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 Total Downloads : 18
 Authors : Shaveen Garg, M.M. Sinha, H.C. Gupta
 Paper ID : IJERTCONV1IS01002
 Volume & Issue : AMRP – 2013 (Volume 1 – Issue 01)
 Published (First Online): 30072018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Lattice Dynamical Investigations for Insulating Cs_{2}CdBr_{4} in Orthorhombic Phase
Shaveen Garg
Department of Physics, Sant Longowal Institute of Engineering and Technology, Longowal148106, India
email: gargshaveen@gmail.com
M.M. Sinha
Department of Physics, Sant Longowal Institute of Engineering and Technology, Longowal148106, India
H.C. Gupta
Department of Physics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi110016, India
Abstract
Cs2CdBr4 is widely used for acoustooptic 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 ferroelastic 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 nonpolar ferroelastic (centrosymmetric) [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 acoustooptic figure of merit and hence were found to be good acoustooptic 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 nonpolar ferroelastic [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 GF 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 (cm1) 
Experimental frequencies (cm1) [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%, K621%, 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%, H127% 
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%, K638%, 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 submatrices of a finite size. By summing these submatrices 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 forthe 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 cm1 of Ag, 204 cm1 of B2g, 204 cm1 of B3u, 204 cm1 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 cm1 of Ag, 193 cm1 of B2g, 194 cm1 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 cm1 of Ag, 165 cm1 of B2g, 163 cm1 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 cm1 of B3g, 185 cm1 of B1g and 185 cm1 of B2u). Force constant K10 (CsBr2) is much dominating force constant for the next higher frequency of these modes (i.e. 168 cm1 of B3g, 168 cm1 of B1g and 168 cm1 of B2u). Force constant K3 (CdBr3) and K4 (CsBr3) are the main contributors for the 3rd higher frequency of these modes (i.e. 109 cm1 of B3g, 102 cm1 of B1g and 101 cm1 of B2u). For the frequency region of 11070 cm1 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, CsBr 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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