Many-body interactions and high-pressure equations of state in rare-gas solids
The T 0K equations of state (EOS) of rare-gas solids (RGS) (He, Ne, Ar, Kr, and Xe) are calculated in the experimentally studied ranges of pressures accounting for two- and three-body interatomic forces. Solid-state corrections to the pure two-body Aziz et al. potentials included the long-range...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1217732017-06-17T03:02:54Z Many-body interactions and high-pressure equations of state in rare-gas solids Freiman, Yu.A. Tretyak, S.M. Classical Cryocrystals The T 0K equations of state (EOS) of rare-gas solids (RGS) (He, Ne, Ar, Kr, and Xe) are calculated in the experimentally studied ranges of pressures accounting for two- and three-body interatomic forces. Solid-state corrections to the pure two-body Aziz et al. potentials included the long-range Axilrod–Teller three-body interaction and short-range three-body exchange interaction. The energy-scale and length-scale parameters of the latter were taken as adjustable parameters of theory. The calculated T 0K EOS for all RGS are in excellent agreement with experiment in the whole range of pressures. The calculated EOS for Ar, Kr, and Xe exhibit inflection points where the isothermal bulk moduli have non-physical maxima indicating that account of only three-body forces becomes insufficient. These points lie at pressures 250, 200, and 175 GPa (volume compressions of approximately 4.8, 4.1, and 3.6) for Ar, Kr, and Xe, respectively. No such points were found in the calculated EOS of He and Ne. The relative magnitude of the three-body contribution to the ground-state energy with respect to the two-body one as a function of the volume compression was found to be non-monotonic in the sequence Ne–Ar–Kr–Xe. In a large range of compressions, Kr has the highest value of this ratio. This anomally high three-body exchange forces contributes to the EOS so large negative pressure that the EOS for Kr and Ar as a function of compression nearly coincide. At compressions higher approximately 3.5, the curves intersect and further on the EOS of Kr lies lower than that of Ar. 2007 Article Many-body interactions and high-pressure equations of state in rare-gas solids / Yu.A. Freiman, S.M. Tretyak // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 719-727. — Бібліогр.: 48 назв. — англ. 0132-6414 PACS: 64.60.Cn; 67.80.–s; 67.90.+z http://dspace.nbuv.gov.ua/handle/123456789/121773 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
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Classical Cryocrystals Classical Cryocrystals |
| spellingShingle |
Classical Cryocrystals Classical Cryocrystals Freiman, Yu.A. Tretyak, S.M. Many-body interactions and high-pressure equations of state in rare-gas solids Физика низких температур |
| description |
The T 0K equations of state (EOS) of rare-gas solids (RGS) (He, Ne, Ar, Kr, and Xe) are calculated in
the experimentally studied ranges of pressures accounting for two- and three-body interatomic forces.
Solid-state corrections to the pure two-body Aziz et al. potentials included the long-range Axilrod–Teller
three-body interaction and short-range three-body exchange interaction. The energy-scale and length-scale
parameters of the latter were taken as adjustable parameters of theory. The calculated T 0K EOS for all
RGS are in excellent agreement with experiment in the whole range of pressures. The calculated EOS for Ar,
Kr, and Xe exhibit inflection points where the isothermal bulk moduli have non-physical maxima indicating
that account of only three-body forces becomes insufficient. These points lie at pressures 250, 200, and 175
GPa (volume compressions of approximately 4.8, 4.1, and 3.6) for Ar, Kr, and Xe, respectively. No such
points were found in the calculated EOS of He and Ne. The relative magnitude of the three-body contribution
to the ground-state energy with respect to the two-body one as a function of the volume compression
was found to be non-monotonic in the sequence Ne–Ar–Kr–Xe. In a large range of compressions, Kr has the
highest value of this ratio. This anomally high three-body exchange forces contributes to the EOS so large
negative pressure that the EOS for Kr and Ar as a function of compression nearly coincide. At compressions
higher approximately 3.5, the curves intersect and further on the EOS of Kr lies lower than that of Ar. |
| format |
Article |
| author |
Freiman, Yu.A. Tretyak, S.M. |
| author_facet |
Freiman, Yu.A. Tretyak, S.M. |
| author_sort |
Freiman, Yu.A. |
| title |
Many-body interactions and high-pressure equations of state in rare-gas solids |
| title_short |
Many-body interactions and high-pressure equations of state in rare-gas solids |
| title_full |
Many-body interactions and high-pressure equations of state in rare-gas solids |
| title_fullStr |
Many-body interactions and high-pressure equations of state in rare-gas solids |
| title_full_unstemmed |
Many-body interactions and high-pressure equations of state in rare-gas solids |
| title_sort |
many-body interactions and high-pressure equations of state in rare-gas solids |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
Classical Cryocrystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121773 |
| citation_txt |
Many-body interactions and high-pressure
equations of state in rare-gas solids / Yu.A. Freiman, S.M. Tretyak // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 719-727. — Бібліогр.: 48 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT freimanyua manybodyinteractionsandhighpressureequationsofstateinraregassolids AT tretyaksm manybodyinteractionsandhighpressureequationsofstateinraregassolids |
| first_indexed |
2025-07-08T20:30:05Z |
| last_indexed |
2025-07-08T20:30:05Z |
| _version_ |
1837112078903541760 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 719–727
Many-body interactions and high-pressure
equations of state in rare-gas solids
Yu.A. Freiman and S.M. Tretyak
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
Received December 18, 2006
The T � 0 K equations of state (EOS) of rare-gas solids (RGS) (He, Ne, Ar, Kr, and Xe) are calculated in
the experimentally studied ranges of pressures accounting for two- and three-body interatomic forces.
Solid-state corrections to the pure two-body Aziz et al. potentials included the long-range Axilrod–Teller
three-body interaction and short-range three-body exchange interaction. The energy-scale and length-scale
parameters of the latter were taken as adjustable parameters of theory. The calculated T � 0 K EOS for all
RGS are in excellent agreement with experiment in the whole range of pressures. The calculated EOS for Ar,
Kr, and Xe exhibit inflection points where the isothermal bulk moduli have non-physical maxima indicating
that account of only three-body forces becomes insufficient. These points lie at pressures 250, 200, and 175
GPa (volume compressions of approximately 4.8, 4.1, and 3.6) for Ar, Kr, and Xe, respectively. No such
points were found in the calculated EOS of He and Ne. The relative magnitude of the three-body contribu-
tion to the ground-state energy with respect to the two-body one as a function of the volume compression
was found to be non-monotonic in the sequence Ne–Ar–Kr–Xe. In a large range of compressions, Kr has the
highest value of this ratio. This anomally high three-body exchange forces contributes to the EOS so large
negative pressure that the EOS for Kr and Ar as a function of compression nearly coincide. At compressions
higher approximately 3.5, the curves intersect and further on the EOS of Kr lies lower than that of Ar.
PACS: 64.60.Cn Order-disorder transformations; statistical mechanics of model systems;
67.80.–s Solid helium and related quantum crystals;
67.90.+z Other topics in quantum fluids and solids; liquid and solid helium.
Keywords: rare-gas solids, equations of state, two and tree-body interatomic forces.
1. Introduction
The availability of equation of state (EOS) and structure
measurements in the megabar pressure range imparted a
new momentum to the problem of the application of the
many-body potentials to the analysis of the experimental
results [1–8]. It is well-known [9] that the potential consist-
ing from the pure two-body potential and the long-range
Axilrod–Teller (AT) three-body correction makes it possible
to reproduce the zero-temperature zero-pressure properties
and low-pressure EOS data of solid Ne, Ar, Kr, and Xe. As
known, the AT forces is a three-body analogue of the Lon-
don two-body dispersion forces. Loubeyre [6,7] calculated
the EOS of solid He, Ar, Kr, and Ne by using the potential
which included also the three-body exchange correction
term and obtained good agreement with experiment. This
term originates from the Pauli exclusion principle and
means that the charge distributions of two atoms change in
the presence of a third atom. Kim et al. [8] represented the
exact three-body exchange interaction by a short-range pair
correction term and found the adjusting parameters of this
term from the condition that it corrects deviations between
experimental pressures and pressures calculated without
this term. The resulting EOS for solid Kr and Xe were in fair
agreement with experiment to the highest experimental
pressures. Thus it has been demonstrated that the EOS of
these systems may be well described using the effective
two-body potential with a specially adjusted short-range
correcting term instead of the real three-body exchange in-
teraction. Subsequently, this approach was used to describe
materials other than RGS, for example, in the case of com-
pressed solid hydrogen the short-range behavior of the
Silvera–Goldman potential [10] was corrected by Hemley et
al. [11] with a help of the same two-term polinomial correct-
ing term. Thus EOS calculations are generally believed to be
not very sensitive to the difference between the many-body
© Yu.A. Freiman and S.M. Tretyak, 2007
potential and the effective two-body one. At the same time,
the structural implications of these two types of potentials
may be different [1,3]. As was shown in Refs. 1 and 3, the
lattice distortion parameter in compressed solid helium and
hydrogen is very sensitive to the real nature of the inter-
molecular potential and thus can be used as a probe of the
many-body forces. The same is true about other directional
characteristics, for example, elastic moduli. Nontheless
solid-state calculations based on many-body potentials are
scarce. Not much is known about the role of many-body
forces in such high-pressure phenomena as high-pressure
polymorphic transitions, pressure-driven metallization,
pressure-induced dissociation, etc. Many issues concerning
the EOS and phase diagrams of RGS remain still to be un-
derstood.
In this article the T � 0 K EOS of He, Ne, Ar, Kr, and
Xe are calculated in the experimentally studied ranges of
pressures accounting for two- and three-body interatomic
forces. Solid-state corrections to the pure two-body Aziz
et al. potentials included the long-range Axilrod–Teller
three-body interaction and short-range three-body ex-
change interaction. Following Loubeyre [6,7], we took
the latter for all the RGS family in a Slater–Kirkwood
form. The main differences of our approach from that of
Refs. 6 and 7 are the following: (a) we took into account
zero-point contribution of the many-body forces to the
free energy of the systems; (b) we took into account the
contribution of the three-body exchange interaction from
all the different triplets formed by two neighbors of a cen-
tral atom lying inside the sphere of five nearest neighbor
distances; (c) the energy-scale and length-scale parame-
ters of the three-body exchange interaction were taken as
adjustable parameters of theory. The calculated T � 0 K
EOS for all RGS are in excellent agreement with experi-
ment in the whole range of pressures.
2. Intermolecular potential
The best available pair potentials for rare gas solids are
the Aziz pair potentials (He) [12], (Ne) [13], (Ar and Kr)
[14], (Xe) [15], which reproduce a variety of experimental
gas phase data. as well as zero-temperature, zero-pressure
properties of solid phase. These potentials can be repre-
sented in the form of the Silvera–Goldman potential [10]
U R R R f R C Rp c n
n( ) exp( ) ( )
, ,
� � � � � �� � � 2
6 8 10
. (1)
Here R is the interatomic distance, � � �, , are coefficients of
the exponential repulsive term, C n are the dispersion coef-
ficients. f Rc ( ) is a damping or attenuation function that
smoothly joints the multipole terms with the short-range
exponential repulsion term. It prevents from singularity at
the origin by the long-range multipole terms.
f R
R /R R R
R R
c ( )
exp[ ( ) ] , ,
, ,
* *
*
�
� � �
�
�
��
1
1
2
(2)
where R DRm
* � , Rm is the position of the potential mini-
mum of the potential. Parameters of the Aziz potentials
(in a.u.) are given in Table 1.
However, the pure pair potential does not describe
properly the properties of solid-state phase and many-
body corrections should be taken into account. At small
pressures the main correction comes from the three-body
dispersion interaction. At high pressures the many-body
exchange effects should be taken into account. The equa-
tion-of-state-calculations based on the Aziz pair poten-
tials give pressures which are systematically above the
experimental equation of state; this illustrates the fact that
at short distances the Aziz potential is too stiff and should
be soften by adding the three-body exchange correction
term. The short-range correction to the pair potentials
originates from alteration of the charge clouds of two
molecules in the presence of neighboring molecules due
to the Pauli exclusion principle. Ab initio calculations of
Bulski and Chalasinski [16] showed that the contribution
of the three-body overlap potentials are important for
rare-gas atoms. Here we restrict ourselves to the
three-particle interaction, which consists of three-body
dispertion forces (Axilrod–Teller forces) and tree-body
exchange forces.
720 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
Yu.A. Freiman and S.M. Tretyak
Table 1. Parameters (in a.u.) of pair potential
Substance � � �
103 C6 C8 C10 D Rm
He [12] 2.925 2.381405 0 1.45995 14.21950 187.1926 1.28 5.4379
N2 [13] 4.786192 2.373569 3.808392 6.44696 96.4992 1519.985 1.36 5.841140
Ar [14] 4.630839 1.518400 35.961866 64.2995 1639.9865 50999.49 1.36 7.0987520
Kr [14] 4.251076 1.240412 40.54818 130.89515 3719.97 166938.3 1.28 7.574018
Xe [15] 2.93519 0.65698 72.8075 288.758 10999.9 56 105.
1.45 8.24430
Let three paticles form a triangle with sides r1, r2, r3
and angles �1, �2, �3. Then the total three-body potential
U tr is given by a Slater–Kirkwood-type expression [17,6]
U A r r r
C
r r r
tr tr
tr� � � � � �
�
��
�
�
�
��
�exp[ ( )]� 1 2 3
1
3
2
3
3
3
� � � � �( cos cos cos )1 3 1 2 3 . (3)
Table 2. Parameters (in a.u.) of three-body potential (Eq. 3)
Substance A �tr Ctr
He 30.644
a
1.0245
a
1.4807a
Ne 566.969
b
1.1896b 11.835d
Ar 1331.369
c
0.905888c 526.5d
Kr 1717.154
c
0.842728c 1578,3d
Xe 1964.058
c
0.770203c 5283d
N o t e s : a
aRef. 6; bthis work (slightly modified parameters from Ref. 4);
cthis work (slightly modified parameters from Ref. 7, see the
text); dRef. 18.
The first exponential term of this equation represents the
exchange three-body interaction which at small interatomic
distances descibes the alterations of the charge densities of
two interacting molecules by the presence of a third one.
The second term, known as the Axilrod–Teller one, is the
large–distance asymptote of the triple-dipole interaction. At
large interatomic separations, the main contribution to the
U tr term comes from the Axilrod–Teller term and at smaller
internuclear separations the three-body exchange overlap
interaction is the dominant term. Parameters of the three-
body potential (in a. u.) are given in Table 2.
It could be argued about the need of taking into account
of the three-body interactions since an effective pair poten-
tial which includes their isotropic average can well repro-
duce the experimental equation of state. However the ef-
fective pair potential which best reproduces the equation of
state worst reproduces the elastic constants, which are di-
rection-dependent quantities. But what is more important,
the three-body forces and the effective pair forces are ac-
ting differently when considering such phenomena as
fcc-hcp phase transition, and pressure-driven metallization
and molecular-nonmolecular transitions. In the case of hcp
simple solids (He or H 2) the effective two-body potential
and a real one, consisting of a sum of the pair and
three-body terms foretold qualitatively different pressure
dependence of the c a/ ratio [3].
3. T = 0 K. Equation of states
To use the low- and room-temperature measurements
in inferring correct magnitudes of the three-body interac-
tion it is convenient to reduce all the experimental data to
0 K. To this end we will extract the thermal pressure Pth
from from available experimental data. This term was cal-
culated by the use of the Mie–Gr��uneisen model
P RT V Dth � ( ) / (3 � � � � �� , (4)
where R is the gas constant, D x( ) is the Debye function, �
is the Debye temperature, and �, the Gr��uneisen parameter,
is the logarithmic volume derivative of �:
� � �d d Vln / ln .� (5)
By integrating the latter equation with a specific form for
the volume dependence of the Gr��uneisen constant we ob-
tain an explicit dependence of the Debye temperature on
volume. Analyzing different models, Holt and Ross [19]
concluded that the linear approximation
� � �� �0 1 0( / )V V , (6)
whereV0 is the zero-pressure volume, is appropriative for
solids such as the RGS for which the closed shell repul-
sive forces determine the EOS at high compressions.
They suggested that � 0 is approximately 0.5. As a result,
we get
� �/ / exp[ ( / )]0 0 1 01� �V V V V� . (7)
Using the described procedure, we obtain P0K by sub-
stracting the thermal contribution from experimental
pressures. In these calculations the following parameters
were used: Ne: � 0 0 5� . , �1 2 05� . [20]; Kr: � 0 0 5� . ,
�1 2 3� . [8]; Xe: � 0 0 5� . , �1 2 4� . [21]. Ross et al. [22]
calculated the dependence �( )V for solid Ar. Approximat-
ing their �( )V by the linear function Eq. (6) we obtained
�1 1416� . , � 0 1 066� . .
The ground-state energy Egs of the system can be written
in the form
E U U Ugs p zp� � �tr , (8)
where Uzp is the energy of zero-point oscillations.
In the following, the pressure region where zero-point
effects play a decisive role will be excluded from consider-
ation, so in calculations of the zero-point contribution
standart methods of lattice dynamics have been used, in par-
ticular, the Einstein approximation. In this approximation
the zero-point energy can be found in the following way:
Uzp x y z� � �
�
2
( )� � � , (9)
where �� (� � x y z, , ) are frequences of the harmonic os-
cillations of helium atoms near their equilibrium posi-
tions.
Many-body interactions and high-pressure equations of state in rare-gas solids
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 721
3.1. Solid He
An accurate determination of the EOS of 4He has been
an important research objective for decades [2,23–29].
Using techniques originally developed for high-pressure
x-ray diffraction of solid hydrogen, the crystal structure
of solid helium was determined by single-crystal synchro-
tron methods up to 23 GPa at 300 K [27] and demon-
strated that solid helium has hexagonal-closed-packed
structure over this P T� range, in contrast to theoretical
predictions. The highest compression reached in the EOS
experiments is 8.4.
The calculated dependence of P V( ) for solid helium is
shown in Fig. 1 along with the experimental results from
Refs. 23,27,29. It is seen that the pair potential is too stiff
and overestimates the pressure values calculated with its
help considerably. The negative contribution from the
three-body interaction becomes appreciable starting ap-
proximately at three-fold compression and increases rap-
idly with pressure. At pressures over approximately 5
GPa (four-fold compression), the pair interaction approx-
imation becomes inadequate. The dotted curve at the
same Figure shows the pressure values calculated disre-
garding the contribution from zero-point oscillations. It is
seen that the effect of the zero-point motion remains sub-
stantial up to the highest pressure reached, though it
gradually decreases with rising pressure.
The pressure of solid helium on the logarithmic scale
as a function of the relative volume change V V/ 0 is
shown in the Inset in Fig. 1,a. The EOS in this form
readily illustrates limiting pressures or compressions
where the approximation, which takes into account two-
and three-body interactions and neglects more-body
(four-body first of all) interactions, fails. As we will see
below, a signature of such inedequacy is an inflection
point, which can be seen at the high-pressure region of the
EOS on the logarithmic scale of Ar, Kr, and Xe. The in-
flection point at the EOS means that there is a non-physi-
cal maximum at the derivative � �P V/ , i.e., at the isother-
mal bulk modulus B VT ( ). In the case of helium, in the
studied region (P up to 60 GPa or near ten-fold compres-
sions) no inflection point was found at the curve
ln ( / )P V V0 , which we consider as evidence that we are
still within the limits of the adopted approximation.
The contributions of the pair and triple forces, and
zero-point oscillations to the ground-state energy of solid
He are shown in Fig 1,b. As can be seen, the relationship be-
tween the contributions is rather complicated and is essen-
tially different in different regions of the molar volume. At
relatively small compressions V V0 / � 3, the ground-state
energy Egs is dominated by the contribution of zero-point
oscillations E zp . At V V0 3/ � the contributions of the pair
and triple forces compensate each other E Ep � �tr 0 and
thus E Egs zp� . In the range of compressions 3 �V V0 / � 4
the inequality holds E E Egs zp p� � . At V � 3 5. cm
3
/mol
(six-fold compression), E Ezp � �tr 0 and E Egs p� . At still
higher compresions the static lattice contributions E Ep , tr
dominate E zp and consequently the EOS.
3.2. Solid Ne
The behavior of neon at high densities has been the
subject of comparatively few experimental and theoreti-
cal investigations in relation to studies of other rare-gas
solids. Early high-pressure measurements of the heavy
RGS by Anderson and Swenson [30] by using the pis-
ton-displacement technique provided low-temperature
EOS data for solid neon up to 2.1 GPa. Diamond-an-
722 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
Yu.A. Freiman and S.M. Tretyak
He
V, cm /mol3
2 6 10 14
b
11
9
7
5
3
1
–1
–3
–5
Ep
Ezp
Etr
1
0
–1
4 6 8 10 12 14
E
1
0
,
a.
u
.
g
s
–
3
P,
G
P
a
70
60
50
40
30
20
10
0
He
V/V0
P,
G
P
a
0.1 0.3 0.5 0.7
10
2
10
1
10
0
10
–1
10
–2
2 4 6 8 10 12 14
V, cm /mol3
Egs
a
V, cm /mol3
E
g
s
1
0
,
a.
u
.
–
3
Fig. 1. The calculated T � 0 K equation of state for solid He.
Solid symbols are experimental data from Refs. 23,27,29. Solid
curve was calculated taking into account the pair Aziz potential,
three-body potential, and zero-point contribution, dashed curve
was obtained disregarding the three-body potential; dotted curve
was obtained without regard for the zero-point oscillations. The
inset shows the zero-temperature pressures on the logarithmic
scale as a function of compression (a). Contributions of the pair
and triple forces, and zero-point oscillations to the ground-state
energy (b).
vil-cell studies by Hazen et al. [31] showed that at room
temperature fluid neon freezes at 14.4 GPa. Finger et al.
[32] in single-crystal x-ray diffraction studies determined
the crystal structure and equation of state at room temper-
atures from the freezing presure up to 14.4 GPa. It was
shown that solid neon retains the cubic closed-packed
structure. Hemley et al. [20] using energy-dispersive syn-
chrotron x-ray diffraction techniques investigated the
crystal structure and equation of state up to 110 GPa at
room temperatures. They showed that solid neon remains
an insulator with the fcc structure in this P T� range. The
highest volume compression reached in experiment is
3.56, more than two times less than for solid helium.
Hemley et al. [20,33] compared their experimental data
with results of their own lattice dynamics calculations us-
ing several different pair potentials. As it usually occurs
with the EOS calculated with pure two-body potentials, the
EOS calculated with the Aziz–Slaman potential [13] ex-
hibits good agreement with experiment at low pressure, but
at higher pressures the calculated pressures are signifi-
cantly overestimated (see also comments to this point in
Ref. 4). On the other hand, an effective exp-6 potential can
be found that gives a good fit at the high-pressure region at
the sacrifice of agreement at low compressions.
Path-integral Monte Carlo simulations of solid neon
[4] confirmed inadequacy of pure two-body potentials for
the high-pressure range. The inclusion of the three-body
interaction in the Slater–Kirkwood form with two
adjustible parameters (A and � tr ) brought the results of
simulations into a good agreement with the experimental
results. Our calculations with the intermolecular potential
consisting of the Aziz–Slaman pair potential [13] plus the
three-body interaction with the parameters of the three-
body short-range term from Ref. 4 (A = 566.94 a.u., � tr =
= 1.1636 a.u.) confirm results of the simulation. Using
slightly modified parameters (see Table 2) we obtained
excellent agreement with the experimental results in all
the studied range of pressures (Fig. 2).
There were several theoretical predictions for behavior
at much higher densities [34–37]. In the high-pressure re-
gion, the Birch–Murnaghan EOS based on the experimen-
tal results agrees well with the pressures calculated by
Boettger and Trickey [36] and by Boettger [37] whereas
the pressures calculated by Zharkov and Trubitsyn [34]
and by Hama [35] are systematically lower. Our calcula-
tion for the terapascal range is given in the Inset in Fig. 2,a.
As in the case of solid helium, no inflection point was
found in the high-pressure range implying that we did not
fall outside the limits of application of the adopted approx-
imation. As show the comparison with the theoretical re-
sults for the terapascal range, our results are in excellent
agreement both with the Birch–Murnaghan EOS based on
the experimental results of Hemley et al. [20] and that of
Refs. 36 and 37.
The contributions of the pair and triple forces, and zero-
point oscillations to the ground-state energy for solid Ne are
shown in Fig 2,b. This figure is typical for the heavy RGS (Ne,
Ar, Kr, Xe) and diverges considerably from that for solid He.
At the low compressions (V /V0 higher � 1.5), the ground state
Egs and consequently the EOS is dominated by the static lat-
tice contributions E Ep, tr . At the highest compression
( .V � 3 5 3cm /mol) the zero-point contribution to the ground
state of Ne is about 10%. The figures for Ne, Ar, Kr, and Xe
look alike except that the respective zero point contributions to
the ground-state energy for a given V/V0 are progressively go
down as one passes from Ne to Xe.
Many-body interactions and high-pressure equations of state in rare-gas solids
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 723
��
1 5 9
0
4
8
12
16
P,
G
P
a
P,
G
P
a
V, cm /mol3
V/V0
100
120
80
60
40
20
0
3 5 7 9 11 13
0.1 0.3 0.5 0.7 0.9
103
102
101
100
10–1
10–2
V, cm /mol3
NeEp
E
Ezp
Etr
Ne a
b
–4
–8
E
1
0
,
a.
g
s
–
4
u
.
Fig. 2. The calculated T � 0 K equation of state for solid Ne.
Solid symbols are experimental data: low pressure data are from
Ref. 30, high-pressure data are from Ref. 20. The inset shows the
zero-temperature pressures on the logarithmic scale extended to
the terapascal range as a function of compression (a). Contribu-
tions of the pair and triple forces, and zero-point oscillations to
the ground-state energy (b).
3.3. Solid Ar, Kr, and Xe
The low-pressure low-temperature piston-displace-
ment measurements by Anderson and Swenson [30] gave
EOS of solid argon to pressures of 2.0 GPa. Room tem-
perature high-pressure measurements of Ross et al. [22]
extended the EOS data to pressures of 80 GPa. The high-
est volume compression reached in experiment is 2.83,
near three times less than in solid helium.
The calculated T � 0 dependence of P V( ) for solid argon is
shown in Fig. 3 along with the experimental results. In our cal-
culations we used slightly modified parameters A and � tr of
the short-range three-body potential (see Table 2) compared to
that from Ref. 7 (A = 1328.67 a.u., � tr = 0.87314 a.u.). Agree-
ment between theory and experiment is excellent both in the
low- and high-pressure range. Our calculation extended for the
terapascal range is given in the Inset in Fig. 3. In contrast to
solid helium and neon, there is an inflection point at the P V( )
curve for Ar at V/V0 0 21� . (P � 250 GPa). Kr and Xe also
exhibit analogous inflection points at their EOS (see Insets in
Figs. 4 and 5) at V/V0 � 0.245 and 0.28 (P � 200 GPa and
175 GPa) respectively. As was said above, at this points the
isothermal bulk modulus B VT ( ) has a non-physical maximum
indicating that the adopted approximation taking into account
pair- and triple-body forces fails.
In addition to low-pressure data of Anderson and
Swenson [30] there are two sets of high-pressure data for
solid krypton. EOS measurements by Polian et al. [38,39]
up to 30 GPa were performed by x-ray energy-dispersive
techniques using synchrotron radiation and the elastic
properties were determined using Brillouin scattering.
X-ray measurements by Aleksandrov et al. [40] gave the
EOS data up to 52 GPa. The highest volume compression
reached in experiment is 2.375, the lowest value among
the RGS solids.
The calculated zero-temperature dependence of P V( )
for solid krypton is shown in Fig. 4 along with the experi-
mental results. In our calculations parameters A and � tr
of the short-range three-body potential were slightly
modified (see our values in Table 2) compared to that
from Ref. 7 (A = 1717.19 a.u., � tr = 0.81811 a.u.). As can
be seen from Fig. 4, except two uppermost experimental
points at 52 and 47.5 GPa, which most likely are in error,
agreement between our calculations and all available ex-
perimental results (including low-pressure results by An-
derson and Swenson not shown in Figure) are excellent.
Our calculation extended for the terapascal range is given
in the Inset in Fig. 4.
The EOS of kripton was calculated in Refs. 7,8, and 41.
Loubeyre [7] using the self-consistent phonon approxima-
tion and Barker [41] using Monte Carlo method on the base
of the same pair plus three-body intermolecular potential
obtained that the pressure is somewhat underestimated
compared to experiment. On this ground Barker concluded
that the three-body exchange term is too high and that the
best agreement with experiment can be obtained using the
Axilrod–Teller interaction as the only many-body interac-
tion. This conclusion is in variance with results of statisti-
cal mechanical calculations by Kim et al. [8]. They intro-
duced an effective two-term polinomial short-range term
to the pair plus Axilrod–Teller potential and found parame-
ters of this term from the condition that it minimizes the de-
viation between theory and experiment. The resulted EOS
is in fair agreement between theory and experiment to the
highest experimental pressures. In fact, we found parame-
ters of the short-range exchange term from the same condi-
724 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
Yu.A. Freiman and S.M. Tretyak
P,
G
P
a
P,
G
P
a
Ar
V, cm /mol3
V/V0
103
102
101
100
10–1
10–2
8 10 12 14 16 18 20
100
80
60
40
20
0
0.2 0.4 0.6 0.8 1.0
Fig. 3. The calculated T � 0 K equation of state for solid Ar. Solid
symbols are experimental data of Ross et al. [22]. The inset shows
the zero-temperature pressures on the logarithmic scale extended
to the terapascal range as a function of compression.
0
�
P,
G
P
a
P,
G
P
a
V, cm /mol3
103
102
101
100
10–1
10–2
V/V0
0.2 0.6 0.8 1.00.4
Kr
11 13 15 17 19 21 23 25
50
40
30
20
10
Fig. 4. The calculated T � 0 K equation of state for solid Kr. Solid
symbols are experimental data of Polian et al. [38,39] (circules)
and of Aleksandrov et al. [40] (squares). The inset shows the
zero-temperature pressures on the logarithmic scale extended to
the terapascal range as a function of compression.
tion, but our results are in markedly better agreement with
experiment. Additional studies are needed to answer the
question: whether this difference is connected with the fact
that we used the real many-body potential whereas Kim et
al. used the effective two-body potential or resulted from
other details of calculations.
In comparison to other RGS, there are numerous P V�
measurements of solid Xe. In early piston-displacement
low-temperature measurements of Anderson and
Swenson [30] the pressure range up to 2.5 GPa was inves-
tigated, and Syassen and Holzapfel [42] in tungsten- and
boron-carbide anvil studies extended P V� measurements
to 11 GPa. Schiferl et al. [43], Asaumi [21], Zisman et al.
[44] brought EOS measurements up to 23, 32 and 53 GPa,
respectively. Jephcoat et al. [45], and Reichlin et al. [47]
extended EOS experiments to 137 and 172 GPa, respec-
tively. The highest volume compression reached in exper-
iment is 3.55, the same limiting value which was attained
in experiments with Ne. Jephcoat et al. found that xenon
transforms at 14 GPa from fcc to a still unidentified inter-
mediate structure that transforms to hcp at 75 GPa. Ac-
cording to optical studies, insulator-to-metal transition in
Xe occurs at pressures 132 GPa or 150 GPa according to
Ref. 46 or Ref. 47 respectively.
The calculated T � 0 dependence of P V( ) for solid xe-
non is shown in Fig. 5 along with the experimental
results. In our calculations we used slightly modified
parameters A and � tr of the short-range three-body poten-
tial (see Table 2) compared to that from Ref. 7 (A =
= 1964.08 a. u., � tr = 0.75249 a.u.). Agreement between
theory and experiment is excellent for the whole pressure
range. Our results extended for the terapascal range are
given in the Inset in Fig. 5.
The EOS of solid xenon was calculated by Loubeyre
[7] and Kim et al. [8] . The comparison of our results with
that from Refs. 7 and 8 exactly repeats the situation which
was analyzed for solid Kr.
3.4. Comparison of P V( ) relations
The comparison of P V( ) relations is usually performed
between the reduced pressure P/B V0 0( ) and relative volume
change V/V0, where V0 and B V0 0( ) are the T � 0, P � 0 vol-
ume and bulk modulus, respectively. At low pressures, the
reduced T � 0 EOS P/B V f V/V0 0 0( ) ( )� are identical to a
high accuracy for the three heaviest rare gases and are only
slightly different for solid Ne [48]. At higher pressures this
relation between the reduced pressures and volumes does
not hold and we compared the calculated T � 0 EOS for
solid He, Ne, Ar, Kr, and Xe as a function of the relative
compression V/V0 (Fig. 6). As can be seen, much higher
pressure is needed in the heavier rare gases compared with
He to reach a given relative compression, which is due to the
repulsive effect of the core electrons [37]. The differences
between the heavier rare gases are not so large, with the
curves for Ar and Kr near coinside, which means that the lat-
ter lies anomalously low. Moreover, at higher compressions
(V/V0 0 35� . ), the curve for Ar crosses that for Kr and at
higher compressions lie above the Kr curve.
This anomaly is a consequence of the fact that the
three-body exchange forces and the negative contribution
to the pressure along with it increase with volume com-
pressionV/V0 faster for Kr than for Ar (Fig. 7). Analyzing
the relative magnitude of the three-body contribution to
the ground-state energy with respect to the two-body in-
teraction as a function of V /V0 we found that this value is
non-monotonic in the sequence Ne–Ar–Kr–Xe. In the
range of compressions two- to five-fold the curve for Kr
Many-body interactions and high-pressure equations of state in rare-gas solids
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 725
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
P,
G
P
a
P,
G
P
a
V, cm /mol3
V/V0
103
102
101
100
10–1
10–2
0.2 0.6 0.8 1.00.4
8 12 16 20 24 28 32
180
160
140
120
100
80
60
40
20
0
Xe
Fig. 5. The calculated T � 0 K equation of state for solid Xe.
Solid circules are low-pressure experimental data from Ref.
30. High-pressure data: open squares Ref. 45; open circules
Ref. 47. The inset shows the calculated zero-temperature pres-
sures on the logaritmic scale extended to the terapascal range
as a function of compression.
P,
G
P
a
V/V0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ar
Xe
Kr
ArHe
Ne
100
80
60
40
20
0
Fig. 6. The calculated T � 0 K equations of state for solid He, Ne,
Ar, Kr, and Xe as a function of the relative compression V/V0.
lies higher than those for other RGS. Surprisingly, Xe has
the lowest value at the two-fold compressions but its
curve goes steeper and intersects susccessively the curves
for Ne and Ar. The intersection point with the Kr curve
lies outside the five-fold compression where the adopted
approximation becomes inappropriate.
4. Conclusions
The EOS of rare gas solids (He, Ne, Ar, Kr, and Xe) are
calculated in the experimentally studied ranges of pressures
taking into account two- and three- body interatomic forces.
In the capacity of the two-body potentials we used potentials
developed for the whole group of substances by Aziz and col-
laborations [12–15]. They are derived from two-body proper-
ties and represent pair interaction between two isolated at-
oms. The solid-phase intermolecular potentials which we
used to describe the EOS in solids up to very high volume
compressions contained the long-range Axilrod–Teller
three-body term and short-range three-body exchange inter-
action. The latter we used in the Slater–Kirkwood form and
treated its energy-scale and length-scale parameters as ad-
justed parameters of theory. The long-range three-body term
is repulsive and the resulted positive correction to the pres-
sure is essential at small pressures, while the short-range
three-body term is attractive and contributes the negative cor-
rection to the pressure, which means that the three-body ex-
change interaction softens the two-body repulsion at high
pressure. The calculated T � 0 K EOS are in excellent agree-
ment with experiment in the whole range of pressures for all
RGS. The calculated EOS for Ar, Kr, and Xe extended for the
terapascal range exhibit inflection points where the isother-
mal bulk moduli have non-physical maxima indicating that
the limitation of the many-body forces by the three-body in-
teractions becomes invalid. These points lie at pressures 250,
200, and 175 GPa (the volume compressions 4.76, 4.08, and
3.57) for Ar, Kr, and Xe, respectively. No such points were
found for He and Ne. Let us note that this level of compres-
sions is reached in experiment for Ne and Xe but, for un-
known reasons, the reached compressions for Ar and Kr are
markedly smaller with the least for Kr (less than 2.4).
The zero-point energy and respectively T � 0 K EOS
for the heavier EOS (Ne, Ar, Kr, and Xe) starting from
relatively small compressions (1.5-fold and lower for the
heaviest RGS) is dominated by the static lattice contribu-
tions. At the highest compression reached (3.5-fold) the
zero-point contribution to the ground state of Ne is about
10%. In the case of solid He, at compressions approxi-
mately 3-fold and smaller the ground-state is dominated
by the zero-point oscillations. The static lattice contribu-
tions becomes dominating starting approximately from
6-fold compressions. At the highest pressure reached
(about 60 GPa) the zero-point contribution to the ground
state of He is still about 25%.
The relative magnitude of the three-body contribution
to the ground energy with respect to the two-body interac-
tion as a function of V V0 / was found to be non-mo-
notonic in the sequence Ne–Ar–Kr–Xe. In the range of
compressions two- to five-fold the highest value has Kr.
Surprisingly, Xe at the two-fold compression has the least
value but it increases with compression faster than others.
This anomally high three-body contribution where domi-
nates the attractive short-range exchange forces results in
so large negative pressure that the EOS for Kr and Ar as a
function of compression near coincide with that for Ar
even intersecting the Kr curve at the compressions higher
approximately 3.5.
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Kr
XeNe
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Many-body interactions and high-pressure equations of state in rare-gas solids
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 727
|