Analysis of thermal expansivity of solids at extreme compression
Thermodynamics of solids in the limit of infinite pressure formulated by Stacey reveals that the thermal expansivity (alpha) of solids tends to zero at infinite pressure. The earlier models for the volume dependence of thermal expansivity do not satisfy the infinite pressure behaviour of thermal e...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1195802017-06-08T03:07:37Z Analysis of thermal expansivity of solids at extreme compression Shanker, J. Singh, B.P. Jitendra, K. Thermodynamics of solids in the limit of infinite pressure formulated by Stacey reveals that the thermal expansivity (alpha) of solids tends to zero at infinite pressure. The earlier models for the volume dependence of thermal expansivity do not satisfy the infinite pressure behaviour of thermal expansivity. The expressions for the volume dependence of the isothermal Anderson- Gruneisen parameter (delta T) considered in the derivation of earlier formulations for alpha (V) have been found to be inadequate. A formulation for the volume dependence of delta T is presented here which is similar to the model due to Burakovsky and Preston for the volume dependence of the Gruneisen parameter. The new formulation for alpha (V) reveals that delta T infinity must be greater than zero for satisfying the thermodynamic result according to which alpha tends to zero at in nite pressure. It is found that our model fits well the experimental data on thermal expansivity alpha (V) for hcp iron corresponding to a wide range of pressures (0 360 GPa). Згiдно умови, сформульованої Стейсi для термодинамiки твердих тiл в границi нескiнченого тиску, їх термiчне розширення прямує до нуля при нескiнченому тиску. Попереднi моделi для опису залежностi термiчного розширення вiд об’єму не задовiльняють цiй умовi. Вирази для залежностi iзотермiчного параметра Андерсона-Грюнайзена вiд об’єму, отриманi в попререднiх формулюваннях, виявилися невiдповiдними. Нами представлено об’ємну залежнiсть параметра Андерсона-Грюнайзена, яка є подiбною до отриманої ранiше для однiє з моделей. Отриманi нами результати демонструють, що параметр Андерсона-Грюнайзена при нескiнченому тиску мусить бути бiльшим нiж нуль для того, щоб задовiльнити термодинамiчну умову, згiдно якої термiчне розширення прямує до нуля при нескiнченому тиску. Знайдено, що наша модель узгоджується добре з експериментальними даними, що стосуються об’ємної залежностi для залiза з гексагональною щiльною упаковкою в широкiй областi тискiв (0–360 GPa). 2008 Article Analysis of thermal expansivity of solids at extreme compression / J. Shanker, B.P. Singh, K. Jitendra // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 681-686. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 64.30.+y, 65.70.+y DOI:10.5488/CMP.11.4.681 http://dspace.nbuv.gov.ua/handle/123456789/119580 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
Thermodynamics of solids in the limit of infinite pressure formulated by Stacey reveals that the thermal expansivity
(alpha) of solids tends to zero at infinite pressure. The earlier models for the volume dependence
of thermal expansivity do not satisfy the infinite pressure behaviour of thermal expansivity. The expressions
for the volume dependence of the isothermal Anderson- Gruneisen parameter (delta T) considered in the
derivation of earlier formulations for alpha (V) have been found to be inadequate. A formulation for the volume
dependence of delta T is presented here which is similar to the model due to Burakovsky and Preston for the
volume dependence of the Gruneisen parameter. The new formulation for alpha (V) reveals that delta T infinity
must be greater than zero for satisfying the thermodynamic result according to which alpha tends to zero at
in nite pressure. It is found that our model fits well the experimental data on thermal expansivity alpha (V) for
hcp iron corresponding to a wide range of pressures (0 360 GPa). |
| format |
Article |
| author |
Shanker, J. Singh, B.P. Jitendra, K. |
| spellingShingle |
Shanker, J. Singh, B.P. Jitendra, K. Analysis of thermal expansivity of solids at extreme compression Condensed Matter Physics |
| author_facet |
Shanker, J. Singh, B.P. Jitendra, K. |
| author_sort |
Shanker, J. |
| title |
Analysis of thermal expansivity of solids at extreme compression |
| title_short |
Analysis of thermal expansivity of solids at extreme compression |
| title_full |
Analysis of thermal expansivity of solids at extreme compression |
| title_fullStr |
Analysis of thermal expansivity of solids at extreme compression |
| title_full_unstemmed |
Analysis of thermal expansivity of solids at extreme compression |
| title_sort |
analysis of thermal expansivity of solids at extreme compression |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119580 |
| citation_txt |
Analysis of thermal expansivity of solids at extreme compression / J. Shanker, B.P. Singh, K. Jitendra // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 681-686. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shankerj analysisofthermalexpansivityofsolidsatextremecompression AT singhbp analysisofthermalexpansivityofsolidsatextremecompression AT jitendrak analysisofthermalexpansivityofsolidsatextremecompression |
| first_indexed |
2025-07-08T16:12:37Z |
| last_indexed |
2025-07-08T16:12:37Z |
| _version_ |
1837095883863228416 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 4(56), pp. 681–686
Analysis of thermal expansivity of solids at extreme
compression
J.Shanker, B.P.Singh∗, K.Jitendra
Department of Physics, Institute of Basic Sciences, Khandari, Agra (U.P.) INDIA 282002
Received August 28, 2006, in final form October 16, 2007
Thermodynamics of solids in the limit of infinite pressure formulated by Stacey reveals that the thermal ex-
pansivity (alpha) of solids tends to zero at infinite pressure. The earlier models for the volume dependence
of thermal expansivity do not satisfy the infinite pressure behaviour of thermal expansivity. The expressions
for the volume dependence of the isothermal Anderson- Grüneisen parameter (delta T) considered in the
derivation of earlier formulations for alpha (V) have been found to be inadequate. A formulation for the volume
dependence of delta T is presented here which is similar to the model due to Burakovsky and Preston for the
volume dependence of the Grüneisen parameter. The new formulation for alpha (V) reveals that delta T infinity
must be greater than zero for satisfying the thermodynamic result according to which alpha tends to zero at
infinite pressure. It is found that our model fits well the experimental data on thermal expansivity alpha (V) for
hcp iron corresponding to a wide range of pressures (0–360 GPa).
Key words: thermal expansivity, Anderson-Grüneisen parameter, thermodynamics, hcp iron
PACS: 64.30.+y, 65.70.+y
1. Introduction
The Anderson-Grüneisen parameter is an important physical quantity for understanding the
thermoelastic properties of solids at high pressures and high temperatures [1]. In the present paper
we study the volume dependence of the Anderson-Grüneisen parameter and derive an improved
formula for thermal expansivity of solids which is found to be consistent with the thermodynamic
constraint at extreme compression. The isothermal Anderson-Grüneisen δTis defined as [1]
δT = −
1
αKT
(
∂KT
∂T
)
P
, (1)
where α is the thermal expansivity or volume thermal expansion coefficient
α =
1
V
(
∂V
∂T
)
P
(2)
and KT, the isothermal bulk modulus
KT = −V
(
∂P
∂V
)
T
. (3)
Using the thermodynamic identity
(
∂α
∂P
)
T
=
1
K2
T
(
∂KT
∂T
)
P
(4)
in equation (1) we get
δT =
α
V
(
∂α
∂V
)
T
. (5)
∗E-mail: drbps.ibs@gmail.com
c© J.Shanker, B.P.Singh, K.Jitendra 681
J.Shanker, B.P.Singh, K.Jitendra
Equation (5) can be integrated to obtain α as a function of volume V , provided we know the
dependence of the Anderson-Grüneisen parameter δT on V . It has been found by Anderson and
Isaak [2] that δT depends on V in the following manner
δT = δ0
T
(
V
V0
)k
, (6)
where δT = δ0
T
at V = V0, the reference state (P = 0). k is a dimensionless thermoelastic parameter
defined as [2]
k =
(
∂ ln δT
∂ ln V
)
T
. (7)
An alternative form for δT(V ) has been considered by Chopelas and Boehler [3] as follows
δT = (δ0
T + 1)
V
V0
− 1. (8)
Equation (8) is the basis for an equation of state (EOS) formulated by Kumar [4,5] which turns
out to be the same as the usual Tait EOS [6]. Equation (6) with k = 1 is also used for developing
an EOS and for investigating the thermoelastic properties of solids [7–10].
The Chopelas-Boehler formulation (equation 8) when used in equation (5) gives the following
relationship [7,8]
α
α0
=
(
V
V0
)
−1
exp
[
−
(
δ0
T + 1
)
(
1 −
V
V0
)]
, (9)
where α0is the thermal expansivity at P = 0. On the other hand, when the Anderson- Isaak
formulation (equation 6) is used in equation (5), we get
α
α0
= exp
{
−
δ0
T
k
[
1 −
(
V
V0
)k
]}
. (10)
Anderson et al. [8] have made a comparative study of equations (9) and (10). It should be
emphasized here that the Chopelas-Boehler relationship (equation 9) and the Anderson-Isaak for-
mulation (equation 10) are not consistent with the infinite pressure behaviour based on thermo-
dynamics. The thermal expansivity α should tend to zero at extreme compression, V approaching
zero [11,12]. But equation (9) predicts α → ∞, and equation (10) gives a finite value of α at
extreme compression (V → 0, P → ∞). We therefore present a revised formulation for α(V).
2. Formulation based on the Burakovsky-Preston model
Burakovsky and Preston [13] have recently formulated a model for the volume dependence of
the Grüneisen parameter γ based on the following expression
γ = γ∞ + a
(
V
V0
)1/3
+ b
(
V
V0
)n
, (11)
where γ∞ is the value of γ at extreme compression V → 0. The value of γ∞= 1/2 or 2/3 based
on the Thomas-Fermi model [13,14]. γ∞ is treated as a universal constant, i.e., the same for all
materials, whereas a, b and n(> 1) are material-dependent parameters. It was found [15] that
the Burakovsky-Preston model equation (11) satisfies the thermodynamic constraints γ → γ∞,
q = (d ln γ/d lnV )T → 0, and λ = (d ln q/d ln V )T → λ∞, where γ∞ and λ∞ are finite positive
values.
It is appropriate to consider that the Anderson-Grüneisen parameter δT follows a volume de-
pendence similar to equation (11). A similarity for the volume dependence of γ and δT was pointed
out earlier by Tallon [16]. We can thus write
δT = δT∞ + c1
(
V
V0
)1/3
+ c2
(
V
V0
)m
, (12)
682
Analysis of thermal expansivity of solids at extreme compression
where δT∞ represents the value of δT at V → 0. In analogy with equation (11), δT∞ can be
considered as a universal constant and c1, c2 and m as constants for a given material. When we
use equation (12) in equation (5), and then on integrating we find the following expression for the
thermal expansivity α,
α
α0
=
(
V
V0
)δT∞
exp
{
−3c1
[
1 −
(
V
V0
)1/3
]
−
c2
m
[
1 −
(
V
V0
)m]
}
. (13)
On comparing equation (13) with equations (9) and (10) we note that δT∞ = −1 in the
Chopelas-Boehler formulation (equation 9), and δT∞= 0 in the Anderson-Isaak formulation. The
infinite pressure condition for α based on thermodynamics (α → 0 at V → 0) is satisfied only when
δT∞ is greater than zero. This is a result similar to that (δS∞ > 0) obtained by Stacey and Davis
[11]. Here δS is the adiabatic Anderson-Grüneisen parameter related to the temperature derivative
of adiabatic bulk modulus KS [1]
δS = −
1
αKS
(
∂KS
∂T
)
P
. (14)
It should be mentioned that KS and KT are related by the thermodynamic identity
KS = KT(1 + γαT ) . (15)
Stacey and Davis [11] emphasized that isothermal and adiabatic properties become identical in
the limit of infinite pressure. So, when δS∞ > 0, we should also have δT∞ > 0.
An independent expression for δT∞ can be obtained from the thermodynamic identity [1]
δT = K ′
T − 1 + q +
(
∂ ln CV
∂ ln V
)
T
, (16)
which gives at P → ∞
δT∞ = K ′
∞
− 1 +
[(
∂ ln CV
∂ ln V
)
T
]
∞
. (17)
To know more about δT∞, we use the following thermodynamic identities [1,12]
[
∂ ln(αKT)
∂ ln V
]
T
= δT − K ′
T (18)
and
[
∂ ln(αKT)
∂ ln V
]
S
= q − 1. (19)
3. Results and discussions
The Grüneisen parameter is related to the thermal and elastic properties of the material by the
formula
γ =
αKTV
CV
=
αKSV
CP
, (20)
where CV and CP are the specific heats at constant volume and constant pressure, respectively.
It follows from equation (20) that α should decrease with a decreasing volume or an increasing
pressure since gamma decreases and bulk modulus increases faster than 1/V . It is desirable to
judge the suitability of equation (13) for α (V) which is based on a model for δT(V) (equation 12)
similar to that (equation 11) formulated by Burakovsky and Preston [13] for γ(V).
We have in all five parameters viz. α0, δT∞, c1, c2 and m in equation (13). Equation (12) at
V = V0 gives
δ0
T = δT∞ + c1 + c2 . (21)
683
J.Shanker, B.P.Singh, K.Jitendra
We take δT∞= 2/3 based on equation (17) using K ′
∞
= 5/3 derived from the Thomas-Fermi
model [13,14], and neglecting the last term in equation (17) for the volume derivative of CV since
it is very small at high pressure and high temperature [1,11]. The value 2/3 for δT∞ can also be
supported from the identities (18) and (19). Equation (18) can be integrated along an isotherm
whereas equation (19) can be integrated along an adiabat. We integrate equation (19) between the
limits V = V0 to V → 0 to obtain
[ln(αKT)]
0
V0
=
0
∫
V0
(q − 1)
dV
V
. (22)
Since q = (d ln γ/d lnV )T becomes zero at V → 0 [11], it is found from equation (22) that the
product αKT tends to infinity at P → ∞ or V → 0. Equation (18) was used by Anderson [17] and
others [18, 19] to discuss the nature of variation of αKT with volume. According to equation (18),
αKT → ∞ only when δT − K ′
T
is negative at P → ∞. This reveals that δT∞ must be less than
K ′
∞
. Thus the value of δT∞ should be constrained as follows:
0 < δT∞ < K ′
∞
. (23)
The value of 2/3 for δT∞ taken in the present study satisfies the above constraint. δT∞ should
be considered as a universal constant in the same sense as γ∞ and K ′
∞
. The other parameters
δ0
T
, α0, c1, c2 and m depend on the material chosen for the study. To judge the suitability of
equation (13) for α(V ) we use the experimental data for hcp iron which was well studied for a wide
range of pressures [20,21]. For hcp iron we take δ0
T
= 5.32 and α0 = 7.83 · 10−5 K−1 from Isaak
and Anderson [20]. Using δ0
T
= 5.32 and δT∞ = 2/3 in equation (21) we have
c1 + c2 = 4.65. (24)
The parameters c1, c2 and m are now fitted to experimental data [20,21] for hcp iron in the
pressure range 0–360 GPa given in table 1. The fitted parameters are found to have the values c1=
3.60, c2 = 1.05, and m = 1.5. With the help of these parameters, values of α(V ) are determined
using equation (13) and then compared with the experimental values reported by Isaak and An-
derson [20] in figure 1. We find that our model fits the experimental data well particularly in view
of the fact that the experimental data become increasingly imprecise as pressure increases due to
non-hydrostaticity, comparably low quality of pressure standards, recrystallization etc.
Table 1. Experimental data for the thermal expansivity α (P, V) in 10−5K−1 for hcp iron [20,21].
V/V0 P (GPa) α (10−5 K−1)
1.0000 0 7.83
0.8767 30.0 3.88
0.8470 41.8 3.42
0.8172 56.2 2.94
0.7875 74.0 2.47
0.7578 95.9 2.02
0.7281 123.1 1.61
0.6984 156.8 1.27
0.6686 199.0 1.00
0.6389 252.1 0.82
0.6092 319.3 0.71
0.6048 330.0 0.70
0.5944 359.5 0.68
684
Analysis of thermal expansivity of solids at extreme compression
Figure 1. Thermal expansivity α(V ) for hcp iron, continuous curve calculated in the present
study (equation 13), and experimental data [20,21].
4. Conclusions
It was emphasized by Tallon [16] that the volume dependence of δTshould be similar to that of
γ. The volume dependence of δT is required for investigating the variation of thermal expansivity
α with volume V . We have presented a formulation for α(V ) (equation 13) using a model for δT (V )
(equation 12) which is similar to the model for γ(V ) (equation 11) originally due to Burakovsky
and Preston [13]. In both the models (equation (11) and equation (12)) the first term on the right is
a universal constant (γ∞ or δT∞), and the remaining two terms depend on the volume representing
the concave up and concave down behaviour [13]. The experimental data for thermal expansivity
of hcp iron [20,21] for a wide pressure range up to 360 GPa have been fitted well with the help
of equation (13) using the reasonable values of parameters, m > 1 and c1/c2 = 3.4, in agreement
with the original model due to Burakovsky and Preston [13].
Acknowledgement
We are thankful to the reviewer for his valuable comments which have been very useful in
revising the manuscript. Thanks are also due to Mrs. Sudha Singh for her help in the computational
work.
References
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11. Stacey F.D., Davis P.M., Phys. Earth Planet. Inter., 2004, 142, 137.
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J.Shanker, B.P.Singh, K.Jitendra
12. Stacey F.D., Rep. Prog. Phys., 2005, 68, 341.
13. Burakovsky L., Preston D.L., J. Phys. Chem. Solids, 2004, 65, 1581.
14. Holzapfel W.B., Hartwig M., Sievers W., J. Phys. Chem. Ref. Data, 2001, 30, 515.
15. Shanker J., Singh B.P, Baghel H.K., Physica B, 2007, 387, 409.
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17. Anderson O.L., J. Phys. Chem. Solids, 1997, 58, 335.
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Аналiз термiчного розширення твердих тiл при
екстремальному стисненнi
Дж.Шанкер, Б.П.Сiнг, K.Джiтендра
Фiзичний факультет, Iнститут природничих наук, Кхандарi, Агра Iндiя 282002
Отримано 28 серпня 2006 р., в остаточному виглядi – 16 жовтня 2007 р.
Згiдно умови, сформульованої Стейсi для термодинамiки твердих тiл в границi нескiнченого тиску, їх
термiчне розширення прямує до нуля при нескiнченому тиску. Попереднi моделi для опису залежно-
стi термiчного розширення вiд об’єму не задовiльняють цiй умовi. Вирази для залежностi iзотермi-
чного параметра Андерсона-Грюнайзена вiд об’єму, отриманi в попререднiх формулюваннях, вияви-
лися невiдповiдними. Нами представлено об’ємну залежнiсть параметра Андерсона-Грюнайзена,
яка є подiбною до отриманої ранiше для однiє з моделей. Отриманi нами результати демонструють,
що параметр Андерсона-Грюнайзена при нескiнченому тиску мусить бути бiльшим нiж нуль для то-
го, щоб задовiльнити термодинамiчну умову, згiдно якої термiчне розширення прямує до нуля при
нескiнченому тиску. Знайдено, що наша модель узгоджується добре з експериментальними дани-
ми, що стосуються об’ємної залежностi для залiза з гексагональною щiльною упаковкою в широкiй
областi тискiв (0–360 GPa).
Ключовi слова: термiчне розширення, параметр Андерсона-Грюнайзена, термодинамiка
PACS: 64.30.+y, 65.70.+y
686
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