Structural study of liquid rare earth metals from charged hard sphere reference fluid
The present article deals with the investigation of structure factor, s(q) ; radial distribution function, g(r) and interatomic distance, r₁ of liquid rare earth metals, Nd, Dy, Ho, Er and Lu by adopting Charged Hard Sphere (CHS) reference fluid. To describe electron-ion interaction, our well est...
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Інститут фізики конденсованих систем НАН України
2002
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irk-123456789-1206702017-06-13T03:03:18Z Structural study of liquid rare earth metals from charged hard sphere reference fluid Thakor, P.B. Gajjar, P.N. Jani, A.R. The present article deals with the investigation of structure factor, s(q) ; radial distribution function, g(r) and interatomic distance, r₁ of liquid rare earth metals, Nd, Dy, Ho, Er and Lu by adopting Charged Hard Sphere (CHS) reference fluid. To describe electron-ion interaction, our well established model potential along with the dielectric function due to Taylor is used. Good agreement between present and experimental findings is concluded. В даній статті вивчаються структурний фактор s(q) , радіальна функція розподілу g(r) і міжатомна відстань r₁ рідкоземельних рідких металів Nd, Dy, Ho, Er і Lu на основі флюїду заряджених твердих сфер. Для опису взаємодії електрон-іон використовується наш модельний потенціал разом з діелектричною функцією Тейлора. Робиться висновок про добре узгодження отриманих результатів з експериментальними. 2002 Article Structural study of liquid rare earth metals from charged hard sphere reference fluid / P.B. Thakor, P.N. Gajjar, A.R. Jani // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 493-501. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 71.15.H, 61.25.M DOI:10.5488/CMP.5.3.493 http://dspace.nbuv.gov.ua/handle/123456789/120670 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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| description |
The present article deals with the investigation of structure factor, s(q) ;
radial distribution function, g(r) and interatomic distance, r₁ of liquid rare
earth metals, Nd, Dy, Ho, Er and Lu by adopting Charged Hard Sphere
(CHS) reference fluid. To describe electron-ion interaction, our well established
model potential along with the dielectric function due to Taylor is
used. Good agreement between present and experimental findings is concluded. |
| format |
Article |
| author |
Thakor, P.B. Gajjar, P.N. Jani, A.R. |
| spellingShingle |
Thakor, P.B. Gajjar, P.N. Jani, A.R. Structural study of liquid rare earth metals from charged hard sphere reference fluid Condensed Matter Physics |
| author_facet |
Thakor, P.B. Gajjar, P.N. Jani, A.R. |
| author_sort |
Thakor, P.B. |
| title |
Structural study of liquid rare earth metals from charged hard sphere reference fluid |
| title_short |
Structural study of liquid rare earth metals from charged hard sphere reference fluid |
| title_full |
Structural study of liquid rare earth metals from charged hard sphere reference fluid |
| title_fullStr |
Structural study of liquid rare earth metals from charged hard sphere reference fluid |
| title_full_unstemmed |
Structural study of liquid rare earth metals from charged hard sphere reference fluid |
| title_sort |
structural study of liquid rare earth metals from charged hard sphere reference fluid |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120670 |
| citation_txt |
Structural study of liquid rare earth metals from charged hard sphere reference fluid / P.B. Thakor, P.N. Gajjar, A.R. Jani // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 493-501. — Бібліогр.: 18 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT thakorpb structuralstudyofliquidrareearthmetalsfromchargedhardspherereferencefluid AT gajjarpn structuralstudyofliquidrareearthmetalsfromchargedhardspherereferencefluid AT janiar structuralstudyofliquidrareearthmetalsfromchargedhardspherereferencefluid |
| first_indexed |
2025-07-08T18:19:01Z |
| last_indexed |
2025-07-08T18:19:01Z |
| _version_ |
1837103833338085376 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 3(31), pp. 493–501
Structural study of liquid rare earth
metals from charged hard sphere
reference fluid
P.B.Thakor, P.N.Gajjar, A.R.Jani
Department of Physics, Sardar Patel University,
Vallabh Vidyanagar 388 120, Gujarat, India
Received August 27, 2001
The present article deals with the investigation of structure factor, s(q) ;
radial distribution function, g(r) and interatomic distance, r1 of liquid rare
earth metals, Nd, Dy, Ho, Er and Lu by adopting Charged Hard Sphere
(CHS) reference fluid. To describe electron-ion interaction, our well estab-
lished model potential along with the dielectric function due to Taylor is
used. Good agreement between present and experimental findings is con-
cluded.
Key words: pseudopotential, structure factor, radial distribution function,
rare earth metals
PACS: 71.15.H, 61.25.M
1. Introduction
The structure factor S(q), is one of the important properties to study the various
electronic, magnetic, static and dynamic properties of a material, in liquid states,
which is a measure of particle correlations in the reciprocal space. The charged hard
sphere (CHS) model is very useful to evaluate the structure factor of metals in liq-
uid state [1–6]. Such a system of CHS in a uniform background of electrons has
been solved exactly in a mean spherical approximation by Palmer and Weeks [5].
According to CHS model, the reference system consists of Coulombically interacting
positively charged point charges in a uniform background of conduction electrons.
Though the CHS method has proved to be very useful for explaining structural prop-
erties of liquid metals, the study of liquid rare earth metals using CHS is limited [3].
Gopala Rao and Bandyopadhyay [3] have reported the structure factor, S(q); radial
distribution function, g(r); and interatomic distance, r1 of the nearest neighbour
atoms for Nd, Dy, Ho, Er and Lu by employing CHS reference fluid with Ashcroft
empty core model potential. They have fitted the input parameters like effective
valency (Z), potential parameter (rc) and hard sphere diameter (σ) in such a way
that a good agreement with experimental findings should be obtained.
c© P.B.Thakor, P.N.Gajjar, A.R.Jani 493
P.B.Thakor, P.N.Gajjar, A.R.Jani
The present paper deals with the structural studies of some liquid rare earth
metals, which reports the structure factor, S(q); radial distribution function, g(r)
and interatomic distance, r1 of the nearest neighbour atoms for Nd, Dy, Ho, Er and
Lu. The important aspect of the present investigation is to make the computation
free from any fitting procedure to reproduce correct experimental data as it was
done by Gopala Rao and Bandyopadhyay [3].
Here, our well established single parametric local pseudopotential is used to
represent the electron-ion interaction. The present model potential in real space is
of the form [6–12]
V (r) =
{
0, r < rc,
−(Ze2/r)[1 − exp(−r/rc)], r > rc .
(1.1)
The corresponding bare-ion form factor in the reciprocal space is given as [6–12]
VB(q) =
(
−4πZe2
Ωq2
) [
cos(qrc) −
{
(qrc) exp(−1)
1 + q2r2
c
}
{sin(qrc) + (qrc) cos(qrc)}
]
.
(1.2)
Here Z, Ω, q and rc are the valency, atomic volume, wave vector and the parameter
of the potential, respectively. In the present investigations, the parameter of the
potential is determined using [13],
rc = 0.51(Z)−1/3Ra , (1.3)
where Ra is the atomic radius.
This model potential is the modified version of the Ashcroft’s empty core model.
It is continuous in r space. Here 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 [6–12].
2. Theory
The CHS model was studied within the framework of a mean spherical approxi-
mation inside the core and outside the core, a perturbation in the form of Coulomb
interaction is assumed to act by Palmer and Weeks [5]. In the CHS approximation,
the direct correlation function is given by [3,4,6]
C0(r) =
A + B
(
r
σ
)
+ C
(
r
σ
)2
+ D
(
r
σ
)3
+ E
(
r
σ
)5
, r < σ,
−γ /
(
r
σ
)
, r > σ.
(2.1)
The coefficients involved in equation (2.1) are given by,
A = −
(1 + 2η)2
(1 − η)4
+
Q2
4(1 − η)2
−
(1 + η)QK
12η
−
(5 + η2)2
60η
,
B = 6ηM2, C =
K2
6
, D =
(η
2
)
(A − K2U), E =
ηK2
60
(2.2)
494
Structural study of liquid rare earth metals. . .
with
Q =
(1 + 2η)
(1 − η)
[
1 −
{
1 + 2(1 − η)3K
(1 + 2η)2
}1/2
]
,
M =
Q2
24η
−
(1 + 0.5η)
(1 − η)2
, U =
(1 + η − η2/5)
12η
−
(1 − η) Q
12ηK
,
γ = β
(Ze)2
ε0σ
, K = (24ηγ)1/2, η =
(π
6
)
ρσ3. (2.3)
Here, γ, K and η are the dimensionless variables, which represent the Coulomb
interaction potential, the inverse screening length due to Debye and Huckel and
packing fraction, respectively. Here, σ is the charged hard sphere diameter, Ze is
the ionic charge, β = 1/kBT , kB is Boltzmann constant, T is absolute temperature of
the system, ε0 is the dielectric constant of the medium. Since the electron background
is uniform, its dielectric constant is unity.
The static structure factor S0(q) of the reference system is related to its direct
correlation function in the following form [3,4,6]:
S0(q) =
1
[1 − ρC0(q)]
. (2.4)
The mathematical expression for ρC0(q) is given by [3,4,6]
ρC0(q) =
(
24η
q6
)
[
Aq3(sin q − q cos q)
+ Bq2
{
2q sin q − (q2 − 2) cos q − 2
}
+ Cq
{
(3q2 − 6) sin q − (q2 − 6)
}
+ D
{
(4q2 − 24)q sin q − (q4 − 12q2 + 24) cos q + 24
}
+ E
{
6(q4 − 20q2 + 120)q sin q − (q6 − 30q4 + 360q2 − 720) cos q − 720
}
/q2
− γq4 cos q
]
(2.5)
Here q is expressed in units of σ−1.
The effect of responding electrons on the ionic motion is taken into account by
assuming a weak coupling between valence electrons and ions which is also the basis
of a standard pseudopotential approach. Within a linear screening approximation,
the structure factor of a liquid metal is given by [3,4,6]
S(q) =
S0(q)
[
1 + ρβV̄ (q)S0(q)
] (2.6)
with
V̄ (q) =
V 2
B
(q)
φ(q)
[
1
ε(q)
− 1
]
, (2.7)
which is the attractive screening correlation to the direct ion-ion potential, φ(q) =
4πe2/q2 is the Fourier transform of bare Coulombic interaction between two electrons
495
P.B.Thakor, P.N.Gajjar, A.R.Jani
and VB(q) is the bare ion pseudopotential. The modified dielectric function ε(q) has
the following form
ε(q) = 1 + [1 − f(q)] [εH(q) − 1] , (2.8)
with the static Hartree dielectric function εH(q) represented by
εH(q) = 1 +
me2
2πh2kFY 2
[
1 +
(1 − Y 2)
2Y
ln
∣
∣
∣
∣
1 + Y
1 − Y
∣
∣
∣
∣
]
, (2.9)
where m is the ionic mass, h the Planck’s constant, kF the Fermi wave vector, e is
the electronic charge and Y = q/2kF .
The local field correction f(q) due to Taylor [14] is used to incorporate the
exchange and correlation among the conduction electrons in the dielectric screening
f(q) =
q2
4k2
F
[
1 +
0.1534
πkF
]
. (2.10)
The expression for the radial distribution function is given by [3,6]
g (r) = 1 +
(
1
2π2ρr
)
∞
∫
0
q {S(q) − 1} sin (qr)dq. (2.11)
Using this radial distribution function we obtain the interatomic distance, r1 of the
nearest neighbour atoms. The interatomic distance r1 corresponds to the maximum
peak of g(r) curve.
3. Results and discussion
The constants and parameters used in the present computations of structure
factor, S(q) and radial distribution function, g(r) for the liquid rare earth metals
are tabulated in table 1.
Figures 1–5 show the computed values of S(q) and g(r) of Nd, Dy, Ho, Er and
Lu, respectively along with the experimental findings [15].
Table 1. Parameters and constants used in present calculation.
Metal T (K) ρ (gm/cm3) Z η kF (Å−1) rc (Å)
Nd 1473 6.78 1.5 0.40 1.0793 0.8113
Dy 1703 8.14 1.5 0.43 1.1025 0.7899
Ho 1753 8.25 1.5 0.43 1.1020 0.7868
Er 1793 8.37 1.5 0.44 1.1022 0.7828
Lu 1953 9.18 1.5 0.44 1.1197 0.7725
496
Structural study of liquid rare earth metals. . .
Figure 1. Structure factor, S(q) and radial distribution function, g(r) for Nd at
1473 K.
Figure 2. Structure factor, S(q) and radial distribution function, g(r) for Dy at
1703 K.
Figure 3. Structure factor, S(q) and radial distribution function, g(r) for Ho at
1753 K.
497
P.B.Thakor, P.N.Gajjar, A.R.Jani
Figure 4. Structure factor, S(q) and radial distribution function, g(r) for Er at
1793 K.
Figure 5. Structure factor, S(q) and radial distribution function, g(r) for Lu at
1953 K.
Table 2. Position of first and second peak in S(q) (Å−1) and interatomic distance
(r1).
Metal
Peak positions in S(q) (Å−1)
Interatomic distance r1(Å)
First Second
Present Expt.[15] Present Expt.[15] Present Other[3] Expt.[15]
Nd 2.116 2.10 4.081 4.23 3.227 3.50 3.45
Dy 2.173 2.12 4.138 4.03 3.227 3.40 3.43
Ho 2.173 2.21 4.138 4.35 3.227 3.40 3.32
Er 2.173 2.24 4.138 4.38 3.227 3.50 3.28
Lu 2.210 2.28 4.195 4.43 3.175 3.40 3.22
498
Structural study of liquid rare earth metals. . .
In table 2, we have compared the position of the first and the second peaks in
S(q) with experimental results of Waseda and Miller [15].
The deviation of the presently generated results from experimental data at the
first peaks of S(q) are of the order of 0.16 to 0.27. Similarly, for g(r) these deviations
are of the order of 0.52 to 0.74. For all the five liquid rare earth metals, the magnitude
of the first peak in S(q) and g(r) is slightly higher than the experimental data, but
the position of the first and second peaks in S(q) is well estimated.
The interatomic distance r1 of the nearest neighbour atoms is also investigat-
ed and compared with other theoretical [3] as well as experimental findings [15]
in table 2. The excellent qualitative agreement between present and experimental
findings is obtained.
Gopala Rao and Bandyopadhyay [3] have chosen Z, rc and σ in such a way that
it generates satisfactory structural data. They have fitted these three parameters to
obtain correct experimental predictions. However, the present investigation is free
from such kind of a fitting procedure. In the previously reported study of liquid
rare earth metals, the uncertainty in the data Z is observed. Delley et al. [16] have
estimated Z = 1.3, Duthie and Pettifor [17] have estimated the value Z = 1.1 to 1.5.
Waseda and Miller [15] have estimated Z = 1.33 to 2.09 for Lu. For Lu, Delley et
al. [16] have suggested Z closer to two. While Johansson [18] have assumed Z = 3.0.
Gopala Rao and Bandyopadhyay [3] have taken Z = 1.54 to 2.03 to obtain a better
agreement with the experimental findings. Thus, instead of making an adjustment
in Z, we have considered Z = 1.5 for all the five liquid rare earth metals and the
uncertainty in the parameters is totally avoided, consistently.
And finally we conclude that, though the present computation is free from any
artificial fitting procedure to predict correct experimental data, it is capable of ex-
plaining very good results for the structural data of liquid rare earth metals. Hence,
the reported data are more meaningful and will provide a better source for further
comparison either with theoretical or with experimental data. This confirms the
applicability of our model potential and CHS method for predicting the structural
studies of the liquid rare earth metals.
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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500
Structural study of liquid rare earth metals. . .
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