Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste
The temperature and volume dependences of the thermal conductivity of the Kr₁–ξ Xeξ solid solution are analyzed in a model in which heat is transferred by low-frequency phonons; above the phonon mobility edge this is done by "diffusive" modes migrating randomly from site to site. The phono...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe system / V.A. Konstantinov, E.S. Orel, V.P. Revyakin // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 1007-1011. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859957974969090048 |
|---|---|
| author | Konstantinov, V.A. Orel, E.S. Revyakin, V.P. |
| author_facet | Konstantinov, V.A. Orel, E.S. Revyakin, V.P. |
| citation_txt | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe system / V.A. Konstantinov, E.S. Orel, V.P. Revyakin // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 1007-1011. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | The temperature and volume dependences of the thermal conductivity of the Kr₁–ξ Xeξ solid solution are analyzed in a model in which heat is transferred by low-frequency phonons; above the phonon mobility edge this is done by "diffusive" modes migrating randomly from site to site. The phonon mobility edge w₀ is determined from the condition that the phonon mean free path limited by umklapp processes and scattering on point defects cannot be smaller than one-half the phonon wavelength. The Bridgman coefficientg g = - (∂ln Λ/∂lnV)T is the weighted mean over these modes, whose volume dependences differ strongly. It is shown that the amount of heat transferred by the "diffusive" modes above 100 K is quite large even in pure Kr and it increases with rising temperature and impurity concentration.
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Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10, p. 1007–1011
Heat transfer by low-frequency phonons and «diffusive»
modes in cryocrystal solutions: the Kr–Xe system
V.A. Konstantinov, E.S. Orel, and V.P. Revyakin
B. Verkin Institute for Low Temperature Physics and Engineering
of the Naional Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: konstantinov@ilt.kharkov.ua
The temperature and volume dependences of the thermal conductivity of the Kr1–� Xe� solid so-
lution are analyzed in a model in which heat is transferred by low-frequency phonons; above the
phonon mobility edge this is done by «diffusive» modes migrating randomly from site to site. The
phonon mobility edge �0 is determined from the condition that the phonon mean free path limited
by umklapp processes and scattering on point defects cannot be smaller than one-half the phonon
wavelength. The Bridgman coefficient g V T� � � �( ln ln )� � is the weighted mean over these
modes, whose volume dependences differ strongly. It is shown that the amount of heat transferred
by the «diffusive» modes above 100 K is quite large even in pure Kr and it increases with rising
temperature and impurity concentration.
PACS: 66.70. +f, 63.20.Ls
Introduction
The solidified inert gases Ar, Kr, and Xe are among
the simplest objects in the physics on the solids and
are therefore used traditionally for comparison of ex-
perimental and calculated data [1]. At temperatures
close to or above the Debye temperature (T � D) the
thermal conductivity of perfect crystals is determined
solely by phonon–phonon scattering and it is expected
to follow the law �
1/T [2]. To obey the law, the
volume of the crystal should remain invariable, be-
cause the modes would otherwise change and so would
the temperature dependence of the thermal conducti-
vity [2,3].
However, isochoric studies of the thermal conduc-
tivity of heavy solid inert gases show a considerable
deviation from the above dependence due to the ap-
proach of the thermal conductivity to its lower limit
[4,5]. The concept of the lower limit of the thermal
conductivity proceeds from the assumption that all
the excitations are weakly localized in the regions
whose sizes are half the wavelength, �/2. As a result,
the excitations can hop from site to site through ther-
mal diffusion [6]. In this case the lower limit of ther-
mal conductivity �min of the lattice at T � D can be
written as:
�min ( )� �
�
�
�
� �
1
2 6
2
1 3
2 3�
k l vtBn v , (1)
where vl and vt are the longitudinal and transverse
sound velocities, n = 1/a3 is the number of atoms per
unit volume, and kB is the Boltzmann constant.
Further studies of heat transfer in the solid solutions
Kr1–�(CH4)� (0 � � � 1) [7] and Kr1–� Xe� (0 � � � 0,14)
[8] have detected a gradual change from the thermal
conductivity typical of a perfect crystal to the lower
limit of the thermal conductivity �min as the crystal
becomes increasingly disordered. More recently the
volume dependence of the thermal conductivity of the
Kr1–�(CH4)� solid solution was analyzed in the frame-
work of a model which assumes that the phonon mean
free path cannot decrease infinitely [9]. In present
study, the temperature and volume dependences of the
thermal conductivity of the Kr1–�Xe� solid solution
are analyzed using the model mentioned above.
Model
We use Debye’s expression for the thermal conduc-
tivity [10,11]
� � �
k
v
l dB
D
2 2 2
2
0
�
� � �
�
( ) , (2)
© V.A. Konstantinov , E.S. Orel, and V.P. Revyakin, 2003
where v is the sound velocity; �D is the Debye fre-
quency (�D = (6�2)1/3v/a); l(�) is the phonon
mean free path determined by the U-processes and by
scattering on point defects:
l l lu i( ) ( ( ) ( ))� � �� �� � �1 1 1. (3)
The phonon mean free paths corresponding to each
mechanism of scattering are described as [2,10,11]
l v A Tu( )� �� 2 , A
k
Ma
B
D
�
18
2
3 2
2 3
� �
�
; (4)
l v Bi ( )� �� 4, B
D
�
3
2 3
�
�
�
; (5)
where the Grüneisen parameter � = – (�ln D/�lnV)T,
M is the average atomic weight of the solution: M =
= (1 – �) MKr+ �MXe; a = (1 – �) aKr + �aXe .
Taking into account the difference �M between the
atomic (molecular) masses of the impurity and the
matrix and the lattice dilatation, the coefficient � can
be written as [7]
�
� �
� � ��
�
�
�
�� � �( )1 6
2M
M
a
a
, (6)
where �M = M– MXe ; �a = a – aXe.
Expression (3) is not valid if l(�) is of the order of
one-half the phonon wavelength �/2 = �v/� or
smaller. A similar situation was considered previously
for the case of U-processes alone [11]. Let us assume
that in the general case
l
v A T B( ) ( ), ,
,
�
� � � �
�� � � � � � �
�
� � �
� � �
�
�
�
2 4
0
0
2,
0
v D
(7)
where � is a numerical coefficient of the order of
unity. There is evidence that the Ioffe–Regel crite-
rion, which suggests localization, is not applicable for
a phonon gas [12]. Nevertheless, we will refer to the
excitations whose frequencies are above the photon
mobility edge �0 as «localized» or «diffusive». Since
completely localized states do not contribute to the
thermal conductivity, we assume that the localization
is weak and the excitations can hop from site to site
diffusively, as was suggested by Cahill and Pohl [6].
The frequency �0 can be found from the condition
v A T B( )� � �� �0
2
0
4
0� � v , (8)
as
�
��
0 1
3
3 31
2
1 1 1 1� � � � � ��
�
!
"#
( )B
u u , (9)
where the dimensionless parameter u is
u
A T
B
�
4
27
2 2 3 3� �
, (10)
If �0 $ �D, the mean free path of all the modes ex-
ceeds �/2, and at T � D we obtain the well-known
expression [11]:
�ph arctan�
k
ATB
B
AT
B
D
2
1
2�
�
v
. (11)
At �0 � �D the thermal conductivity integral sepa-
rates into two parts describing the contributions to
heat transfer from the low-frequency phonons and the
«diffusive» high-frequency modes:
� = �ph + �loc , (12)
In the high-temperature limit (T � D) these contri-
butions are:
�ph arctan�
k
ATB
B
AT
B
2
1
2 0
�
�
v
, (13)
�loc v
� �
�
�
� �
kB
D4
2
0
2( ). (14)
The dependence of the thermal conductivity on the
specific volume is characterized by the Bridgman co-
efficient [3,13]:
g V T� � � �( ln ln )� . (15)
Taking into account that ( ln ln )� �A V T = 3� +
+ 2q – 2/3, where q V T� � �( ln ln )� , and that
( ln ln )� � �B V T 3� (as follows from Eqs. (4), (5))
and (�ln�/�lnV)T % 0, we have:
g � �
�
�
�
�
ph
ph
loc
locg g , (16)
where
g
V
q
T
ph
ph
� �
�
�
�
�
�
�
�
�
� � � �
ln
ln
�
2�
�
��
�
�
�
�
� ��
�
�
�
�
B
AT
B
AT
B
AT
q
�
� �
�
0
0
2
0
0
1
1
3
arctan
, (17)
g
V T
loc
loc� �
�
�
�
�
�
�
� � � � �
ln
ln
�
�
1
3
�
�
�
2
2
0
2
2
0
2
0
� �
� � � �
D
D( ), (18)
1008 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10
V.A. Konstantinov, E.S. Orel, and V.P. Revyakin
�
�
�0
0
1
3
6 1
� �
�
�
�
�
�
�
� � �
�
&
ln
lnV
u
uT
& � � � � ��
�
!
"#
� �1 1 1 1 6 6 23 3u u q( )� . (19)
Results and discussion
The isochoric thermal conductivity of the Kr1-� Xe�
(� = 0.034, 0.072, 0.14) solid solution was studied on
samples of different densities in the interval of tem-
peratures from 80 K to the onset of melting. The
choice of the system, concentrations, and temperature
interval was dictated by the following.
The phase diagram of the Kr1–� Xe� solid solu-
tion is well known [14]. The liquid and solid phases
have a point of equal concentrations at 114.1 K and
� = 0.15. Between 75 and 114 K the components form
an fcc solid solution for all 1 � � � 0. When samples
are grown with a temperature gradient along the mea-
suring cell, the solid solution can become layered at
� > 0.15. The highest Xe concentration was therefore
limited to 14%.
The isochoric thermal conductivities of pure Kr and
the Kr1–� Xe� solution, for which the isochoric condi-
tion comes into play at 80 K are shown in Figs. 1–4
(black squares). The computer fitting of thermal con-
ductivity using Eqs. (12)—(14) was performed by the
least squares method, varying the coefficients A and �.
The parameters of the Debye model for thermal con-
ductivity used in the fitting (a, v [1,14]; � — coeffi-
cients calculated by Eq. (6)), and the fitted values A
and � are listed in Table along with the Bridgman co-
efficients obtained in the experiment and calculated
by Eqs. (16)–(19). The calculation was done using
the values � = 2.5 and q = 1 [1,14].
The fitting results are shown in Figs. 1–4 (solid
lines). The same figures show the contributions
(dash-dot lines) to the thermal conductivity from the
low-frequency phonons, �ph, and from the «diffusive»
modes, �loc. The dashed line in the figures indicate the
lower limits of the thermal conductivity �min, which
were obtained as asymptotes of the �(Ò) dependence
at Ò ' (.
It is seen in Fig. 1 that in pure Kr the «localiza-
tion» of the high-frequency modes starts above 90 K.
As the temperature rises, the amount of heat trans-
ferred by the «diffusive» modes increases, and at 160 K
it becomes equal to the heat transferred by the low-
frequency phonons. In the solution with � = 0.034 (see
Fig. 2) the «localization» of the high-frequency modes
starts at 30 K and above 100 K most of the heat is
transferred by the «diffusive» modes. As the tempe-
Heat transfer by low-frequency phonons and «diffusive» modes in cryocrystal solutions: the Kr–Xe system
Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 1009
0
–
1 K
cm
m
W
�
,
2,0
1,0
3,0
–
1
)
)
T, K
pure Kr
80 120 160
minloc
ph
�
�
�
�
Fig. 1. Fitting results for the isochoric thermal conductiv-
ity and calculated relative contributions of low-frequency
phonons and «diffusive» modes to the thermal conductivi-
ty of pure Kr (molar volume is 28,5 ñm3/mole).
T, K
80 120 160
0
–
1 K
cm
m
W
�
,
2,0
1,0
3,0
–
1
)
)
min
loc
ph
�
�
�
�
� = 0.034
Fig. 2. Fitting results for the isochoric thermal conductiv-
ity and calculated relative contributions of low-frequency
phonons and «diffusive» modes to the thermal conductiv-
ity of the Kr1–� Xe� (� = 0.034) solid solution (molar vol-
ume is 29.1 cm3/mole).
T, K
80 120 1600
0,5
–
1 K
cm
m
W
�
,
2,0
1,0
1,5
–
1
)
)
min
loc
ph
�
�
�
�
� = 0.072
Fig. 3. Fitting results for the isochoric thermal conductiv-
ity and calculated relative contributions of mobile low-fre-
quency phonons and «diffusive» modes to the thermal con-
ductivity of the Kr1–� Xe� (� = 0.072) solid solution
(molar volume is 29.4 cm3/mole).
rature and the impurity concentration increase, so
does the amount of heat transferred by the «diffusive»
modes. For � = 0.14 (see Fig. 4) practically all the
heat at Ò � D is transferred by the «diffusive» modes.
The lower limit of thermal conductivity found by fitting
is 1.1–1.2 times higher than �min calculated by Eq. (1).
As is seen in Table, the experimental and calculated
Bridgman coefficients are in fairly good agreement if one
notes that the value of g is estimated with large uncer-
tainty and the model disregards phonon dispersion and
the real density of states. The temperature dependence of
the Bridgman coefficients g V T� � � �( ln ln )� of the
Kr1–� Xe� solid solution calculated by Eqs. (16)–(19) is
shown in Fig. 5. Equations (16)–(19) describe the gen-
eral tendency of the Bridgman coefficient g to decrease as
the crystal becomes increasingly disordered and most of
the heat transferred by the «diffusive» modes.
Table
Parameters of the Debye model for thermal conductivity
used in the fitting: a, v, and �, obtained through fitting A
and �; calculated gth and experimental gexp Bridgman
coefficients
�
à*10–8,
cm
v,
km/s
�
À*10–16,
s/K
� g
exp
g
th
0 3.62 0.86 0 3.1 1.2 9.4 9.2
0.034 3.64 0.86 0.1 3.8 1.2 8.0 5.7
0.072 3.65 0.87 0.19 6.4 1.2 5.5 4.6
0.14 3.67 0.87 0.29 9.8 1.1 4.0 3.8
Conclusions
It is shown that the temperature and volume de-
pendences of the thermal conductivity of the Kr1–� Xe�
(� � 0.14) solid solution can be described in the frame-
work of a model in which heat is transferred by
low-frequency phonons; above the phonon mobility
edge, heat is transferred by the «diffusive» modes mi-
grating randomly from site to site. The phonon mobil-
ity edge �0 is found from the condition that the
phonon mean free path determined by the umklapp
processes and scattering on point defects cannot be-
come smaller than one-half the phonon wavelength.
The Bridgman coefficient g V T� � � �( ln ln )� is
the weighted mean over these modes, which differ
strongly in their volume dependence. It is shown that
the amount of heat transferred by the «diffusive»
modes is quite large above 100 K even in pure Kr and
it increases with rising temperature and impurity con-
centration.
The authors are indebted to Prof. V.G. Manzhelii,
Full Member of NAS of Ukraine, and Prof. R.O. Pohl
(Cornell University) for fruitful discussions.
This study was supported by the Ukrainian Minis-
try of Education and Science, Project F7/286-2001.
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1010 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10
V.A. Konstantinov, E.S. Orel, and V.P. Revyakin
0
–
1
T, K
K
cm
m
W
min
loc
ph
�
�
�
�
�
,
� = 0.14
0,5
1,0
1,5
80 120 160
–
1
)
)
Fig. 4. Fitting results for the isochoric thermal conductiv-
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quency phonons and «diffusive» modes to the thermal con-
ductivity of the Kr1–� Xe� (� = 0.14) solid solution (molar
volume is 29.8 cm3/mole).
g
80
2
4
Kr Xe
6
120 160
T, K
� = 0.072
� = 0.14
� = 0.034
1– � �
Fig. 5. Calculated temperature dependence of the Bridg-
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solid solution.
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Heat transfer by low-frequency phonons and «diffusive» modes in cryocrystal solutions: the Kr–Xe system
Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 1011
|
| id | nasplib_isofts_kiev_ua-123456789-128908 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T16:20:43Z |
| publishDate | 2003 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Konstantinov, V.A. Orel, E.S. Revyakin, V.P. 2018-01-14T12:43:11Z 2018-01-14T12:43:11Z 2003 Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe system / V.A. Konstantinov, E.S. Orel, V.P. Revyakin // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 1007-1011. — Бібліогр.: 14 назв. — англ. 0132-6414 PACS: 66.70., 63.20.Ls https://nasplib.isofts.kiev.ua/handle/123456789/128908 The temperature and volume dependences of the thermal conductivity of the Kr₁–ξ Xeξ solid solution are analyzed in a model in which heat is transferred by low-frequency phonons; above the phonon mobility edge this is done by "diffusive" modes migrating randomly from site to site. The phonon mobility edge w₀ is determined from the condition that the phonon mean free path limited by umklapp processes and scattering on point defects cannot be smaller than one-half the phonon wavelength. The Bridgman coefficientg g = - (∂ln Λ/∂lnV)T is the weighted mean over these modes, whose volume dependences differ strongly. It is shown that the amount of heat transferred by the "diffusive" modes above 100 K is quite large even in pure Kr and it increases with rising temperature and impurity concentration. The authors are indebted to Prof. V.G. Manzhelii, Full Member of NAS of Ukraine, and Prof. R.O. Pohl (Cornell University) for fruitful discussions. This study was supported by the Ukrainian Ministry of Education and Science, Project F7/286-2001. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Low-Temperature Thermodynamics and Structure Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste Article published earlier |
| spellingShingle | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste Konstantinov, V.A. Orel, E.S. Revyakin, V.P. Low-Temperature Thermodynamics and Structure |
| title | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste |
| title_full | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste |
| title_fullStr | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste |
| title_full_unstemmed | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste |
| title_short | Heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the Kr–Xe syste |
| title_sort | heat transfer by low-frequency phonons and "diffusive" modes in cryocrystal solutions: the kr–xe syste |
| topic | Low-Temperature Thermodynamics and Structure |
| topic_facet | Low-Temperature Thermodynamics and Structure |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/128908 |
| work_keys_str_mv | AT konstantinovva heattransferbylowfrequencyphononsanddiffusivemodesincryocrystalsolutionsthekrxesyste AT oreles heattransferbylowfrequencyphononsanddiffusivemodesincryocrystalsolutionsthekrxesyste AT revyakinvp heattransferbylowfrequencyphononsanddiffusivemodesincryocrystalsolutionsthekrxesyste |