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Mixed temperature and pressure derivative of isothermal bulk modulus for Debye like solid
Solid State Sciences 11 (2009) 918–921
Contents lists available at ScienceDirect
Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie
Mixed temperature and pressure derivative of isothermal bulk modulus for Debye like solid S.K. Sharma* Shivalik Institute of Engineering and Technology, Aliyaspur, Ambala 133206, Haryana, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 July 2008 Received in revised form 26 September 2008 Accepted 15 October 2008 Available online 1 November 2008
In the present paper we propose three relationships to predict the values of the mixed temperature and pressure derivative of isothermal bulk modulus (v2KT/vPvT) at zero pressure based on different assumptions. It is found that (v2KT/vPvT) changes linearly with increase in temperature above Debye temperature at zero pressure. The relationships developed in this study are applied to MgO and the predicted values of (v2KT/vPvT) compare favorably with the experimental results. Ó 2008 Elsevier Masson SAS. All rights reserved.
PACS: 91.60Gf 66.70 Keywords: Isothermal bulk modulus Anderson–Gruneisen parameter Mixed temperature and pressure derivative of isothermal bulk modulus MgO
1. Introduction The temperature and pressure dependence of bulk modulus is an essential factor which determines the mechanical behavior of materials in the area of high temperature and high pressure field. A precise knowledge of these parameters is therefore essential in devising reliable equation of state of materials that is applicable over an extended range of temperature and pressure. While there have been many phenomenological treatments of the individual pressure or temperature dependence of bulk modulus, giving rise to diverse isothermal or isobaric equations of state, a comprehensive thermodynamic analysis of the mixed temperature and pressure derivative of isothermal bulk modulus, namely (v2KT/vPvT), is rarely comparative. The parameter (v2KT/vPvT) can be interpreted as the temperature derivative of KT0 i.e., ðvKT0 =vTÞP . It is often the case that the numerical value of accurate experimental determination is also fraught with many difficulties [1]. Various researchers [2,3] have recently studied the thermodynamics of (v2KT/vPvT) and suggested somewhat implicitly that this quantity may be considered as nearly zero for all practical purposes. Further, it also emerges from their work that the temperature dependence of bulk
* Tel.: þ91 9416670488. E-mail address: [email protected] 1293-2558/$ – see front matter Ó 2008 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2008.10.004
modulus can be effectively treated as a function of only volume. This means that the pure thermal dependence of bulk modulus namely (vKT/vT)V is zero. As discussed in detail by Anderson [1], this is a very useful and proven high temperature approximation for many minerals of geophysical interest. Raju et al. [3] derived a thermodynamic framework for evaluating the mixed pressure (P) and temperature (T) derivative of isothermal bulk modulus (v2KT/vPvT) under the assumption that the pure thermal derivative of isothermal bulk modulus (vKT/vT)V is zero. Raju et al. [3] formulated the following three relationships (Eqs. (1)–(3)) to evaluate the values of (v2KT/vPvT) in the following manners
v2 K T vPvT v2 K T vPvT
!
¼
!
¼
vdT vT
g bP;T
1 vV ¼ ¼ dT a bP;T vT P P
vCV vP
(1)
(2) T
and
v2 K T vPvT
! ¼ kCP
(3)
S.K. Sharma / Solid State Sciences 11 (2009) 918–921
where bP,T is related to the isothermal Anderson–Gruneisen parameter dT, through following relation [4]
bP;T ¼
2. Method of analysis Anderson and Isaak [8] used the following relationship
V
(4)
dT
dT ¼ dT0
V k V0
and a is thermal expansivity which is defined as
1 vV V vT P
a ¼
(5)
g, CP, CV stand, respectively, for thermal Gruneisen parameter, isobaric and isochoric heats and k is a temperature independent thermoelastic constant and correlated with isothermal bulk modulus (KT) and enthalpy (H) as vln KT k ¼ vH P
(6)
Various researchers [5–7] used the following thermodynamic identity
0
vKT vT
P
¼ adT dT KT0 þ
vln dT vln V
dT ¼ dT0
vKT0 v2 KT v vKT v ¼ ¼ ¼ ðaKT dT Þ vPvT vT vP vT vP
vKT vT
¼ aKT dT
(9)
and
vP KT ¼ V vV
(10)
vKT vT
¼ P
vKT vT
aKT
V
vKT vP
vKT vT
¼ aV P
P
v2 KT vPvT
vKT vV
V vKT KT vV T
(16)
¼ dT0 aKT
(17)
! ¼ adT0
V V0
(18)
Eq. (18) suggests that it is possible to evaluate (v2KT/vPvT) from a knowledge of a, dT0 and (V/V0). In literature [12–14] the interrelationship between isothermal bulk modulus and volume expansion ratio is also shown as following
KT ¼ K0
V V0
m
(19)
where m is a constant. In Ref. [15] m is p. Differentiating Eq. (19) with respect to temperature, we get
(11)
vKT vT
¼ maKT
(20)
P
Inserting Eqs. (1) and (20) into Eq. (9), we have
v2 KT vPvT
(12)
! ¼ ma
(21)
T
Comparing Eqs. (9) and (12) now we have
dT ¼
T
(15)
Inserting Eqs. (1) and (17) into Eq. (9), we get
Using (v2KT/vT)V ¼ 0 and Eq. (10) into Eq. (11), we get
vKT vT
T
Raju et al. [3] used the following thermodynamic identity to formulate Eq. (1)
Differentiating Eq. (16) with respect to temperature, we obtain
(8)
P
V V0
V KT ¼ K0 exp dT0 1 V0
where
Eq. (15) is known as Tallon model [12]. Now following the Tallon model [12], isothermal bulk modulus can be written as
(7)
T
(14)
where k is a dimensionless thermoelastic parameter. k ¼ 1 has been widely used for developing an equation of state and for investigating thermoelastic properties of solids [3,5,9–11], Eq. (14) now becomes
The term (vln dT/vln V)T in the above equation is a dimensionless thermoelastic parameter i.e., k. The term (v2KT/vPvT) may also be expressed as
919
(13)
In the present study, we have derived thermodynamic relationships to evaluate the values of (v2KT/vPvT). These relationships are based on different assumptions which are to be given in next section. Since MgO is a material of key importance to Earth Sciences and solid state physics: it is one of the most abundant minerals in the Earth (especially its lower mantle) and a prototype material for a large group of ionic oxides and the experimental data on (v2KT/vPvT) are easily available for solid under consideration. Therefore we have taken an example of Debye like solid MgO to test the validity of present relationships. The method of analysis is given in Section 2. Results are discussed and compared in Section 3.
Using the values of m and temperature dependent a, the values of (v2KT/vPvT) are estimated. Following the work of Chopelas and Boehler [16], we consider dT to depend on volume as
V V0
dT þ 1 ¼ dT0 þ 1
(22)
isothermal bulk modulus can be written as
V V 1 exp dT0 þ 1 V0 V0
KT ¼ K0
(23)
Differentiating Eq. (23) with respect to temperature, we get
vKT vT
V ¼ aKT 1 dT0 þ 1 V0 P
Combining Eqs. (1), (9) and (24), we obtain
(24)
920
v2 KT vPvT
S.K. Sharma / Solid State Sciences 11 (2009) 918–921
! ¼ a
V V0
dT0 þ 1
1
MgO
2.90
(25)
2.65
3. Results and discussion
Table 1 The values of (v2KT/vPvT) for MgO estimated using different Eqs. (18), (21) and (25). T (K)
V/V0 [5]
a (105 K1) [5]
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Average over 300–1800 K
1.0000 1.0034 1.0073 1.0113 1.0153 1.0196 1.0240 1.0284 1.0331 1.0379 1.0428 1.0476 1.0529 1.0582 1.0635 1.0689 –
3.12 3.57 3.84 4.02 4.14 4.26 4.38 4.47 4.56 4.65 4.71 4.80 4.89 4.98 5.04 5.13 –
(v2KT/vPvT) (104 K1) Eq. (18)
Eq. (21)
Eq. (25)
[3]
1.49 1.71 1.84 1.94 2.00 2.07 2.13 2.19 2.24 2.30 2.34 2.39 2.45 2.51 2.55 2.61 2.17
1.54 1.76 1.89 1.98 2.04 2.10 2.15 2.20 2.24 2.29 2.32 2.36 2.41 2.45 2.48 2.52 2.17
1.49 1.71 1.84 1.94 2.01 2.08 2.15 2.20 2.26 2.31 2.36 2.42 2.48 2.54 2.58 2.65 2.19
1.53 1.71 1.84 1.93 2.01 2.07 2.14 2.20 2.26 2.32 2.37 2.44 2.50 2.56 2.63 2.70 2.20
2.40
d2KT/dPdT
Herein we have derived three relationships, based on different assumptions [12–14] which are cited in the form of Eqs. (18), (21) and (25). As we have written dT0 in two equations (Eqs. (21) and (25)). The value of dT0 ¼ 4:76 for MgO is taken from Singh and Chandra [17]. And the value of m ¼ 4.92 used in Eq. (21) to estimate the values of (v2KT/vPvT) for the solid under consideration is adopted from Ref. [15]. The calculated results are enlisted in Table 1 along with volume ratio based on density measurements [5] and thermal expansivity data [5]. We have applied the derived equations (Eqs. (18), (21) and (25)) to calculate (v2KT/vPvT) in temperature range 300–1800 K at zero pressure. Raju et al. [3] viewed that the approximation given by Eq. (1) and originating from the main assumption of the study may be considered as the basic one; others arise as a result of subsequent simplifications brought on to the main result. Eq. (1) is identical with Eq. (7) for dT ¼ KT0 and (v2ln dT/ vln V)T ¼ 1. Since Eq. (1) is proved identical for thermodynamic identity (Eq. (7))under some constraints. So we have chosen Eq. (1) for extending the work of Raju et al. [3]. And for the sake of comparison, we have taken the results based on Eq. (1) in the last column of Table 1 and the corresponding curve is also plotted in Fig. 1 for the sake of direct vision comparison. Fig. 1 reflects that the value of (v2KT/vPvT) is increased with increasing temperature, i.e. (v2KT/vPvT) is positive [5]. It has also been found that the mixed pressure and temperature derivative of isothermal bulk modulus (v2KT/vPvT) at zero pressure is to be linear above Debye temperature i.e., T > QD. This study gives us an idea about the nature of (v2KT/ vPvT) that all derived relationships are directly dependent on thermal expansivity. This in turn leads to nature of (v2KT/vPvT) as shown in Fig. 1. Roberts and Smith [18] showed that (v2KT/vT)V arises solely from the change in KT at constant volume, calling it an intrinsic change, whereas (vKT/vT)P has an effect arising from thermal expansivity (a). Smith and Cain [19] pointed out that (vKT/vT)V measures the explicit dependence of KT on T, since the effect of thermal expansion has been removed. For MgO, a description of KT0 vs T and P and dT vs T and P has been obtained by using the potential induced breathing (PIB) electron gas model, which is a type of first principles approach [17]. By this method the Helmholtz energy, F vs V at constant T at selected temperatures is computed. Isaak et al. [20] assumed the
2.15 Eq.(18) Eq.(21) Eq.(25)
1.90
Raju et al [3]
1.65
1.40 300
500
700
900
1100
1300
1500
1700
1900
Temperature (K) Fig. 1. Plots of the (v2KT/vPvT) (104 K1) vs temperature (K) for MgO using different relationships.
quasiharmonic approximation i.e., explicit effects of anharmonicity are not accounted for. However, the Isaak et al. [20] results are in very good agreement with measurements of the temperature dependence of several thermoelastic properties. As we can see from Table 1, there is a close agreement between different estimates of (v2KT/vPvT), especially in view of the possible uncertainties in the experimental thermal property data. According to Anderson [5] the estimated value of (v2KT/vPvT) is 4.3 104 K1. The experimental value of (v2KT/vPvT) is 3.9 104 K1 given by Isaak [21]. The value of (v2KT/vPvT) or ðvKT0 =vTÞP is 3.0 104 K1 [20]. It is found that the calculated results show better consistency with 3.0 104 K1. 4. Conclusion In the present paper, we have derived three relationships based on different assumptions. We have estimated values of (v2KT/vPvT) at zero pressure using Eqs. (18), (21) and (25). To testify these relationships we have taken an example of Debye like solid MgO. The (v2KT/vPvT) estimated through all relations show linearity above Debye temperature. All presented relationships directly depend on thermal expansivity due to which (v2KT/vPvT) confirm the nature as conveyed in Fig. 1. If we see average value over 300– 1800 K temperature range then we find that the values of (v2KT/ vPvT) are slightly deviated from each other and show better agreement with 3.0 104 K1. However, the results predicted through Eq. (25) are very close to Raju et al. [3] which are cited in last column of Table 1. Acknowledgement I am very thankful to the anonymous reviewers for their helpful comments, which have been useful for revising the manuscript. Thanks are also due to Professor Jai Shanker, Department of Physics, IBS, Khandari, Agra, India. References [1] O.L. Anderson, Thermoelastic parameters of solids, in: M. Levy (Ed.), Handbook of Elastic Properties of Solids, Liquids and Gases, Vol. III, Academic Press, San Diego, 2001. [2] H. Jeon, B. Cho, J. Appl. Phys. 88 (2001) 6084. [3] S. Raju, K. Sivasubramanian, E. Mohandas, Physica B 324 (2002) 312.
S.K. Sharma / Solid State Sciences 11 (2009) 918–921 [4] S. Raju, K. Sivasubramanian, E. Mohandas, Phys. Chem. Chem. Phys. 3 (2001) 1391. [5] O.L. Anderson, Equations of State of Solids for Geophysics and Ceramic Science, Oxford University Press, New York, 1995. [6] S. Rekha, B.S. Sharma, HT-HP 35/36 (2004) 337. [7] F.D. Stacey, Rep. Prog. Phys. 68 (2005) 341. [8] O.L. Anderson, D.G. Isaak, J. Phys. Chem. Solids 54 (1993) 221. [9] M. Kumar, Solid State Commun. 92 (1994) 463. [10] K. Sushil, Physica B 367 (2005) 114. [11] J. Shanker, B.P. Singh, K. Jitendra, Earth & Life 2/2 (2007) 3.http://www.geofinds.com.
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
J.L. Tallon, J. Phys. Chem. Solids 41 (1980) 837. F.D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A. 30 (1944) 244. A. Vijay, Physica B 349 (2004) 62. S.K. Srivastava, S.K. Sharma, Physica B 373 (2006) 258. A. Chopelas, R. Boehler, Phys. Res. 94 (1989) 5873. B.P. Singh, H. Chandra, Physica B 358 (2005) 1. R.W. Roberts, C.S. Smith, J. Phys. Chem. Solids 31 (1970) 619. C.S. Smith, L.S. Cain, J. Phys. Chem. Solids 41 (1980) 199. D.G. Isaak, R.E. Cohen, M.J. Mehl, J. Geophys. Res. 95 (1990) 7055. D.G. Isaak, Phys. Earth Planet. Inter. 80 (1993) 37 as quoted in Ref. [5].
921
Contents lists available at ScienceDirect
Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie
Mixed temperature and pressure derivative of isothermal bulk modulus for Debye like solid S.K. Sharma* Shivalik Institute of Engineering and Technology, Aliyaspur, Ambala 133206, Haryana, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 July 2008 Received in revised form 26 September 2008 Accepted 15 October 2008 Available online 1 November 2008
In the present paper we propose three relationships to predict the values of the mixed temperature and pressure derivative of isothermal bulk modulus (v2KT/vPvT) at zero pressure based on different assumptions. It is found that (v2KT/vPvT) changes linearly with increase in temperature above Debye temperature at zero pressure. The relationships developed in this study are applied to MgO and the predicted values of (v2KT/vPvT) compare favorably with the experimental results. Ó 2008 Elsevier Masson SAS. All rights reserved.
PACS: 91.60Gf 66.70 Keywords: Isothermal bulk modulus Anderson–Gruneisen parameter Mixed temperature and pressure derivative of isothermal bulk modulus MgO
1. Introduction The temperature and pressure dependence of bulk modulus is an essential factor which determines the mechanical behavior of materials in the area of high temperature and high pressure field. A precise knowledge of these parameters is therefore essential in devising reliable equation of state of materials that is applicable over an extended range of temperature and pressure. While there have been many phenomenological treatments of the individual pressure or temperature dependence of bulk modulus, giving rise to diverse isothermal or isobaric equations of state, a comprehensive thermodynamic analysis of the mixed temperature and pressure derivative of isothermal bulk modulus, namely (v2KT/vPvT), is rarely comparative. The parameter (v2KT/vPvT) can be interpreted as the temperature derivative of KT0 i.e., ðvKT0 =vTÞP . It is often the case that the numerical value of accurate experimental determination is also fraught with many difficulties [1]. Various researchers [2,3] have recently studied the thermodynamics of (v2KT/vPvT) and suggested somewhat implicitly that this quantity may be considered as nearly zero for all practical purposes. Further, it also emerges from their work that the temperature dependence of bulk
* Tel.: þ91 9416670488. E-mail address: [email protected] 1293-2558/$ – see front matter Ó 2008 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2008.10.004
modulus can be effectively treated as a function of only volume. This means that the pure thermal dependence of bulk modulus namely (vKT/vT)V is zero. As discussed in detail by Anderson [1], this is a very useful and proven high temperature approximation for many minerals of geophysical interest. Raju et al. [3] derived a thermodynamic framework for evaluating the mixed pressure (P) and temperature (T) derivative of isothermal bulk modulus (v2KT/vPvT) under the assumption that the pure thermal derivative of isothermal bulk modulus (vKT/vT)V is zero. Raju et al. [3] formulated the following three relationships (Eqs. (1)–(3)) to evaluate the values of (v2KT/vPvT) in the following manners
v2 K T vPvT v2 K T vPvT
!
¼
!
¼
vdT vT
g bP;T
1 vV ¼ ¼ dT a bP;T vT P P
vCV vP
(1)
(2) T
and
v2 K T vPvT
! ¼ kCP
(3)
S.K. Sharma / Solid State Sciences 11 (2009) 918–921
where bP,T is related to the isothermal Anderson–Gruneisen parameter dT, through following relation [4]
bP;T ¼
2. Method of analysis Anderson and Isaak [8] used the following relationship
V
(4)
dT
dT ¼ dT0
V k V0
and a is thermal expansivity which is defined as
1 vV V vT P
a ¼
(5)
g, CP, CV stand, respectively, for thermal Gruneisen parameter, isobaric and isochoric heats and k is a temperature independent thermoelastic constant and correlated with isothermal bulk modulus (KT) and enthalpy (H) as vln KT k ¼ vH P
(6)
Various researchers [5–7] used the following thermodynamic identity
0
vKT vT
P
¼ adT dT KT0 þ
vln dT vln V
dT ¼ dT0
vKT0 v2 KT v vKT v ¼ ¼ ¼ ðaKT dT Þ vPvT vT vP vT vP
vKT vT
¼ aKT dT
(9)
and
vP KT ¼ V vV
(10)
vKT vT
¼ P
vKT vT
aKT
V
vKT vP
vKT vT
¼ aV P
P
v2 KT vPvT
vKT vV
V vKT KT vV T
(16)
¼ dT0 aKT
(17)
! ¼ adT0
V V0
(18)
Eq. (18) suggests that it is possible to evaluate (v2KT/vPvT) from a knowledge of a, dT0 and (V/V0). In literature [12–14] the interrelationship between isothermal bulk modulus and volume expansion ratio is also shown as following
KT ¼ K0
V V0
m
(19)
where m is a constant. In Ref. [15] m is p. Differentiating Eq. (19) with respect to temperature, we get
(11)
vKT vT
¼ maKT
(20)
P
Inserting Eqs. (1) and (20) into Eq. (9), we have
v2 KT vPvT
(12)
! ¼ ma
(21)
T
Comparing Eqs. (9) and (12) now we have
dT ¼
T
(15)
Inserting Eqs. (1) and (17) into Eq. (9), we get
Using (v2KT/vT)V ¼ 0 and Eq. (10) into Eq. (11), we get
vKT vT
T
Raju et al. [3] used the following thermodynamic identity to formulate Eq. (1)
Differentiating Eq. (16) with respect to temperature, we obtain
(8)
P
V V0
V KT ¼ K0 exp dT0 1 V0
where
Eq. (15) is known as Tallon model [12]. Now following the Tallon model [12], isothermal bulk modulus can be written as
(7)
T
(14)
where k is a dimensionless thermoelastic parameter. k ¼ 1 has been widely used for developing an equation of state and for investigating thermoelastic properties of solids [3,5,9–11], Eq. (14) now becomes
The term (vln dT/vln V)T in the above equation is a dimensionless thermoelastic parameter i.e., k. The term (v2KT/vPvT) may also be expressed as
919
(13)
In the present study, we have derived thermodynamic relationships to evaluate the values of (v2KT/vPvT). These relationships are based on different assumptions which are to be given in next section. Since MgO is a material of key importance to Earth Sciences and solid state physics: it is one of the most abundant minerals in the Earth (especially its lower mantle) and a prototype material for a large group of ionic oxides and the experimental data on (v2KT/vPvT) are easily available for solid under consideration. Therefore we have taken an example of Debye like solid MgO to test the validity of present relationships. The method of analysis is given in Section 2. Results are discussed and compared in Section 3.
Using the values of m and temperature dependent a, the values of (v2KT/vPvT) are estimated. Following the work of Chopelas and Boehler [16], we consider dT to depend on volume as
V V0
dT þ 1 ¼ dT0 þ 1
(22)
isothermal bulk modulus can be written as
V V 1 exp dT0 þ 1 V0 V0
KT ¼ K0
(23)
Differentiating Eq. (23) with respect to temperature, we get
vKT vT
V ¼ aKT 1 dT0 þ 1 V0 P
Combining Eqs. (1), (9) and (24), we obtain
(24)
920
v2 KT vPvT
S.K. Sharma / Solid State Sciences 11 (2009) 918–921
! ¼ a
V V0
dT0 þ 1
1
MgO
2.90
(25)
2.65
3. Results and discussion
Table 1 The values of (v2KT/vPvT) for MgO estimated using different Eqs. (18), (21) and (25). T (K)
V/V0 [5]
a (105 K1) [5]
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Average over 300–1800 K
1.0000 1.0034 1.0073 1.0113 1.0153 1.0196 1.0240 1.0284 1.0331 1.0379 1.0428 1.0476 1.0529 1.0582 1.0635 1.0689 –
3.12 3.57 3.84 4.02 4.14 4.26 4.38 4.47 4.56 4.65 4.71 4.80 4.89 4.98 5.04 5.13 –
(v2KT/vPvT) (104 K1) Eq. (18)
Eq. (21)
Eq. (25)
[3]
1.49 1.71 1.84 1.94 2.00 2.07 2.13 2.19 2.24 2.30 2.34 2.39 2.45 2.51 2.55 2.61 2.17
1.54 1.76 1.89 1.98 2.04 2.10 2.15 2.20 2.24 2.29 2.32 2.36 2.41 2.45 2.48 2.52 2.17
1.49 1.71 1.84 1.94 2.01 2.08 2.15 2.20 2.26 2.31 2.36 2.42 2.48 2.54 2.58 2.65 2.19
1.53 1.71 1.84 1.93 2.01 2.07 2.14 2.20 2.26 2.32 2.37 2.44 2.50 2.56 2.63 2.70 2.20
2.40
d2KT/dPdT
Herein we have derived three relationships, based on different assumptions [12–14] which are cited in the form of Eqs. (18), (21) and (25). As we have written dT0 in two equations (Eqs. (21) and (25)). The value of dT0 ¼ 4:76 for MgO is taken from Singh and Chandra [17]. And the value of m ¼ 4.92 used in Eq. (21) to estimate the values of (v2KT/vPvT) for the solid under consideration is adopted from Ref. [15]. The calculated results are enlisted in Table 1 along with volume ratio based on density measurements [5] and thermal expansivity data [5]. We have applied the derived equations (Eqs. (18), (21) and (25)) to calculate (v2KT/vPvT) in temperature range 300–1800 K at zero pressure. Raju et al. [3] viewed that the approximation given by Eq. (1) and originating from the main assumption of the study may be considered as the basic one; others arise as a result of subsequent simplifications brought on to the main result. Eq. (1) is identical with Eq. (7) for dT ¼ KT0 and (v2ln dT/ vln V)T ¼ 1. Since Eq. (1) is proved identical for thermodynamic identity (Eq. (7))under some constraints. So we have chosen Eq. (1) for extending the work of Raju et al. [3]. And for the sake of comparison, we have taken the results based on Eq. (1) in the last column of Table 1 and the corresponding curve is also plotted in Fig. 1 for the sake of direct vision comparison. Fig. 1 reflects that the value of (v2KT/vPvT) is increased with increasing temperature, i.e. (v2KT/vPvT) is positive [5]. It has also been found that the mixed pressure and temperature derivative of isothermal bulk modulus (v2KT/vPvT) at zero pressure is to be linear above Debye temperature i.e., T > QD. This study gives us an idea about the nature of (v2KT/ vPvT) that all derived relationships are directly dependent on thermal expansivity. This in turn leads to nature of (v2KT/vPvT) as shown in Fig. 1. Roberts and Smith [18] showed that (v2KT/vT)V arises solely from the change in KT at constant volume, calling it an intrinsic change, whereas (vKT/vT)P has an effect arising from thermal expansivity (a). Smith and Cain [19] pointed out that (vKT/vT)V measures the explicit dependence of KT on T, since the effect of thermal expansion has been removed. For MgO, a description of KT0 vs T and P and dT vs T and P has been obtained by using the potential induced breathing (PIB) electron gas model, which is a type of first principles approach [17]. By this method the Helmholtz energy, F vs V at constant T at selected temperatures is computed. Isaak et al. [20] assumed the
2.15 Eq.(18) Eq.(21) Eq.(25)
1.90
Raju et al [3]
1.65
1.40 300
500
700
900
1100
1300
1500
1700
1900
Temperature (K) Fig. 1. Plots of the (v2KT/vPvT) (104 K1) vs temperature (K) for MgO using different relationships.
quasiharmonic approximation i.e., explicit effects of anharmonicity are not accounted for. However, the Isaak et al. [20] results are in very good agreement with measurements of the temperature dependence of several thermoelastic properties. As we can see from Table 1, there is a close agreement between different estimates of (v2KT/vPvT), especially in view of the possible uncertainties in the experimental thermal property data. According to Anderson [5] the estimated value of (v2KT/vPvT) is 4.3 104 K1. The experimental value of (v2KT/vPvT) is 3.9 104 K1 given by Isaak [21]. The value of (v2KT/vPvT) or ðvKT0 =vTÞP is 3.0 104 K1 [20]. It is found that the calculated results show better consistency with 3.0 104 K1. 4. Conclusion In the present paper, we have derived three relationships based on different assumptions. We have estimated values of (v2KT/vPvT) at zero pressure using Eqs. (18), (21) and (25). To testify these relationships we have taken an example of Debye like solid MgO. The (v2KT/vPvT) estimated through all relations show linearity above Debye temperature. All presented relationships directly depend on thermal expansivity due to which (v2KT/vPvT) confirm the nature as conveyed in Fig. 1. If we see average value over 300– 1800 K temperature range then we find that the values of (v2KT/ vPvT) are slightly deviated from each other and show better agreement with 3.0 104 K1. However, the results predicted through Eq. (25) are very close to Raju et al. [3] which are cited in last column of Table 1. Acknowledgement I am very thankful to the anonymous reviewers for their helpful comments, which have been useful for revising the manuscript. Thanks are also due to Professor Jai Shanker, Department of Physics, IBS, Khandari, Agra, India. References [1] O.L. Anderson, Thermoelastic parameters of solids, in: M. Levy (Ed.), Handbook of Elastic Properties of Solids, Liquids and Gases, Vol. III, Academic Press, San Diego, 2001. [2] H. Jeon, B. Cho, J. Appl. Phys. 88 (2001) 6084. [3] S. Raju, K. Sivasubramanian, E. Mohandas, Physica B 324 (2002) 312.
S.K. Sharma / Solid State Sciences 11 (2009) 918–921 [4] S. Raju, K. Sivasubramanian, E. Mohandas, Phys. Chem. Chem. Phys. 3 (2001) 1391. [5] O.L. Anderson, Equations of State of Solids for Geophysics and Ceramic Science, Oxford University Press, New York, 1995. [6] S. Rekha, B.S. Sharma, HT-HP 35/36 (2004) 337. [7] F.D. Stacey, Rep. Prog. Phys. 68 (2005) 341. [8] O.L. Anderson, D.G. Isaak, J. Phys. Chem. Solids 54 (1993) 221. [9] M. Kumar, Solid State Commun. 92 (1994) 463. [10] K. Sushil, Physica B 367 (2005) 114. [11] J. Shanker, B.P. Singh, K. Jitendra, Earth & Life 2/2 (2007) 3.http://www.geofinds.com.
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
J.L. Tallon, J. Phys. Chem. Solids 41 (1980) 837. F.D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A. 30 (1944) 244. A. Vijay, Physica B 349 (2004) 62. S.K. Srivastava, S.K. Sharma, Physica B 373 (2006) 258. A. Chopelas, R. Boehler, Phys. Res. 94 (1989) 5873. B.P. Singh, H. Chandra, Physica B 358 (2005) 1. R.W. Roberts, C.S. Smith, J. Phys. Chem. Solids 31 (1970) 619. C.S. Smith, L.S. Cain, J. Phys. Chem. Solids 41 (1980) 199. D.G. Isaak, R.E. Cohen, M.J. Mehl, J. Geophys. Res. 95 (1990) 7055. D.G. Isaak, Phys. Earth Planet. Inter. 80 (1993) 37 as quoted in Ref. [5].
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