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Volume dependence of the Grüneisen parameter for MgO
Solid State Communications 152 (2012) 414–416
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Volume dependence of the Grüneisen parameter for MgO S.S. Kushwah a,∗ , M.P. Sharma b a
Department of Physics, Rishi Galav College, Morena, 476001, M.P., India
b
Department of Chemistry, Rishi Galav College, Morena, 476001, M.P., India
article
info
Article history: Received 14 April 2011 Received in revised form 19 September 2011 Accepted 28 November 2011 by Xincheng Xie Available online 3 December 2011 Keywords: D. Thermal expansion D. Thermodynamic properties D. Equation of state E. High pressure
abstract Unified analyses for P–V –T equations of state of MgO recently presented by several workers in order to obtain a solution for pressure-scale problems in high P–T experiments make use of a new functional form for volume dependence of the Grüneisen parameter. In the present study we point out that this formulation is not consistent with variation of the third order Grüneisen parameter with the change in pressure. We have emphasized that the Stacey–Davis formulation for the volume dependence of the Grüneisen parameter is more appropriate than the other functional form used in the recent literature. The results have been obtained and reported for the volume dependence of the Grüneisen parameter and its higher order derivatives. © 2011 Elsevier Ltd. All rights reserved.
For understanding the high pressure–high temperature behaviour of solids, it is necessary to have a reliable knowledge of pressure P–volume V –temperature T relationships [1]. For this purpose an equation of state (EOS) can be used with the help of different approaches. First, we can use an isothermal EOS to determine P–V relationships along different isotherms at selected temperatures using input parameters corresponding to each temperature [2,3]. The other approach is based on the Mie–Grüneisen–Debye (MGD) model for evaluating the thermal effects in order to determine P–V relationships at high temperatures [4–9]. In this model we need to have an accurate knowledge of the volume dependence of the Grüneisen parameter, γ (V ) [4]. In the present study we emphasize that the relationship for volume dependence of the Grüneisen parameter, γ (V ), originally due to Al’tshuler et al. [10] has been widely used by various workers in the recent literature [4–9,11,12]. An analysis of this relationship performed here reveals its inadequacy for predicting the variation of third-order Grüneisen parameter with the change in pressure [13–15]. We find that the Stacey–Davis formulation for the volume dependence of the Grüneisen parameter is more appropriate than the relationship due to Al’tshuler et al. [10]. The results have been obtained and reported for the volume dependences of γ and higher-order derivatives in the case of MgO.
The Grüneisen parameter γ is an important thermoelastic property related to other thermal and elastic properties as follows [1]:
γ =
α KT V CV
α KS V
=
CP
,
where α is the thermal expansivity, V the volume, KT (KS ) the isothermal (adiabatic) bulk modulus, and CV (CP ) the constant volume (pressure) specific heat. The second and third-order Grüneisen parameters are defined as [16]:
q=
d ln γ d ln V
= T
V
γ
Correspondence to: HIG-933, Housing Board Colony, Morena, 476001, India. Tel.: +91 7532232271; fax: +91 7532400767. E-mail address: [email protected] (S.S. Kushwah). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.11.041
dγ dV
(2) T
and
λ=
d ln q d ln V
= T
V
q
dq dV
.
(3)
T
The parameters γ , q and λ appear frequently in the thermodynamic identities for higher order thermoelastic properties of solids [14–16], and their knowledge is not only desirable, but also necessary. Al’tshuler et al. [10] have presented the following relationship for the volume dependence of the Grüneisen parameter
γ = γ∞ + (γ0 − γ∞ ) ∗
(1)
V V0
m
,
(4)
where γ0 and γ∞ are the values of γ at V = V0 and V → 0, respectively and m is a constant. Eq. (4) has been used recently
S.S. Kushwah, M.P. Sharma / Solid State Communications 152 (2012) 414–416
415
by numerous workers [4–9,11,12]. Eq. (4), on differentiation with respect to V , yields the following expressions for higher order parameters q=
m
γ
(γ − γ∞ )
(5)
γ∞ . γ
(6)
and
λ=m
In the limit V → 0 or P → ∞, γ → γ∞ and therefore q → 0 (Eq. (5)), and from Eq. (6)
γ∞ = m
(7)
and
γ λ = γ∞ λ∞ .
(8)
Thus Eq. (4) satisfies the thermodynamic constraints viz. finite values of γ∞ and λ∞ , and q∞ → 0. Eqs. (5) and (6) give the following relations also q+λ=m
Fig. 1. Values of volume dependence of the Grüneisen parameter γ for MgO.
(9)
and q0 + λ0 = q∞ + λ∞ = λ∞ .
(10)
Since γ and q both decrease with the increase in pressure, λ must increase with the increase in pressure according to Eq. (8) as well as Eq. (10). However, this finding is not supported by the seismic data for the lower mantle and the core of the Earth [17]. It has been found [13–17] that λ decreases with the increase in pressure, and therefore Eq. (4), on which Eqs. (8) and (10) are based, is inadequate. Thermodynamic identities for higher derivative properties [15,16] also reveal that λ decreases with the increase in pressure or compression. This shortcoming can be removed by using the model due to Stacey and Davis [17] who have used a linear relationship between λ and q supported by the seismic data. Eqs. (103)–(105) given in Ref. [17] yield the following relationships
γ∞ = γ0
λ0 λ∞
− λ
q0 0 −λ∞
(11)
Fig. 2. Values of volume dependence of the second order Grüneisen parameter q for MgO.
and
(λ0 − λ∞ ) 1− λ0
γ = γ∞
V V0
q0 λ∞ − λ0 −λ ∞
.
(12)
We have used Eq. (12) to determine the values of Grüneisen parameter as a function of volume ratio (V /V0 ) for MgO. The value of γ0 (1.52) has been taken from the recent literature [4–6,9]. The value of q0 (1.41) has been determined from the modified free volume theory [15] using the Stacey reciprocal K -primed equation of state [16,17]. It should be mentioned that the value of q0 thus determined is in close agreement with the value of q0 = 1.406 for MgO taken by Wu et al. [18]. The value of λ∞ is determined using the relationship [13].
λ∞ =
′2 K∞
K0′
,
(13)
′ where K0′ and K∞ are the values of K ′ = dK /dP, pressure derivative of bulk modulus at P = 0 and P → ∞, respectively. We have ′ taken K0′ = 4.15 and K∞ = 2.49 so as to satisfy the relationship [16,17,19] ′ K∞
K0′
=
3 5
.
(14)
Eqs. (13) and (14) then yield λ∞ = 1.49 for MgO. The value of γ∞ can be determined from the thermodynamic identity [16,17]
γ∞ =
′ K∞
2
1
− . 6
(15)
Eq. (15) gives γ∞ = 1.08 for MgO. The value of λ0 satisfying Eq. (11) turns out to be equal to 8.84. These values of various parameters viz. γ0 , γ∞ , q0 , λ0 and λ∞ are used to evaluate the volume dependences of γ , q and λ. The results for MgO are shown in Figs. 1–3. It should be mentioned that the results obtained in the present study using the Stacey–Davis formulation satisfy the thermodynamic constraints for the solids in the limit of extreme compression [16,17]. It is found that λ0 equal to 8.84 is significantly larger than λ∞ equal to 1.49 suggesting that λ decreases rapidly with the increase in pressure, and attains a constant value in the limit of infinite pressure. The corresponding results for γ (V ), q(V ) and λ(V ) obtained from the Al’tshuler formulation, Eqs. (4)–(10), are also included in Figs. 1–3 for the sake of comparison. For the Al’tshuler model we need only three parameters γ0 , γ∞ and q0 . These are γ0 = 1.52, γ∞ = 1.08 and q0 = 1.41 in the case of MgO. The results based on the two formulations differ substantially from each other particularly for λ. Extrapolation of the analysis to MgO
416
S.S. Kushwah, M.P. Sharma / Solid State Communications 152 (2012) 414–416
parameter obtained by Stacey and Davis give the correct trend of variation of λ. References
Fig. 3. Values of volume dependence of the third order Grüneisen parameter λ for MgO.
is appropriate, as it abundant in the mantle. Experimental data or accurate calculations for λ are not available for presenting a comparison. The values of λ, the third derivative of Grüneisen
[1] O.L. Anderson, Equation of State of Solids for Geophysics and Ceramic Science, Oxford University Press, New York, 1995. [2] S.S. Kushwah, J. Shanker, Physica B 253 (1998) 90. [3] S.S. Kushwah, H.C. Shrivastava, K.S. Singh, Physica B 388 (2007) 20. [4] S. Speziale, C.S. Zha, T.S. Duffy, R.J. Hemley, H.K. Mao, J. Geophys. Res. 106B (2001) 515. [5] P.I. Dorogokupets, A. Dewaele, High Pressure Res. 27 (2007) 431. [6] P.I. Dorogokupets, A.R. Oganov, Phys. Rev. B 75 (2007) 24115. [7] A.B. Belonoshko, P.I. Dorogokupets, B. Johansson, S.K. Saxena, L. Koci, Phys. Rev. B 78 (2008) 104107. [8] Y. Tange, Y. Nishihara, T. Tsuchiya, J. Geophys. Res. 114 (2009) B03208. doi:10.1029/2008JB005813. [9] P.I. Dorogokupets, Phys. Chem. Minerals (2010) doi:10.1007/s00269-0100367-2. [10] L.V. Al’tshuler, S.E. Brusnikin, E.A. Kuz’menkov, J. Appl. Mech. Tech. Phys. 28 (1987) 129. [11] W.B. Holzapfel, M. Hartwig, W. Sievers, J. Phys. Chem. Ref. Data 30 (2001) 515. [12] J. Shanker, B.P. Singh, H.K. Baghel, Physica B 387 (2007) 409. [13] J. Shanker, B.P. Singh, Physica B 370 (2005) 78. [14] J. Shanker, B.P. Singh, K. Jitendra, Condens. Matter Phys. 12 (2009) 205. [15] S.S. Kushwah, N.K. Bhardwaj, Int. J. Mod. Phys. B 24 (2010) 1187. [16] F.D. Stacey, Rep. Progr. Phys. 68 (2005) 341. [17] F.D. Stacey, P.M. Davis, Phys. Earth Planet. Inter. 142 (2004) 137. [18] Z. Wu, R.M. Wentzcovitch, K. Umemoto, B. Li, K. Hirose, J.C. Zheng, J. Geophys. Res. 113 (2008) B06204. doi:10.1029/2007 JB005275. [19] S.S. Kushwah, N.K. Bhardwaj, J. Phys. Chem. Solids 70 (2009) 700.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Volume dependence of the Grüneisen parameter for MgO S.S. Kushwah a,∗ , M.P. Sharma b a
Department of Physics, Rishi Galav College, Morena, 476001, M.P., India
b
Department of Chemistry, Rishi Galav College, Morena, 476001, M.P., India
article
info
Article history: Received 14 April 2011 Received in revised form 19 September 2011 Accepted 28 November 2011 by Xincheng Xie Available online 3 December 2011 Keywords: D. Thermal expansion D. Thermodynamic properties D. Equation of state E. High pressure
abstract Unified analyses for P–V –T equations of state of MgO recently presented by several workers in order to obtain a solution for pressure-scale problems in high P–T experiments make use of a new functional form for volume dependence of the Grüneisen parameter. In the present study we point out that this formulation is not consistent with variation of the third order Grüneisen parameter with the change in pressure. We have emphasized that the Stacey–Davis formulation for the volume dependence of the Grüneisen parameter is more appropriate than the other functional form used in the recent literature. The results have been obtained and reported for the volume dependence of the Grüneisen parameter and its higher order derivatives. © 2011 Elsevier Ltd. All rights reserved.
For understanding the high pressure–high temperature behaviour of solids, it is necessary to have a reliable knowledge of pressure P–volume V –temperature T relationships [1]. For this purpose an equation of state (EOS) can be used with the help of different approaches. First, we can use an isothermal EOS to determine P–V relationships along different isotherms at selected temperatures using input parameters corresponding to each temperature [2,3]. The other approach is based on the Mie–Grüneisen–Debye (MGD) model for evaluating the thermal effects in order to determine P–V relationships at high temperatures [4–9]. In this model we need to have an accurate knowledge of the volume dependence of the Grüneisen parameter, γ (V ) [4]. In the present study we emphasize that the relationship for volume dependence of the Grüneisen parameter, γ (V ), originally due to Al’tshuler et al. [10] has been widely used by various workers in the recent literature [4–9,11,12]. An analysis of this relationship performed here reveals its inadequacy for predicting the variation of third-order Grüneisen parameter with the change in pressure [13–15]. We find that the Stacey–Davis formulation for the volume dependence of the Grüneisen parameter is more appropriate than the relationship due to Al’tshuler et al. [10]. The results have been obtained and reported for the volume dependences of γ and higher-order derivatives in the case of MgO.
The Grüneisen parameter γ is an important thermoelastic property related to other thermal and elastic properties as follows [1]:
γ =
α KT V CV
α KS V
=
CP
,
where α is the thermal expansivity, V the volume, KT (KS ) the isothermal (adiabatic) bulk modulus, and CV (CP ) the constant volume (pressure) specific heat. The second and third-order Grüneisen parameters are defined as [16]:
q=
d ln γ d ln V
= T
V
γ
Correspondence to: HIG-933, Housing Board Colony, Morena, 476001, India. Tel.: +91 7532232271; fax: +91 7532400767. E-mail address: [email protected] (S.S. Kushwah). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.11.041
dγ dV
(2) T
and
λ=
d ln q d ln V
= T
V
q
dq dV
.
(3)
T
The parameters γ , q and λ appear frequently in the thermodynamic identities for higher order thermoelastic properties of solids [14–16], and their knowledge is not only desirable, but also necessary. Al’tshuler et al. [10] have presented the following relationship for the volume dependence of the Grüneisen parameter
γ = γ∞ + (γ0 − γ∞ ) ∗
(1)
V V0
m
,
(4)
where γ0 and γ∞ are the values of γ at V = V0 and V → 0, respectively and m is a constant. Eq. (4) has been used recently
S.S. Kushwah, M.P. Sharma / Solid State Communications 152 (2012) 414–416
415
by numerous workers [4–9,11,12]. Eq. (4), on differentiation with respect to V , yields the following expressions for higher order parameters q=
m
γ
(γ − γ∞ )
(5)
γ∞ . γ
(6)
and
λ=m
In the limit V → 0 or P → ∞, γ → γ∞ and therefore q → 0 (Eq. (5)), and from Eq. (6)
γ∞ = m
(7)
and
γ λ = γ∞ λ∞ .
(8)
Thus Eq. (4) satisfies the thermodynamic constraints viz. finite values of γ∞ and λ∞ , and q∞ → 0. Eqs. (5) and (6) give the following relations also q+λ=m
Fig. 1. Values of volume dependence of the Grüneisen parameter γ for MgO.
(9)
and q0 + λ0 = q∞ + λ∞ = λ∞ .
(10)
Since γ and q both decrease with the increase in pressure, λ must increase with the increase in pressure according to Eq. (8) as well as Eq. (10). However, this finding is not supported by the seismic data for the lower mantle and the core of the Earth [17]. It has been found [13–17] that λ decreases with the increase in pressure, and therefore Eq. (4), on which Eqs. (8) and (10) are based, is inadequate. Thermodynamic identities for higher derivative properties [15,16] also reveal that λ decreases with the increase in pressure or compression. This shortcoming can be removed by using the model due to Stacey and Davis [17] who have used a linear relationship between λ and q supported by the seismic data. Eqs. (103)–(105) given in Ref. [17] yield the following relationships
γ∞ = γ0
λ0 λ∞
− λ
q0 0 −λ∞
(11)
Fig. 2. Values of volume dependence of the second order Grüneisen parameter q for MgO.
and
(λ0 − λ∞ ) 1− λ0
γ = γ∞
V V0
q0 λ∞ − λ0 −λ ∞
.
(12)
We have used Eq. (12) to determine the values of Grüneisen parameter as a function of volume ratio (V /V0 ) for MgO. The value of γ0 (1.52) has been taken from the recent literature [4–6,9]. The value of q0 (1.41) has been determined from the modified free volume theory [15] using the Stacey reciprocal K -primed equation of state [16,17]. It should be mentioned that the value of q0 thus determined is in close agreement with the value of q0 = 1.406 for MgO taken by Wu et al. [18]. The value of λ∞ is determined using the relationship [13].
λ∞ =
′2 K∞
K0′
,
(13)
′ where K0′ and K∞ are the values of K ′ = dK /dP, pressure derivative of bulk modulus at P = 0 and P → ∞, respectively. We have ′ taken K0′ = 4.15 and K∞ = 2.49 so as to satisfy the relationship [16,17,19] ′ K∞
K0′
=
3 5
.
(14)
Eqs. (13) and (14) then yield λ∞ = 1.49 for MgO. The value of γ∞ can be determined from the thermodynamic identity [16,17]
γ∞ =
′ K∞
2
1
− . 6
(15)
Eq. (15) gives γ∞ = 1.08 for MgO. The value of λ0 satisfying Eq. (11) turns out to be equal to 8.84. These values of various parameters viz. γ0 , γ∞ , q0 , λ0 and λ∞ are used to evaluate the volume dependences of γ , q and λ. The results for MgO are shown in Figs. 1–3. It should be mentioned that the results obtained in the present study using the Stacey–Davis formulation satisfy the thermodynamic constraints for the solids in the limit of extreme compression [16,17]. It is found that λ0 equal to 8.84 is significantly larger than λ∞ equal to 1.49 suggesting that λ decreases rapidly with the increase in pressure, and attains a constant value in the limit of infinite pressure. The corresponding results for γ (V ), q(V ) and λ(V ) obtained from the Al’tshuler formulation, Eqs. (4)–(10), are also included in Figs. 1–3 for the sake of comparison. For the Al’tshuler model we need only three parameters γ0 , γ∞ and q0 . These are γ0 = 1.52, γ∞ = 1.08 and q0 = 1.41 in the case of MgO. The results based on the two formulations differ substantially from each other particularly for λ. Extrapolation of the analysis to MgO
416
S.S. Kushwah, M.P. Sharma / Solid State Communications 152 (2012) 414–416
parameter obtained by Stacey and Davis give the correct trend of variation of λ. References
Fig. 3. Values of volume dependence of the third order Grüneisen parameter λ for MgO.
is appropriate, as it abundant in the mantle. Experimental data or accurate calculations for λ are not available for presenting a comparison. The values of λ, the third derivative of Grüneisen
[1] O.L. Anderson, Equation of State of Solids for Geophysics and Ceramic Science, Oxford University Press, New York, 1995. [2] S.S. Kushwah, J. Shanker, Physica B 253 (1998) 90. [3] S.S. Kushwah, H.C. Shrivastava, K.S. Singh, Physica B 388 (2007) 20. [4] S. Speziale, C.S. Zha, T.S. Duffy, R.J. Hemley, H.K. Mao, J. Geophys. Res. 106B (2001) 515. [5] P.I. Dorogokupets, A. Dewaele, High Pressure Res. 27 (2007) 431. [6] P.I. Dorogokupets, A.R. Oganov, Phys. Rev. B 75 (2007) 24115. [7] A.B. Belonoshko, P.I. Dorogokupets, B. Johansson, S.K. Saxena, L. Koci, Phys. Rev. B 78 (2008) 104107. [8] Y. Tange, Y. Nishihara, T. Tsuchiya, J. Geophys. Res. 114 (2009) B03208. doi:10.1029/2008JB005813. [9] P.I. Dorogokupets, Phys. Chem. Minerals (2010) doi:10.1007/s00269-0100367-2. [10] L.V. Al’tshuler, S.E. Brusnikin, E.A. Kuz’menkov, J. Appl. Mech. Tech. Phys. 28 (1987) 129. [11] W.B. Holzapfel, M. Hartwig, W. Sievers, J. Phys. Chem. Ref. Data 30 (2001) 515. [12] J. Shanker, B.P. Singh, H.K. Baghel, Physica B 387 (2007) 409. [13] J. Shanker, B.P. Singh, Physica B 370 (2005) 78. [14] J. Shanker, B.P. Singh, K. Jitendra, Condens. Matter Phys. 12 (2009) 205. [15] S.S. Kushwah, N.K. Bhardwaj, Int. J. Mod. Phys. B 24 (2010) 1187. [16] F.D. Stacey, Rep. Progr. Phys. 68 (2005) 341. [17] F.D. Stacey, P.M. Davis, Phys. Earth Planet. Inter. 142 (2004) 137. [18] Z. Wu, R.M. Wentzcovitch, K. Umemoto, B. Li, K. Hirose, J.C. Zheng, J. Geophys. Res. 113 (2008) B06204. doi:10.1029/2007 JB005275. [19] S.S. Kushwah, N.K. Bhardwaj, J. Phys. Chem. Solids 70 (2009) 700.