Thermodynamic properties of liquid alkali metals
The internal energy, entropy and Helmholtz free energy of liquid alkali metals, viz. Na, K, Rb and Cs are investigated using pseudopotential perturbation scheme based on Gibbs-Bogoliubov variational technique. A local pseudopotential is applied to describe the electron-ion interaction in the liq...
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irk-123456789-1204652017-06-13T03:04:35Z Thermodynamic properties of liquid alkali metals Thakor, P.B. Gajjar, P.N. Jani, A.R. The internal energy, entropy and Helmholtz free energy of liquid alkali metals, viz. Na, K, Rb and Cs are investigated using pseudopotential perturbation scheme based on Gibbs-Bogoliubov variational technique. A local pseudopotential is applied to describe the electron-ion interaction in the liquid alkali metals. To introduce the exchange and correlation effects, the local field correction function proposed by Taylor is employed. The computed values for internal energy, entropy and Helmholtz free energy for the liquid alkali metals are in excellent agreement with the experimental data. Внутрішня енергія, ентропія та вільна енергія Гельмгольца рідких лужних металів (Na, K, Rb і Cs) вивчається в рамках підходу збурених псевдопотенціалів, що базується на варіаційній техніці Гіббса-Боголюбова. Для опису електрон-іонної взаємодії в рідких лужних металах застосовано локальний псевдопотенціал. Обмінні та кореляційні ефекти представлено з використанням запропонованої Тейлором функції, що враховує поправку на локальне поле. Отримані значення внутрішньої енергії, ентропії та вільної енергії Гельмгольца для рідких лужних металів чудово узгоджуються з експериментальними даними. 2001 Article Thermodynamic properties of liquid alkali metals / P.B. Thakor, P.N. Gajjar, A.R. Jani // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 473-480. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 65.50, 71.15.H DOI:10.5488/CMP.4.3.473 http://dspace.nbuv.gov.ua/handle/123456789/120465 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The internal energy, entropy and Helmholtz free energy of liquid alkali metals,
viz. Na, K, Rb and Cs are investigated using pseudopotential perturbation
scheme based on Gibbs-Bogoliubov variational technique. A local
pseudopotential is applied to describe the electron-ion interaction in the
liquid alkali metals. To introduce the exchange and correlation effects, the
local field correction function proposed by Taylor is employed. The computed
values for internal energy, entropy and Helmholtz free energy for the
liquid alkali metals are in excellent agreement with the experimental data. |
| format |
Article |
| author |
Thakor, P.B. Gajjar, P.N. Jani, A.R. |
| spellingShingle |
Thakor, P.B. Gajjar, P.N. Jani, A.R. Thermodynamic properties of liquid alkali metals Condensed Matter Physics |
| author_facet |
Thakor, P.B. Gajjar, P.N. Jani, A.R. |
| author_sort |
Thakor, P.B. |
| title |
Thermodynamic properties of liquid alkali metals |
| title_short |
Thermodynamic properties of liquid alkali metals |
| title_full |
Thermodynamic properties of liquid alkali metals |
| title_fullStr |
Thermodynamic properties of liquid alkali metals |
| title_full_unstemmed |
Thermodynamic properties of liquid alkali metals |
| title_sort |
thermodynamic properties of liquid alkali metals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/120465 |
| citation_txt |
Thermodynamic properties of liquid alkali metals / P.B. Thakor, P.N. Gajjar, A.R. Jani // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 473-480. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
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AT thakorpb thermodynamicpropertiesofliquidalkalimetals AT gajjarpn thermodynamicpropertiesofliquidalkalimetals AT janiar thermodynamicpropertiesofliquidalkalimetals |
| first_indexed |
2025-07-08T17:55:46Z |
| last_indexed |
2025-07-08T17:55:46Z |
| _version_ |
1837102370338635776 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 3(27), pp. 473–480
Thermodynamic properties of liquid
alkali metals
P.B.Thakor, P.N.Gajjar, A.R.Jani
Department of Physics, Sardar Patel University,
Vallabh Vidyanagar-388120, Gujarat, India
Received September 18, 2000, in final form December 13, 2000
The internal energy, entropy and Helmholtz free energy of liquid alkali met-
als, viz. Na, K, Rb and Cs are investigated using pseudopotential pertur-
bation scheme based on Gibbs-Bogoliubov variational technique. A local
pseudopotential is applied to describe the electron-ion interaction in the
liquid alkali metals. To introduce the exchange and correlation effects, the
local field correction function proposed by Taylor is employed. The com-
puted values for internal energy, entropy and Helmholtz free energy for the
liquid alkali metals are in excellent agreement with the experimental data.
Key words: internal energy, entropy, Helmholtz free energy,
pseudopotential
PACS: 65.50, 71.15.H
1. Introduction
The Gibbs-Bogoliubov variational approach to investigating the thermodynamic
properties of liquid alkali metals using pseudopotentials theory is well established
and very effectively used in the past [1–13]. In such type of studies, one is concerned
with choosing an appropriate reference system for the structure and the pseudopo-
tential for electron-ion interactions. In all the above studies, people have used hard
sphere (HS) [4–8] or one-component plasma (OCP) [9] or optimized random phase
approximation [10] or charged hard sphere (CHS) [11,12] or soft sphere (SS) [13,14]
as a reference system. For the electron-ion interactions they have employed either
Ashcroft empty core model (AS) [15] or Heine-Abarenkov model (HA) [16] or Har-
monic model potential (HMP) [17] or Generalized nonlocal model potential (GN-
MP) [13] or energy independent nonlocal model potential (EINMP) [7, 8]. Singh
and Singh [4] have reported the thermodynamic properties of alkali metals at 373K
and noticed that for the alkali metals, the results of Helmholtz free energy due to
AS model deviate from the experimental data 1.55% to 4.07%. The outcome due to
HA model gives 0.24% to 2.47% deviation from the experimental findings. In the
case of HMP, this deviation is 6.62% to 14.65%. They have employed hard-sphere
c© P.B.Thakor, P.N.Gajjar, A.R.Jani 473
P.B.Thakor, P.N.Gajjar, A.R.Jani
reference system for structure contribution. In comparison to AS and HA model,
the results due to HMP are poor. All such studies reveal that the accuracy of the
results depends on the application of model potential. The different forms of the
model potential have generated different values of the thermodynamic properties.
Hence, we thought it worthwhile to investigate thermodynamic properties of alkalies
using our model potential [18–20].
This paper deals with the computation of thermodynamic properties of liquid
alkali metals Na, K, Rb and Cs at 373 K based on Gibbs-Bogoliubov (GB) variational
technique. In the present work we choose the hard sphere reference system which
provides analytical representation in term of single parameter, σ known as hard
sphere diameter [21,22] and instead of working with historical model potentials like
AS model and HA model, we have used our own single parametric model to describe
electron-ion interaction [18–20].
The expression for our well established model potential in real space is [18–20],
V 0 (r) = 0, r < rc ,
= −
(
2Z
r
) [
1− exp
(
−r
rc
)]
, r > rc. (1.1)
The Fourier transform in reciprocal space is [18–20],
V 0(q) =
(
−8πZ
Ωq2
){
cos(qrc)−
qrc exp(−1)
1 + q2r2c
[sin(qrc) + qrc cos(qrc)]
}
. (1.2)
Here Z, Ω, q and rc are the valency, atomic volume, wave vector and the potential
parameter, respectively. The parameter rc is evaluated using zero pressure condition.
This method of determination of parameter is independent of any fitting procedure
with the observed quantities. As parameter is determined with the zero pressure
condition, it leads the system into the equilibrium position with minimum energy.
This model potential is the modified version of the Ashcroft’s empty core model.
It is continuous in r space. In comparison with the other historical model potential
[15,16] we have introduced some repulsive part outside the core which vanishes faster
than only Coulomb potential −Ze2/r as r → ∞. Moreover, it may be noted that
the inclusion of this repulsive term outside the core makes the effective core smaller
than the ionic radius of a free ion.
2. Theory
For the investigation of thermodynamic properties of liquid metals, the Helm-
holtz free energy, F , lies at the heart of the pseudopotential perturbation scheme.
The standard thermodynamic relation for the free energy is [2–4]
F = E − TS, (2.1)
where, E is the internal energy and S is the entropy of the system at a temperature
T .
474
Thermodynamic properties of liquid alkali metals
Under the usual perturbation theory, the internal energy E can be expressed as
[2–4],
E = Eion + Eelec + Eelec−ion. (2.2)
Here the first term Eion is composed of kinetic energy {(3/2)kBT} of ions plus the
contribution due to ion-ion interactions, usually known as Madelung contribution.
Thus, we write [2–4],
Eion =
(
3
2
)
kBT +
(
Z2
π
)
∞
∫
0
{a(q)− 1}dq, (2.3)
where Z is the valency and a(q) is the structure factor. For the structure dependent
contribution, the structure factor, a(q), for liquid metals is calculated from the
Percus-Yevick solution for hard sphere fluids which is characterized by the hard
sphere diameter (σ) or, equivalently, by the packing fraction η = (πσ 3/6Ω). Here we
have considered η = 0.45 as described by Ashcroft and Langreth [21,22].
The term Eelec in equation (2.2) is the energy of the homogeneous electron gas
which is the sum of kinetic energy of electrons, exchange energy, correlation energy,
and the low temperature specific heat contribution for the electron gas. Hence, the
expression for Eelec becomes [2–4]
Eelec = NZ
{
(
3
10
)
k2
F −
(
3
4π
)
kF − 0.0474− 0.0155 lnkF −
(
1
2
)(
πkB
kF
)2
T 2
}
,
(2.4)
with kF is the Fermi wave vector and N is the total number of atoms.
In the framework of pseudopotential second ordered perturbation theory Eelec−ion
has been obtained by [2–4],
Eelec−ion = lim
q→0
{
V 0 (q) +
8πZ
q2
}(
Z
Ω
)
+
1
16π3
∞
∫
0
{
V 0 (q)
}2
a (q)
{
1
ε (q)
− 1
}
q4dq,
(2.5)
where, the first term on right hand side represents the first order energy and the
second is the band structure energy. Here, V 0(q) is bare ion pseudopotential and
ε(q) is the modified Hartree dielectric function [23]. ε(q), which takes into account
the conduction electrons interaction, is of the form
ε (q) = 1 + {εH (q)− 1} {1 +G (q)}. (2.6)
In this expression εH(q) is the Hartree dielectric function and G(q) is the correc-
tion factor for the exchange and correlated motion of the conduction electrons. In
the present study we consider the local field correction G(q) due to Taylor [24].
G (q) =
(
q2
4k2
F
) [
1 +
0.1534
πkF
]
. (2.7)
475
P.B.Thakor, P.N.Gajjar, A.R.Jani
Table 1. Input parameters used in the computation.
Metal T (K) Z rc (au) Ω (au) kF(au) σ (au)
Na 373 1 1.0729 277.932 0.4740 6.2046
K 373 1 1.4408 535.332 0.3810 7.7199
Rb 373 1 1.5747 656.168 0.3560 8.2618
Cs 373 1 1.7474 830.565 0.3291 8.9370
The second most essential part of investigating the free energy is the entropy,
S, of the hard sphere fluids. Using the information of hard sphere diameter (σ) or
packing fraction η, one can evaluate the entropy, S, as [2–4].
S = Sgas + Sη + Selec, (2.8)
with
Sgas =
(
5
2
)
kB + kB ln
{
Ω
(
kBT
4π
)
1
2
}
, (2.9)
Sη = kBη (3η − 4) (1− η)−2 , (2.10)
Selec =
(
π2k2
BT
k2
F
)
. (2.11)
3. Results and discussion
We have calculated various contributions to the internal energy, entropy and
Helmholtz free energy of Na, K, Rb and Cs at 373 K. In expressions (2.3) and (2.5),
the integration has been carried out up to 40kF to avoid any artificial cutoff in the
calculation and to achieve proper convergence. The input parameters used in present
calculation are shown in table 1.
The presently calculated values of internal energy, entropy and Helmholtz free
energy are compared with experimental [25] as well as other theoretical data [4] in
tables (2)–(4). Singh and Singh [4] have reported internal energy and entropy using
AS model, HA model and HMP. In the case of Na reported data of Helmholtz free
energy, there are deviations 4.07%, 2.47% and 6.62% due to AS model, HA model,
HMP, respectively. For K these deviations are 4.02%, 0.24% and 9.08%. For Rb
1.55%, 1.041% and 12.72% while for Cs these are 3.64%, 0.42% and 14.65%.
The deviation of the Helmholtz free energy from the experimental values in the
present study are 1.4% for Na, 0.41% for K, 0.71% for Rb and 2.84% for Cs, which
one can consider within the experimental error bar. Thus it is confirmed from tables
2 and 3 that the presently calculated values of internal energy and entropy are in
good agreement with experimental data as well as one of the best results among the
other reported data [4]. This confirms the applicability of our model potential in the
476
Thermodynamic properties of liquid alkali metals
Table 2. Various contributions to the internal energy.
Metal
Eelec
×10−3 (au)
Eion
×10−3 (au)
Eelec−ion
×10−3 (au)
Internal energy
(E)× 10−3 (au)
Present Others Present Others Present Others Present Others Expt.
[4] [4] [4] [4] [25]
–81.70 –81.61 –211.48 –212.9 62.95 76.50 –230.24 –217.31 –226.0
Na –81.61 –211.92 72.57 –220.96
–81.61 –210.16 50.00 –241.77
–80.04 –79.89 –169.28 –169.03 59.12 66.49 –190.20 –182.43 –190.0
K –79.89 –168.39 58.47 –189.81
–79.89 –166.41 38.29 –208.01
–78.57 –78.41 –157.94 –156.77 57.66 58.54 –178.86 –176.64 –180.0
Rb –78.41 –156.49 53.37 –181.53
–78.41 –151.45 26.98 –202.88
–76.50 –76.31 –145.74 –143.53 56.14 56.33 –166.10 –163.51 –170.0
Cs –76.31 –144.19 50.84 –169.66
–76.31 –134.38 14.97 –195.72
Table 3. Various contributions to the entropy.
Entropy
Metal
Sgas/kB −Sη/kB Selec/kB S/kB
Present Others Present Others Present Others Present Others Expt.
[4] [4] [4] [4] [25]
11.222 11.2238 3.9421 4.2284 0.0519 0.0518 7.3317 7.047 7.79
Na 11.2238 4.1400 0.0518 7.136
11.2238 3.6523 0.0518 7.623
12.6742 12.6756 3.9421 3.7968 0.0803 0.0803 8.8123 8.959 9.45
K 12.6756 3.6037 0.0803 9.152
12.6756 3.1211 0.0803 9.635
14.0507 14.0522 3.9421 3.4692 0.0920 0.0919 10.200 10.67 10.35
Rb 14.0522 3.3929 0.0919 10.75
14.0522 2.4472 0.0919 11.69
14.9486 14.9501 3.9421 3.1283 0.1076 0.1076 11.114 11.92 12.12
Cs 14.9501 3.2983 0.1076 11.75
14.9501 1.8473 0.1076 13.21
477
P.B.Thakor, P.N.Gajjar, A.R.Jani
Table 4. Contribution to the Helmholtz free energy.
Helmholtz free energy
F × 10−3 (au)Metal
Present Others [4] Expt. [25]
–238.9049 –225.6262 –235.2061
Na –229.3932
–250.7787
–200.6147 –193.0176 –201.1167
K –200.6257
–219.3965
–190.9164 –189.2497 –192.2315
Rb –194.2342
–216.6951
–179.239 –177.5969 –184.3233
Cs –183.5460
–211.3314
investigation of thermodynamic properties of liquid alkali metals. The investigations
with the other liquid metals, alloys and metallic glasses are in progress [26].
4. Acknowledgement
The work is supported under the special assistance programme at the level of
Departmental Research Support by the University Grants Commission, New Delhi,
INDIA.
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479
P.B.Thakor, P.N.Gajjar, A.R.Jani
Термодинамічні властивості рідких лужних металів
П.Б.Факор, П.Н.Гедджар, А.Р.Дженай
Університет ім. Сердара Пейтела, фізичний факультет
Валлабх Відянагар–388120, Гуджарат, Індія
Отримано 18 вересня 2000 р., в остаточному вигляді –
13 грудня 2000 р.
Внутрішня енергія, ентропія та вільна енергія Гельмгольца рідких
лужних металів (Na, K, Rb і Cs) вивчається в рамках підходу збурених
псевдопотенціалів, що базується на варіаційній техніці Гіббса-Бого-
любова. Для опису електрон-іонної взаємодії в рідких лужних мета-
лах застосовано локальний псевдопотенціал. Обмінні та кореляцій-
ні ефекти представлено з використанням запропонованої Тейлором
функції, що враховує поправку на локальне поле. Отримані значен-
ня внутрішньої енергії, ентропії та вільної енергії Гельмгольца для рід-
ких лужних металів чудово узгоджуються з експериментальними да-
ними.
Ключові слова: внутрішня енергія, ентропія, вільна енергія
Гельмгольца, псевдопотенціал
PACS: 65.50, 71.15.H
480
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