The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures
The thermal conductivity of solid CO was investigated in the temperature range 1–20 K. The experimental temperature dependence of thermal conductivity of solid CO was described using the time-relaxation method within the Debye model. The comparison of the experimental temperature dependences of the...
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irk-123456789-1171332017-05-21T03:03:13Z The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures Sumarokov, V. Jeżowski, A. Stachowiak, P. 7th International Conference on Cryocrystals and Quantum Crystals The thermal conductivity of solid CO was investigated in the temperature range 1–20 K. The experimental temperature dependence of thermal conductivity of solid CO was described using the time-relaxation method within the Debye model. The comparison of the experimental temperature dependences of the thermal conductivity of N₂ and CO shows that in the case of CO there is an additional large phonon scattering at temperatures near the maximum. The analysis of the experimental data indicates that this scattering is caused by the frozen disordered dipole subsystem similar to a dipole glass. The scattering is described by the resonant phonon scattering on tunnelling states and on low-energy quasi-harmonic oscillations within the soft potential model. 2009 Article The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures / V. Sumarokov, A. Jeżowski, P. Stachowiak // Физика низких температур. — 2009. — Т. 35, № 4. — С. 442-447. — Бібліогр.: 31 назв. — англ. 0132-6414 PACS: 63.20–e, 66.70.–f, 44.10.+i http://dspace.nbuv.gov.ua/handle/123456789/117133 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals |
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7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals Sumarokov, V. Jeżowski, A. Stachowiak, P. The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures Физика низких температур |
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The thermal conductivity of solid CO was investigated in the temperature range 1–20 K. The experimental temperature dependence of thermal conductivity of solid CO was described using the time-relaxation method within the Debye model. The comparison of the experimental temperature dependences of the thermal conductivity of N₂ and CO shows that in the case of CO there is an additional large phonon scattering at temperatures near the maximum. The analysis of the experimental data indicates that this scattering is caused by the frozen disordered dipole subsystem similar to a dipole glass. The scattering is described by the resonant phonon scattering on tunnelling states and on low-energy quasi-harmonic oscillations within the soft potential model. |
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Sumarokov, V. Jeżowski, A. Stachowiak, P. |
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Sumarokov, V. Jeżowski, A. Stachowiak, P. |
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Sumarokov, V. |
| title |
The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures |
| title_short |
The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures |
| title_full |
The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures |
| title_fullStr |
The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures |
| title_full_unstemmed |
The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures |
| title_sort |
influence of the disordered dipole subsystem on the thermal conductivity of the co solid at low temperatures |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2009 |
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7th International Conference on Cryocrystals and Quantum Crystals |
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http://dspace.nbuv.gov.ua/handle/123456789/117133 |
| citation_txt |
The influence of the disordered dipole subsystem on the thermal conductivity of the CO solid at low temperatures / V. Sumarokov, A. Jeżowski, P. Stachowiak // Физика низких температур. — 2009. — Т. 35, № 4. — С. 442-447. — Бібліогр.: 31 назв. — англ. |
| series |
Физика низких температур |
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2025-07-08T11:42:12Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, No. 4, p. 442–447
The influence of the disordered dipole subsystem
on the thermal conductivity of the CO solid
at low temperatures
V. Sumarokov
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: sumarokov@ilt.kharkov.ua
A. Je¿owski and P. Stachowiak
W. Trzebiatowski Institute of Low Temperature and Structure Research of Polish Academy of Sciences
P.O. Box 1410, 50-950 Wroclaw, Poland
E-mail: a.jezowski@int.pan.wroc.pl;
p.stachowiak@int.pan.wroc.pl
Received January 4, 2009
The thermal conductivity of solid CO was investigated in the temperature range 1–20 K. The experimen-
tal temperature dependence of thermal conductivity of solid CO was described using the time-relaxation
method within the Debye model. The comparison of the experimental temperature dependences of the ther-
mal conductivity of N2 and CO shows that in the case of CO there is an additional large phonon scattering at
temperatures near the maximum. The analysis of the experimental data indicates that this scattering is
caused by the frozen disordered dipole subsystem similar to a dipole glass. The scattering is described by the
resonant phonon scattering on tunnelling states and on low-energy quasi-harmonic oscillations within the
soft potential model.
PACS: 63.20–e Phonons in crystal lattices;
66.70.–f Nonelectronic thermal conduction and heat-pulse propagation in solids; thermal waves;
44.10.+i Heat conduction.
Keywords: thermal conductivity, heat transfer, molecular cryocrystals, dipolar disordered system.
Introduction
Solid carbon monoxide (CO) belongs to a group of
molecular cryocrystals with linear molecules, N2 type
crystals (N2, CO, CO2, and N2O) [1,2]. Solid CO was in-
vestigated in details by various methods: structural meth-
ods, Raman, IR spectroscopy, NQR and NMR methods,
theoretical methods and others (see, e.g., Refs. 1–14).
Heat capacity [15–21], thermal expansion [5,7,22,23] and
propagation of sound [24] have been researched in details
(see also parts 2 in books [1,2] and references therein).
Solid CO, being in equilibrium with its vapour, exists
in two crystallographic phases. Under equilibrium vapour
pressure, the structure of the low-temperature orienta-
tionally ordered �-phase of carbon monoxide (�� =
= 61.57 Ê [19,25]) is the same as for � -N2 (Ò�� = 35.61 K
[19,25]), namely, the fcc structure with an arrangement of
molecular axes along the spatial diagonals of the elemen-
tary cell (the Pa3 space group) [2] with four molecules in
the cell. As the CO molecule has a nonzero permanent di-
pole moment, with decreasing temperature a dipole-or-
dering (ordering on the ends of molecules) and a phase
transition into a lower temperature equilibrium phase of a
structure P213 (which differs from the Pa3 structure
mainly by that the CO molecules are ordered with respect
to the ends) should occur. The theoretical estimation of
temperature of such phase transition gives 5 K [26]. Al-
though in the work by Vegard [3] the low-temperature
structure of solid CO is determined as P213, researches on
NQR [11,12], NMR [13], dielectric susceptibility [14],
calorimetric [17,18,20,21], structural [5,6] studies spec-
ify that a non-equilibrium structure is observed. The au-
© V. Sumarokov, A. Je¿owski, and P. Stachowiak, 2009
thors of the x-ray studies [5] concluded that in solid CO
an «average» Pa3 structure is observed, with the electron
charge centres of the molecules localized in the lattice
sites with the displaced mass centre of the molecule. Elec-
tron-diffraction results [6] and conclusions of lattice dy-
namics studies [10] are consistent with this conclusion. In
the monograph on cryocrystals [2] (Chapter 12) the pref-
erence is also given to the structure with disordering with
respect the ends of molecules. The estimation [12] of fre-
quency (~109 s–1) of the end-to-end reorientations near
the temperature of �–� phase transition (using NMR data
[11]) testifies that the end-to-end ordering is practically
absent in the high temperature region of �-CO. The esti-
mations [20,23] of the residual entropy of CO indicated
that the majority of the CO molecules is in the disordered
end-to-end reorientation state. Atake et al. [21] detected
relaxation phenomena in the temperature drift in the tem-
perature interval between 14 K and 19 K in low tempera-
ture calorimetric study of �-CO, and concluded that the
end-to-end reorientation freezes in this temperature inter-
val, and defined the temperature of a transition to the
glassy state (Tg ~ 18 K). However, this relaxation effect
has not been observed in the thermal conductivity stu-
dies [27].
Since the disordered dipole subsystem of CO freezes,
at low temperatures, local low-energy excitations, spe-
cific for glass-like systems, appear in the energy spectrum
of the system. The phonon interaction with them should
be reflected in the temperature dependence of the thermal
conductivity.
The aim of this work is to investigate the influence of
the disordered dipole subsystem on the heat transfer in
CO solid at the low-temperature region of existence of
�-phase.
Experiment
The studies of thermal conductivity of the solid CO
were performed by the stationary method with the axial
thermal flux [28] in the temperature range 1–20 K. Crys-
talline samples were grown directly in a measuring cylin-
drical glass cell of the dimensions: 67 mm in height, inner
diameter of 6.4 mm and thickness of walls of 0.95 mm.
Two germanium resistance thermometers, attached to the
walls of the ampoule, were used for the measurement of
temperature and temperature gradient. The distance be-
tween thermometers was equalled to 33 mm. The lower
thermometer was on the distance about 14 mm from the
bottom of the ampoule.
The samples were obtained from gas of 99.996 % pu-
rity. The gas had a natural isotope composition. CO crys-
tals have been grown directly from the gaseous phase,
passing the liquid phase, at the temperature slightly below
the triple point. After growing and annealing of the sam-
ple at 59.5 K for 12 hours, the crystal was cooled down to
the temperature of liquid helium at the rate ~1 mm/h. The
time of passing of the �–� phase transition was about
70 h.
The random error of low-temperature measurements
did not exceed 1.5%. The other details of the experiment
were described in Ref. 27.
Results and discussion
The thermal conductivity of �-CO has been investi-
gated in the temperature interval 1–20 K. Previous exper-
imental results are presented in Ref. 27. The experimental
temperature dependence of the thermal conductivity �(Ò)
of CO is shown in Fig. 1. The behavior of �(Ò) qualita-
tively is typical for dielectric crystals. The thermal con-
ductivity increases with temperature, reaching a maximal
value 28 mW/(cm·K) at Tmax � 6 K. For comparison, in
the same Figure the experimental curve of the thermal
conductivity of nitrogen [29] is also shown.
Solid carbon monoxide and nitrogen have much in
common, and in many aspects solid CO is an analogue of
N2. They have close molecular and crystal parameters
[1,2] (molecular mass, lattice parameter, structure of both
The influence of the disordered dipole subsystem on the thermal conductivity
Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 443
1 10
10
2
10
1
T
h
er
m
al
co
n
d
u
ct
iv
it
y,
m
W
/(
cm
·K
)
T, K
N2
CO
Fig. 1. Low-temperature thermal conductivity of crystalline
CO and N2. Experiment (�) — CO, (�) – N2 from Ref. 29.
Fitting: solid line — CO according Eq. (5).
solid-state phases, etc.). As a result, one could expect that
experimental curves of thermal conductivity for both
crystals should be close to each other. However, like in
the case of CO2 and N2O [30], low temperature thermal
conductivities of CO and N2 differ appreciably. In the
high temperature region thermal conductivity of CO is
much higher than that of nitrogen. For example, thermal
conductivity of carbon monoxide at 20 K is twice as large
as that of nitrogen. However, with decreasing temperature
the character of temperature dependence of thermal con-
ductivity of CO changes sharply and the CO curve crosses
the nitrogen curve near 9 K. The maximum of the temper-
ature dependence of thermal conductivity for carbon
monoxide is an order of magnitude lower than that of ni-
trogen, and is shifted to a higher temperature region. The
maxima differ considerably in the shape, being much
broader for CO than N2 . The comparison of their temper-
ature dependences shows that in the case of carbon mon-
oxide there is an additional large scattering of phonons in
comparison with nitrogen.
Recently, it was found out that the influence of the dis-
ordered dipole subsystem on the heat transfer in solid
N2O in the low temperature region results in the large re-
duction of thermal conductivity of N2O in comparison
with CO2 [30]. In Ref. 30 this large reduction was ex-
plained within the framework of the model of soft po-
tentials (SPM) [31] as being the result of the resonant
phonon scattering at the tunnel two-level systems and
low-energy soft quasi-harmonic oscillations.
The principal difference between CO and N2 lies in the
fact that the CO molecule has a non-zero permanent di-
pole moment. Since the disordered dipole subsystem of
CO is frozen at low temperatures, there are local low-en-
ergy excitations specific for glass-like systems. The pho-
non interaction with them is reflected in the temperature
dependence of the thermal conductivity.
Experimental results were analyzed using the time-re-
laxation method [28] within the Debye model. The ther-
mal conductivity of dielectric solids can be written in the
form
� �
�
� �GT f x dxR
T
3
0
/
( ) , (1)
w h e r e G k vB� 4 2 3/ 2 � , f x x x x( ) / ( )�
4 21e e , x �
� �� / k TB , kB is the Boltzmann constant, � is the Planck
constant, � is the phonon frequency, � is the charac-
teristic Debye temperature, �R is the relaxation time
for the «resistive» interaction processes, v vl� �
[( 3
�
2 33 1 3vt ) / ] / is the sound velocity averaged over the
longitudinal vl and transverse vt polarizations [28].
Experimental results of thermal conductivity were an-
alyzed disregarding both the contribution of the librons to
the heat transfer and the phonon scattering by librations,
since the lowest libron excitation level is about 44.5 cm–1
(65 K) [2].
Assuming that different scattering mechanisms are in-
dependent, the relaxation time �R can be written as
� �R i
�
1 1. (2)
Here � i
1 (i = b, p, d, u) are the relaxation rates in different
mechanisms of the phonon scattering processes. The tem-
perature and frequency dependences of the relaxation
rates [28] for the phonon scattering on the grain bound-
aries, stress fields of dislocations, isotopic impurities and
in the U-processes are as follows:
� b ba
�1 ; � d da xT
�1 ; � p pa x T
�1 4 4 ;
� u a x T a T
�
1 2 5
1u 2uexp ( / ) .
(3)
The obtained experimental data were approximated
using our procedure described in Ref. 29. The parameters
aj (j=b, p, d, 1u, 2u) were estimated by minimizing the
functional
[( ) / ]� � �ci ei ei
2, where �ci and �ei are the
calculated and experimental thermal conductivity coeffi-
cients, respectively, at the i th point. The calculation was
performed using the values from Ref. 2: � = 103 K, vl =
= 2014.5 m/s, vt = 1103.5 m/s.
An approximation of the experimental temperature de-
pendence of thermal conductivity of carbon monoxide
fails to be described in the low temperature region by
Eq. (1) taking into account Eqs. (2) and (3).
To account for the extra (in comparison with N2) large
phonon scattering in solid CO it is necessary to introduce
an additional summand into the expression for the relax-
ation time � R
1 (2), as it was done in the case of N2O [30].
We described this additional scattering mechanism in the
model of soft potentials [31]. The expression for the re-
laxation rate � sp
1 of acoustic phonons can be written as
follows:
� sp c xT
x
c xT c xT
� � �1
1 2
4
3
3
2
tanh ( ) . (4)
The first summand describes the resonant scattering of
phonons on the tunnel states (TLS), the second and the
third terms describe the scattering on low-energy soft
quasi-harmonic oscillators [31].
The fitting procedure gives the combined relaxation time:
� R x T xT
� � � � � �1 7 4 4 52 86 10 12 9 2 41 10. . .
� �
� � �177 10 12 55 2 63 10
2
4 2 5 7. ( . / ) .x T T xT
x
exp tanh
� � �258 307 104 6 3. ( ) .xT xT . (5)
Figure 1 shows the fitting curve (solid line) describing
the thermal conductivity of solid CO. It can be seen that
below 17 K the description of the experimental results is
444 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4
V. Sumarokov, A. Je¿owski, and P. Stachowiak
quite good. On this ground, we concluded that the influ-
ence of the disorder dipole subsystem on thermal conduc-
tivity of crystal CO can be described within the frame-
work of the SPM model.
Using the obtained values (5) of the intensity of the
phonon scattering and Eq. (3) for relaxation times we esti-
mated [28] the crystallite size, density of dislocations,
and density of isotope defects. The size of crystalline
grains, 4.3·10–2 mm, is close to that for nitrogen. The den-
sity of dislocations, 4.3·108 cm–2, in CO is smaller than in
N2 . The intensity of the phonon–phonon interaction in
solid CO is six times less than in N2 .
Figure 2 shows the relaxation rates of phonons versus
phonon energy for different temperatures from 1.5 K up
The influence of the disordered dipole subsystem on the thermal conductivity
Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 445
1 110 10
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.0
1.0
Eph, K Eph, K
b
sp
i
T = 1.5 K
T = 4 K
sp
b
U
T = 8 K
sp
U
b
T = 15 K
sp
U
b
10
10
10
10
10
10
10
10
10
7
10
7
10
7
10
7
10
4
10
4
10
4
10
4
10
1
10
1
10
1
10
1
T = 1.5K
A
B
C
T = 4 K
A
C
C
T = 8 K
A
B
T = 15 K
A
B
C
B
(s
p
)
� i–
1
(s
p
)
� i–
1
(s
p
)
� i–
1
(s
p
)
� i–
1
0
0
0
0
�
�
R
i
/
�
�
R
i
/
�
�
R
i
/
�
�
R
i
/
a b
Fig. 2. Ratio of the relaxation rate for the individual scattering processes � i
1 (i = b, d, i, u, sp) to the «resistive» relaxation rate � r
1
versus the phonon energy. Scattering of phonons: b — on grain boundaries, d — on dislocation stress fields, i — on point defects,
u — on phonons (the U-processes), sp — the resonant phonon scattering on tunnel states and on low-energy vibrations with the
framework of the model of soft potentials (a). The relaxation rate for the individual scattering processes � isp
1 (i = A, B, C) versus
the phonon energy. The resonant phonon scattering: on tunnel states (A) and on low-energy quasi-harmonic oscillations (B, C) (b).
to 15 K. The ratio of the relaxation rate of the individual
scattering processes � i
1 (i = b, d, i, u, sp) to the «resis-
tive» relaxation rate � R
1 versus the phonon energy is de-
picted in Fig. 2,a. The contribution of the boundary scat-
tering is essential at 1.5 K for low-energy phonons and
tends to decrease with increasing temperature. The � sp
1 is
the main contribution at temperatures 1.5–15 K, it is the
second largest exceeded only by the boundary scattering
for low-energy phonons below 1.5 K, and above 15 K it is
the second largest exceeded by the contribution from
U-processes for phonons with energy over 25 K. Contri-
butions of other scattering mechanisms are negligible.
Figure 2,b shows the relaxation rate for the individual
scattering processes � isp
1 (i = A, B, C) as a function of the
phonon energy (see (4)), where A, B, and C are the reso-
nant phonon scattering on the tunnel states A, and low-en-
ergy quasi-harmonic oscillators B and C. The relaxation
rate A from TLS is close to the relaxation rate B from the
second summand at 1.5 K. When temperature increases
the relaxation rate B becomes dominant in comparison
with the rates A and C. The relaxation rate C is smaller
than others by the several orders of magnitude.
The resonant phonon scattering on quasi-local elemen-
tary excitations not only makes possible to describe the
thermal conductivity k(T) of solid CO, but also allows
to explain the large difference in the thermal conductivity
of solid CO and N2 in the low temperature region. To
this end we shall replace � R N
1
2( ), allowing to describe
k(T) for N2 (curve 1, Fig. 3), by � � �R RN N
� �1
2
1
2( ( )
�
� sp
1( )CO . As a result we received the curve 2 (Fig. 3),
which describes well the thermal conductivity of solid CO
below the maximum point. The curve 2 transforms into the
curve 3, taking into account distinctions in the parameters
of U-processes.
Thus, the relaxation time � sp
1 (4) allows to describe
both the temperature dependence of the thermal conduc-
tivity of CO, and the large difference in thermal conduc-
tivities of solid CO and N2 at low temperatures.
Conclusions
The thermal conductivity of solid CO was studied in
the temperature range 1–20 K. The experimental tempera-
ture dependence of the thermal conductivity of solid CO
was described using the time-relaxation method within
the Debye model. Analysis shows that for solid CO the
following can be stated: the size of crystalline grains is
close to that of nitrogen, the density of dislocations is
smaller than in nitrogen, and the intensity of the pho-
non–phonon interaction is six times less than in nitrogen.
The comparison of the experimental temperature de-
pendences of the thermal conductivity of solid N2 and CO
shows that in the case of CO there is an additional large
phonon scattering at temperatures near the maximum.
The analysis of the experimental data indicates that this
scattering is caused by the frozen disordered dipole sub-
system similar to the dipole glass. This scattering is
described by the resonant phonon scattering on tunnelling
states and on low-energy quasi-harmonic oscillations
within the soft potential model.
The authors gratefully thank Yu.A. Freiman for fruit-
ful discussion.
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1
2
3
1 10
10
2
10
1
T
h
er
m
al
co
n
d
u
ct
iv
it
y
m
W
/(
cm
·K
)
T, K
N2
CO
Fig. 3. The temperature dependence of thermal conductivity.
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