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On the high-pressure equation of state for solids
PflYSICAi ELSEVIER
Physica B 229 (1997) 419 420
Comment On the high-pressure equation of state for solids J. Shanker*, B. Singh, S.S.
Kushwah 1
Department of Physics, Institute of Basic Sciences, Khandari, Agra 282 002, India Received 21 June 1996
Abstract It is found that the high-pressure equation of state recently reported by K umar [-Physica B 212 (1995) 391, 217 (1996) 143] is not a new equation. This is the same as the usual Tait equation of state which is well known in the literature and frequently used by several investigators. It is emphasised that the usual Tait equation of state is consistent with the Chopelas-Boehler relationship for the variation of the Anderson Gruneisen parameter d r with compression (V/Vo).
Keywords: High pressure; Equation of state Recently, Kumar [1, 2] derived an equation of state to study the pressure-volume relationship for different classes of solids. Using some thermodynamic approximations, he obtained the following equation:
(UTE) frequently used in the literature [3 8] and given below [5, 8]:
(1)
where AV = V 0 - V. Eq. (3) can be rewritten as follows:
P
where P is pressure, V/Vo is compression, Bo is the isothermal bulk modulus at zero pressure and A = B~ + 1. Here B~ is the pressure derivative of isothermal bulk modulus at zero pressure. Eq. (1) can be rewritten as follows:
It would be interesting to compare this equation with the well known usual Tait equation of state
AV_ 1 In I 1 + (1 - +B~) - P Vo (B; + 1) Bo
(B~+l)(1-~o>=ln[1-+
( B °l '+BI o) P
(3)
(4)
On comparing Eqs. (2) and (4) it is found that these are identical. Thus, Eq. (1) reported by Kumar [1, 2] is same as the usual Tait equation (3). The expression for isothermal bulk modulus based on the UTE has been reported in the literature [3] and given below
*Corresponding author. LPermanent address: Rishi Galav College, Morena (MP), lndia. 0921-4526/'97,,'$17.00 Copyright PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 5 2 8 -
],
1997 Elsevier Science B.V. All rights reserved 5
420
a( Shanker et al. / Physica B 229 (1997) 419 420
On the other hand, Kumar reported the following expression for isothermal bulk modulus B [2]: B=
O~ooeXp (B•+I)
1-~oo
"
(6)
Eqs. (5) and (6) are same in view of Eq. (2). Substituting Eq. (2) in (5) we get (6). Thus, Eqs. (5) and (6) both are based on the UTE [Eq. (3)]. Now, if we use the well-known thermodynamic approximation [9 12] according to which the product of thermal expansivity ~ and isothermal bulk modulus B remains constant for a given material under the effect of pressure, we can write :~B = ~oBo,
(7)
where :~o and Bo are the values of :~ and B at zero pressure. If Eq. (7) holds good we have another useful approximation for the Anderson-Gruneisen parameter C~r given below [13] l(dB~ 6r -
--
(8)
1+
(9)
Bo
and --=
If one makes use of the approximation given by (8), then Eq. (11) takes the form V (07-+ 1) = (c~ro + 1)~oo,
(12)
which is the relationship investigated by Chopelas and Boehler [15] on the basis of an analysis of experimental data. In fact, Kumar [1, 2] started his analyses on the basis of (12) and obtained an equation of state which is the same as UTE. The details of UTE including its nomenclature and derivation can be found in Refs. [3, 4].
dB
c~B\dr]e = ~'
At P = 0 we have, therefore, 6ro = B;. Eqs. (5) and (6) with the help of (7) yield the following expressions:
~o
by Kumar further support the validity of the UTE. Finally, it should be pointed out that the UTE is consistent with the relationship given by Chopelas and Boehler [15]. Eqs. (3) and (5) yield the following expression: dB V + 1 = (B; + I ) Vo. (11) d~
exp
-(6to+
1) 1 -
.
(10)
~o
Eqs. (9) and (10) are useful relations for predicting the pressure dependence of ~ along isotherms. It is pertinent to mention here that Eq. (10) is identical with the formula for e reported by Kumar [14]. Thus, the equations reported by Kumar in his recent papers [1, 2, 14] are, in fact, not new but same as those given by the well-known UTE. It has already been found and reported in the literature [7, 8] that the UTE yields good agreement with experimental data. The numerical results obtained
References [lJ M. Kumar, Physica B 212 (1995) 391. [2J M. Kumar, Physica B 217 (1996) 143. [3] J.R. MacDonald, Rev. Mod. Phys. 38 (1996) 669; 41 (1969) 316. I-4] A.T.J. Hayward, Brit. J. Appl. Phys. 18 (1967) 965. [5] J. Freund and R. Ingalls, J. Phys. Chem. Solids 50 (1989) 263. [6] H. Schlosser and J. Ferrante, Phys. Rev. B40 (1989) 6405, and references therein. [7] H. Schlosser and J. Ferrante, J. Phys.: Condens. Matter l (1989) 1941. [8] E.M. Thomas and J. Shanker, Phys. Star. Sol. (B) 189 (1995) 363. [9] J.L. Tallon, J. Phys. Chem. Sol. 41 (1980) 837. [10] F. Birch, J. Geophys. Res. 91 (1986) 4949. [111 T. Yagi, J. Phys. Chem. Solids 39 (1978) 563. [12] J. Shanker and M. Kumar, Phys. Star. Sol. (B) 179 (1993) 351. [131 O.L. Anderson, D.G. Isaak and H. Oda, Rev. Geophys. 30 (1992) 57. [14] M. Kumar, Solid State Commun. 92 (1994) 463. [15] A. Chopelas and R. Boehler, Geophys. Res. Lett. 19 (1992) 1983.
Physica B 229 (1997) 419 420
Comment On the high-pressure equation of state for solids J. Shanker*, B. Singh, S.S.
Kushwah 1
Department of Physics, Institute of Basic Sciences, Khandari, Agra 282 002, India Received 21 June 1996
Abstract It is found that the high-pressure equation of state recently reported by K umar [-Physica B 212 (1995) 391, 217 (1996) 143] is not a new equation. This is the same as the usual Tait equation of state which is well known in the literature and frequently used by several investigators. It is emphasised that the usual Tait equation of state is consistent with the Chopelas-Boehler relationship for the variation of the Anderson Gruneisen parameter d r with compression (V/Vo).
Keywords: High pressure; Equation of state Recently, Kumar [1, 2] derived an equation of state to study the pressure-volume relationship for different classes of solids. Using some thermodynamic approximations, he obtained the following equation:
(UTE) frequently used in the literature [3 8] and given below [5, 8]:
(1)
where AV = V 0 - V. Eq. (3) can be rewritten as follows:
P
where P is pressure, V/Vo is compression, Bo is the isothermal bulk modulus at zero pressure and A = B~ + 1. Here B~ is the pressure derivative of isothermal bulk modulus at zero pressure. Eq. (1) can be rewritten as follows:
It would be interesting to compare this equation with the well known usual Tait equation of state
AV_ 1 In I 1 + (1 - +B~) - P Vo (B; + 1) Bo
(B~+l)(1-~o>=ln[1-+
( B °l '+BI o) P
(3)
(4)
On comparing Eqs. (2) and (4) it is found that these are identical. Thus, Eq. (1) reported by Kumar [1, 2] is same as the usual Tait equation (3). The expression for isothermal bulk modulus based on the UTE has been reported in the literature [3] and given below
*Corresponding author. LPermanent address: Rishi Galav College, Morena (MP), lndia. 0921-4526/'97,,'$17.00 Copyright PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 5 2 8 -
],
1997 Elsevier Science B.V. All rights reserved 5
420
a( Shanker et al. / Physica B 229 (1997) 419 420
On the other hand, Kumar reported the following expression for isothermal bulk modulus B [2]: B=
O~ooeXp (B•+I)
1-~oo
"
(6)
Eqs. (5) and (6) are same in view of Eq. (2). Substituting Eq. (2) in (5) we get (6). Thus, Eqs. (5) and (6) both are based on the UTE [Eq. (3)]. Now, if we use the well-known thermodynamic approximation [9 12] according to which the product of thermal expansivity ~ and isothermal bulk modulus B remains constant for a given material under the effect of pressure, we can write :~B = ~oBo,
(7)
where :~o and Bo are the values of :~ and B at zero pressure. If Eq. (7) holds good we have another useful approximation for the Anderson-Gruneisen parameter C~r given below [13] l(dB~ 6r -
--
(8)
1+
(9)
Bo
and --=
If one makes use of the approximation given by (8), then Eq. (11) takes the form V (07-+ 1) = (c~ro + 1)~oo,
(12)
which is the relationship investigated by Chopelas and Boehler [15] on the basis of an analysis of experimental data. In fact, Kumar [1, 2] started his analyses on the basis of (12) and obtained an equation of state which is the same as UTE. The details of UTE including its nomenclature and derivation can be found in Refs. [3, 4].
dB
c~B\dr]e = ~'
At P = 0 we have, therefore, 6ro = B;. Eqs. (5) and (6) with the help of (7) yield the following expressions:
~o
by Kumar further support the validity of the UTE. Finally, it should be pointed out that the UTE is consistent with the relationship given by Chopelas and Boehler [15]. Eqs. (3) and (5) yield the following expression: dB V + 1 = (B; + I ) Vo. (11) d~
exp
-(6to+
1) 1 -
.
(10)
~o
Eqs. (9) and (10) are useful relations for predicting the pressure dependence of ~ along isotherms. It is pertinent to mention here that Eq. (10) is identical with the formula for e reported by Kumar [14]. Thus, the equations reported by Kumar in his recent papers [1, 2, 14] are, in fact, not new but same as those given by the well-known UTE. It has already been found and reported in the literature [7, 8] that the UTE yields good agreement with experimental data. The numerical results obtained
References [lJ M. Kumar, Physica B 212 (1995) 391. [2J M. Kumar, Physica B 217 (1996) 143. [3] J.R. MacDonald, Rev. Mod. Phys. 38 (1996) 669; 41 (1969) 316. I-4] A.T.J. Hayward, Brit. J. Appl. Phys. 18 (1967) 965. [5] J. Freund and R. Ingalls, J. Phys. Chem. Solids 50 (1989) 263. [6] H. Schlosser and J. Ferrante, Phys. Rev. B40 (1989) 6405, and references therein. [7] H. Schlosser and J. Ferrante, J. Phys.: Condens. Matter l (1989) 1941. [8] E.M. Thomas and J. Shanker, Phys. Star. Sol. (B) 189 (1995) 363. [9] J.L. Tallon, J. Phys. Chem. Sol. 41 (1980) 837. [10] F. Birch, J. Geophys. Res. 91 (1986) 4949. [111 T. Yagi, J. Phys. Chem. Solids 39 (1978) 563. [12] J. Shanker and M. Kumar, Phys. Star. Sol. (B) 179 (1993) 351. [131 O.L. Anderson, D.G. Isaak and H. Oda, Rev. Geophys. 30 (1992) 57. [14] M. Kumar, Solid State Commun. 92 (1994) 463. [15] A. Chopelas and R. Boehler, Geophys. Res. Lett. 19 (1992) 1983.