Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state
We have derived formulations for the pressure derivatives of bulk modulus up to the third order and for higher order Gr¨uneisen parameters using the generalized free volume theory, and the generalized Rydberg equation of state. The properties derived in the present study are directly related to th...
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| Zitieren: | Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state / J. Shanker, B.P. Singh, K. Jitendra // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 205-213. — Бібліогр.: 33 назв. — англ. |
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irk-123456789-1199882017-06-11T03:04:44Z Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state Shanker, J. Singh, B.P. Jitendra, K. We have derived formulations for the pressure derivatives of bulk modulus up to the third order and for higher order Gr¨uneisen parameters using the generalized free volume theory, and the generalized Rydberg equation of state. The properties derived in the present study are directly related to the understanding of thermoelastic properties of solids. The third order Gr¨uneisen parameter (lambda λ) in the limit of in nite pressure has been found to approach a positive finite value for lambda in nity (λ∞) equal to 1/3. This is a result shown to be independent of the value of K-prime in nity, i. e., the pressure derivative of the bulk modulus at infinite pressure. The results based on other equations of state have also been reported and discussed. We find a relationship between λ∞ and pressure derivatives of bulk modulus at infinite pressure which is satisfied by different types of equations of state. Ми отримали формулювання для похiдних третього порядку за тиском вiд об’ємних модулiв i для параметрiв Грюнайзена вищого порядку, використовуючи узагальнену теорiю вiльного об’єму та узагальнене рiвняння стану Рiдберга. Отриманi властивостi є безпосередньо пов’язаними iз розумiнням термоелектричних властивостей твердих тiл. Показано, що параметр Грюнайзена третього порядку (λ) в границi нескiнченного тиску (λ∞) наближається до скiнченного позитивного значення, рiвного 1/3. Показано, що цей результат не залежить вiд значення похiдної за тиском вiд об’ємного модуля при нескiнченному тиску. Також обговорюються результати, отриманi на основi iнших рiвнянь стану. Ми знайшли спiввiдношення, що зв’язує λ∞ та похiднi за тиском вiд об’ємних модулiв при нескiнченному тиску, яке задовольняється для рiзних типiв рiвнянь стану. 2009 Article Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state / J. Shanker, B.P. Singh, K. Jitendra // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 205-213. — Бібліогр.: 33 назв. — англ. 1607-324X PACS: 65, 64.10.+h, 91.60.Fe, 46.25.4f, 62.20.D, 81.40.Jj, 62.50.-p DOI:10.5488/CMP.12.2.205 http://dspace.nbuv.gov.ua/handle/123456789/119988 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
We have derived formulations for the pressure derivatives of bulk modulus up to the third order and for higher
order Gr¨uneisen parameters using the generalized free volume theory, and the generalized Rydberg equation
of state. The properties derived in the present study are directly related to the understanding of thermoelastic
properties of solids. The third order Gr¨uneisen parameter (lambda λ) in the limit of in nite pressure has
been found to approach a positive finite value for lambda in nity (λ∞) equal to 1/3. This is a result shown
to be independent of the value of K-prime in nity, i. e., the pressure derivative of the bulk modulus at infinite
pressure. The results based on other equations of state have also been reported and discussed. We find a
relationship between λ∞ and pressure derivatives of bulk modulus at infinite pressure which is satisfied by
different types of equations of state. |
| format |
Article |
| author |
Shanker, J. Singh, B.P. Jitendra, K. |
| spellingShingle |
Shanker, J. Singh, B.P. Jitendra, K. Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state Condensed Matter Physics |
| author_facet |
Shanker, J. Singh, B.P. Jitendra, K. |
| author_sort |
Shanker, J. |
| title |
Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state |
| title_short |
Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state |
| title_full |
Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state |
| title_fullStr |
Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state |
| title_full_unstemmed |
Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state |
| title_sort |
extreme compression behaviour of higher derivative properties of solids based on the generalized rydberg equation of state |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119988 |
| citation_txt |
Extreme compression behaviour of higher derivative properties of solids based on the generalized Rydberg equation of state / J. Shanker, B.P. Singh, K. Jitendra // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 205-213. — Бібліогр.: 33 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shankerj extremecompressionbehaviourofhigherderivativepropertiesofsolidsbasedonthegeneralizedrydbergequationofstate AT singhbp extremecompressionbehaviourofhigherderivativepropertiesofsolidsbasedonthegeneralizedrydbergequationofstate AT jitendrak extremecompressionbehaviourofhigherderivativepropertiesofsolidsbasedonthegeneralizedrydbergequationofstate |
| first_indexed |
2025-07-08T17:02:00Z |
| last_indexed |
2025-07-08T17:02:00Z |
| _version_ |
1837098987846369280 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 2, pp. 205–213
Extreme compression behaviour of higher derivative
properties of solids based on the generalized Rydberg
equation of state
J.Shanker, B.P.Singh, K.Jitendra
Department of Physics. Institute of Basic Sciences, Khandari, Agra, 282002, India
Received September 24, 2008, in final form April 2, 2009
We have derived formulations for the pressure derivatives of bulk modulus up to the third order and for higher
order Grüneisen parameters using the generalized free volume theory, and the generalized Rydberg equation
of state. The properties derived in the present study are directly related to the understanding of thermoelastic
properties of solids. The third order Grüneisen parameter (lambda λ) in the limit of infinite pressure has
been found to approach a positive finite value for lambda infinity (λ∞) equal to 1/3. This is a result shown
to be independent of the value of K-prime infinity, i. e., the pressure derivative of the bulk modulus at infinite
pressure. The results based on other equations of state have also been reported and discussed. We find a
relationship between λ∞ and pressure derivatives of bulk modulus at infinite pressure which is satisfied by
different types of equations of state.
Key words: pressure derivatives of bulk modulus, Grüneisen parameters, infinite pressure behaviour,
thermoelastic properties
PACS: 65, 64.10.+h, 91.60.Fe, 46.25.4f, 62.20.D, 81.40.Jj, 62.50.-p
1. Introduction
The Vinet equation of state (EOS) [1] based on the Rydberg potential function [2] has been
widely used [3–11] in spite of its main shortcoming regarding the extreme compression behaviour of
solids in the limit of infinite pressure [12,13]. This shortcoming was rectified [10,14] by generalizing
the Vinet-Rydberg EOS. The generalized Rydberg EOS formulated by Stacey [14] can be written
as
P = 3K0x
−K′
∞
(
1 − x1/3
)
exp
[
f(1 − x1/3)
]
, (1)
where K0 is the value of bulk modulus at P = 0, x = V/V0, and K ′
∞
is the value of pressure
derivative of bulk modulus at P → ∞. The parameter f appearing in the exponential term in
equation (1) is given below
f =
3
2
K ′
0 − 3K ′
∞
+
1
2
= −3K0K
′′
0 −
3
4
K ′ 2
0 +
1
12
, (2)
where K ′
0 and K ′′
0 are the values of the first and the second pressure derivatives of bulk modulus,
both at P = 0. According to Stacey [12–14], K ′
∞
is an equation of state parameter depending
on the material in the same sense as K ′
0 and K ′′
0 are. On the basis of thermodynamic principles,
it has been found [13,14] that K ′
∞
> 5/3. Stacey [14] emphasized that the generalized Rydberg
EOS has the advantage of being successfully applicable to the solids under compressions as well
as to the solids when they are dilated. Singh [15] has also demonstrated the usefulness of the
generalized Rydberg EOS in investigating the high pressure properties of solids. In the present
study we produce formulations for the higher derivative thermoelastic parameters and for the
pressure derivatives of bulk modulus using the generalized Rydberg EOS and our earlier approach
[16] based on the modified free volume theory of Grüneisen parameter [13,17]. Then we discuss
and compare the results based on other equations of state.
c© J.Shanker, B.P.Singh, K.Jitendra 205
J.Shanker, B.P.Singh, K.Jitendra
2. Grüneisen parameter and thermoelastic properties
The most important parameter providing connection between thermal and elastic properties is
the Grüneisen parameter [18]
γ =
αKT
ρCV
=
αKS
ρCP
, (3)
where α is thermal expansivity, ρ is density, KT and KS are isothermal and isentropic bulk moduli,
CV and CP are specific heats at constant volume and constant pressure, respectively. According
to the generalized free-volume formula [13,17], γ is related to pressure P , bulk modulus K and its
pressure derivative K ′ = dK/dP , as follows
γ =
(K ′/2) − (1/6)− (t/3)(1 − P/3K)
1− 2t(P/3K)
. (4)
The parameter t takes different values for different formulations of gamma based on different
approximations reviewed by Stacey and Davis [13]. Thus t = 0 for the Slater formula [19], t = 1 for
the formulation developed by Dugdale and MacDonald [20], t = 2 for the free-volume formula [17],
and t = 2.35 resulted in a molecular dynamical calculation by Barton and Stacey [21]. Equation (4)
can be applied to different types of solids, metals as well as insulators, because it is derived from
the fundamental relationship between thermal pressure and thermal energy [18,21]. The pressure
dependence or volume dependence of the Grüneisen parameter can be studied with the help of
equation (4) using different equations of state [6,22,23]. The value of t in equation (4) can be
determined empirically by adjusting it to some known condition such as γ = γ0 at P = 0. The
value of t determined in this way has been found to be equal to 1.436 for the lower mantle and
outer core [13,16]. Expressions for higher order Grüneisen parameters q and λ can be derived by
taking t as a constant. The final result for the third order Grüneisen parameter obtained in the
present study for solids in the limit of extreme compression is independent of the parameter t.
By rewriting equation (4) in the following equivalent form [22]
γ =
K ′
2
−
1
6
− ε , (5)
where
ε =
t(K − K ′P )
(3K − 2tP )
. (6)
The higher order Grüneisen parameters are defined as [16,24]
q =
(
d ln γ
d ln V
)
T
= −
K
γ
(
dγ
dP
)
T
(7)
and
λ =
(
d ln q
d ln V
)
T
= −
K
q
(
dq
dP
)
T
. (8)
Then, the following expressions are obtained from the differentiation of equation (5)
qγ = −
KK ′′
2
+ K
dε
dP
(9)
and
γq(q + λ) =
K ′KK ′′
2
+
K2K ′′′
2
− KK ′
dε
dP
− K2
d2ε
dP 2
. (10)
Equations (9) and (10) yield
(q + λ) = −K ′
−
[
(K2K ′′′/KK ′′) − (2K/K ′′)(d2ε/dP 2)
]
1 − (2/K ′′)dε/dP )
, (11)
206
Extreme compression behaviour of higher derivative properties of solids
where the pressure derivatives of ε are obtained from equation (6) as follows
dε
dP
= −
[tK ′′P + ε(3K ′
− 2t)]
(3K − 2tP )
(12)
and
d2ε
dP 2
= −
[tK ′′′P + tK ′′ + 3εK ′′ + 2(3K ′
− 2t)(dε/dP )]
(3K − 2tP )
. (13)
Equations (12) and (13) can be rearranged as follows according to the terms appearing in
equation (11)
1
K ′′
(
dε
dP
)
= −
[t + (ε/K ′′P )(3K ′
− 2t)]
[(3K/P ) − 2t)]
(14)
and
K
K ′′
(
d2ε
dP 2
)
= −
[t(K2K ′′′/KK ′′) + (t + 3ε)K/P + (3K ′
− 2t)(K/P )(2/K ′′)(dε/dP )]
(3 − 2tP/K)(K/P )
. (15)
Equation (6) is rearranged as follows and then used in equation (14)
ε
K ′′P
=
t(K − K ′P )
(3 − 2tP/K)K2K ′′P/K
. (16)
The most important results based on thermodynamics of solids [13, 25] are that (1) γ → γ∞, a
finite positive value, (2) q → 0, and (3) λ → λ∞ > 0, all at infinite pressure. For all the equations
of state which satisfy K ′
∞
> 0, we can write [14]
K ′
∞
=
(
P
K
)
−1
∞
. (17)
3. Pressure derivatives of bulk modulus
Here we derive the pressure derivatives of bulk modulus up to the third order using the gen-
eralized Rydberg EOS (equation (1)), and then study the relationship between the third order
Grüneisen parameter λ and the pressure derivatives of bulk modulus. Equation (1), on differen-
tiating with respect to x = V/V0 and using the definition of bulk modulus, K = −x(dP/dx),
gives
K
P
−
x1/3
3(1 − x1/3)
= K ′
∞
+
f
3
x1/3 , (18)
where f is given by equation (2). At extreme compression, x → 0, equation (18) reduces to equa-
tion (17). By successive differentiations of equation (18) with respect to pressure we obtain the
following expressions for K ′ = dK/dP , K ′′ = d2K/dP 2 and K ′′′ = d3K/dP 3
K ′
(
K
P
)
−
(
K
P
)2
= −
1
3
y − y2
−
f
9
x1/3 , (19)
(
KK ′′ + K ′2
)
(
K
P
)
− 3K ′
(
K
P
)2
+ 2
(
K
P
)3
=
1
9
y + y2 + 2y3 +
f
27
x1/3 (20)
and
(
K2K ′′′ + 4KK ′′K ′ + K ′3
)
(
K
P
)
− (4KK ′′ + 7K ′2)
(
K
P
)2
+ 12K ′
(
K
P
)3
− 6
(
K
P
)4
= −
1
27
y −
7
9
y2
− 4y3
− 6y4
−
f
81
x1/3 , (21)
207
J.Shanker, B.P.Singh, K.Jitendra
where y = (1/3)x1/3/(1 − x1/3). It may be noted that y → 0 as x → 0 at extreme compression.
Then, it is also revealed from equations (20) and (21) with the help of equation (17) that KK ′′
→ 0
as well as K2K ′′′
→ 0 in the limit of infinite pressure.
Equations (19) and (20) give
KK ′′
K ′
− (K/P )
+ K ′
− 2
(
K
P
)
= −
(3y + 27y2 + 54y3 + fx1/3)
3(3y + 9y2 + fx1/3)
. (22)
By rearranging equation (21) and then dividing by equation (20), we find
N
D
= −
(3y + 63y2 + 324y3 + 486y4 + fx1/3)
3(3y + 27y2 + 54y3 + fx1/3)
, (23)
where the numerator N and denominator D on the left hand side of equation (23) are given as
follows
N =
K2K ′′′
KK ′′
+ 4
(
K ′
−
K
P
)
+
K ′2
KK ′′
(
K ′
−
K
P
)
−
6
KK ′′
(
K
P
) (
K ′
−
K
P
)2
(24)
and
D = 1 +
(
1
KK ′′
) (
K ′
−
K
P
) (
K ′
−
2K
P
)
. (25)
Equations given above are presented such that they can be simplified and evaluated conveniently
at infinite pressure.
4. Extreme compression behaviour
For solids at extreme compression (V → 0, and P → ∞ ) we derive the following relations.
Equation (22) at P → ∞ gives
[
K2K ′′
(K − K ′P )
]
∞
= −K ′
∞
(
K ′
∞
−
1
3
)
. (26)
Using this result (equation (26)) in equation (16) and applying the condition at P → ∞, we
get
( ε
K ′′P
)
∞
=
−tK ′
∞
(3K ′
∞
− 2t)
(
K ′
∞
−
1
3
) . (27)
Equation (14) gives the following result at infinite pressure
[
1
K ′′
(
dε
dP
)]
∞
=
−t
3(3K ′
∞
− 2t)
(
K ′
∞
−
1
3
) . (28)
Equations (23)–(25) with the help of equation (26) yield
(
K2K ′′′
KK ′′
)
∞
= −
(
K ′
∞
+
1
3
)
. (29)
With the help of Equations (28) and (29), equation (15) is reduced to the following result at
P → ∞
[
1
K ′′
(
d2ε
dP 2
)]
∞
= −
t
(
K ′
∞
+ 1
3
)
3(3K ′
∞
− 2t)
(
K ′
∞
−
1
3
) . (30)
The final result for the third order Grüneisen parameter λ at P → ∞ is obtained from equa-
tion (11) using equations (28)–(30). It should be mentioned that the second order Grüneisen
parameter q (equation (9)) tends to zero at P → ∞ since KK ′′, ε and dε/dP all tend to zero as
it is evident from equations (20), (6) and (12), respectively. Thus, we find from equation (11) at
infinite pressure
λ∞ =
1
3
. (31)
208
Extreme compression behaviour of higher derivative properties of solids
5. Discussions and conclusions
The third order Grüneisen parameter λ is an important physical quantity related to pressure
derivatives of bulk modulus up to the third order and, therefore, depends on the derivatives of
potential energy up to the fifth order [13]. Any fundamental knowledge about λ would provide
a new powerful lead to high pressure equation of state. Thus, λ∞ is an important equation of
state parameter like K ′
∞
. In addition to this, λ∞ plays a significant role in describing the volume
dependence of the Grüneisen parameter and other thermoelastic properties. We have the following
relationship
λ =
V
q
(
dq
dV
)
=
(
γ
dγ/dV
)
d
dV
[
V
γ
(
dγ
dV
)]
=
(
γ
dγ/dV
)
[
1
γ
(
dγ
dV
)
−
V
γ2
(
dγ
dV
)2
+
V
γ
(
d2γ
dV 2
)
]
= 1 −
V
γ
(
dγ
dV
)
+
V (d2γ/dV 2)
(dγ/dV )
. (32)
Burakovsky and Preston [23] have formulated the following model for the volume dependence
of γ
γ = γ∞ + c1
(
V
V0
)m
+ c2
(
V
V0
)n
, (33)
where c1 and c2 are constants for a given material. The exponent m > 1, and n = 1/3. At extreme
compression V → 0, we have γ = γ∞. Equations (32) and (33) when used at infinite pressure reveal
that λ∞ = n = 1/3. Thus, the result for λ∞ obtained in the present study using the generalized
Rydberg EOS is consistent with the model for γ(V ) due to Burakovsky and Preston [23]. It should
be mentioned that the parameters in equation (33) are likely to change for a material undergoing
phase transitions. However, equation (33) can be extrapolated to extreme compression by assuming
that the material remains in the same phase.
It should be pointed out that in the present formulation based on equations (5) and (6) we
have taken ε → 0 at P → ∞ which is possible only when the denominator of equation (6) does
not become zero, i.e. K ′
∞
remains different from 2t/3. When ε → 0 at P → ∞, equation (5) gives
γ∞ =
K ′
∞
2
−
1
6
. (34)
Equation (34), which represents the Slater formula at P → ∞, is valid according to the analysis
presented by Stacey [25]. However, in the Burakovsky-Preston model, we have t = 5/2 in the
extreme compression limit, and K ′
∞
= 5/3 and, therefore, K ′
∞
= 2t/3. Thus, according to this
model [23, 26], ε does not become zero, and equation (34) is not a valid result. In fact, ε becomes
indeterminate from equation (6) at P → ∞ when K ′
∞
= 2t/3 = 5/3. It should be mentioned
that Burakovsky and Preston [23] have taken the parameter t to change with pressure in their
formulation based on the generalized free volume theory. Thus, the values of t taken by them
remain between zero and 5/2 respectively at P = 0 and at P → ∞.
Stacey [12–14] has mentioned that equation (34) for γ∞ results when we substitute equation (17)
in equation (4). However, this is only incomplete information. Using equation (17) in equation (4)
we get the following complete expression [26]
(
K ′
∞
−
2t
3
) (
γ∞ −
K ′
∞
2
−
1
6
)
= 0. (35)
Thus, it is evident that equation (35) reduces to equation (34) under the condition when K ′
∞
is not equal to 2t/3. This point is very important emphasizing the need for an analysis involving ε
based on equations (5) and (6). Such an analysis is also necessary for determining higher derivative
parameters q and λ at finite pressures. According to Stacey and Davis [13], the Slater formula
209
J.Shanker, B.P.Singh, K.Jitendra
assumes the status of an identity supporting the validity of equation (34). There is enough evidence
as reviewed by Burakovsky and Preston [23] that the maximum value of t can be 5/2, and, therefore,
2t/3 cannot be more than 5/3. Since the Stacey thermodynamic constraint reveals K ′
∞
> 5/3, so
K ′
∞
is also greater than 2t/3, and hence equation (34) is fundamentally correct.
However, we have to follow an alternative approach to find λ∞ when K ′
∞
= 5/3. In this case
the generalized Rydberg EOS (equation (1)) reduces to the following form at strong compressions
when x = V/V0 is close to zero,
P = aV −5/3 exp(−bV 1/3), (36)
where a and b are constants greater than zero. Equation (36) was used in the literature [27,28] as an
EOS yielding accurate results at ultrahigh pressures. An analysis based on equation (36), for which
K ′
∞
= 5/3, was presented earlier [26] where the same result, λ∞ = 1/3, was obtained. It should
be mentioned that the melting curves for Al, Ar, Cu, Pd and Pt determined using n = λ∞ = 1/3
in equation (33) present close agreement with the experimental data [23,29,30].
It should be emphasized that the result λ∞ = 1/3 is specific to the generalized Rydberg EOS
(equation (1)) formulated by Stacey [14]. This value of λ∞ is in agreement with the Burakovsky-
Preston model [23] but substantially deviates from the result λ∞ ≈ 3 for the Earth core derived
from an analysis of seismic data [13]. The seismic value of λ∞ is significantly higher than λ∞ =
K
′
2
∞
/K ′
0 = 1.8 obtained by Shanker and Singh [16] from the Stacey reciprocal K-primed EOS [24]
taking K ′
0 = 5.0 and K ′
∞
= 3.0 for the Earth’s core. It shows that λ∞ is very sensitive to the
assumed equations and that any secure fix on it will be a very powerful constraint on equations
of state in general. To achieve this objective, we need to compare the results with those based on
other equations of state.
Stacey [12–14,25] has emphasized in a very convincing manner that analytical equations of
state describing the behaviour of materials through all the phase transitions from zero pressure to
the Thomas-Fermi state are not valid mainly because the parameters appearing in such equations
cannot remain unchanged through phase transitions. The same criticism applies to equation (33)
for γ(V ) due to Burakovsky and Preston [23]. However, various equations can be generalized in the
same way as the Rydberg EOS, and can be extrapolated to the extreme compression by considering
the material to remain in the same phase so as to satisfy equation (17). In this case the infinite
pressure parameters such as K ′
∞
, γ∞, λ∞ are not the Thomas-Fermi state parameters. They are
material – dependent and phase-dependent in the same sense as the zero pressure parameters (K0,
K ′
0, K ′′
0 ) depend on the material in a given phase.
We have generalized the Holzapfel EOS (AP2) and the Hama-Suito EOS [3,22] by taking the
factor (V/V0)
−K′
∞ in place of (V/V0)
−5/3, in the same way as Stacey [14] generalized the Rydberg
EOS. These two equations after generalization yield the same result λ∞ = 1/3 with the help of
formulation based on the modified free volume theory [16]. The generalized fourth-order Poirier-
Tarantola logarithmic EOS [31,32] yields a different result, λ∞ = 0, which is not consistent with
the thermodynamic considerations (λ∞ > 0). The Keane EOS [33], as discussed in the Appendix,
gives λ∞ = K ′
∞
. This result is in agreement with λ∞ ≈ 3 [13] because K ′
∞
is also nearly 3 for the
earth’s core.
The most important finding based on different equations of state discussed above is that all
of them satisfy the following relationship at infinite pressure for the ratio of third- and second
pressure derivatives of bulk modulus
(
K2K ′′′
KK ′′
)
∞
= −(K ′
∞
+ λ∞). (37)
Equation (37) is a result of detailed analysis involving ε and its pressure derivatives dε/dP and
d2ε/dP 2. The conclusion drawn by Stacey and Davis [13] that the Slater formula becomes valid
at infinite pressure gets strong support from the validity of equation (37). In fact, equation (37)
can be directly obtained by differentiating twice the Slater formula for γ (ε = 0 in equation (5)).
Equation (11) is also directly reduced to equation (37) at infinite pressure when the pressure
derivatives of ε are zero. But we did not arrive at equation (37) in this manner.
210
Extreme compression behaviour of higher derivative properties of solids
Stacey [12–14] has given two alternative statements: (i) the Slater formula becomes valid at
P → ∞, and (ii) all the formulae for γ represented by equation (4) reduce to equation (34). The
two statements are not equivalent. The second statement appears to be more appropriate. It is
clear from equation (6) that ε → 0 at P → ∞ due to equation (17) for any value of t = 0, 1, 2
etc. This favours the second statement. It is further substantiated by the fact that equation (11)
reduces to equation (37) even when dε/dP and d2ε/dP 2 are not zero at P → ∞. It has been found
in the present study that different equations of state satisfy the relationship
(
K2K ′′′
KK ′′
)
∞
=
[(K/K ′′)(d2ε/dP 2)]∞
[(1/K ′′)(dε/dP )]∞
. (38)
Equation (38) when used in equation (11) gives equation (37). Equation (38) is valid for all
values of t, and, therefore, various formulae based on different values of t converge to equation (37)
for λ∞ .
Appendix
Here we present some expressions based on the K-primed equations. Using the Stacey EOS we
obtained [16]
λ∞ =
K ′ 2
∞
K ′
0
(39)
and
(
K2K ′′′
KK ′′
)
∞
= −
K ′
∞
K ′
0
(K ′
0 + K ′
∞
) . (40)
Equations (39) and (40) exactly satisfy equation (37). Equations (25), (26) and (32) of [16]
are identical with equations (13), (11) and (15) respectively, which have been written in compact
forms in the present paper.
There has been a mistake in the interpretation of results [16] based on the Keane EOS [33].
The correct interpretation is as follows. For the Keane EOS we have
K2K ′′ = −K0K
′(K ′
0 − K ′
∞
). (41)
In the limit P → ∞, K ′
→ K ′
∞
, and, therefore, K2K ′′ remains finite. Using the Keane EOS,
we can also write
(K − K ′P )
K0
= 1 − (K ′
0 − K ′
∞
) ln x + (K ′
0 − K ′
∞
)
P
K
. (42)
Equation (42) gives
(K − K ′P )∞ → ∞ (43)
at x = V/V0 → 0 (P → ∞). In spite of equation (43), ε∞ → 0 because equation (6) can be
rewritten as
ε =
t(1 − K ′P/K)
(3 − 2tP/K)
. (44)
Equation (44) gives ε∞ → 0 due to equation (17). The results expressed in equations (36) and
(37) of [16] are modified as follows
(
K2K ′′
K − K ′P
)
→ 0. (45)
Equation (45) is based on equations (41) and (43). Equation (45) substituted in equation (16)
gives
( ε
K ′′P
)
→ ∞. (46)
With the help of equations (46) and (14) we get
1
K ′′
(
dε
dP
)
→ ∞. (47)
211
J.Shanker, B.P.Singh, K.Jitendra
The Keane EOS gives [16]
(
K2K ′′′
KK ′′
)
∞
= −2K ′
∞
. (48)
Equations (47) and (48) when substituted in equation (15) yield
[(K/K ′′)(d2ε/dP 2)]∞
[(1/K ′′)(dε/dP )]∞
= −2K ′
∞
. (49)
Finally we find from equation (11) the following result
λ∞ = K ′
∞
. (50)
It should be mentioned that equations (48), (49) and (50) satisfy Equations (37) and (38)
given in the present paper. Equations (31), (33) and (34) of [16] based on the Stacey EOS also
satisfy equations (37) and (38). Not only the Keane EOS and the Stacey EOS, but also other
equations, such as the Rydberg EOS, Holzapfel EOS, Hama-Suito EOS, Poirier-Tarantola EOS
satisfy equations (37) and (38). It should be mentioned that equations (17), (34), (37) and (38)
are quite general and they do not depend on the empirical equations of state.
Equation (38) can further be solved using the results based on equations (14)–(16) at P → ∞.
Thus we get from equation (38)
(
K2K ′′′
KK ′′
)
∞
= −2K ′
∞
−
1
K ′
∞
(
K2K ′′
K − K ′P
)
∞
. (51)
Equations (37) and (51) give
λ∞ = K ′
∞
+
1
K ′
∞
(
K2K ′′
K − K ′P
)
∞
. (52)
We have verified that equations (51) and (52) are satisfied by the Stacey EOS, Keane’s EOS and
generalized Rydberg EOS. This is a remarkable finding, particularly in view of the fact that λ∞,
(K2K ′′′/KK ′′)∞ and
(
K2K ′′/ (K − K ′P )
)
∞
determined from one EOS differ much from those
based on the other EOS.
References
1. Vinet P., Ferrante J., Rose J.H., Smith J.R., Geophys. Res., 1987, 92, 9319; J. Phys. Condens. Matter,
1989, 1, 1941.
2. Rydberg R., Z. Phys., 1932, 73, 376.
3. Hama J., Suito K., J. Phys. Condens Matter, 1996, 8, 67.
4. Shanker J., Kushwah S.S., Sharma M.P., Physica B, 1999, 271, 158.
5. Cohen R.E., Gulsern O., Hemley R.J., American Mineralog., 2000, 85, 338.
6. Vocadlo L., Poirier J.-P., Price G.D., American Mineralog., 2000, 85, 390.
7. Gaurav S., Sharma B.S., Sharma S.B., Upadhyaya S.C., Physica B, 2002, 322, 328; J. Phys. Chem.
Solids, 2004, 65, 1635.
8. Sushil K., Arunesh K., Singh P.K., Sharma B.S., Physica B, 2004, 352, 134.
9. Singh V., Gautam A.K., Singh K.S., Physica B, 2004, 352, 164.
10. Arunesh K., Dharmendra K., Physica B, 2005, 364, 130.
11. Chauhan R.S., Singh C.P., Physica B, 2007, 387, 352.
12. Stacey F.D., Phys. Earth Planet. Inter., 2001, 128, 179.
13. Stacey F.D., Davis P.M., Phys. Earth Planet. Inter., 2004, 142, 137.
14. Stacey F.D. Rep. Prog. Phys., 2005, 68, 341.
15. Singh B.P., Physica B, 2005, 369, 11l.
16. Shanker J., B.P Singh, Physica B, 2005, 370, 78.
17. Vashchenko V.Ya., Zubarev V.N., Soy. Phys. Solid State, 1963, 5, 653.
18. Anderson O.L. Equation of State of Solids for Geophysics and Ceramic Sciences. Oxford University
Press, New York, 1995.
212
Extreme compression behaviour of higher derivative properties of solids
19. Slater J.C. Introduction to Chemical Physics. McGraw, New York, 1939.
20. Dugdale J.S., MacDonald D.K.C., Phys. Rev., 1953, 89, 832.
21. Barton M.A., Stacey F.D., Phys. Earth Planet. Inter., 1985, 39, 167.
22. Holzapfel W.B., Hartwig M., Sievers W., J. Phys. Chem. Ref. Data, 2001, 30, 515.
23. Burakovsky L., Preston D.L., J. Phys. Chem. Solids, 2004, 65, 1581.
24. Stacey F.D., Isaak D.G., Geophys. J. Int., 2001, 146, 143.
25. Stacey F.D., Geophys. J. Int., 2000, 143, 621.
26. Shanker J., Singh B.P., Baghel H.K., Physica B, 2007, 387, 409.
27. Barnes J.F., Phys. Rev., 1967, 153, 153.
28. Holzapfel W.B. – In. Molecular Systems Under High Pressure, eds. Pucci R., Piccitto G. North-Holland,
Amsterdam, 1991, p. 61.
29. Cheng V.M., Daniel W.B., Crawford R.K., Phys. Lett. A, 1973, 43, 109.
30. Jeong J.-W., Chang K.J., J. Phys. Condens. Matter, 1999, 11, 3799.
31. Poirier J.-P., Tarantola A., Phys. Earth Planet. Inter., 1998, 109, 1.
32. Kushwah S.S., Shrivastava H.C., Singh K.S., Physica B, 2007, 388, 20.
33. Keane A., Aust. J. Phys., 1954, 7, 322.
Властивостi твердих тiл при екстремальному стисненнi на
основi узагальненого рiвняння стану Рiдберга
Дж.Шанкер, Б.П.Сiнг, K.Джiтендра
Фiзичний факультет, Iнститут природничих наук, Кхандарi, Агра Iндiя 282002
Отримано 24 вересня 2008 р., в остаточному виглядi – 2 квiтня 2009 р.
Ми отримали формулювання для похiдних третього порядку за тиском вiд об’ємних модулiв i для
параметрiв Грюнайзена вищого порядку, використовуючи узагальнену теорiю вiльного об’єму та
узагальнене рiвняння стану Рiдберга. Отриманi властивостi є безпосередньо пов’язаними iз розу-
мiнням термоелектричних властивостей твердих тiл. Показано, що параметр Грюнайзена третього
порядку (λ) в границi нескiнченного тиску (λ∞) наближається до скiнченного позитивного значення,
рiвного 1/3. Показано, що цей результат не залежить вiд значення похiдної за тиском вiд об’ємного
модуля при нескiнченному тиску. Також обговорюються результати, отриманi на основi iнших рiв-
нянь стану. Ми знайшли спiввiдношення, що зв’язує λ∞ та похiднi за тиском вiд об’ємних модулiв
при нескiнченному тиску, яке задовольняється для рiзних типiв рiвнянь стану.
Ключовi слова: похiднi за тиском вiд об’ємних модулiв, параметри Грюнайзера, поведiнка при
нескiнченному тиску, термоелектричнi властивостi
PACS: 65, 64.10.+h, 91.60.Fe, 46.25.4f, 62.20.D, 81.40.Jj, 62.50.-p
213
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