Vacuum 196 (2022) 110725
Contents lists available at ScienceDirect
Vacuum
journal homepage: www.elsevier.com/locate/vacuum
Equation-of-state and melting curve of solid neon and argon up to 100 GPa
Nguyen Van Nghia a ,∗, Ho Khac Hieu b,c ,∗, Duong Dai Phuong d
a
Faculty of Electrical and Electronics Engineering, Thuyloi University, 175 Tay Son, Dong Da, Ha Noi 116705, Viet Nam
Institute of Research and Development, Duy Tan University, 03 Quang Trung, Hai Chau, Da Nang 550900, Viet Nam
c
Faculty of Environmental and Natural Sciences, Duy Tan University, 03 Quang Trung, Hai Chau, Da Nang 550900, Viet Nam
d
Military College of Tank Armour Officer, Vinh Phuc 15900, Viet Nam
b
ARTICLE
INFO
Keywords:
Noble gases
Neon
Argon
Lattice parameter
Melting
High pressure
ABSTRACT
Equation-of-state and melting curve of neon and argon solids at high pressure are investigated based on the
statistical moment method taking into account the anharmonicity contributions of thermal lattice vibrations.
We have derived the analytical expression of equation-of-state and proposed a procedure to calculate melting
temperature of materials at high pressure. We perform numerical calculations for neon and argon solids up to
100 GPa by using the Buckingham function (exp-6) to model the interaction between atoms. Our theoretical
calculations reveal the strong dependence of the lattice constant and melting temperature on pressure of raregas solids, in particular at below 10 GPa. The derived equation-of-states of neon and argon go along with
earlier experiments up to pressure 100 GPa. The present argon melting curve is in good agreement with
previous measurements up to pressure 80 GPa. Our melting line of neon can reproduce well melting data
points using W absorbers with thickness 8 μm and 6–8 μm up to pressure nearly 80 GPa, and differs from the
measurements using W absorbers with thickness 4–6 μm.
1. Introduction
The investigations of rare-gas solids are of great interest in materials
science and high-pressure physics because of their inert nature as
well as their closed-shell electronic configuration [1–5]. In geophysics,
high-pressure properties of rare-gas solids give us important data for
geophysical investigation (i.e., for study the evolution and dynamics
of the Earth, stars and planets, especially where they appear in a
condensed and purified state) [6,7]. It is well-known that noble gas
solids are widely used as a hydrostatic pressure media in diamond
anvil cell (DAC) experiments [8]. They are also appropriate candidates
for internal pressure standard in X-ray diffraction (XRD) measurements [4]. These applications come from the facts that solid gases
remain stable at high pressure and high temperature [2]. Therefore,
thermodynamic properties of noble gas solids under extreme density
and temperature conditions play an important role to understand and
design DAC experiments. Additionally, because noble gas solids have
simple cubic structure and closed-shell electronic configuration like
to that of free atoms, they become excellent candidates for physical
model verification. Recently, the combination of DACs with various
optical and synchrotron X-ray techniques allows researchers to perform
experiments at extreme pressure–temperature conditions of the Earth’s
deep interior [9]. Many experimental works have been done to measure
the equation-of-state (EOS) and melting temperature of solid neon
(Ne) and argon (Ar) crystals [10]. Notwithstanding some prominent
discrepancies due to the difficulties of diagnostic equipment under
extreme conditions still exist.
Neon is the second element of the rare gas family. At ambient
pressure, Ne crystallizes to a face-centred cubic (FCC) structure at
temperature 24.4 K. Using energy-dispersive XRD with microcollimated
synchrotron radiation, Hemley et al. showed that the structure of neon
remains unchanged up to 110 GPa at room temperature [11]. Because
of the quite large energy band gap between the filled 2p-valence states
to the 3d-conduction state in electronic structure, one may expect the
physical properties including high-pressure melting behaviour of neon
will be different from those of the heavier noble gas solids [12]. Experimentally, melting point of neon was measured in a heated DAC up to
5.5 GPa by Vos et al. [13]. On the theoretical side, Koči et al. performed
ab initio calculations (using the Vienna Ab initio Simulation Package
(VASP)) and classical molecular dynamics (MD) simulations to derive
the melting curve of neon up to 150 GPa [14]. Recently, SantamaríaPérez et al. reported new measured melting temperature data of Ne
up to 70 GPa using the laser-heated DAC technique [12]. They showed
that their melting data are in good agreement with MD simulations and
ab initio calculations of Koči et al. However, their melting curve does
not agree with the prediction from the corresponding states law in the
range from 15 to 70 GPa.
∗ Corresponding authors.
E-mail addresses: nghia_nvl@tlu.edu.vn (N. Van Nghia), hieuhk@duytan.edu.vn (H.K. Hieu).
https://doi.org/10.1016/j.vacuum.2021.110725
Received 16 September 2021; Received in revised form 28 October 2021; Accepted 30 October 2021
Available online 13 November 2021
0042-207X/© 2021 Elsevier Ltd. All rights reserved.
Vacuum 196 (2022) 110725
N. Van Nghia et al.
The thermal averages ⟨𝑢𝑖𝛽 𝑢𝑖𝜂 ⟩𝑝 and ⟨𝑢𝑖𝛽 𝑢𝑖𝜂 𝑢𝑖𝜁 ⟩𝑝 are, respectively,
the second-order moment and third-order moment. These two power
moments can be expressed through the first-order moment ⟨𝑢𝑖𝛽 ⟩𝑝 as
follows [26]
⟨ ⟩
(
) 𝜃𝛿𝛽𝜂
𝜕 𝑢𝑖𝛽 𝑝 ℏ𝛿𝛽𝜂
⟩
⟨
⟨ ⟩ ⟨ ⟩
ℏ𝜔
+
𝑢𝑖𝛽 𝑢𝑖𝜂 𝑝 = 𝑢𝑖𝛽 𝑝 𝑢𝑖𝜂 𝑝 + 𝜃
coth
−
,
(4)
𝜕𝑎𝜂
2𝑚𝜔
2𝜃
𝑚𝜔2
According to accretion models, argon is the most abundant rare gas
in the Earth’s atmosphere [15]. It forms simple FCC crystal structure
at low pressures. By angle-dispersive XRD measurements, Errandonea
et al. showed the appearance of the hexagonal close packed (HCP)
structure in argon at 49.6 GPa, but the structural phase transition FCCto-HCP still does not complete at pressure 114 GPa [16]. They predicted
this structural phase transition would be achieved at 310 GPa. Experimentally, argon melting temperature was measured in an external
DAC using a new interferometric technique to 6 GPa [17]. Recent
measurement of melting curve of argon has been executed by Boehler
et al. in a laser-heated DAC to nearly 80 GPa reaching melting point
around 3300 K [18]. On the theoretical side, Belonoshko performed
MD simulations to study EOS and melting temperature of argon up
to 400 GPa. Up to about 40 GPa, the MD simulated melting data
are in good agreement with laser-heated DAC measurement as well
as prediction from corresponding states scaling from the neon melting curve [18]. However, above 40 GPa, there occurs a significant
difference from the laser-heated DAC measurement.
In this paper, the equation-of-state and pressure-dependent melting temperature of solid neon and argon crystals are investigated
based on the statistical moment method (SMM) [19–21]. For this purpose, the pressure dependence of the lattice parameter and the atomic
mean-square displacement (MSD) are firstly considered. Afterwards,
the derived results are used to study the EOS and melting temperature
(based on the Lindemann melting criterion) of neon and argon solids.
Numerical calculations for these two solid rare-gases will be performed
up to pressure of 100 GPa and compared, where possible, with previous
measurements and calculations to verify the proposed model.
⟨ ⟩
⟨ ⟩
𝜕 2 𝑢𝑖𝛽 𝑝
⟨
⟩
⟨ ⟩ ⟨ ⟩ ⟨ ⟩
⟨ ⟩ 𝜕 𝑢𝑖𝜂 𝑝
2
+𝜃
𝑢𝑖𝛽 𝑢𝑖𝜂 𝑢𝑖𝜁 𝑝 = 𝑢𝑖𝛽 𝑝 𝑢𝑖𝜂 𝑝 𝑢𝑖𝜁 𝑝 + 𝜃𝑃𝛽𝜂𝜁 𝑢𝑖𝛽 𝑝
𝜕𝑎𝜂
𝜕𝑎𝜂 𝜕𝑎𝜁
⟨ ⟩
⟨ ⟩
(
) 𝜃 𝑢𝑖𝜁 𝑝 𝛿𝛽𝜂
ℏ 𝑢𝑖𝜁 𝑝 𝛿𝛽𝜂
ℏ𝜔
+
,
(5)
coth
−
2𝑚𝜔
2𝜃
𝑚𝜔2
where 𝜃 = 𝑘𝐵 𝑇 , 𝑘𝐵 is the Boltzmann’s constant, 𝑚 is the atom mass, 𝜔
is the atomic vibration frequency, the Kronecker delta 𝛿𝛽𝜂 = 1 (𝛽 = 𝜂)
or 0 (otherwise), 𝑃𝛽𝜂𝜁 = 1 (𝛽 = 𝜂 = 𝜁) or 0 (otherwise) depending on
the Cartesian components 𝛽, 𝜂 and 𝜁.
Then Eq. (3) can be re-written into an equivalent nonlinear differential equation as
𝛾𝜃 2
)
(
) ⎫
⎧ ⎡(
⎤
4
𝜕 4 𝜑𝑖0
1 ⎪∑ ⎢ 𝜕 𝜑𝑖0
⎥⎪
𝛾=
+6
.
⎨
2 𝜕𝑢2
⎥⎬
12 ⎪ 𝑖 ⎢ 𝜕𝑢4
𝜕𝑢
𝑖𝑥
𝑖𝑥 𝑖𝑦 𝑒𝑞 ⎦⎪
𝑒𝑞
⎭
⎩ ⎣
The SMM was developed by Tang & Hung and their collaborators [19,22,23]. Many published papers show its efficiency on the
study of structural, mechanical as well as thermodynamic properties
of crystalline materials under pressure [24,25]. Readers can easily find
more details of SMM elsewhere. Below we summarize the main results
of SMM, the SMM equation-of-state and Lindemann melting criterion
of crystals.
Assuming the system has a Hamiltonian as
⟨𝑉̂ ⟩𝛼 𝑑𝛼 ,
where
𝑦0 (𝑇 ) ≈
(1)
⟨𝑉̂ ⟩𝛼 = −𝜕𝜓∕𝜕𝛼,
(9)
𝑦 (𝑇 ) ≈ 𝑦0 (𝑇 ) + 𝐴1 𝑝,
√
2𝛾𝜃 2
𝐴,
3𝑘3
𝛾 2 𝜃2
(10)
𝛾 3 𝜃3
𝛾 4 𝜃4
𝛾 5 𝜃5
𝛾 6 𝜃6
𝑎2 +
𝑎3 +
𝑎4 +
𝑎5 +
𝑎6 ,
𝐴 = 𝑎1 +
𝑘4
𝑘6
𝑘8
𝑘10
𝑘12
]
[
)
2𝛾 2 𝜃 2 (
1
𝑋
𝐴1 =
1+
1+
(𝑋 + 1) ,
𝑘
2
𝑘4
𝛼
∫0
(8)
The solution of the nonlinear differential equation (6) is the mean
atomic displacement 𝑦 (𝑇 ) at temperature 𝑇 . The displacement 𝑦 (𝑇 ) can
be written in term of the supplemental force 𝑝 as [19,26]
̂ 0 denotes the harmonic lattice Hamiltonian, the latter term is
where 𝐻
the anharmonic contributions caused by thermal lattice vibrations. The
Helmholtz free energy of this system at temperature 𝑇 is then derived
by [19]
𝜓 = 𝜓0 −
(6)
where 𝑋 = (ℏ𝜔∕2𝜃) coth(ℏ𝜔∕2𝜃), 𝑦 ≡ ⟨𝑢𝑖 ⟩𝑝 , 𝑘 and 𝛾 parameters
correspond to the second-order derivative and fourth-order derivative
of potential function 𝜑𝑖𝑗 (𝑟) as the following formulas
(
)
2
1 ∑ 𝜕 𝜑𝑖0
𝑘=
≡ 𝑚𝜔2 ,
(7)
2 𝑖
𝜕𝑢2𝑖𝑥 𝑒𝑞
2. Theoretical approach
̂ =𝐻
̂ 0 − 𝛼 𝑉̂ ,
𝐻
𝑑2𝑦
𝑑𝑦
𝜃
+ 3𝛾𝜃𝑦
+ 𝑘𝑦 + 𝛾 (𝑋 − 1) − 𝑝 = 0,
𝑑𝑝
𝑘
𝑑𝑝2
(11)
(12)
and
(2)
𝑋
,
2
13 47
23 2 1 3
𝑎2 =
+
𝑋+
𝑋 + 𝑋 ,
3
6
6
2
)
(
25 121
50 2 16 3 1 4
𝑎3 = −
+
𝑋+
𝑋 +
𝑋 + 𝑋 ,
3
6
3
3
2
43 93
169 2 83 3 22 4 1 5
𝑎4 =
+
𝑋+
𝑋 +
𝑋 +
𝑋 + 𝑋 ,
(13)
3
2
3
3
3
2
)
(
103 749
363 2 391 3 148 4 53 5 1 6
𝑎5 = −
+
𝑋+
𝑋 +
𝑋 +
𝑋 +
𝑋 + 𝑋 ,
3
6
2
3
3
6
2
561
1489 2 927 3 733 4 145 5 31 6 1 7
𝑎6 = 65 +
𝑋+
𝑋 +
𝑋 +
𝑋 +
𝑋 +
𝑋 + 𝑋 .
2
3
2
3
2
3
2
Additionally, from the moment expansion formula (4) one can find
out the atomic MSD function of FCC crystal as [26]
𝑎1 = 1 +
where the free energy of the system 𝜓0 corresponds to the Hamiltonian
̂ 0 and ⟨...⟩𝛼 presents the expected values at thermal equilibrium.
𝐻
Let us describe the interaction between 𝑖 and 𝑗 atoms in the crystal
by pair potential 𝜑𝑖𝑗 (𝑟) function. It is known that by taking derivatives
of the interatomic potential, we can derive the force acting on the
zeroth central atom. Because of thermal lattice vibrations, the 0th
central atom in the lattice is exerted on by a supplementary force 𝑝𝜂
and the net force acting on the 0th atom should be zero. Then we obtain
the force balance relation as the following
)
(
)
( 2
∑
𝜕 3 𝜑𝑖0
𝜕 𝜑𝑖0
1 ∑
⟨𝑢𝑖𝛽 ⟩𝑝 +
⟨𝑢 𝑢 ⟩ +
(3)
𝜕𝑢𝑖𝛽 𝜕𝑢𝑖𝜂 𝑒𝑞
2 𝛽,𝜂,𝛾 𝜕𝑢𝑖𝛽 𝜕𝑢𝑖𝜂 𝜕𝑢𝑖𝛾 𝑒𝑞 𝑖𝛽 𝑖𝛾 𝑝
𝛽,𝜂
(
)
𝜕 4 𝜑𝑖0
1 ∑
+
⟨𝑢 𝑢 𝑢 ⟩ − 𝑝𝜂 = 0,
6 𝛽,𝛾,𝜁 𝜕𝑢𝑖𝛽 𝜕𝑢𝑖𝜂 𝜕𝑢𝑖𝛾 𝜕𝑢𝑖𝜁 𝑒𝑞 𝑖𝛽 𝑖𝛾 𝑖𝜁 𝑝
⟨ 2 ⟩ 2𝛾𝜃 2
𝜃
𝑢 =
(14)
𝐴 + 𝜃𝐴1 + (𝑋 − 1) .
𝑘
3𝑘3
The Helmholtz free energy 𝜓 of FCC system can be evaluated from
the internal energy and has the form as [19,26]
{[
)]
2𝛾 (
3𝑁𝜃 2
𝑋
𝜓 = 𝑈0 + 𝜓0 +
𝛾2 𝑋 2 − 1 1 +
+
3
2
𝑘2
where 𝛽, 𝜂, 𝛾, 𝜁 = 𝑥, 𝑦, 𝑧; 𝑢𝑖𝜂 is the 𝜂 component of the 𝑖th atom displacement; ⟨𝑢𝑖𝛽 ⟩𝑝 , ⟨𝑢𝑖𝛽 𝑢𝑖𝜂 ⟩𝑝 and ⟨𝑢𝑖𝛽 𝑢𝑖𝜂 𝑢𝑖𝜁 ⟩𝑝 are the thermal averages of
the atomic displacements; and the subscript eq indicates the evaluation
at equilibrium.
2
Vacuum 196 (2022) 110725
N. Van Nghia et al.
](
)}
)
(
2𝜃 4 2
𝑋
,
(15)
+
𝛾2 𝑋 − 2 𝛾12 + 2𝛾1 𝛾2 (1 + 𝑋) 1 +
2
2
𝑘 3
[
(
)]
where 𝜓0 = 3𝑁𝜃 𝑥 + ln 1 − 𝑒−2𝑥 (with 𝑥 = ℏ𝜔∕2𝜃) is the harmonic
( )
∑
part of free energy 𝜓; 𝑈0 = 𝑖 𝜑𝑖0 𝑟𝑖 is the total interaction energy on
the zeroth atom; 𝛾1 and 𝛾2 are defined, respectively, as
[
𝛾1 =
1 ∑
48 𝑖
(
Neon
Argon
𝐸𝑐 ∕𝑘𝐵 (K)
𝑟0 (Å)
𝛼
36.486
122
3.121
3.85
13.2
13.5
(16a)
𝛾 = 4(𝛾1 + 𝛾2 ),
and
Table 1
Buckingham potential parameters of solid neon [13] and argon [34].
𝜕 4 𝜑𝑖0
𝜕𝑢4𝑖𝑥
)
; 𝛾2 =
𝑒𝑞
6 ∑
48 𝑖
(
𝜕 4 𝜑𝑖0
𝜕𝑢2𝑖𝑥 𝜕𝑢2𝑖𝑦
)
.
(16b)
𝑒𝑞
In Thermodynamics, pressure 𝑃 and free energy 𝜓 of the crystal
have a relation as
)
(
)
(
𝜕𝜓
𝜕𝜓
𝑟
=−
.
(17)
𝑃 =−
𝜕𝑉
3𝑉
𝜕𝑟
From Eqs. (15) and (17), Tang and Hung derived the relation of
pressure and volume of the crystal as [20]
[
]
1 𝜕𝑈0 𝜃𝑋 𝜕𝑘
𝑃 𝑣 = −𝑟
,
(18)
+
6 𝜕𝑟
2𝑘 𝜕𝑟
where 𝑣 = 𝑉 ∕𝑁 is the atomic volume of a crystal having volume 𝑉 .
The above equation (18) is so-called the SMM equation-of-state
(EOS). By numerically solving this SMM EOS we can obtain the atomic
nearest-neighbour distance (NND) 𝑟1 (𝑃 , 𝑇 ) at 𝑃 and 𝑇 conditions. The
pressure-induced volume change at a given temperature 𝑇 can be
obtained as
𝑟3 (𝑃 , 𝑇 )
𝑉
= 1
.
𝑉0
𝑟3 (0, 𝑇 )
Fig. 1. The change of volume crystal of solid neon under compression. Experimental
data [11,35,36] have been shown for comparison.
(19)
1
of many physical properties of rare-gases such as heat and entropy
of melting, shock wave adiabat [14,31,32]. Nevertheless, the pair
potential can well describe neither negative Cauchy discrepancy nor a
realistic model of noble gases [33]. These specific phenomena requires
more precise potential, i.e., the many-body potential type. In this work,
for simplicity of numerical calculations, we choose the Buckingham
pair potential to describe the interaction between two intermediate
atoms of neon and argon crystals. This our choice of Buckingham
potential and its parameters for neon and argon is double-checked
by making a comparison between the calculated pressure-dependent
lattice parameter and those of previous measurements.
Using this Buckingham function and the coordination sphere
method, the potential energy of system 𝑈0 (on the first five coordination spheres) can be calculated as
(
)
𝑁 ∑
𝑈0 =
𝜑 |𝑟 + 𝑢𝑖 | ,
(23)
2 𝑖 𝑖0 𝑖
{
[ (
)]
( )6 }
𝑎
𝑟0
𝑁
6
𝛼
=
𝐸
𝐴 exp 𝛼 1 − 0
−
𝐴
,
2 𝑐 𝛼−6 𝛼
𝑟0
𝛼 − 6 6 𝑎0
About melting problem, one of the oldest and the most widely
used methods to approximately predict melting point of material is
the Lindemann criterion [27–29]. This criterion states that melting of
a solid occurs when the ratio between the square root of the atomic
MSD from the equilibrium position and the interparticle distance tends
to a threshold value. In this work, melting points of neon and argon
solids at various pressure will be calculated by using the modified
Lindemann criterion as follows [30]: We assume the Lindemann ratio
remains unchanged for all studied pressure range. Then firstly using
the SMM approach [19,20] we evaluate the Lindemann ratio at widelyknown experimental melting point at ambient pressure (through the
⟨ ⟩
MSD 𝑢2 and NND 𝑟1 ). After that we numerically solve the melting
temperature for neon and argon solids at different pressure based on
this ratio.
Here it should be noted that, at pressure 𝑃 and temperature 𝑇 ,
Lindemann ratio 𝜉(𝑃 , 𝑇 ) is
√⟨ ⟩
𝑢2 (𝑃 , 𝑇 )
𝜉(𝑃 , 𝑇 ) =
(20)
𝑟1 (𝑃 , 𝑇 )
⟨ 2⟩
where MSD function 𝑢 (𝑃 , 𝑇 ) has the form as
⟨ 2⟩
𝑢 (𝑃 , 𝑇 ) = ⟨𝑢⟩2 (𝑃 , 𝑇 ) + 𝜃𝐴1 (𝑃 , 𝑇 ) +
𝜃
(𝑋 − 1) .
𝑘(𝑃 , 0)
where 𝐴𝛼 and 𝐴6 have the forms as
]
[
]
[
(
)𝑎
(
)𝑎
𝐴𝛼 =𝑧1 + 𝑧2 exp 𝛼 1 − 𝜈2 0 + 𝑧3 exp 𝛼 1 − 𝜈3 0 +
𝑟
𝑟
[
] 0
[
] 0
(
) 𝑎0
(
) 𝑎0
𝑧4 exp 𝛼 1 − 𝜈4
+ 𝑧5 exp 𝛼 1 − 𝜈5
,
𝑟0
𝑟0
(21)
3. Results and discussion
(24)
and
For numerical calculations, we need to know the potential function
𝜑(𝑟). The interatomic interaction between atoms in rare-gas solids
can be characterized precisely by the Buckingham (exp-6) pairwise
potential as [14]
{
[ (
)]
( 𝑟 )6 }
6
𝑟
𝛼
0
𝜑 (𝑟) = 𝐸𝑐
exp 𝛼 1 −
−
,
(22)
𝛼−6
𝑟0
𝛼−6 𝑟
𝐴6 = 𝑧1 +
𝑧2
𝜈26
+
𝑧3
𝜈36
+
𝑧4
𝜈46
+
𝑧5
𝜈56
,
(25)
where 𝑟𝑖 is the radius of the 𝑖th coordination sphere (𝑖 = 1, 5); 𝑟𝑖 =
𝜈𝑖 𝑎0 , in which 𝑎0 = 𝑟1 ; 𝑧𝑖 is the number of atoms being on the √
𝑖th
coordination sphere.√For FCC system: 𝜈1 = 1 and 𝑧1 = 12, 𝜈2 = 1∕√2
and 𝑧2 = 6, 𝜈3 = 1∕ 3 and 𝑧3 = 24, 𝜈4 = 1∕2 and 𝑧4 = 12, 𝜈5 = 1∕ 5
and 𝑧5 = 24.
First of all, using the above total potential energy 𝑈0 , we can
solve SMM EOS (18) to derive the average distance 𝑟(𝑃 , 𝑇 ), and then
where 𝐸𝑐 denotes the cohesion energy, 𝑟0 is the equilibrium NND, and
𝛼 parameter relates to the bulk modulus of material. The potential parameters of neon and argon crystals are listed in Table 1. Here it should
be clearly noted that, the pair potential type is suitable for investigation
3
Vacuum 196 (2022) 110725
N. Van Nghia et al.
Fig. 2. The dependence of melting temperature of solid neon on pressure. Our
theoretical melting curve is presented along with previous works [12–14]. The open
purple diamonds, open green circles and golden stars are, respectively, melting data
points using W absorbers with thickness 8 μm, 4–6 μm, and 6–8 μm; filled maroon
hexagrams represent melting data points obtained using diamond heaters and filled
triangles are loosely packed Ir black absorbers [12]. The low pressure melting data
of Vos et al. are represented by filled green squares [13]. The blue dash and green
dash–dot lines are, respectively, the results from one-phase and two-phase molecular
dynamics simulations [14]. The solid aqua line and filled aqua squares are the melting
points calculated by VASP from visual inspection of the atomic diffusion and the
MSD [14].
Fig. 3. The change of volume crystal of solid argon under compression. Experimental
data [38,39] have been shown for comparison.
calculated by VASP from visual inspection of the atomic diffusion and
the MSD [14]. From this figure we can see that our theoretical melting
curve can reproduce well the measurements of Vos et al. [13] (at low
pressure) and of Santamaría-Pérez et al. [12] up to pressure nearly
80 GPa. Comparing to experiment of Santamaría-Pérez et al. [12] the
error of melting data in SMM approach is within the experimental
uncertainty. At higher pressure, the discrepancy starts to be appeared,
with the theoretical melting line being increasingly slower than the
experimental points. Moreover, our results are between two melting
curves calculated by MD simulations with the Lennard-Jones pair potential of Koči et al. [14]. Notably, our theoretical melting curve is
in reasonable agreement with two-phase MD simulations and ab initio
one-phase calculations [14] at high pressure. The deviation from the
two-phase MD simulations curve could be seen as the limitation of
precision of the present method. This phenomenon may originate from
the
of Lindemann melting criterion. The Lindemann ratio
√⟨ nature
√⟨𝜉 =
⟩
⟩
𝑢2 ∕𝑟 between square root of atomic mean-square vibration
𝑢2
and NND 𝑟 reaching a threshold value 𝜉0 [37] means that materials
start melting only when the thermal vibrations of atoms tend to a
limit. Therefore, Lindemann criterion can be seen as an empirical onephase approach. The initial slope of our melting curve is 𝑑𝑇𝑚 ∕𝑑𝑃 ∼
88.85 K/GPa. When pressure increases, the slope of melting line reduces
accordingly. At 50 GPa the melting slope predicted by present method
has the value 𝑑𝑇𝑚 ∕𝑑𝑃 ∼ 23.55 K/GPa, while the average melting slope
in the pressure range 0–100 GPa is 𝑑𝑇𝑚 ∕𝑑𝑃 ∼ 28.24 K/GPa.
With the same procedure of solid neon, by solving SMM EOS (18) we
can derive the lattice parameter and pressure-induced volume change
of solid argon. Our calculation 𝑉 ∕𝑉0 of argon crystal up to 100 GPa at
room temperature is shown in Fig. 3. We also present experimental data
points measured by XRD in a diamond-window static high-pressure
cell [38] and under quasi-hydrostatic conditions in a DAC [39] for
comparison. Here, it should be noted that Errandonea et al. reported
angle-dispersive XRD data exposing the appearance of HCP phase in
Ar at 49.6 GPa, but the structural phase transition FCC to HCP does
not complete at 114 GPa [16]. As observed from Fig. 3, our SMM EOS
goes along with previous measurements [38,39], where our theoretical
calculations well reproduce the pressure-induced volume change 𝑉 ∕𝑉0
of Ar up to pressure 80 GPa.
Applying the combination of SMM approach and Lindemann criterion, we obtain melting data points of argon crystal up to 100 GPa.
Our theoretical melting line is displayed along with previous works [18,
40,41] in Fig. 4. The closed circles are melting data points of LH-DAC
measurements by Boehler et al. [18]. The low pressure melting data of
the pressure-induced volume change 𝑉 ∕𝑉0 of solid neon at different
compression. This calculation can be seen as a verification of present
theory. By compressing solid neon using DAC techniques, Hemley
et al. [11] showed that the structure of solid neon remains stable
with FCC to pressure above 100 GPa at 300 K. Fig. 1 is the pressure
dependence of volume compression 𝑉 ∕𝑉0 of FCC solid neon up to
pressure 100 GPa at room temperature. In this figure, our theoretical
calculations are presented along with experimental data measured by
XRD in a single-crystal cell dimensions by Finger et al. [35] and in a
resistively heated (RH) DAC by Dewaele et al. [36]. More than that the
measurement by Hemley et al. [11] using the energy dispersive XRD
with microcollimated synchrotron radiation has also been added. As it
can be seen from Fig. 1, the pressure-induced volume change 𝑉 ∕𝑉0 of
neon crystal is significantly sensitive to pressure. It drops rapidly when
pressure increases, especially, at below 10 GPa. From this result we can
infer that solid neon is relatively soft at low pressure. And pressure–
volume relation of solid neon in our theoretical SMM approach is in
good agreement with those of previous experimental data [11,35,36].
The good illustration of this agreement can be observed in Fig. 1, where
our calculations reproduce the pressure-induced volume change 𝑉 ∕𝑉0
of Ne up to pressure 100 GPa. The difference among these results is
smaller than 2%. This means that the combination of SMM approach
with Buckingham potential provides a good prediction for lattice constant (and then equation-of-state) of solid neon. In consequence, the
derived pressure-dependent NND between two intermediate atoms can
be used for study melting temperature of neon crystal at high pressure.
In Fig. 2, we show the melting curve of Ne crystal derived from SMM
approach using Buckingham pair potential. Our theoretical melting
curve is presented along with previous works [12–14] for comparison.
The open diamonds, open circles and open stars are, respectively,
melting data points using W absorbers with thickness 8 μm, 4–6 μm,
and 6–8 μm; filled hexagrams represent melting data points obtained
using diamond heaters and filled triangles are loosely packed Ir black
absorbers [12]. The low pressure melting data of Vos et al. are represented by filled green squares [13]. The dash and dash–dot lines
are, respectively, the results from one-phase and two-phase MD simulations [14]. The solid line and filled aqua squares are the melting points
4
Vacuum 196 (2022) 110725
N. Van Nghia et al.
up to pressure 100 GPa. The present Ar melting curve is in good
agreement with previous DAC measurements up to pressure 80 GPa
(Refs. [18] and [40]) and differs significantly from the MD melting
data of Belonoshko [41] above 40 GPa. For neon solid gas, the present
melting line reproduces the earlier experimental data (Ref. [12]) up to
pressure 70 GPa. Furthermore, our melting curve also reasonably agrees
with two-phase MD simulation and ab initio calculations (Ref. [14]).
However, it does not agree with melting data points using W absorbers with thickness 4–6 μm (Ref. [12]) as well as one-phase MD
simulations (Ref. [14]). In order to resolve these discrepancies, further
experimental and theoretical studies need to be performed.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data availability
Fig. 4. The dependence of melting temperature of solid argon on pressure. Our
theoretical melting curve is presented along with previous works [18,40,41]. The closed
circles are melting data points of LH-DAC measurements by Boehler et al. [18]. The
low pressure melting data of Zha et al. are represented by open squares, and open up
triangles [40]. The open diamonds are the results from MD simulations [41].
Data available on request from the authors.
Acknowledgement
The authors would like to acknowledge Prof. Vu Van Hung for his
useful comments and suggestions.
Zha et al. are represented by open squares and open up triangles [40].
The open diamonds are the results from MD simulations [41]. As it can
be seen in Fig. 4, SMM melting curve agrees well with experimental
data points of low pressure measurements [40] as well as high pressure
LH-DAC (up to 80 GPa) [18]. Nevertheless, our theoretical line is
only consistent with MD simulated data at pressure below 30 GPa.
Beyond 30 GPa, the growing discrepancy between present theory and
MD simulation starts to appear, with the theoretical melting line being
increasingly lower than the MD simulated data. Initially, our melting
curve has a large slope 𝑑𝑇𝑚 ∕𝑑𝑃 ∼ 237.80 K/GPa. At 30 GPa the slope
of melting predicted by SMM is 𝑑𝑇𝑚 ∕𝑑𝑃 ∼ 33.30 K/GPa, while the
average melting slope in the pressure range 30–80 GPa is 𝑑𝑇𝑚 ∕𝑑𝑃 ∼
24.44 K/GPa.
Before making conclusions we discuss some reasons causing the
difference between present melting calculations and previous measurements. The first reason is the effect of used empirical potential on
numerical calculations. In this work, the interaction potential between
two atoms is assumed could be described by Buckingham (exp-6) pair
function. However, at high pressure, this simple pair potential might
not be rigorously suitable for the complex interactions in solid neon
and argon crystals. The many-body potential function could be more
appropriate for noble-gas solids [33]. The second reason is the neglect
of electron configurations of solid neon and argon. As reported by
Ross et al. [42], the hybridization of the 5p-like valence and 5d-like
conduction states plays a significant role on the steep lowering of the
slope of melting. As a consequence of it, to determine more exactly the
high-pressure melting temperature of the material, the current method
should take into account the contribution of its electronic properties.
References
[1] W.A. Caldwell, J.H. Nguyen, B.G. Pfrommer, F. Mauri, S.G. Louie, R.
Jeanloz, Science (ISSN: 0036-8075) 277 (1997) 930–933, https://science.
sciencemag.org/content/277/5328/930.full.pdf, URL https://science.sciencemag.
org/content/277/5328/930.
[2] S. Ono, Sci. Rep. (ISSN: 2045-2322) 10 (1) (2020) 1393, URL https://doi.org/
10.1038/s41598-020-58252-8.
[3] A.N. Singh, J.C. Dyre, U.R. Pedersen, J. Chem. Phys. 154 (13) (2021) 134501,
URL https://doi.org/10.1063/5.0045398.
[4] A. Dewaele, A.D. Rosa, N. Guignot, D. Andrault, J.E.F.S. Rodrigues, G. Garbarino,
Sci. Rep. (ISSN: 2045-2322) 11 (1) (2021) 15192, URL https://doi.org/10.1038/
s41598-021-93995-y.
[5] D. Errandonea, B. Schwager, R. Boehler, M. Ross, Phys. Rev. B 65 (2002) 214110,
URL https://link.aps.org/doi/10.1103/PhysRevB.65.214110.
[6] A.P. Jephcoat, Nature 393 (1998) 355, URL http://dx.doi.org/10.1038/30712.
[7] Y.J. Gu, W.L. Quan, G. Yang, M.J. Tan, L. Liu, Q.F. Chen, Phys. Rev. E 102
(2020) 043214, URL https://link.aps.org/doi/10.1103/PhysRevE.102.043214.
[8] A. Jayaraman, Rev. Modern Phys. 55 (1983) 65–108, URL https://link.aps.org/
doi/10.1103/RevModPhys.55.65.
[9] W. Wei, X. Li, N. Sun, S.N. Tkachev, Z. Mao, Am. Mineral. (ISSN: 0003-004X) 104
(11) (2019) 1650–1655, https://pubs.geoscienceworld.org/msa/ammin/articlepdf/104/11/1650/4856704/am-2019-7033.pdf, URL https://doi.org/10.2138/
am-2019-7033.
[10] O.R. Smits, P. Jerabek, E. Pahl, P. Schwerdtfeger, Phys. Rev. B 101 (2020)
104103, URL https://link.aps.org/doi/10.1103/PhysRevB.101.104103.
[11] R.J. Hemley, C.S. Zha, A.P. Jephcoat, H.K. Mao, L.W. Finger, D.E. Cox, Phys. Rev.
B 39 (1989) 11820–11827, URL https://link.aps.org/doi/10.1103/PhysRevB.39.
11820.
[12] D. Santamaría-Pérez, G.D. Mukherjee, B. Schwager, R. Boehler, Phys. Rev. B 81
(2010) 214101, URL https://link.aps.org/doi/10.1103/PhysRevB.81.214101.
[13] W.L. Vos, J.A. Schouten, D.A. Young, M. Ross, J. Chem. Phys. 94 (5) (1991)
3835–3838, URL https://doi.org/10.1063/1.460683.
[14] L. Koči, R. Ahuja, A.B. Belonoshko, Phys. Rev. B 75 (2007) 214108, URL
https://link.aps.org/doi/10.1103/PhysRevB.75.214108.
[15] S.S. Shcheka, H. Keppler, Nature (ISSN: 1476-4687) 490 (7421) (2012) 531–534,
URL https://doi.org/10.1038/nature11506.
[16] D. Errandonea, R. Boehler, S. Japel, M. Mezouar, L.R. Benedetti, Phys. Rev. B
73 (2006) 092106, URL https://link.aps.org/doi/10.1103/PhysRevB.73.092106.
[17] C.-S. Zha, R. Boehler, Phys. Rev. B 31 (1985) 3199–3201, URL http://link.aps.
org/doi/10.1103/PhysRevB.31.3199.
[18] R. Boehler, M. Ross, P. Söderlind, D.B. Boercker, Phys. Rev. Lett. 86 (2001)
5731–5734, URL https://link.aps.org/doi/10.1103/PhysRevLett.86.5731.
[19] N. Tang, V.V. Hung, Phys. Stat. Sol. (B) (ISSN: 1521-3951) 149 (2) (1988)
511–519, URL http://dx.doi.org/10.1002/pssb.2221490212.
[20] N. Tang, V.V. Hung, Phys. Stat. Sol. (B) (ISSN: 1521-3951) 162 (2) (1990)
371–377, URL http://dx.doi.org/10.1002/pssb.2221620206.
4. Conclusions
In present work, we investigated the equation-of-state and pressuredependent melting temperature of solid neon and argon up to pressure
100 GPa by means of the statistical moment method and interatomic
Buckingham (exp-6) pair potential function. This approach has taken
into account the anharmonicity contributions caused by thermal lattice
vibrations. Our present work shows that lattice parameters and melting temperatures of solid rare gases depend strongly on pressure. In
particular, at pressure below 10 GPa, lattice parameter drops rapidly
while melting temperature increases robustly with pressure. The SMM
approach has reproduced the equation-of-states of neon and argon
5
Vacuum 196 (2022) 110725
N. Van Nghia et al.
[31] N. Tang, V.V. Hung, Phys. Status Solidi. (B) 161 (1) (1990) 165–171,
https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.2221610115, URL https:
//onlinelibrary.wiley.com/doi/abs/10.1002/pssb.2221610115.
[32] C.P. Herrero, Phys. Rev. B 65 (2001) 014112, URL https://link.aps.org/doi/10.
1103/PhysRevB.65.014112.
[33] E. Pechenik, I. Kelson, G. Makov, Phys. Rev. B 78 (2008) 134109, URL https:
//link.aps.org/doi/10.1103/PhysRevB.78.134109.
[34] M. Ross, B. Alder, J. Chem. Phys. 46 (11) (1967) 4203–4210, URL https:
//doi.org/10.1063/1.1840523.
[35] L.W. Finger, R.M. Hazen, G. Zou, H.K. Mao, P.M. Bell, Appl. Phys. Lett. 39 (11)
(1981) 892–894, URL https://doi.org/10.1063/1.92597.
[36] A. Dewaele, F. Datchi, P. Loubeyre, M. Mezouar, Phys. Rev. B 77 (2008) 094106,
URL https://link.aps.org/doi/10.1103/PhysRevB.77.094106.
[37] F. Lindemann, Physik Z11 (1910) 609–612.
[38] M. Ross, H.K. Mao, P.M. Bell, J.A. Xu, J. Chem. Phys. 85 (2) (1986) 1028–1033,
URL https://doi.org/10.1063/1.451346.
[39] P. Richet, J.-A. Xu, H.-K. Mao, Phy. Chem. Mineral. (ISSN: 1432-2021) 16 (3)
(1988) 207–211, URL https://doi.org/10.1007/BF00220687.
[40] C. Zha, R. Boehler, D.A. Young, M. Ross, J. Chem. Phys. 85 (2) (1986)
1034–1036, URL https://doi.org/10.1063/1.451295.
[41] A.B. Belonoshko, High Press. Res. 10 (4) (1992) 583–597, URL https://doi.org/
10.1080/08957959208202841.
[42] M. Ross, R. Boehler, P. Söderlind, Phys. Rev. Lett. 95 (2005) 257801, URL
https://link.aps.org/doi/10.1103/PhysRevLett.95.257801.
[21] V.V. Hung, T.X. Linh, V.T. Thanh Ha, D.D. Phuong, H.K. Hieu, Eur. Phys. J. B
(ISSN: 1434-6036) 91 (2) (2018) 44, URL https://doi.org/10.1140/epjb/e201880540-0.
[22] K. Masuda-Jindo, S.R. Nishitani, V. Van Hung, Phys. Rev. B 70 (2004) 184122,
URL http://link.aps.org/doi/10.1103/PhysRevB.70.184122.
[23] V.V. Hung, D.D. Phuong, N.T. Hoa, H.K. Hieu, Thin Solid Films (ISSN: 00406090) 583 (2015) 7–12, URL http://www.sciencedirect.com/science/article/pii/
S0040609015002552.
[24] V.V. Hung, D.T. Hai, H.K. Hieu, Vacuum (ISSN: 0042-207X) 114 (2015) 119–123,
URL https://www.sciencedirect.com/science/article/pii/S0042207X15000287.
[25] T.T. Ha, D.D. Phuong, P.T.M. Hanh, H.K. Hieu, Chem. Phys. (ISSN: 03010104) 539 (2020) 110928, URL https://www.sciencedirect.com/science/article/
pii/S0301010420308107.
[26] K. Masuda-Jindo, V.V. Hung, P.D. Tam, Phys. Rev. B 67 (2003) 094301, URL
http://link.aps.org/doi/10.1103/PhysRevB.67.094301.
[27] D. Errandonea, Phys. B: Condens. Matter. (ISSN: 0921-4526) 357 (3–
4) (2005) 356–364, URL http://www.sciencedirect.com/science/article/pii/
S0921452604012396.
[28] H.K. Hieu, N.N. Ha, AIP Adv. 3 (11) (2013) 112125, URL http://scitation.aip.
org/content/aip/journal/adva/3/11/10.1063/1.4834437.
[29] H.K. Hieu, Vacuum (ISSN: 0042-207X) 109 (2014) 184–186, URL http://www.
sciencedirect.com/science/article/pii/S0042207X14002462.
[30] H.K. Hieu, J. Appl. Phys. 116 (16) (2014) URL http://scitation.aip.org/content/
aip/journal/jap/116/16/10.1063/1.4899511.
6