Theory of rare gas alloys: heat capacity
Low-temperature heat capacity of cryocrystals, which contain impurity clusters has been investigated theoretically and experimentally. Such defects might essentially enrich low-frequency part of the phonon spectrum by introducing both localized and delocalized vibrations. The effect of both types...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Theory of rare gas alloys: heat capacity / M.I. Bagatskii, S.B. Feodosyev, I.A. Gospodarev, O.V. Kotlyar, E.V. Manzhelii, A.V. Nedzvetskiy, E.S. Syrkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 741-746. — Бібліогр.: 33 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1217762025-06-03T16:29:07Z Theory of rare gas alloys: heat capacity Bagatskii, M.I. Feodosyev, S.B. Gospodarev, I.A. Kotlyar, O.V. Manzhelii, E.V. Nedzvetskiy, A.V. Syrkin, E.S. Classical Cryocrystals Low-temperature heat capacity of cryocrystals, which contain impurity clusters has been investigated theoretically and experimentally. Such defects might essentially enrich low-frequency part of the phonon spectrum by introducing both localized and delocalized vibrations. The effect of both types of the vibrations on the temperature dependence of the heat capacity is analyzed. Heat capacity of the disordered solid solution Kr–Ar (Ar concentration is ~25%) is studied as an example of the effect of the light weakly connected impurities on the low-temperature thermodynamic characteristics of the system. The mass defect of such an impurity induces «phonon pumping» from the low-frequency part of the spectrum into the high-frequency part and decreasing the low-temperature heat capacity, while the weakened interaction between the impurity and the host atoms combined with even weaker interaction between the impurities leads to the formation of the low-temperature maximum on the heat capacity temperature dependence. The analysis performed shows that at rather high Ar concentrations, the nonmonotonous temperature dependence of the relative change in the heat capacity of solid Kr₁₋pArp solutions is determined by excitation of delocalized high-dispersion low-frequency phonons. 2007 Article Theory of rare gas alloys: heat capacity / M.I. Bagatskii, S.B. Feodosyev, I.A. Gospodarev, O.V. Kotlyar, E.V. Manzhelii, A.V. Nedzvetskiy, E.S. Syrkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 741-746. — Бібліогр.: 33 назв. — англ. 0132-6414 PACS: 63.20.–e; 63.20.Mt; 63.50.+x; 63.70.+h https://nasplib.isofts.kiev.ua/handle/123456789/121776 en Физика низких температур application/pdf 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 Bagatskii, M.I. Feodosyev, S.B. Gospodarev, I.A. Kotlyar, O.V. Manzhelii, E.V. Nedzvetskiy, A.V. Syrkin, E.S. Theory of rare gas alloys: heat capacity Физика низких температур |
| description |
Low-temperature heat capacity of cryocrystals, which contain impurity clusters has been investigated
theoretically and experimentally. Such defects might essentially enrich low-frequency part of the phonon
spectrum by introducing both localized and delocalized vibrations. The effect of both types of the vibrations
on the temperature dependence of the heat capacity is analyzed. Heat capacity of the disordered solid solution
Kr–Ar (Ar concentration is ~25%) is studied as an example of the effect of the light weakly connected
impurities on the low-temperature thermodynamic characteristics of the system. The mass defect of such an
impurity induces «phonon pumping» from the low-frequency part of the spectrum into the high-frequency
part and decreasing the low-temperature heat capacity, while the weakened interaction between the impurity
and the host atoms combined with even weaker interaction between the impurities leads to the formation of
the low-temperature maximum on the heat capacity temperature dependence. The analysis performed shows
that at rather high Ar concentrations, the nonmonotonous temperature dependence of the relative change in
the heat capacity of solid Kr₁₋pArp solutions is determined by excitation of delocalized high-dispersion
low-frequency phonons. |
| format |
Article |
| author |
Bagatskii, M.I. Feodosyev, S.B. Gospodarev, I.A. Kotlyar, O.V. Manzhelii, E.V. Nedzvetskiy, A.V. Syrkin, E.S. |
| author_facet |
Bagatskii, M.I. Feodosyev, S.B. Gospodarev, I.A. Kotlyar, O.V. Manzhelii, E.V. Nedzvetskiy, A.V. Syrkin, E.S. |
| author_sort |
Bagatskii, M.I. |
| title |
Theory of rare gas alloys: heat capacity |
| title_short |
Theory of rare gas alloys: heat capacity |
| title_full |
Theory of rare gas alloys: heat capacity |
| title_fullStr |
Theory of rare gas alloys: heat capacity |
| title_full_unstemmed |
Theory of rare gas alloys: heat capacity |
| title_sort |
theory of rare gas alloys: heat capacity |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
Classical Cryocrystals |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121776 |
| citation_txt |
Theory of rare gas alloys: heat capacity / M.I. Bagatskii, S.B. Feodosyev, I.A. Gospodarev, O.V. Kotlyar, E.V. Manzhelii,
A.V. Nedzvetskiy, E.S. Syrkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 741-746. — Бібліогр.: 33 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT bagatskiimi theoryofraregasalloysheatcapacity AT feodosyevsb theoryofraregasalloysheatcapacity AT gospodarevia theoryofraregasalloysheatcapacity AT kotlyarov theoryofraregasalloysheatcapacity AT manzheliiev theoryofraregasalloysheatcapacity AT nedzvetskiyav theoryofraregasalloysheatcapacity AT syrkines theoryofraregasalloysheatcapacity |
| first_indexed |
2025-11-28T01:11:36Z |
| last_indexed |
2025-11-28T01:11:36Z |
| _version_ |
1849994566906675200 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 741–746
Theory of rare gas alloys: heat capacity
M.I. Bagatskii, S.B. Feodosyev, I.A. Gospodarev, O.V. Kotlyar, E.V. Manzhelii,
A.V. Nedzvetskiy, and E.S. Syrkin
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: feodosiev@ilt.kharkov.ua
Received October 20, 2006
Low-temperature heat capacity of cryocrystals, which contain impurity clusters has been investigated
theoretically and experimentally. Such defects might essentially enrich low-frequency part of the phonon
spectrum by introducing both localized and delocalized vibrations. The effect of both types of the vibrations
on the temperature dependence of the heat capacity is analyzed. Heat capacity of the disordered solid solu-
tion Kr–Ar (Ar concentration is ~25%) is studied as an example of the effect of the light weakly connected
impurities on the low-temperature thermodynamic characteristics of the system. The mass defect of such an
impurity induces «phonon pumping» from the low-frequency part of the spectrum into the high-frequency
part and decreasing the low-temperature heat capacity, while the weakened interaction between the impurity
and the host atoms combined with even weaker interaction between the impurities leads to the formation of
the low-temperature maximum on the heat capacity temperature dependence. The analysis performed shows
that at rather high Ar concentrations, the nonmonotonous temperature dependence of the relative change in
the heat capacity of solid Kr Ar1�p p solutions is determined by excitation of delocalized high-dispersion
low-frequency phonons.
PACS: 63.20.–e Phonons in crystal lattice;
63.20.Mt Phonon-defect interaction;
63.50.+x Vibrational states of disordered systems;
63.70.+h Statistical mechanics of lattice vibrations and displacive phase transitions.
Keywords: disordered solid solution, impurity cluster, boson peak, Jacoby matrix, Green function.
Introduction
For many years the influence of defects upon the phys-
ical properties of cryocrystals has been one of the most
important problems of low temperature physics of solids
that stimulated intense theoretical and experimental re-
search (e.g., see monograph [1] and references therein).
The results obtained on such objects are explainable with
high accuracy within quite simple crystal lattice models
and can be generalized for a very wide class of crystalline
structures.
At present there is a practically completed theory in-
terpreting the variations of the crystal properties caused
by the so-called isolated defects, whose influence upon
one another is negligible. It is certainly interesting to in-
vestigate crystals with defects that can be defined as com-
plex ones, for example, impurities located near a sample
boundary or a vacancy. When the concentration of impu-
rity atoms in the lattice grows, complex defects can ap-
pear in addition to the isolated ones. Complex defects are
formed by closely-spaced impurities. In some cases they
may be considered as isolated defects [2–5] and can be
described in terms of the regular perturbation theory.
Special interest is focused on systems consisting of
a cryocrystalline matrix containing randomly–dissolved
impurity atoms. At a growing concentration the spectral
characteristics of such solutions exhibit properties that
are typical of amorphous compounds, glasses, biopoly-
mers and so on. One of the features characterizing such
systems is the anomalous frequency distribution of phon-
ons in the long-wavelength (low-frequency) region. This
is evident in Raman and neutron scattering experiments
[6–8] and in the behavior of the low-temperature heat ca-
pacity and thermal conductivity [9–11].
This anomaly of the frequency distribution of phonons
can be described as a maximum of the magnitude
I � g( ) /� �2, where g( )� is the phonon density of states
and � is the frequency. The maximum was called a «boson
© M.I. Bagatskii, S.B. Feodosyev, I.A. Gospodarev, O.V. Kotlyar, E.V. Manzhelii, A.V. Nedzvetskiy, and E.S. Syrkin, 2007
peak». It is typical for glasses, amorphous media, and
other disordered systems in which the influence of de-
fects goes beyond local disturbances. The change in the
vibrational and other physical properties of such systems
cannot be interpreted as regular degenerate perturbation.
In this study we investigate theoretically how the low-
temperature heat capacity of solid Kr Ar1�p p solutions
changes at growing concentration p of Ar atoms. Argon
and krypton are highly soluble in each other and the pa-
rameter p can take any value varying from zero to unity
[12]. The calculated results are compared with the cur-
rently available experimental data. It is shown that the
change in the low-temperature heat capacity at p � 20%
and lower (~ 5 10� %) concentrations of argon has qualita-
tive features induced by delocalized excitations analo-
gous with boson peaks that appear in addition to the local
(including complex) defects.
Changes in heat capacity of krypton caused
by argon impurity
We develop the model crystal lattice of the Kr Ar1�p p
system on the basis of the fcc structure with the lattice pa-
rameter of pure krypton (a � 5.59 � [1]). In our descrip-
tion the interatomic interaction in solidified inert gases is
reduced to the interaction between the nearest neighbors.
In the general case this interaction is characterized by
three force constants �, �, and �:
ik
a a
2 2
0
0
0
0 0
; ;
�
�
�
� �
�
�
�
�
�
�
�
�
� �
� �
�
. (1)
The other force constant matrices can be found from
Eq. (1) by Oh — group symmetry operations, and the ma-
trix of self-action is
ik ik( ; ; ) ( )0 0 0 8 4� �� � � . (2)
Since a cubic-symmetry crystal has three independent
elastic moduli C11, C12, and C 66, we can use their values
and estimate unambiguously the force constants �, �, and
�. For krypton [13] we obtain � Kr–Kr � 7260.3 dyn/cm,
� Kr–Kr � 310.75 dyn/cm, � Kr–Kr � 7797.0 dyn/cm. The
force constants characterizing the Kr–Ar and Ar–Ar in-
teractions can be calculated with help of the Lennard–
Jones potentials (parameters � and � for Kr and Ar see e.g.
[1]):
� Kr–Ar � 7804.0 , � Kr–Ar � 127.00 , � Kr–Ar � 7310.0 ;
� Ar–Ar � 1551.83 , � Ar–Ar � 220 677. , � Ar–Ar � 1331.15.
We assume that the Ar–Ar interaction is the same for the
nearest Ar pairs and the larger clusters of defects (trian-
gles, tetrahedrons, etc.) that can form as the Ar concentra-
tion increases.
The calculations performed in this study are based on
the � -matrix technique (e.g., see [14–16]). The method
does not use explicitly the translational symmetry of the
crystal lattice and permits a straightforward calculation of
the spectral densities corresponding to the displacements
of different atoms of the system along displacements of
different crystallographic directions i:
� �
�
�i i( , ) ( , )r r�
1
Im � . (3)
Here r is the radius-vector characterizing the position of
the particular atom, � is the frequency; the Green function
� i ( , )� r is expressed in terms of the matrix element of op-
erator-resolvent as follows:
� � �i
i iu u
( , ) ( � �)� � �r
r r
� � � �2 2 1 (4)
where the symbol � r
u i
| has the meaning of the displace-
ment of the atom with the radius-vector r along the crys-
tallographic direction i (a certain vector in the space of
atom displacements in the system H); �� is the unit opera-
tor, �� is the operator describing the atomic oscillations in
the system, its eigenvalues are squared eigenfrequencies.
The matrix of this operator can be expressed in terms of
the force constant matrix and the masses of the interacting
atoms m( )r and m( )r� .
The spectral density of the system is
� � � �
� �
�� �
�
�
�( , ) lim Im ( � �) ,p
NN
2 2 1Sp � � (5)
According to [17–20], it is a self-averaging quantity and
can be estimated through averaging the functions over all
r-positions and displacement directions i.
The random distribution of the Ar impurity atoms was
performed using a random-number generator of pseu-
dorandom numbers distributed uniformly in the interval
(0,1). The random-number generator operates on the mul-
tiplying congruent principle [21]. We calculated the spec-
tral densities � �� �( , )p for different concentration of im-
purity atoms. At each concentration, the averaging was
made over several thousands of random configurations
of the impurity distribution. For each configuration the
density of states was determined through averaging over
several tens of spectral densities corresponding to the dis-
placements along different crystallographic directions of
several tens of sequentially arranged atoms.
The vibrational heat capacity C
v
(heat capacity at con-
stant volume) is expressed in terms of the phonon density
as (e.g., see [22]) of solid solutions is expressed in terms
of � �( ) as
C T p R
kT kT
d
v
( , ) sinh ( ) ,�
�
�
�
�
�
�
�
�� �3
2
2� �� �
� � �
�
(6)
742 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
M.I. Bagatskii et al.
where the integral is calculated in the whole region of the
atomic oscillation frequencies � both over the quasi-con-
tinuous spectrum band and over the discrete levels (if
there are).
We obtain � � � �( ) ( , )� � �i p for solid solutions and
� � � �( ) ( , )� �i r for a perfect single-atom crystalline lat-
tice of cubic symmetry. The function !� �� �i , r is the
spectral density caused by the displacement of an arbi-
trary atom in any of the crystallographic directions.
It is known that at low concentrations the interaction
between the impurity atoms is negligible. It is also as-
sumed that the impurity-induced change in the additive
thermodynamic characteristics is linear in impurity con-
centration p. Thus, at p "" 1
# #C T p
C T
p
C T
C T
l
v
v
v
v
( , )
( )
( )
( )(Kr) (Kr)
� ,
where #C Tl
v
( ) is the change in the heat capacity per local
defect. The disturbance of the lattice vibrations generated
by a local defect is as a rule, localized around this defect.
The #C Tl
v
( )-value can be calculated using, for exam-
ple, a shift function if the disturbance is regular and de-
generate [23,24]. According to the traditional interpreta-
tion of crystal lattice vibrations as a superposition of
plane waves, only the disturbance generated by an isoto-
pic impurity can be considered as degenerate (in three-di-
mensional vector models). The �-matrix method treats
the disturbance as degenerate if it is generated by an im-
purity with changed force constants in the case of the
noncentral interaction between the atoms [25]. The pres-
ence of the noncentral forces allows the disturbance to be
degenerated only in the subspace that can be transformed
in one-dimensional representations of the symmetry
group of the particular lattice.
At the same time, local disturbances do not affect the
band width of the quasi-continuous spectrum of the crys-
tal. They can only form discrete local levels beyond the
band. Therefore, the asymptotic behavior of the � -matrix
elements representing the operators �� that describe the
lattice vibrations do not change under the influence of a
local defect. It is natural to call such disturbances asymp-
totically degenerate.
Since the arbitrary matrix element �mn ( )�2 of the
resolvent operator �( ) ( � �)� � �� �2 2 1� � � can be expressed
strait forwardly through the element � �00
2 2( )� �� ( )
(Green function), for m n" we obtain [5]:
� �
� � � � �
mn m n
m n m n
( ) ( , �( ) )
( ) ( ) ( ) ( ) ( ) ,
$ $
$ $ $ $ $
� �
� � �
h h
00 (7)
then the change in the heat capacity for such asymptoti-
cally degenerated local disturbance can be written in the
form of the integral of Eq. (6) with the function � �( ) sub-
stituted by
#� � � � � � � � �
�
( ) [
~
( ) ~( ) ( ) ( )]
(
� � �
�
�
� �
%
%
2 2 2 2 2 2 2
0
� �n n
n
r
l
� �� l ) .
(8)
The second sum in Eq. (8) describes the contribution of
local frequencies if they appear in the disturbed system;
�n ( )�2 are the Jacobian matrix-generated polynomials
(e.g., see [14–16]); the tilded variables refer to a dis-
turbed system and can be calculated using the correspon-
ding Jacobian matrix; the nontilded variables describe an
ideal perfect lattice. The accuracy of computation by Eqs.
(6), (8) using the finite rank � -matrix corresponds to the
accuracy of calculation of the perfect-lattice heat capacity
with the aid of the matrix of the same rank.
Results and discussion
By the methods above mentioned we received the fol-
lowing results. Figure 1 illustrates the temperature de-
pendencies of the relative change
#C T
C T
C T p C T
C T
v
v
v v
v
( )
( )
( , ) ( )
( )(Kr)
(Kr)
(Kr)
�
� � �
in the heat capacity of solid Kr Ar1�p p solutions with vari-
ous Ar concentrations p in reference to the heat capacity
of pure Kr.
The dependence is smooth at moderate p-values (5 and
10%; curves 1 and 2, respectively). At p �15% (curve 3)
the dependence exhibits some «flattening» in the interval
3.5 K & &T 10 K. On a further increase in the Ar concen-
tration: p � 24 4. % (curve 4) and 50 % (curve 5), two
extrema appear in #C T p C T
v v
( , ) / ( )(Kr) — a maximum at
T � 3 5. K and a minimum at T � 9 K (24.4%) and
T �12 5. K (50% ).
The measured temperature dependence of the relative
change in the heat capacity at p � 24 4. % [26] (�) is
Theory of rare gas alloys:heat capacity
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 743
–0,05
0,05
0,10
5 10 15
4
3
2
1
#
C
v
v
/C
T, K
5
Fig. 1. Temperature dependences of the relative change in the
heat capacity at a growing concentration of Ar impurity p, %:
5 (1), 10 (2), 15 (3), 24.4 (4), 50 (5).
shown in Fig. 2. There are two extrema in this case too: a
maximum at T � 3 5. K and a minimum at T � 9 K. The
solid line is a portion of the curve of Fig. 1, which illus-
trates a theoretical calculation of the same magnitude at
the same concentration. It is seen that the experimental
and theoretical results are in good agreement — the
extrema temperatures coincide with a high degree of ac-
curacy and the deviations of #C T p C T
v v
( , ) / ( )(Kr) are ac-
tually within the accuracy of the experiment. True, the
character of dilation at these p-values is more compli-
cated than that allowed for in the model.
Let us analyze the possible reasons for the qualitative
behavior of the relative change in the heat capacity at
growing concentration.
Figure 3 illustrates the temperature dependencies of the
#C T C T
v v
1Ar (Kr)( ) / ( ) — parameter describing the change
in the heat capacity of Kr caused by an isolated atom of
the Ar impurity (curve 1) and the #C T C T
v v
2Ar (Kr)( ) / ( ) —
parameter describing the change in the heat capacity
caused by an isolated pair of adjacent Ar atoms (curve 2).
The smooth rise of curve 1 with temperature and the very
weak temperature dependence of curve 2 permit us to ex-
plain the heat capacity variation at low (down to 10%)
concentrations of Ar impurity at the expense of a super-
position of local disturbances localized near the de-
fects* that generate them. This explanation is totally
unsuitable for the double extrema behavior of the
#C T p C T
v v
( , ) / ( )(Kr) — parameter at higher concentra-
tions.
Since in the Kr Ar1�p p solutions the linkage between
the Ar atoms is much weaker than the Kr–Kr or Kr–Ar
bonds, at p � 20%, when practically each impurity atom
has identical impurity atoms among its nearest neighbors,
some quasi-continuous distribution of the weak bonds oc-
curs in the lattice. In this case each region having the
characteristic size l � $ (sound wavelength) can be char-
acterized by its own set of elastic constants, its own longi-
tudinal and transverse sound velocities which acquire a
meaning of randomly distributed parameters.
No quasi-local low-frequency oscillations can occur in
such systems. The low frequency regions of the density of
states of the Kr Ar1�p p systems are shown in Fig. 4, the Ar
concentrations being as in Fig. 1 (the curves are num-
bered as in Fig. 1, the dashed line shows the density of
states of ideal Kr [13]). In the frequency interval � �& *
(the frequency of the first van Hove singularity) none of
the curves has any sign of the characteristic quasi-local
maximum. The curves have typical «quasi-Debye»
shapes. In other words, the oscillations at these frequen-
cies, which are responsible for the behavior of the heat ca-
pacity in the temperature interval 1–20 K, are completely
delocalized.
At the same time, such systems with randomly distrib-
uted force and elastic parameters are noted for more in-
tensive sound wave dispersion. As a result, the curve � �( )
starts to deviate from the quadratic (Debye) behavior at
lower temperatures. For this reason the ratio � � �( ) ' 2
reaches a maximum in glasses and some other disordered
systems (e.g., the system in [27]). This is the so-called
«boson maximum» (or «boson peak» [6–11,27–32]). Note
that in these studies this peak was investigated mainly for
the phonons with frequencies at which the sound velocity
becomes dependent on the wave vector (from «propa-
gons» to «diffusons», according to the terms of [33]). The
absence of a distinct boundary between the propagons
744 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
M.I. Bagatskii et al.
* The contributions from impurity clusters such as equilateral triangles and tetrahedrons, are qualitatively similar to the contri-
butions made by impurity pairs; besides, the concentration of such localized defects is extremely low.
2 4 6 8 10
–0,08
–0,06
–0,04
–0,02
0
#
C
v
v
/C
T, K
Fig. 2. Temperature dependences of the relative change in the
heat capacity of solid Kr Ar0 756 0 244. . solution: theoretical calcu-
lation (((), experiment (�).
2 4 6 8 10
–0,9
–0,8
–0,7
–0,6
–0,5
1
2
#
C
v
v
/C
T, K
Fig. 3. Temperature dependences of the relative change in the
heat capacity caused by substitution Ar impurity: curve 1 —
isolated Ar atom, curve 2 — isolated pair of adjacent Ar atoms.
and diffusons is usually identified with the Ioffe-Regel ef-
fect (e.g., see [32,33]).
However, even higher-frequency acoustic phonons
whose dispersion ceases being linear (the so-called «dif-
fusons») can form a similar maximum if its frequency is
not higher than �* .
Figure 5 illustrates the ratios � � �( ) ' 2 for the
Kr Ar1�p p . There is a maximum only at p � 50%. At other
concentrations it has no time to form the maximum of the
frequency of the first van Hove singularity -– opening of
isofrequency surfaces of transverse acoustic oscillations.
However, as soon as the propagation of the sound wave
slowing down, the number of the low-frequency phonons
(especially those with k � 0) increases and so does the
low temperature heat capacity. On the other hand, the
number of the low frequency phonons decreases because
of the small mass of the impurity atoms. It is natural that
the two mechanisms — the increase and the decrease in
the number of phonons-compete in Ar–Kr solutions. Note
that at p � 24 4. % the maximum of the relative change in
the heat capacity (Figs. 1, 2) is negative and becomes pos-
itive only at higher Ar concentrations (e.g., see Fig. 1,
curve 5).
Thus, the analysis performed shows that at rather high
Ar concentrations, the nonmonotonous temperature de-
pendence of the relative change in the heat capacity of
solid Kr Ar1�p p solutions is determined by excitation of
delocalized high-dispersion low-frequency phonons. In
turn, the high-dispersion low-frequency phonons are due
to the random distribution of weak bonds in the lattice,
which are typical for the Ar–Ar interaction in this solu-
tion. Note that the distributed weak bonds cause a severe
local anisotropy of the oscillations both of Ar and Kr at-
oms. The local anisotropy leads to some pushing the
phonons towards the ends of the band of the quasi-contin-
uous spectrum, which increases the number of low
frequency phonons.
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0,2 0,4 0,6 0,8
0,2
0,4
0,6
0,8
1,0
1,2
5
4
4
2
31
g
–
1
,
T
H
z
�, THz
Fig. 4. Evolution of low frequency regions (until the first van
Hove singularity appears — the thin dashed vertical line) of the
phonon densities of the Kr Ar1�p p solution at a growing Ar con-
centration p. Heavy dashed line — phonon density of pure Kr.
The curves are numbered as in Fig. 1.
0,2 0,4 0,6 0,8
0,6
0,8
1,0
1,2
1,4
1,6
1,8
2,0
4
4
2
3
1
5
g
/
2
–
3
�
,
T
H
z
�, THz
Fig. 5. Evolution of low-frequency g( )� ) �2 — regions at a
growing Ar concentration p in the solid Kr Ar1�p p solution.
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