Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature
A self-consistent statistical method is used to calculate the Gibbs free energy of vacancy formation in heavy rare gas crystals at high temperature. It is shown that the vacancy formation free energy rapidly falls in the vicinity of the melting point of the crystal. Such behavior is attributed to...
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irk-123456789-1217812017-06-17T03:02:56Z Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature Karasevskii, A.I. Lubashenko, V.V. Classical Cryocrystals A self-consistent statistical method is used to calculate the Gibbs free energy of vacancy formation in heavy rare gas crystals at high temperature. It is shown that the vacancy formation free energy rapidly falls in the vicinity of the melting point of the crystal. Such behavior is attributed to approaching the anharmonic instability point of vibrational subsystem of the solid. 2007 Article Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature / A.I. Karasevskii, V.V. Lubashenko // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 758-764. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 61.72.Bb; 61.72.Ji; 64.70.Dv http://dspace.nbuv.gov.ua/handle/123456789/121781 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Classical Cryocrystals Classical Cryocrystals |
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Classical Cryocrystals Classical Cryocrystals Karasevskii, A.I. Lubashenko, V.V. Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature Физика низких температур |
| description |
A self-consistent statistical method is used to calculate the Gibbs free energy of vacancy formation in
heavy rare gas crystals at high temperature. It is shown that the vacancy formation free energy rapidly falls
in the vicinity of the melting point of the crystal. Such behavior is attributed to approaching the anharmonic
instability point of vibrational subsystem of the solid. |
| format |
Article |
| author |
Karasevskii, A.I. Lubashenko, V.V. |
| author_facet |
Karasevskii, A.I. Lubashenko, V.V. |
| author_sort |
Karasevskii, A.I. |
| title |
Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature |
| title_short |
Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature |
| title_full |
Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature |
| title_fullStr |
Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature |
| title_full_unstemmed |
Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature |
| title_sort |
thermodynamics and microstructure of vacancies in rare gas crystals at high temperature |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
| topic_facet |
Classical Cryocrystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121781 |
| citation_txt |
Thermodynamics and microstructure of vacancies
in rare gas crystals at high temperature / A.I. Karasevskii, V.V. Lubashenko // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 758-764. — Бібліогр.: 34 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT karasevskiiai thermodynamicsandmicrostructureofvacanciesinraregascrystalsathightemperature AT lubashenkovv thermodynamicsandmicrostructureofvacanciesinraregascrystalsathightemperature |
| first_indexed |
2025-07-08T20:30:51Z |
| last_indexed |
2025-07-08T20:30:51Z |
| _version_ |
1837112128500137984 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 758–764
Thermodynamics and microstructure of vacancies
in rare gas crystals at high temperature
A.I. Karasevskii and V.V. Lubashenko
G.V. Kurdyumov Institute for Metal Physics, 36 Vernadsky Str., Kiev 03142, Ukraine
E-mail: akaras@imp.kiev.ua
Received September 7, 2006
A self-consistent statistical method is used to calculate the Gibbs free energy of vacancy formation in
heavy rare gas crystals at high temperature. It is shown that the vacancy formation free energy rapidly falls
in the vicinity of the melting point of the crystal. Such behavior is attributed to approaching the anharmonic
instability point of vibrational subsystem of the solid.
PACS: 61.72.Bb Theories and models of crystal defects;
61.72.Ji Point defects (vacancies, interstitials, color centers, etc.) and defect clusters;
64.70.Dv Solid-liquid transitions.
Keywords: vacancy formation, rare gas crystals.
1. Introduction
Point defects in rare gas crystals (RGC) have been ex-
tensively studied both theoretically and experimentally
for about fifty years. Indeed, the RGC remain popular re-
search objects of the condensed matter physics because
the many-body interactions in these systems may be ef-
fectively described by a sum of two-body interactions,
approximated by various empirical potentials [1] (though
explanation of some fine effects requires many-body in-
teractions to be invoked [2]).
Since there is a great bulk of experimental data accu-
mulated on various physical properties of the RGC, they
are ideal systems for testing various microscopic theories.
For instance, it appeared that the classical lattice dynam-
ics failed in description of strongly anharmonic atomic
motion in the RGC at high temperature [3]. Realization of
this fact became a challenge that stimulated attempts to
work out some methods for adequate self-consisting de-
scription of strongly anharmonic solids. At present, exist-
ing theoretical models [4–7] predict well enough thermal
and elastic properties of the bulk RGC in a wide range of
temperature and pressure, in agreement with the experi-
mental data available.
However, as for properties of point defects in the RGC,
the picture is not so fair. Since the RGC form closed-packed
structure (fcc), vacancies represent the predominant thermal
defect in these solids. To determine directly the equilibrium
concentration cv of vacancies in the RGC, the most accurate
and reliable experimental method is simultaneous measure-
ment of length and x-ray lattice parameter of a specimen
[8,9]. Some authors extracted cv data from comparison of
measured bulk properties of the crystal (length, density,
thermal expansion coefficient) [10–14] with corresponding
x-ray data [8,15–17]. Even measured values of the vacancy
concentration show remarkable divergency. For instance,
reported values of cv near the triple point are for Ar: � �10 3
[12], � � �2 10 4 [9], � � �3 10 3 [15], 10 10 2. � � [14]; for Kr:
7 4 10 3. � � [14], 3 2 10 3. � � [8], 2 9 10 3. � � [13]; for Xe:10 2� [16],
1 2 10 1. � � [14].
Thermodynamical parameters of vacancies, such forma-
tion enthalpy, entropy, volume, etc., are extracted from the
observed data by indirect methods. Vice versa, microscopic
calculations deal with direct evaluation of energetic parame-
ters of the vacancy. At present, there is a variety of theoreti-
cal predictions of vacancy formation parameters made by
different authors via various methods, including Monte
Carlo calculations [18,19] (see, e.g., Refs. 9, 20, 21 where
results of various calculations are compared). Numerous
calculations of vacancy properties have been made, taking
such factors into account as static lattice relaxation, change
of vibrational frequencies, lattice anharmonicity, quantum
and many-body corrections [20]. In spite of some discrep-
ancy between the calculated values of the vacancy parame-
ters reported by different authors, we may, in principle, as-
sert that there is qualitative agreement between calculated
and observed values. For example, the computed values of
© A.I. Karasevskii and V.V. Lubashenko, 2007
the vacancy concentration in Ar near the triple point are
10 4� –10 3� in order [21–24]. It is generally agreed that the
Gibbs free energy of vacancy formation decreases nearly
linearly with temperature, though the idea that it can de-
crease rapidly near the triple point was suggested to explain
anomalous behavior of isohoric specific heat of argon [25].
The aim of the present study is calculation of vacancy
parameters in the RGC at high temperature by means of
the self-consistent statistical method for determination of
thermodynamical properties of anharmonic solids devel-
oped in Ref. 6. The basics of this method are briefly out-
lined in Sec. 2 in relation to the subject of this work. In
Sec. 3 we substantiate the high-temperature approxi-
mation to the self-consistent statistical method. Then in
Sec. 4 we set out the procedure of calculation of the Gibbs
free energy of vacancy formation in the RGC at high tem-
perature. The results of our studies for Ar are presented in
Sec. 5, and in Sec. 6 a brief conclusion is given.
2. Self-consistent statistical method
To describe thermal properties of equilibrium vacan-
cies in a crystal, we should first write down the Gibbs free
energy of vacancy formation as a function of temperature
T and external pressure P,
g T P G T P G T P( , ) ( , ) ( , ),� �1 0 (1)
where G0 and G1 are, respectively, the Gibbs free energies
of a hypothetical perfect crystal and a crystal containing
one vacancy at a fixed lattice site. If the vacancies are as-
sumed to be noninteracting, their equilibrium concentra-
tion is given by
c T P g T P k TB( , ) ( , ) / ]� �exp [ . (2)
Microscopic calculation of g T P( , ) requires both eval-
uation of the Gibbs free energy of a perfect crystal at
given temperature and pressure and proper description of
the system response on creation of a defect. Since the
number of vacancies becomes appreciable near the melt-
ing point only, it is worthwhile to restrict the consider-
ation of their properties with the high-temperature range,
incorporating properly effects of anharmonicity of atomic
vibrations. For this purpose, we follow a recently pro-
posed self-consistent statistical method for calculation of
thermodynamical properties of solids [6]. According to
this approach, the Gibbs free energy of a simple perfect
crystal consisting of N atoms of mass m is written in the
form of the Gibbs–Bogoliubov functional corrected for
the cubic anharmonicity,
G F U U F PVH H0 3� � � � � � � , (3)
where FH and � �U H are, respectively, the Helmholz free
energy and the average potential energy of a reference
harmonic crystal, F3 is a correction for the cubic
anharmonicity of atomic vibrations evaluated within the
second-order perturbation theory, and
U u rij
i j
N
�
�
�1
2
1
( )
,
(4)
is the potential energy of interatomic interaction which is
assumed to be central and pairwise and is approximated
by an empirical potential u r( ). As in Refs. 26, 27, we em-
ploy here a hybrid potential of interatomic interaction.
The interaction of nearest neighbors is described by the
exponential Morse potential,
u r A
r R r R
( ) [ ]
( ) ( )� �� � �
e e
2 0 02
(5)
which is especially convenient for the possibility of ana-
lytical calculations of average values. However, it does
not provide the proper long-range asymptotics of the van
der Waals attraction of atoms in the RGC (� � �r 6), so we
use the attractive part of the Lennard–Jones potential
u r r( ) ( / )� � 4 6
� to approximate interactions of atoms
that are not nearest neihgbors of each other.
The parameters A, , and R0 were determined in
Ref. 27 so that the present model reproduced the observed
values of sublimation energy, interatomic distance, and
bulk modulus of the RGC at T � 0. The parameters
and �
were obtained in a similar way within the model using the
Lennard–Jones (12–6) potential only [6]. The values of
the parameters A, R0, ,
and � for the RGC are listed in
Table 1.
Table 1. Parameters of the Morse and Lennard–Jones potentials
and the de Boer parameters � for the heavy RGC
RGC , �
–1
A, K R0 ,� �,�
, K �
Ar 1.63 117 3.83 3.37 132.52 0.165
Kr 1.52 172 4.09 3.61 182.85 0.088
Xe 1.38 226 4.46 3.92 257.34 0.055
The average potential energy � �U in (3) is calculated
over correlated Gaussian distribution function of atomic co-
ordinates. The parameters of this distribution are deter-
mined by the phonon spectrum of the crystal parametrized
by a single dimensionless parameter c of effective quasi-
elastic bond of neighboring atoms [6]. For the Morse poten-
tial, the average potential energy of interaction of two neigh-
boring atoms can be expressed analytically as
� � � �� � � �u A b q b q[ ]/ /e e2 42 2
2
, (6)
where b R R� � ( )0 is a reduced lattice expansion, R is
the nearest-neighbor distance,
is the inverse square of a
width of the distribution, and q is a dimensionless factor
759 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
A.I. Karasevskii and V.V. Lubashenko
representing a contribution of the correlations to the en-
ergy of interatomic interactions (q � 2 in the absence of
correlation).
The explicit formulae for FH ,U H , F3, and for the distribu-
tion width function of atomic displacements are given in Ref. 6.
Minimization of the functional (3) with respect to the parameters
c and b allows us to compute thermodynamical properties of the
perfect crystal at arbitrary T and P. A limit temperature Tc ,
above which the minimum of G0 with respect to c does not ex-
ist anymore, defines the point of the instability of the solid due to
the strong vibrational anharmonicity. For the perfect heavy
RGC, T Ac � 0 716. at P � 0, i.e., Tc is slightly higher than the
observed melting point.
3. The high-temperature limit
Now let us write down the Gibbs free energy of va-
cancy formation,
g T P u r F Pvi
i
R( , ) ( ) ,� � � � � �
�
�1
2
0
0
� (7)
where the first term describes a change of the crystal en-
ergy due to creation of a vacancy without any regard for
atomic relaxation around it, v is a vacancy volume (as-
sumed to be equal to the atomic volume, R 3 2/ in the fcc
lattice), and �FR is a change of the crystal free energy due
to medium relaxation around the vacancy, including lattice
distortion and changes of atomic vibrational frequencies.
To evaluate the first term in (7), we need the parame-
ters of the perfect crystal only, so it may be considered as
a first-order approximation to the true g T P( , ). However,
the atomic relaxation may contribute substantially to (7),
especially in the vicinity of the point of the high-tempera-
ture instability of the crystalline state. To estimate �FR ,
we should allow for a change of the vibrational spectrum
due to the vacancy formation, which is, generally, rather
complicated.
Since we are concerned with the vacancy properties at
high temperatures, it seems reasonable to benefit from the
one-particle approximation which is known to provide a
good description of vibrational and thermodynamical
properties of solids in this temperature range [20]. It was
shown in Ref. 6 that a contribution of the interatomic cor-
relations to the average potential energy of the crystal can
be discarded at high temperature (q � 2 in (6)), so that we
can use a one-particle distribution function of atomic
displacements, represented by a Gaussian function
f qi i( ) exp ( )
/
q � �
�
�
�
�
� �
�
3 2
2 , (8)
with q i being a displacement of the atom from its site. The
inverse square of the distribution
can be presented as a
power series [6]
�
� �
( , ) ( )c
c
n
c
l
l
l
l
� � �
�
�
�
�
�
�
�
�
2 2
2
0
2
1
�
. (9)
Here c is the parameter of effective quasi-elastic bond of
neighboring atoms, � � k T AB / is reduced temperature, and
� �
�
mk AB
(10)
is the de Boer parameter for the Morse potential, m is
atomic mass. The condition c� / � �� 1 defines the range
of applicability of the high-temperature approximation.
The coefficients n l2 in (9) are determined by integration
over the phonon spectrum of the crystal, and in the case of
the fcc lattice the first four of them are n0 2� , n2 5 6� / ,
n4 0 475� . and n6 0 296� . [6].
In the high-temperature approximation the average po-
tential energy (6) of interaction of neighboring atoms
takes on the form
u
u
A
b b* / * / *�
� �
� �� � � �e e2 2 1 22
(11)
with
* /� 2, while the long-range van der Waals in-
teraction of other pairs of atoms is taken into account
without thermal averaging. Hereafter we will preferably
represent energy in units of A.
The other contributions to G T P0( , ) in the high-tem-
perature limit appear as [6]
� �
�
�
�s
H HF U
A
c c
�
� � �
� � �
�
�
�
�
�
�
�
�
�
�
!
!
� �
�
�
�1
3
1
48
3
4� �
log
�
� ,
(12)
�
�
3
3 3
2
6
2 2 1 2
2
1
4
� � � ��
�
�
�
�
�
� � � �F
A
a
c
b be e/ * / * , (13)
where the dimensionless coefficient a3 2 2� . describing
effective cubic anharmonicity contribution is chosen so
that the calculated instability temperature at P � 0 lies
close to the triple point. Thus, the Gibbs free energy of the
perfect crystal at high temperature is written as
G
AN
z
u
A R
p R
HT
s
0 1
6
3
3
2 2
� � �
�
�
�
�
� � � �*
( )
"
�
� �
, (14)
where z1 12� is the coordination number, " � 4 91. for the
fcc lattice [28], and p P A� /( ) 3 is reduced pressure.
A test of this model shows that it provides an excellent
description of thermodynamical properties of essentially
classical Xe crystal (� � 0 055. ), even at � � 0 15. . (Note
that the instability point � c � 0 72. for all the heavy RGC.)
Moreover, the classical approximation � � 0 appears in
that case satisfactory, too. On the other hand, for Ar
( . )� � 0 165 the high-temperature approximation becomes
valid only at � � 0 45. .
Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 760
4. Vacancy
The high-temperature approximation to the self-con-
sistent statistical method gives us a tool for description of
thermodynamical properties of the vacancies in the heavy
RGC. The following assumptions will be made in consid-
eration of the atomic relaxation around the vacancy.
1. The vacancy is assumed to be at the origin of the
crystal. The relaxation of the atomic distribution in four
coordination spheres of the vacancy is taken into account.
The parameters of the other atoms are assumed to be un-
changed in comparison with those of the perfect crystal.
2. An equilibrium position of each of the relaxing at-
oms is shifted radially by a distance �Rn , where n is the
number of the coordination sphere. We assume �Rn � 0
for outward relaxation.
3. The distribution function of an atom of the va-
cancy’s nth coordination sphere is axially symmetrical
and is written as
f x y z C c x c y zn
n n n
( )( , , ) exp[ ( , ) ( , ) ( )],� � � �
�
�1
2
2
2 2
(15)
where cn1 and cn2 are variational parameters characterizing
longitudinal and transverse widths of distribution, respec-
tively, and the x axis is chosen so that is passes through the
site of the atom and the vacancy. The functional form of the
�( , )c is assumed to be given by Eq. (9).
4. Only pairs of neighboring atoms are assumed to
contribute into the change of the potential energy of inter-
atomic interaction in the vacancy’s surrounding.
Changes of the average vibrational amplitudes of at-
oms around the vacancy affect the potential energy of in-
teratomic interaction (through the parameter
), the en-
tropy term � s , and the cubic correction � 3. A change of
the entropy term is evident,
�� � � �s
n
n
s n s n s
z
c c c� � �
�
� 3
2 3
1
4
1 2 0[ ( ) ( ) ( )] , (16)
where z n is the number of atoms in the nth coordination
sphere of the vacancy, � s c( ) is given by (12), and c0 is the
quasi-elastic bond parameter for the perfect crystal.
A change of the average potential energy of interaction
between two neighboring atoms belonging to the nth and
mth coordination spheres of the vacancy is determined by
angles and sides of a triangle formed by the sites of these
atoms and the vacancy,
�u unm
b bnm nm nm nm* / *� � �� � � �
e e
2 4
2
# #
, (17)
where b R Rnm nm� � ( )0 ,
R R n
m
n
m
n
m
nm n m� � �
��
�
�
�
�
� � �
��
�
�
�
�
�
�
�
�
�
!1
1 1
1 2
� �
/
is a changed distance between the atoms, � �n nR R� / is
a reduced shift of an atom from the vacancy’s nth coordi-
nation sphere,
#
$
$
$
nm
nm
n
nm
n
mn
mc c c
� � � �
cos
( )
sin
( )
cos
( )
s
* * *
2
1
2
2
2
1
in
( )*
2
2
$
mn
mc
,
and cos ( ) /2 21 4$nm n m n� � � . Then the reduced poten-
tial energy contribution to the free energy of the medium
relaxation is given by
�
�
�U z
z u
z un
n
nn nn
nm
m n
nm
*
*
*� �
�
�
�
�
�
!
!� �
� �
1
4
2
, (18)
where z nm is a number of atoms belonging to the mth coor-
dination sphere of the vacancy that are the nearest neigh-
bors of an atom from the nth sphere.
Finally, we have to allow for a cubic anharmonicity
contribution to the vacancy formation free energy. Gener-
alizing Eq. (13) on the situation when atomic vibrational
distributions are not isotropic, we write
�� � � �3
1
4
3 0
1
4
3� � �
� �
� �z c zn
n
nm
m
nm[ ( , ) ] , (19)
where
�
� # #
3
3
2
2 4
2
27
1
4
nm b ba
nm nm nm nm� � ��
��
�
!
%� � � �
e e
/
%
�
��( )c c cni
ijk
nj nk
1
3
2, (20)
with c cn n3 2& .
We should keep in mind that Eq. (13) was derived in the
high-temperature limit from a general expression for the cu-
bic anharmonic contribution to the vibrational part of the
free energy of a perfect crystal [29], with the factor a3 deter-
mined by the phonon spectrum of the perfect crystal. Since
presence of a vacancy distorts substantially the vibrational
spectrum of the crystal, we can use Eq. (19) only for a quali-
tative estimation of the cubic contribution to the free energy
of an imperfect crystal.
Thus, within the present model, the Gibbs free energy
of vacancy formation in the RGC is a function of twelve
variational parameters cnl and � n and is written as
g p
A
z
u
R
( , )
*
�
"
'
�
� � � �
�
�
�
�
� �1
6
2
� � � �
p R
U s
( ) *
� �
3
3
2
� � � . (21)
761 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
A.I. Karasevskii and V.V. Lubashenko
5. Results and discussion
We calculated the equilibrium value of the Gibbs free en-
ergy of vacancy formation for Ar and Xe, minimizing g p( , )�
with respect to a set of variational parameters. It turned out
that inclusion of a contribution of the nearest neighbors of the
vacancy to the cubic anharmonic correction (19) results in
disappearance of the minimum of g with respect to c11 and
c12 at temperature slightly below the bulk instability point � c .
To override this artefact, we had to discard the cubic anhar-
monic contribution of the first coordination sphere of the va-
cancy. We suppose that for the vacancy’s nearest neighbors,
located in essentially nonspherical potential wells, the cubic
correction to g p( , )� is of more sophisticated form than that
given by (19) and (20). Thus, the value of the vacancy forma-
tion free energy calculated with this contribution discarded
represents the upper limit of g p( , )� . However, our estimation
showed that g p( , )� is only slightly sensitive to inclusion of
the anharmonic contribution of the first coordination sphere,
though this may be not a case in the vicinity of the instability
point.
In Fig. 1 we plotted the temperature dependence of the
vacancy formation free energy g at zero pressure calculated
for Ar and Xe (solid lines). In the same plot we show g T( )
computed for a quasi-harmonic crystal, i.e., with a3 0�
(dash lines), and for an anharmonic crystal, but without the
cubic contribution (19) included to the vacancy formation
free energy (dot lines). Near the melting point the calculated
vacancy concentration in both crystals is about 1 6 10 4. � � .
This value agrees with the results of some experimental [9]
and theoretical [18,22,23] studies, though it is less by the or-
der of magnitude than that of others [14,15,21,23]. To com-
pare our results for the vacancy parameters in Ar with others
available in the literature, we collected some experimental
and calculated data in Table 2.
The central result of this study is a drastic reduction of the
vacancy formation free energy near the melting temperature
due to approaching the point of the solid state instability.
Table 2. Vacancy parameters for Ar near the triple point: forma-
tion enthalpy, entropy, Gibbs free energy, and concentration.
Ref. h, K s kB, g , K 104 cv Method
[9] – – – < 2.5 Simultaneous measurement of bulk
and lattice expansion
[15] – – – 30
Measurement of x-ray lattice pa-
rameters along the melting line;
comparison with bulk data
[14] 943.6 6.72 392.6 108
Measurement of coefficient of ther-
mal expansion; comparison with
x-ray data
[22] 957.5 4 – 5.5 Two-body quasi-harmonic model
[18] – – – 4 Monte Carlo two-body simulation
[24] 857 2 689.4 3.5 Self-consistent Einstein model
[23] 805.4 – – 18.6 Correlative method of the unsym-
metrized self-consistent field
[21] 802.3 – 529.4 18.1 Statistical theory of mixtures
This
work
1225 5.9 733 1.6 Self-consistent statistical method
This is also indicative of sharp increase of the vacancy
formation entropy s g T P� � ( (( / ) from about 2 k B at
� � 0 45. to 5 6. k B at � c . At the same time, the vacancy for-
mation enthalpy h g Ts� � also increases near the insta-
bility point, from h �1000 to 1216 K in the considered
temperature range. The idea that the vacancy formation
enthalpy can rapidly increase near the triple point of a
crystal was suggested by Crawford et al. [25] for expla-
nation of anomalous high-temperature behavior of the
isohoric specific heat of Ar. They also presumed that such
behavior of vacancy parameters may be attributed to the
vicinity of the solid state instability point.
In Fig. 2 we show spatial distribution of the equilibrium
values of longitudinal (l �1) and transverse (l � 2) parameters
Thermodynamics and microstructure of vacancies in rare gas crystals at high temperature
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 762
g
,
K
900
800
700
600
500
400
Ar
50 55 60 65 70 75 80 85
Tm
T, K
1800
1600
1400
1200
1000
800
90 100 110 120 130 140 150 160 170
Tm
Xe
g
,
K
T, K
Fig. 1. Temperature dependence of the Gibbs free energy of vacancy formation in Ar and Xe at zero pressure calculated: for a
quasi-harmonic crystal (dash lines); for an anharmonic crystal (solid lines); for an anharmonic crystal without taking into account
the cubic correction to the vacancy formation free energy (dot lines).
cnl in Ar at different temperatures. Note that cn1 and cn2 de-
termine the average amplitudes of radial and tangential vib-
rations of atoms around the vacancy, respectively (the
mean-square average displacement � � �q c2 2
� [6]). As one
may expect, the presence of a vacancy affects mostly the dis-
tributions of displacements of its nearest neighbors, which
appear to be substantially elongated towards the vacancy. The
distributions of the other atoms around the vacancy remain
almost spherical. Unfortunately, our model does not repro-
duce properly the behavior of parameters c11 and c12 near the
instability point, since we ignored the contribution of the va-
cancy’s first coordination sphere to the �� 3. One should
expect that these parameters decrease more steeply as temper-
ature approaches Tc , thus providing additional reduction of
the free energy of vacancy formation.
Figure 3 represents spatial distribution of relative shifts
� �n nR R� / of the equilibrium positions of Ar atoms
around the vacancy at different temperatures. The shifts of
atoms are negative within the considered temperature range,
i.e., the atomic relaxation is inward. A contribution of the
spatial medium relaxation to the vacancy formation energy
varies from 1.6% at � � 0 4. to 4% near � c in comparison
with the value of g calculated taking into account changes of
the amplitudes of atomic vibrations only. These results are
quite consistent with that of Glyde [20,30].
We also considered influence of external pressure on
vacancy formation in the RGC near the melting line. Par-
ticularly, we checked the assumption that the vacancy
concentration is constant along the melting line [31]. We
successfully tested the present high-temperature approxi-
mation to the self-consistent statistical model by calcula-
tion of the equation of state for perfect Ar crystal at 293 K
up to 80 GPa. The agreement with observed data is quite
good within the considered pressure range.
At low pressures ( � 8 kbar) the calculated tempera-
tures of the solid state instability are only slightly higher
that the observed melting point, but at higher pressures
the instability line lies much above the melting line.
Therefore, the condition cv � const along the melting line
is valid at least at low pressures where the curves T Pc ( )
and T PM ( ) almost coincide.
Due to the assumptions we adopted in the present study,
we consider our results for the vacancy formation free en-
ergy as an upper limit for the true g T P( , ). First, to improve
the present model, we should take proper account of the cu-
bic anharmonic contribution of the nearest neighbors of the
vacancy. Second, we used a rather simple model to describe
the long-range interaction of atoms which can also contrib-
ute to the potential energy of the medium relaxation. It is
also worthwhile to seek for the equilibrium value of cv si-
multaneously with that of the bulk variational parameters c
and b, as it was made in Ref. 24. Moreover, an analysis of
this problem suggests that the calculated thermodynamical
properties of the vacancies are much more sensitive to the
input parameters and assumptions than the bulk crystal ther-
modynamics. For example, the long-range many-body ef-
fects are known to decrease the vacancy formation enthalpy
by 6% for Ar and Kr at T � 0 and by 8% for Xe [32].
Another point is that perturbation of the phonon spectrum
due to creation of vacancies is, indeed, more complicated
than that described by the one-particle approximation.
These and other effects may be of crucial importance in the
763 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
A.I. Karasevskii and V.V. Lubashenko
n
� = 0.5
� = 0.6
� = 0.715
� = 0.7
c
n
1
1.65
1.60
1.55
1.50
1.45
1 2 3 4 5 1 2 3 4 5
n
c
n
2
1.80
1.75
1.70
1.65
1.60
1.55
1.50
1.45
� = 0.5
� = 0.6
� = 0.7
� = 0.175
Fig. 2. A profile of the longitudinal (cn1) and transverse parameters (cn2) of the distribution of atomic displacements around the va-
cancy in Ar at different temperatures, n is a coordination sphere number, n � 5 corresponds to the bulk value of c.
1 2 3 4
n
�
n
0
–0.002
–0.004
–0.006
–0.008
–0.010
� = 0.5 � = 0.6
� = 0.7
� = 0.715
Fig. 3. Distribution of the relative shifts of the equilibrium po-
sitions of Ar atoms belonging to the four coordination spheres
of the vacancy at different temperatures.
premelting temperature range, so that the role vacancies
play in the melting transition cannot be discarded.
6. Conclusion
To conclude, let us discuss the role the vibrational
anharmonicity plays in formation of structure defects in a
crystal. An emphasis will be placed on the cubic vibrational
anharmonicity, which is of crucial importance for the evolu-
tion of instability of the phonon subsystem of the crystal. To
get an insight into the nature of such instability, we should
keep in mind that the odd order anharmonicity corresponds
to effective attraction between phonons. In a crystal with the
cubic anharmonicity, the phonon subsystem can be consid-
ered as a nonideal gas of attracting particles, with the num-
ber of particles being an internal parameter of the system.
On the contrary, the even order anharmonicity corresponds
to effective repulsion of phonons and, as known [33], is re-
lated to another type of instability at temperatures much
higher than the melting point of the solid.
At high temperature, the cubic anharmonicity is respon-
sible for the nonlinear reduction of the quasi-elastic bond
parameter c0( )� with temperature, especially in the vicinity
of the critical temperature � c . Such behavior of c0( )� mani-
fests itself in increasing of the widths of atomic distributions
and, as a result, in a nonlinear rise of the average potential
energy of the interatomic interaction. Since creation of
structural defects in the crystal, such as vacancies, is accom-
panied by rupture of a part of interatomic bonds, a dramatic
drop of the binding energy near the instability point paves
the way for the structural disordering of the medium, i.e., for
the melting transition. In our previous works [6,26,27,34]
we show that the premelting effects in the RGC (nonlinear
rise of isobaric specific heat, coefficient of thermal expan-
sion etc.) are associated with the evolution of the solid state
instability. Up to concentrations cv � 10 2� , vacancies have
only little influence on the crystal’s thermodynamical prop-
erties. However, in the direct vicinity of the instability tem-
perature, the vacancy contribution to the system thermody-
namics is comparable with the bulk one [34]. Finally, note
that steep temperature dependence of the free energy of the
vacancy formation near � c can be responsible for large
divergency of observed values of the vacancy concentration
in the RGC [8,9,12–16].
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