Lattice Dynamical Investigations for Insulating Cs₂CdBr₄ in Orthorhombic Phase

Lattice Dynamical Investigations for Insulating Cs₂CdBr₄ in Orthorhombic Phase

Shaveen Garg

Department of Physics, Sant Longowal Institute of Engineering and Technology, Longowal-148106, India

e-mail: gargshaveen@gmail.com

M.M. Sinha

Department of Physics, Sant Longowal Institute of Engineering and Technology, Longowal-148106, India

H.C. Gupta

Department of Physics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-110016, India

Abstract

Cs2CdBr4 is widely used for acousto-optic devices by operating the laser radiation in a wide range of spectra. It undergoes several successive phase transitions upon cooling. Low temperature phases of Cs2CdBr4 were found to be non- polar ferro-elastic but at room temperature and atmospheric pressure, it crystallizes in orthorhombic phase having space group Pnma. In the present paper, we have applied a short- range force constant model to calculate the Raman and infrared wave numbers of Cs2CdBr4 in the orthorhombic phase at zone centre. The calculated Raman and infrared wave numbers are found to be in a satisfactory match with the experimentally observed Raman and infrared wave numbers. Calculations on the zone centre phonons have been made with three bending and thirteen stretching force constants. We have also calculated the significant contribution of each force constant towards the Raman and the infrared wave numbers by investigating the potential energy distribution.

1. Introduction

formula units per unit cell. Upon cooling, it undergoes several successive phase transitions like the other compounds of this	Atom Cs1	Site 4c	x/a 0.1236	y/b 0.25	z/c 0.0960
family and the low temperature phases of Cs2CdBr4 were	Cs2	4c	-0.0170	0.25	0.6762
found to be non-polar ferro-elastic (centro-symmetric) [3].	Cd	4c	0.2225	0.25	0.4236
Shchur et al. [4] assigned some of the Raman and	Br1	4c	-0.0243	0.25	0.4119
infrared modes at zone centre for the compound Cs2CdBr4.	Br2	4c	0.3204	0.25	0.5926
Therefore, in the present paper, we have calculated and	Br3	8d	0.3209	-0.0094	0.3426
assigned all the Raman and infrared zone centre modes of

Cs2BX4 (B= Hg, Cd; X= Cl, Br) crystals belong to the halide family with the -K2S04 structure [1]. Cs2CdBr4 crystals of this family possess high acousto-optic figure of merit and hence were found to be good acousto-optic material [2]. At room temperature, Cs2CdBr4 crystallizes in orthorhombic phase having space group Pnma with four

2. Structure

Cs2CdBr4 undergoes several successive phase transitions upon cooling and the low temperature phases of Cs2CdBr4 were found to be non-polar ferro-elastic [3]. Cs2CdBr4 exhibits the same crystal structure as does the – K2S04 [1] at room temperature and crystallizes in orthorhombic structure having space group Pnma (No. 62-

) having four formula units. The unit cell contains four Cd tetrahedra and eight corresponding Cs+ ions. The lattice parameters (in Ã…) for Cs2CdBr4 are a = 10.235, b = 7.946, c = 13.977 [4]. The site symmetry, atomic coordinates

[4] are given in Table 1. Using factor group analysis, Phonon contribution of each atom at point for Cs2CdBr4 is given below:

Cs1 = Cs2 = Cd = Br1 = Br2 = 2Ag + 2B2g + B3g + B1g + Au + B2u + 2B3u + 2B1u

Br3 = 3Ag + 3B2g + 3B3g + 3B1g + 3Au + 3B2u + 3B3u + 3B1u.

So the total number of zone centre modes present for each species of the space group will be

= 13Ag + 8B1g + 13B2g + 8B3g + 8Au + 13B1u + 8B2u + 13B3u.

Out of these, 1B1u, 1B2u, 1B3u are acoustical, 13Ag, 8B1g, 13B2g are Raman active, 12B1u, 7B2u, 12B3u are infrared active, and 8Au are silent modes.

Table 1. Site symmetry, atomic coordinates

Cs2CdBr4 in their orthorhombic phase having space group Pnma using normal coordinate analysis based on G-F matrix method. The so calculated Raman and infrared wave numbers agrees well with the available experimental results [4].

3. Theory

Lattice dynamical calculations on Cs2CdBr4 in its orthorhombic phase have been carried out by applying Wilson GF matrix method [4]. Wilson's method of obtaining the normal frequencies and the normal coordinates of any system is to obtain eigen values and eigen vectors of the matrix GF, where G is the inverse kinetic energy matrix that involves the

masses and certain spatial relationships of the atoms and consequently, brings the kinetic energies into the equation, and F is the potential energy matrix of the system. F matrix has been constructed using short range force constant model. The magnitude of short range forces generally diminishes after the second neighbor, therefore, interatomic interactions up to only second neighbor are considered in the present calculation. In the present work, two types of short range forces are considered, viz, stretching forces (K1K13) and bending forces (H1H3) in Table 2. To account for the transverse vibrations in the lattice, we have also included the bending forces along with stretching forces for the present calculation.

The secular equation for calculating the normal frequencies is given as

K10	CsBr2	4.0144	0.409
K11	CsBr3	4.0647	0.077
K12	CsBr2	4.0717	0.121
K13	CsBr1	4.1809	0.068
H1	Br1CdBr3	110.718	0.022
H2	Br2CdBr3	105.210	0.028
H3	Br3CdBr3	106.340	0.042

Table 3. Comparison between the calculated and experimental frequencies and corresponding PED

Species	Present Frequencies (cm-1)	Experimental frequencies (cm-1) [4]	Potential Energy Distribution (PED)
Ag,1	204	194	K1- 63%, K2- 29%
Ag,2	193	–	K2- 15%, K3- 39%, K4- 36%
Ag,3	166	173	K1- 16%, K2- 47%, K4- 28%
Ag,4	109	–	K3- 28%, K6-21%, K7- 24%
Ag,5	94	–	K6- 36%, K12- 26%
Ag,6	81	81	K3- 14%, K8- 11%, K9- 22%
Ag,7	69	69	K5- 30%, K12- 16%, H3- 29%
Ag,8	66	–	K11- 62%
Ag,9	47	42	K8- 55%, K13- 12%
Ag,10	43	–	(K7, H2)- 14%, K9- 24%, (K10, K12)- 12%
Ag,11	32	–	K7- 24%, H1-27%
Ag,12	26	28	(K5, K6)- 20%, H2- 18%
Ag,13	15	21	K11- 19%, H3- 38%
B2g,1	204	–	K1- 63%, K2- 30%
B2g,2	193	187	K2- 14%, K3- 40%, K4- 36%
B2g,3	165	172	K1- 16%, K2- 47%, K4- 28%
B2g,4	103	–	K3- 17%, K6-38%, K7- 14%, K12- 11%
B2g,5	92	–	K3- 31%, K4- 17%, K9- 12%
B2g,6	88	–	K7- 21%, K9- 26%, K12- 17%
B2g,7	69	65	K5- 21%, K11- 36%
B2g,8	64	–	K5- 31%, K8- 18%, K11- 26%
B2g,9	51	52	K5- 10%, K8- 33%, H1- 18%
B2g,10	47	–	K6- 17%, K12- 30%,H3- 28%
B2g,11	40	–	K7- 28%, K9- 23%, K10- 13%
B2g,12	25		K7- 18%, H2- 38%, H3- 17%
B2g,13	08	22	K9- 11%, K11- 22%, K12- 14%, H3- 18%
B3g,1	185	181	K3- 66%, K4- 32%

(1)

Here, E is the unit matrix with the same order as that of the G and F matrices, , where c is the light velocity and v is the wave number.

The system such as crystal lattice possesses the translational symmetry. Therefore, the G or F matrix of infinite order will be repetition of sub-matrices of a finite size. By summing these sub-matrices multiplied by a proper set of phase factors, the G or F matrix can be constructed. These phase factors are unity for the lattice vibrations which are active in the infrared or Raman region Therefore, G or F matrix for these optically active vibrations can be obtained by adding the G or F matrix for a Bravais cell to the sum of all the G or F matrices representing the interactions between the Bravais cell and the neighboring Bravais cells.

The input parameters used for the whole calculation are the lattice parameters [4], masses of the atoms, fractional coordinates of the atoms [4], calculated symmetry coordinates

[6] and the experimentally available Raman and infrared wavenumbers [4]. Also, thirteen stretching force constant for

the nearest neighbors, CdBr , CdBr , CdBr , CsBr , Cs

2 1 3 3

Br2, CsBr2, CsBr3, CsBr1, CsBr3, CsBr2, CsBr3, Cs

Br2, and CsBr1 and three bending force constants for angles, Br1CdBr3, Br2CdBr3, and Br3CdBr3 are used in the calculation. The force constants were evaluated by fitting the observed 23 Raman modes and 27 infrared modes [4] and their values are given in Table 2.

4. Results and discussion

Table 2. Stretching and bending force constants and their corresponding values

Force constant	Between atoms	Interatomic distance (Ã…)/angle	Force constant Values (N/cm)
K1	CdBr2	2.5312	0.929
K2	CdBr1	2.5340	0.859
K3	CdBr3	2.5752	0.620
K4	CsBr3	3.6184	0.596
K5	CsBr2	3.6348	0.206
K6	CsBr2	3.6456	0.180
K7	CsBr3	3.6604	0.117
K8	CsBr1	3.6949	0.091
K9	CsBr3	3.8188	0.103

B3g,2	168	172	K10- 96%
B3g,3	109	–	K3- 19%, K4- 43%, K7- 12%
B3g,4	68	68	(K7,K9)- 31%, H2- 17%
B3g,5	61	–	K9- 12%, K13- 68%
B3g,6	45	43	K11- 58%, K13- 14%
B3g,7	33	–	K7- 16%, H1- 60%
B3g,8	15	22	(K9, H1)- 26%, K11- 19%, H2- 17%
B1g,1	185	183	K3- 66%, K4- 32%
B1g,2	168	174	K10- 96%
B1g,3	102	–	K3- 24%, K4- 55%
B1g,4	79	–	K7- 40%, K9- 23%, K13- 21%, H1- 14%
B1g,5	61	66	K7- 11%, K9- 19%, K13- 52%
B1g,6	49	43	K11- 64%, H2- 11%, H3- 17%
B1g,7	22	39	K7- 45%, K9- 50%
B1g,8	16	21	K11- 24%, H2- 65%
B2u,1	185	183	K3- 66%, K4- 32%
B2u,2	168	–	K10- 96%
B2u,3	101	–	K3- 25%, K4- 55%
B2u,4	80	70	K7- 40%, K9- 26%, K13- 17%, H1- 14%
B2u,5	63	61	K9- 20%, K13- 54%, H2- 12%
B2u,6	45	47	K9- 80%, H2- 12%
B2u,7	31	34	K7- 39%, K9- 12%, H2- 33%
B2u,8	0	0	–
B3u,1	204	198	K1- 65%, K2- 26%
B3u,2	194	191	K2- 21%, K3- 37%, K4- 25%
B3u,3	163	–	K1- 19%, K2- 45%, K4- 30%
B3u,4	109	–	K3- 37%, K4- 11%, K7- 19%
B3u,5	100	–	K5- 21%, K6- 37%, K12- 12%
B3u,6	80	85	K6- 12%, K7- 19%, K9- 23%, K12- 16%, H3- 18%
B3u,7	72	75	K5- 10%, (K9, K11)- 21%, H2- 14%
B3u,8	64	65	K11- 46, K12- 18%, H3- 19%
B3u,9	50	49	K8- 62%
B3u,10	44	43	K5- 11%, K9- 17%, H1- 14%, H2- 24%
B3u,11	29	32	(K11, H1)- 11%, H3- 41%
B3u,12	13	–	K6- 11%, K7-

			17%, H1- 44%, H2- 13%
B3u,13	0	0	–
B1u,1	204	199	K1- 65%, K2- 26%
B1u,2	194	184	K2- 22%, K3- 38%, K4- 34%
B1u,3	163	173	K1- 20%, K2- 44%, K4- 30%
B1u,4	102	–	K3- 45%, K4- 21%, K5- 17%
B1u,5	101	–	K6- 41%, K7- 17%, K12- 16%
B1u,6	83	–	K6- 16%, K7- 23%, K9- 31%, H3- 15%
B1u,7	70	71	K5- 18%, K11- 29%, K12- 23%
B1u,8	66	64	K9- 17%, K11- 31%
B1u,9	54	58	K5- 11%, K8- 47%, H1- 17%
B1u,10	41	44	K6- 21%, K8- 12%, K12- 10%, H2- 14%
B1u,11	33	31	H2- 20%, H3- 38%
B1u,12	26	–	K5- 22%, K7- 21%, H2- 16%, H3- 12%
B1u,13	0	0	–

The force constants used in the calculation for the Raman and infrared modes are given in Table 2. The calculated values are compared with the experimentally determined Raman and infrared modes by Shchur et al. [4] in Table 3. It is confirmed from the Table 3 that there is good agreement between the theory and the experiment for all the Raman as well as infrared modes. The potential energy distribution (PED) for each mode is also investigated to determine the contribution of different force constants to various frequencies and the interpretations drawn from the PED are described below.

It is clear from the potential energy distribution that the force constants K1 to K4 contribute significantly to the higher frequencies of each mode. K1 (CdBr2) and K2 (Cd Br1) are the dominating force constants for the highest frequency of the modes which are 13 in number (i.e. 204 cm-1 of Ag, 204 cm-1 of B2g, 204 cm-1 of B3u, 204 cm-1 of B1u) and K2 (CdBr1), K3 (CdBr3) and K4 (CsBr3) are dominant force constants for the next higher frequency of these modes (i.e. 193 cm-1 of Ag, 193 cm-1 of B2g, 194 cm-1 of B3u, 194 cm-

1 of B1u) and K1 (CdBr2), K2 (CdBr1) and K4 (CsBr3) contributes towards the 3rd higher frequency of these modes

(i.e. 166 cm-1 of Ag, 165 cm-1 of B2g, 163 cm-1 of B3u, 163 cm- 1 of B1u). By the contribution of the first three force constants

i.e. K1 (CdBr2), K2 (CdBr1) and K3 (CdBr3) towards the higher frequencies of the modes which are thirteen in number (i.e. Ag, B2g, B3u, and B1u), we may conclude that the tetrahedra group contributes more significantly to the

higher frequencies of these modes. On the other hand, force constants K3 (CdBr3) and K4 (CsBr3) are the dominant force constants for the highest frequency of each mode which are 8 in number (i.e. 185 cm-1 of B3g, 185 cm-1 of B1g and 185 cm-1 of B2u). Force constant K10 (CsBr2) is much dominating force constant for the next higher frequency of these modes (i.e. 168 cm-1 of B3g, 168 cm-1 of B1g and 168 cm-1 of B2u). Force constant K3 (CdBr3) and K4 (CsBr3) are the main contributors for the 3rd higher frequency of these modes (i.e. 109 cm-1 of B3g, 102 cm-1 of B1g and 101 cm-1 of B2u). For the frequency region of 11070 cm-1 of Ag, B2g, B3u and B1u modes, CsBr vibrations dominate. The stretching force constants K4 (CsBr3), K5 (CsBr2), K6 (CsBr2), K7 (Cs Br3), K9 (CsBr3), K11 (CsBr3), and K12 (CsBr2) contribute towards this frequency region. As we move towards the lower frequency region of all the modes, Cs-Br stretching vibrations dominate than any other vibrations and bending force constants also contribute significantly for the much lower frequencies.

5. Conclusions

All the Raman and IR modes for Cs2CdBr4 at the zone centre have been calculated by using the short range force constant model and are compared with the experimental results [4] which are in good agreement. Potential energy distribution for each mode is also investigated to know the contribution of each force constant towards them.

6. Acknowledgments

Author (Shaveen Garg) is highly thankful to MHRD, New Delhi for financial assistance in terms of institute fellowship.

7. References

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2. T. Dudoc, I. Martynyuk-Lototska, I. Trach, G. Romanyuk, M. Romanyuk, and R. Vlokh, Dispersion of Refractive Indices in Cs2CdBr4 and Cs2HgBr4 Crystals Ukr J. Phys Opt., Feb. 2005, pp. 24-26.

3. V I Torgashev, Yu I Yuzyuk, L A Burmistrova, F Smutny, and P Vanek, Raman spectra of Cs2CdBr4 single crystals J. Phys.: Condens. Matter, IOP Science, May 1993, pp. 5761-5772.

4. Ya Shchur, S Kamba, and J Petzelt, Lattice dynamics simulation of Cs2CdBr4 crystal, J. Phys.: Condens. Matter, IOP Science, Feb. 1999, pp. 36153628.

5. T. Shimanouchi, M. Tsuiboi, and T. Miyazawa, Optically active lattice vibrations as treated by the GF-matrix method, J. Chem. Phys., AIP, Nov. 1961, pp. 1597-1612.

6. S. Garg, M. M. Sinha, and H. C. Gupta, Lattice dynamical investigations for Raman and the infrared frequencies of Cs2HgCl4 International Journal of Modern Physics: conf. proc.World Scientific, pp. 694-700.