 34. The agreement between experimental and theoretical values is excellent. Only an obvious and expected discrepancy exists close to the transition temperature. The deviation of the experimental values at higher temperatures is due to the usual discrepancy between the experimental specific heat at constant pressure and the theoretical one at constant volume, which is caused by anharmonic effects. This deviation can be fitted to the usual approximate linear temperature dependence. Thus, model 2reproduces quantitatively up to an excellent degree many of the known physical properties of potassium selenate. It should be noted that the model was not chosen to fit the experimental dynamical properties and only the condition of mechanical stability of the room-temperature structure was used to adjust the model parameters. 9. Model 2(145K) As explained in Sec. II, this model was obtained as a modification of model 2(293 K), using as observed structure the one determined at 145 K(Ref. 17}(see Table II) to adjust the model parameters. In the fitting process (see Sec. II) the selenium charge was kept fixed to the value of model 2(293 K}. An examination of Table IV indicates that the essential difference between this model and model 2(293 K) is acertain decrease of the effective repulsive interactions K-O. This decrease is congruent with the volume contraction of the structure at 145 K. The most conspicuous change in the phonon branches obtained for this model with respect to model 2(293 K) can be seen in Fig. 3, where the lowest X3-Xz branch is shown. The experimental data at 145 K(Ref. 3) are also shown in the figure for comparison. The subtle change in the repulsive K-0 parameters is sufficient to produce aclear softening of the Xz branch, which has now astrong minimum at q=0.25a'. The corresponding mode eigenvector listed in Table Vagrees even better than the one of model 2 TABLE VI. Elastic constants (X10' dyn cm )of KzSe04..experimental (Ref. 32) and calculated values for model 2(293 K). Expt. Calc. 50.5 58.747.9 53.040.7 49.28.1 3.6 Css 14.2 16.3 C„ 15.4 18.1 C&z 15 17.1 Cz 19 16.9 C 17 17.1 8490 I.ETXEBARRIA, J.M. PEREZ-MATO, AND A. CRIADO 42 TABLE VII. Raman frequencies (cm ') of K2Se04. experimental (Ref. 6) and calculated values for model 2(293 K). ~exp 41 75 97 105 121 144 162 cate 43 88 94 106 125 130 162 Discrepancy {%) 4.7 17.3 3.1 1.0 3.3 9.7 0 ~expt 47 74 96 109 120 143 160 ~ca]c 81 88 102 128 146 152 162 Discrepancy (%) 72.3 18.9 6.2 17.4 21.7 6.3 1.2 50 75 95 119 146 57 81 85 114 154 B2g 14.0 8.0 10.5 4.2 5.5 44 75 93 119 142 40 71 82 113 122 B3g 10.0 5.3 11.8 5.0 14.1 (293 K) with the experimental one. The softening of the X2 branch is specially sensitive to avariation of the repulsive parameter A~~ ~0~, 2~. The phonon branch obtained after adecrease of less than 4% (102.6X 10 Kcal/mol) of this parameter with respect to its value in model 2(145 K) is also shown in Fig. 3. What is remarkable is the resulting deep minimum of the branch at apoint (q=0.27a' ), which is now closer to the experimental value of the soft-mode wave vector. These results contrast with those reported in Ref. 12, where the formation of the deep minimum at values close to —, 'a' was attributed to aspecific O-O interaction. We performed additional calculations changing the O-O repulsive force. The results demonstrated that in our 0 Ag Blg B2g 83g 50 FREQUENCY(cm ) FIG. 4. Density of states of K2Se04 corresponding to the external modes calculated with model 2(293 K). Also agraphical cornparison ofthe calculated Rarnan frequencies {dashed lines) and the experimental ones (Ref. 6) is shown.