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High-pressure behavior of calcium chalcogenides
ARTICLE IN PRESS Physica B 405 (2010) 2683–2686
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
High-pressure behavior of calcium chalcogenides Kuldeep Kholiya a,n, Swati Verma b, Kailash Pandey c, B.R.K. Gupta d a
Department of Applied Science, Kumaon Engineering College, Dwarahat 263656, India Department of Computer Science, Kumaon Engineering College, Dwarahat 263656, India Department of Physics, Galgotia College of Engineering & Technology, Gr. Noida 201306, India d Department of Physics, G.B.P.U. of Ag. & Tech., Pantnagar 263145, India b c
a r t i c l e in fo
abstract
Article history: Received 21 May 2009 Received in revised form 7 February 2010 Accepted 20 March 2010
The B1–B2 phase transition and elastic properties of calcium chalcogenides at high pressure have been investigated using the Born–Mayer type potential considered up to nearest and next-nearest neighbor interaction. The computed values of B1–B2 phase transition pressure and equation of state (compression curve) are in excellent agreement with the experimental data. The present study gives the simple and straightforward method to determine the B1–B2 phase transition properties in calcium chalcogenides and shows that for these compounds the short range potential parameters, namely range and hardness parameter, should be different in two different phases, i.e. in B1 and B2 phases. & 2010 Elsevier B.V. All rights reserved.
Keywords: B1–B2 phase transition Calcium chalcogenides Potential model
1. Introduction The calcium chalcogenides viz CaO, CaS, CaSe and CaTe are closed shell ionic systems crystallizing in NaCl-type structure at ambient conditions. High pressure X-ray diffraction investigations of these compounds suggest that they undergo first order phase transition from the sixfold-coordinated NaCl-type (B1 structure) to the eightfold-coordinated CsCl-type (B2 structure) at the respective pressures of 61, 40, 38 and 33 GPa [1–3]. These compounds have technological importance in many applications ranging from catalysis to microelectronics [4]. Recently, Dadsetani and Doosti [5] studied the optical properties of calcium mono-chalcogenides in NaCl phase using the band structure results, obtained through the full potential linearized augmented plane wave (FP-LAPW) method within density functional theory, but the particular interest of these compounds is the B1–B2 structural phase transition they exhibit under pressure. In the past years, many theoretical calculations have been performed to study the phase transition and elastic properties of these compounds within the framework of ab initio calculations [6–11] and different potential models [12–14]. Although some of these calculations predict the phase-transition pressure quite close to the experimental values, ab initio calculation requires rigorous computational work. Beside this the other properties like compression curve and bulk modulus for B2 phase did not match with the experimental values. Thus, in the present
n
Corresponding author. Tel.: + 919310881787. E-mail address: [email protected] (K. Kholiya).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.03.051
study, our aim is to determine (i) the B1–B2 phase transition pressure, (ii) the equation of state (compression curve) and (iii) the mechanical properties (bulk modulus and pressure derivative of bulk modulus), using a suitable potential model that is discussed in the next section.
2. Method of analysis The interionic potential for the calcium chalcogenides can be expressed as X X Zi Zj e2 =rij þ bexpðrij =rÞ ð1Þ UðrÞ ¼ i,j
i,j
In the present study we have considered the same value of potential parameters, i.e. strength parameter b and hardness parameter r for nearest and next-nearest neighbor interaction in same phase while different values of these parameters in different phases (B1 and B2). Thus the cohesive energies for B1 and B2 phases may be given as follows [15–17]: pffiffiffi ! aM Z 2 e2 r 2r þ 12bB1 exp ð2Þ þ 6bB1 exp UB1 ðrÞ ¼ 4pe0 r rB1 rB1 ! 1 2 2 1 a Z e r 2r 1 p ffiffiffi UB2 ðr 1 Þ ¼ M þ8b þ 6b exp exp B2 B2 rB2 4pe0 r 1 3rB2
ð3Þ
where aM and a1M are the Modelung constants for the NaCl and CsCl phases, respectively. rB1,bB1 and rB2, bB2 are, respectively, the hardness parameter and range parameter in B1 and B2 phases.
ARTICLE IN PRESS 2684
K. Kholiya et al. / Physica B 405 (2010) 2683–2686
The stability of the particular crystal structure is given by the minima of the Gibbs free energy, i.e. G ¼ U þ PVTS
ð4Þ
while the hardness parameter (rB2) may be calculated from the minima of the Gibbs free energy whereas the interionic separation (r1) can be calculated with the help of volume collapse at the phase transition pressure.
where U is the internal energy and S is the vibrational entropy at pressure P, volume V and temperature T. We have performed our calculations at T¼0 K and at 0 K; the Gibbs free energy for NaCl (B1) and CsCl (B2) phases is expressed as [18]
GB2 ðr 1 Þ ¼ UB2 ðr 1 Þ þ
1.0
ð5Þ
8ðr 1 Þ3 P pffiffiffi 3 3
ð6Þ
The phase transition pressure is calculated as the pressure at which the difference of Gibbs free energy for two phases, i.e. DG ( ¼GB2–GB1), becomes zero. This requires the determination of hardness and the range parameter in B1 and B2 phases. For B1 phase, the short range parameters may be determined from the thermodynamic condition of bulk modulus and the equilibrium condition [dUB1(r)/dr]r ¼ r0 ¼0. For B2 phase the range parameter may be given as [17,19] 8 bB1 6
bB2 ¼
1.2
ð7Þ
0.8 G(10Δ-19Joule)
GB1 ðrÞ ¼ UB1 ðrÞ þ 2Pr
3
1.4
0.6 0.4 0.2 0.0 -0.2
0
10
20
30
40
50
60
70
80
Pressure(GPa)
-0.4 -0.6 Fig. 2. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaS.
1.4 Table 1a Input parameters. ˚ r0 (A)
CaO CaS CaSe CaTe
2.405 2.845 2.958 3.174
[1] [3] [3] [3]
B0 (GPa)
% volume collapse at transition
111 [2] 64 [3] 51 [3] 42 [3]
10.0 [2] 10.2 [3] 7.7 [3] 4.6 [3]
1.0 0.8 G(10Δ-19Joule)
Crystal
1.2
Table 1b Calculated model parameters. Crystal
bB1 (10
CaO CaS CaSe CaTe
441.18 631.70 442.89 614.25
19
J)
0.4 0.2 0.0 -0.2
˚ rB1 (A)
bB2 (10
0.4257 0.4568 0.5030 0.5025
588.24 842.26 590.53 818.99
19
J)
˚ rB2 (A)
-0.4
0.3927 0.4260 0.4671 0.4702
-0.6
10
20
30
40
50
60
70
80
Pressure(GPa)
1.4 1.2
1.0
1.0
0.8
0.8
ΔG(10-19Joule)
1.2
0.6 0.4 0.2
0.6 0.4 0.2 0.0
0.0 -0.2
0
Fig. 3. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaSe.
1.4
ΔG(10-19Joule)
0.6
0
20
40
60
80
Pressure(GPa)
-0.4 Fig. 1. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaO.
-0.2
0
10
20
30
40
50
60
Pressure(GPa)
-0.4 -0.6 Fig. 4. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaTe.
ARTICLE IN PRESS K. Kholiya et al. / Physica B 405 (2010) 2683–2686
1.05 1.00
The bulk modulus for B2 phase is calculated from the thermodynamic condition B ¼V(d2U/dV2) and the pressure derivative of bulk modulus for both the phases may be calculated by fitting P–V data to the Vinet equation of state [20,21].
Present work
0.95
2685
Experimental
0.90
V/V0
0.85
3. Results and conclusion
0.80 0.75 0.70 0.65 0.60 0.55 0.50 0
10
20
30
40
50
60
70
80
90
Pressure(GPa)
The input data and calculated model parameters are given in Tables 1a and 1b, respectively. From Table 1b it can be observed that the hardness parameter (r) for the B2 phase decreases in comparison to that of the B1 phase. This result is consistent with ˇ ˚ nek and Vacka´rˇ [22], which conclude that the recent study of Simu the higher coordination number of atoms results in lower hardness. The difference of Gibbs free energy (DG) with pressure is given in Figs. 1–4 for CaO, CaS, CaSe and CaTe, respectively. The value of DG decreases with increase in pressure and at a certain pressure it
Fig. 5. Compression curve for CaO. Experimental points are from Ref. [2].
1.0 Present work Present work Experimental
1.0
V/V0
0.9
V/V0
Experimental
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0
10
20
30
40
50
60
Pressure(GPa) 0.5 0
10
20
30
40
50
60
70
80
Fig. 8. Compression curve for CaTe. Experimental points are from Ref. [3].
Pressure(GPa) Table 2 Phase transition and elastic properties.
Fig. 6. Compression curve for CaS. Experimental points are from Ref. [3].
Crystal
Pt (GPa)
B0 in B2 phase (GPa)
B10 In B1 ph.
In B2 ph.
1.0
0.9
V/ V0
CaO Pres. Exp. Oth.
Present work Experimental
CaS Pres. Exp. Oth.
0.8
0.7
CaSe Pres. Exp. Oth.
0.6
0.5
0
10
20
30
40
50
60
70
80
Pressure(GPa) Fig. 7. Compression curve for CaSe. Experimental points are from Ref. [3].
CaTe Pres. Exp. Oth.
60.4 61.0 [2] 61.0 [6] 106.0 [13]
134.62
4.02 4.20 [2] 4.08 [6] 4.11 [7]
3.84
40.6 40.0 [3] 49.8 [8] 39.9 [12]
76.96 64.00 [3] 66.40 [8]
4.11 4.20 [3] 3.90 [8] 4.56 [12]
4.09 4.20 [3] 3.90 [8]
38.4 38.0 35.2 38.5 37.0
[3] [9] [12] [14]
61.03 51.00 [3] 58.29 [9]
4.05 4.20 [3] 4.20 [9] 4.60 [12]
3.96 4.20 [3] 4.12 [9]
33.2 33.0 30.2 34.9 34.5
[3] [9] [12] [14]
50.05 41.80 [3] 42.60 [9]
4.12 4.30 [3] 3.08 [9] 4.81 [12]
4.15 4.30 [3] 3.32 [9]
140.00 [6] 132.80 [7]
4.26 [6] 4.37[7]
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K. Kholiya et al. / Physica B 405 (2010) 2683–2686
becomes zero. This pressure is the phase transition pressure and it comes out to be 60.4 GPa for CaO, 40.6 GPa for CaS, 38.4 GPa for CaSe and 33.2 GPa for CaTe, which are in good agreement with the experimental findings. It is worth mentioning that in one class of solid, the phase transition pressure is related to the equilibrium separation (r0) and bulk modulus (B0) of B1 phase and it increases with increase in r0 and decreases with increase in B0. This is because of the mechanism of B1–B2 structural phase transition, which may be understood as the unfolding of packing and separation of cation and anion layers parallel to (0 0 1). We have also computed the variation of volume (V/V0) with pressure. The results obtained are plotted in Figs. 5–8 along with the experimental data. It is found that the obtained results for both B1 and B2 phases are in good agreement with the experimental data. The simplicity and applicability of the considered potential model encouraged the authors to determine the bulk modulus B0 and the pressure derivative of bulk modulus B10 for B2 phase in which there is scarcity of experimental data. The obtained values of these parameters along with the phase transition pressure and B10 for B1 phase are given in Table 2 with other theoretical results. It is evident from Tables 1(a) and 2 that in all calcium chalcogenides our calculated values of bulk modulus for B2 phase somehow increases compared to the B1 phase. This is consistent with the experimental work of Zimmer et al. [4] supported by their empirical relation for the bulk modulus of B1 and B2 phases given as B02 ¼B01(V02/V01) 1.1. As volume collapses at phase transition, V02 will always be less than V01, implying that B02 will always be greater than B01.The calculated values of the first order pressure derivative of bulk modulus (B10 ) for both the phases are almost the same, which is in agreement with the experimental results [3,4].
The overall discussion and the obtained results for phase transition pressure and compression curve support our potential model and method of calculation, in which we have considered the different values of range and hardness parameter in B1 and B2 phases.
References [1] R.W.G. Wyckoff, Crystals Structures, vol. I, Wiley, New York, 1963. [2] P. Richet, H.K. Mao, P.M. Bell, J. Geophys. Res. 93 (1988) 15279. [3] H. Luo, R.G. Greene, K. Ghandehari, Ting Li, A.L. Ruoff, Phys. Rev. B 50 (1994) 16232. [4] H.G. Zimmer, H. Winzen, K. Syassen, Phys. Rev. B 32 (1985) 4066. [5] M. Dadsetani, H. Doosti, Comput. Mater. Sci. 45 (2009) 315. [6] H. Zhang, M.S.T. Bukowinski, Phys. Rev. B 44 (1991) 2495. [7] H. Baltache, R. Khenata, M. Sahnoun, M. Driz, B. Abbar, B. Bouhafs, Physica B 344 (2004) 334. [8] P.E. van Camp, V.E. van Dorin, J.L. Martins, Phys. Stat. Sol. (b) 190 (1995) 193. [9] R. Khenata, M. Sahnoun, H. Baltache, M. Re´rat, D. Rached, M. Driz, B. Bouhafs, Physica B 371 (2006) 12. [10] P. Cortona, P. Masri, J. Phys.: Condens. Matter 10 (1998) 8947. [11] S. Ekbundit, A. Chizmeshya, R. LaViolette, G.H. Wolf, J. Phys.: Condens. Matter 8 (1996) 8251. [12] B.S. Arya, M. Aynyas, S.P. Sanyal, J. Optoelectron. Advan. Mater. 10 (2008) 1311. [13] K.N. Jog, R.K. Singh, S.P. Sanyal, Phys. Rev. B 31 (1985) 6047. [14] P.K. Jha, S.P. Sanyal, Phys. Status Solidi. (b) 212 (1999) 241. [15] Kuldeep Kholiya, B.R.K. Gupta, Pramana J. Phys. 68 (2007) 649. [16] Kuldeep Kholiya, B.R.K. Gupta, Physica B 387 (2007) 271. [17] Kuldeep Kholiya, B.R.K. Gupta, Phase Transit. 81 (2008) 403. [18] A.J. Cohen, R.G. Gordon, Phys. Rev. B 12 (1975) 3228. [19] M.P. Tosi, Solid State Phys. 16 (1964) 1. [20] P. Vinet, J. Ferrante, J.H. Rose, J.R. Smith, J. Geophys. Res. 92 (1987) 9319. [21] Kamal Devlal, Kuldeep Kholiya, Indian J. Phys. 80 (2006) 801. ˇ ˚ nek, J. Vacka´rˇ, Phys. Rev. Lett. 96 (2006) 085501. [22] A. Simu
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
High-pressure behavior of calcium chalcogenides Kuldeep Kholiya a,n, Swati Verma b, Kailash Pandey c, B.R.K. Gupta d a
Department of Applied Science, Kumaon Engineering College, Dwarahat 263656, India Department of Computer Science, Kumaon Engineering College, Dwarahat 263656, India Department of Physics, Galgotia College of Engineering & Technology, Gr. Noida 201306, India d Department of Physics, G.B.P.U. of Ag. & Tech., Pantnagar 263145, India b c
a r t i c l e in fo
abstract
Article history: Received 21 May 2009 Received in revised form 7 February 2010 Accepted 20 March 2010
The B1–B2 phase transition and elastic properties of calcium chalcogenides at high pressure have been investigated using the Born–Mayer type potential considered up to nearest and next-nearest neighbor interaction. The computed values of B1–B2 phase transition pressure and equation of state (compression curve) are in excellent agreement with the experimental data. The present study gives the simple and straightforward method to determine the B1–B2 phase transition properties in calcium chalcogenides and shows that for these compounds the short range potential parameters, namely range and hardness parameter, should be different in two different phases, i.e. in B1 and B2 phases. & 2010 Elsevier B.V. All rights reserved.
Keywords: B1–B2 phase transition Calcium chalcogenides Potential model
1. Introduction The calcium chalcogenides viz CaO, CaS, CaSe and CaTe are closed shell ionic systems crystallizing in NaCl-type structure at ambient conditions. High pressure X-ray diffraction investigations of these compounds suggest that they undergo first order phase transition from the sixfold-coordinated NaCl-type (B1 structure) to the eightfold-coordinated CsCl-type (B2 structure) at the respective pressures of 61, 40, 38 and 33 GPa [1–3]. These compounds have technological importance in many applications ranging from catalysis to microelectronics [4]. Recently, Dadsetani and Doosti [5] studied the optical properties of calcium mono-chalcogenides in NaCl phase using the band structure results, obtained through the full potential linearized augmented plane wave (FP-LAPW) method within density functional theory, but the particular interest of these compounds is the B1–B2 structural phase transition they exhibit under pressure. In the past years, many theoretical calculations have been performed to study the phase transition and elastic properties of these compounds within the framework of ab initio calculations [6–11] and different potential models [12–14]. Although some of these calculations predict the phase-transition pressure quite close to the experimental values, ab initio calculation requires rigorous computational work. Beside this the other properties like compression curve and bulk modulus for B2 phase did not match with the experimental values. Thus, in the present
n
Corresponding author. Tel.: + 919310881787. E-mail address: [email protected] (K. Kholiya).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.03.051
study, our aim is to determine (i) the B1–B2 phase transition pressure, (ii) the equation of state (compression curve) and (iii) the mechanical properties (bulk modulus and pressure derivative of bulk modulus), using a suitable potential model that is discussed in the next section.
2. Method of analysis The interionic potential for the calcium chalcogenides can be expressed as X X Zi Zj e2 =rij þ bexpðrij =rÞ ð1Þ UðrÞ ¼ i,j
i,j
In the present study we have considered the same value of potential parameters, i.e. strength parameter b and hardness parameter r for nearest and next-nearest neighbor interaction in same phase while different values of these parameters in different phases (B1 and B2). Thus the cohesive energies for B1 and B2 phases may be given as follows [15–17]: pffiffiffi ! aM Z 2 e2 r 2r þ 12bB1 exp ð2Þ þ 6bB1 exp UB1 ðrÞ ¼ 4pe0 r rB1 rB1 ! 1 2 2 1 a Z e r 2r 1 p ffiffiffi UB2 ðr 1 Þ ¼ M þ8b þ 6b exp exp B2 B2 rB2 4pe0 r 1 3rB2
ð3Þ
where aM and a1M are the Modelung constants for the NaCl and CsCl phases, respectively. rB1,bB1 and rB2, bB2 are, respectively, the hardness parameter and range parameter in B1 and B2 phases.
ARTICLE IN PRESS 2684
K. Kholiya et al. / Physica B 405 (2010) 2683–2686
The stability of the particular crystal structure is given by the minima of the Gibbs free energy, i.e. G ¼ U þ PVTS
ð4Þ
while the hardness parameter (rB2) may be calculated from the minima of the Gibbs free energy whereas the interionic separation (r1) can be calculated with the help of volume collapse at the phase transition pressure.
where U is the internal energy and S is the vibrational entropy at pressure P, volume V and temperature T. We have performed our calculations at T¼0 K and at 0 K; the Gibbs free energy for NaCl (B1) and CsCl (B2) phases is expressed as [18]
GB2 ðr 1 Þ ¼ UB2 ðr 1 Þ þ
1.0
ð5Þ
8ðr 1 Þ3 P pffiffiffi 3 3
ð6Þ
The phase transition pressure is calculated as the pressure at which the difference of Gibbs free energy for two phases, i.e. DG ( ¼GB2–GB1), becomes zero. This requires the determination of hardness and the range parameter in B1 and B2 phases. For B1 phase, the short range parameters may be determined from the thermodynamic condition of bulk modulus and the equilibrium condition [dUB1(r)/dr]r ¼ r0 ¼0. For B2 phase the range parameter may be given as [17,19] 8 bB1 6
bB2 ¼
1.2
ð7Þ
0.8 G(10Δ-19Joule)
GB1 ðrÞ ¼ UB1 ðrÞ þ 2Pr
3
1.4
0.6 0.4 0.2 0.0 -0.2
0
10
20
30
40
50
60
70
80
Pressure(GPa)
-0.4 -0.6 Fig. 2. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaS.
1.4 Table 1a Input parameters. ˚ r0 (A)
CaO CaS CaSe CaTe
2.405 2.845 2.958 3.174
[1] [3] [3] [3]
B0 (GPa)
% volume collapse at transition
111 [2] 64 [3] 51 [3] 42 [3]
10.0 [2] 10.2 [3] 7.7 [3] 4.6 [3]
1.0 0.8 G(10Δ-19Joule)
Crystal
1.2
Table 1b Calculated model parameters. Crystal
bB1 (10
CaO CaS CaSe CaTe
441.18 631.70 442.89 614.25
19
J)
0.4 0.2 0.0 -0.2
˚ rB1 (A)
bB2 (10
0.4257 0.4568 0.5030 0.5025
588.24 842.26 590.53 818.99
19
J)
˚ rB2 (A)
-0.4
0.3927 0.4260 0.4671 0.4702
-0.6
10
20
30
40
50
60
70
80
Pressure(GPa)
1.4 1.2
1.0
1.0
0.8
0.8
ΔG(10-19Joule)
1.2
0.6 0.4 0.2
0.6 0.4 0.2 0.0
0.0 -0.2
0
Fig. 3. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaSe.
1.4
ΔG(10-19Joule)
0.6
0
20
40
60
80
Pressure(GPa)
-0.4 Fig. 1. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaO.
-0.2
0
10
20
30
40
50
60
Pressure(GPa)
-0.4 -0.6 Fig. 4. The variation of the difference for Gibbs free energies (DG) in B1 and B2 phases with pressure for CaTe.
ARTICLE IN PRESS K. Kholiya et al. / Physica B 405 (2010) 2683–2686
1.05 1.00
The bulk modulus for B2 phase is calculated from the thermodynamic condition B ¼V(d2U/dV2) and the pressure derivative of bulk modulus for both the phases may be calculated by fitting P–V data to the Vinet equation of state [20,21].
Present work
0.95
2685
Experimental
0.90
V/V0
0.85
3. Results and conclusion
0.80 0.75 0.70 0.65 0.60 0.55 0.50 0
10
20
30
40
50
60
70
80
90
Pressure(GPa)
The input data and calculated model parameters are given in Tables 1a and 1b, respectively. From Table 1b it can be observed that the hardness parameter (r) for the B2 phase decreases in comparison to that of the B1 phase. This result is consistent with ˇ ˚ nek and Vacka´rˇ [22], which conclude that the recent study of Simu the higher coordination number of atoms results in lower hardness. The difference of Gibbs free energy (DG) with pressure is given in Figs. 1–4 for CaO, CaS, CaSe and CaTe, respectively. The value of DG decreases with increase in pressure and at a certain pressure it
Fig. 5. Compression curve for CaO. Experimental points are from Ref. [2].
1.0 Present work Present work Experimental
1.0
V/V0
0.9
V/V0
Experimental
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0
10
20
30
40
50
60
Pressure(GPa) 0.5 0
10
20
30
40
50
60
70
80
Fig. 8. Compression curve for CaTe. Experimental points are from Ref. [3].
Pressure(GPa) Table 2 Phase transition and elastic properties.
Fig. 6. Compression curve for CaS. Experimental points are from Ref. [3].
Crystal
Pt (GPa)
B0 in B2 phase (GPa)
B10 In B1 ph.
In B2 ph.
1.0
0.9
V/ V0
CaO Pres. Exp. Oth.
Present work Experimental
CaS Pres. Exp. Oth.
0.8
0.7
CaSe Pres. Exp. Oth.
0.6
0.5
0
10
20
30
40
50
60
70
80
Pressure(GPa) Fig. 7. Compression curve for CaSe. Experimental points are from Ref. [3].
CaTe Pres. Exp. Oth.
60.4 61.0 [2] 61.0 [6] 106.0 [13]
134.62
4.02 4.20 [2] 4.08 [6] 4.11 [7]
3.84
40.6 40.0 [3] 49.8 [8] 39.9 [12]
76.96 64.00 [3] 66.40 [8]
4.11 4.20 [3] 3.90 [8] 4.56 [12]
4.09 4.20 [3] 3.90 [8]
38.4 38.0 35.2 38.5 37.0
[3] [9] [12] [14]
61.03 51.00 [3] 58.29 [9]
4.05 4.20 [3] 4.20 [9] 4.60 [12]
3.96 4.20 [3] 4.12 [9]
33.2 33.0 30.2 34.9 34.5
[3] [9] [12] [14]
50.05 41.80 [3] 42.60 [9]
4.12 4.30 [3] 3.08 [9] 4.81 [12]
4.15 4.30 [3] 3.32 [9]
140.00 [6] 132.80 [7]
4.26 [6] 4.37[7]
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K. Kholiya et al. / Physica B 405 (2010) 2683–2686
becomes zero. This pressure is the phase transition pressure and it comes out to be 60.4 GPa for CaO, 40.6 GPa for CaS, 38.4 GPa for CaSe and 33.2 GPa for CaTe, which are in good agreement with the experimental findings. It is worth mentioning that in one class of solid, the phase transition pressure is related to the equilibrium separation (r0) and bulk modulus (B0) of B1 phase and it increases with increase in r0 and decreases with increase in B0. This is because of the mechanism of B1–B2 structural phase transition, which may be understood as the unfolding of packing and separation of cation and anion layers parallel to (0 0 1). We have also computed the variation of volume (V/V0) with pressure. The results obtained are plotted in Figs. 5–8 along with the experimental data. It is found that the obtained results for both B1 and B2 phases are in good agreement with the experimental data. The simplicity and applicability of the considered potential model encouraged the authors to determine the bulk modulus B0 and the pressure derivative of bulk modulus B10 for B2 phase in which there is scarcity of experimental data. The obtained values of these parameters along with the phase transition pressure and B10 for B1 phase are given in Table 2 with other theoretical results. It is evident from Tables 1(a) and 2 that in all calcium chalcogenides our calculated values of bulk modulus for B2 phase somehow increases compared to the B1 phase. This is consistent with the experimental work of Zimmer et al. [4] supported by their empirical relation for the bulk modulus of B1 and B2 phases given as B02 ¼B01(V02/V01) 1.1. As volume collapses at phase transition, V02 will always be less than V01, implying that B02 will always be greater than B01.The calculated values of the first order pressure derivative of bulk modulus (B10 ) for both the phases are almost the same, which is in agreement with the experimental results [3,4].
The overall discussion and the obtained results for phase transition pressure and compression curve support our potential model and method of calculation, in which we have considered the different values of range and hardness parameter in B1 and B2 phases.
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