The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures
The thermal conductivities of CO₂ and N₂O solids have been investigated in the low-temperature range 1–40 K. The thermal conductivities of CO₂ and N₂O are large compared with those of simple molecular crystals such as N₂, CO, or O₂ in the whole investigated temperature range. Analysis of the exper...
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Sumarokov, V.V. Stachowiak, P. Jeżowski, A. 2017-06-16T08:07:15Z 2017-06-16T08:07:15Z 2007 The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures / V.V. Sumarokov, P. Stachowiak, A. Jeżowski // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 778-782. — Бібліогр.: 23 назв. — англ. 0132-6414 PACS: 63.20.–e; 66.70.+f; 44.10.+i https://nasplib.isofts.kiev.ua/handle/123456789/121794 The thermal conductivities of CO₂ and N₂O solids have been investigated in the low-temperature range 1–40 K. The thermal conductivities of CO₂ and N₂O are large compared with those of simple molecular crystals such as N₂, CO, or O₂ in the whole investigated temperature range. Analysis of the experimental data by the Callaway method shows that relatively large size of crystalline grains, low density of dislocations and weak phonon–phonon interaction might be the reasons for the good thermal conduction in these crystals at temperatures near the maxima. A comparison between calculated values of the intensity of normal phonon scattering processes and experiment gives evidence that in N₂O there is an additional (in comparison with CO₂) giant scattering of phonons. This scattering is described in the frameworks of soft potential model by the resonance phonon scattering on tunnel states and low-energy vibratons. The authors gratefully thank Yu.A. Freiman, B.Ya. Gorodilov and M.A. Strzhemechny for fruitful discussion. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Classical Cryocrystals The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures Article published earlier |
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| title |
The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures |
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The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures Sumarokov, V.V. Stachowiak, P. Jeżowski, A. Classical Cryocrystals |
| title_short |
The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures |
| title_full |
The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures |
| title_fullStr |
The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures |
| title_full_unstemmed |
The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures |
| title_sort |
peculiarities of heat transfer in co₂ and n₂o solids at low temperatures |
| author |
Sumarokov, V.V. Stachowiak, P. Jeżowski, A. |
| author_facet |
Sumarokov, V.V. Stachowiak, P. Jeżowski, A. |
| topic |
Classical Cryocrystals |
| topic_facet |
Classical Cryocrystals |
| publishDate |
2007 |
| language |
English |
| container_title |
Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The thermal conductivities of CO₂ and N₂O solids have been investigated in the low-temperature range
1–40 K. The thermal conductivities of CO₂ and N₂O are large compared with those of simple molecular crystals
such as N₂, CO, or O₂ in the whole investigated temperature range. Analysis of the experimental data by
the Callaway method shows that relatively large size of crystalline grains, low density of dislocations and
weak phonon–phonon interaction might be the reasons for the good thermal conduction in these crystals at
temperatures near the maxima. A comparison between calculated values of the intensity of normal phonon
scattering processes and experiment gives evidence that in N₂O there is an additional (in comparison with
CO₂) giant scattering of phonons. This scattering is described in the frameworks of soft potential model by
the resonance phonon scattering on tunnel states and low-energy vibratons.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121794 |
| citation_txt |
The peculiarities of heat transfer in CO₂ and N₂O solids at low temperatures / V.V. Sumarokov, P. Stachowiak, A. Jeżowski // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 778-782. — Бібліогр.: 23 назв. — англ. |
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| first_indexed |
2025-11-25T22:54:35Z |
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2025-11-25T22:54:35Z |
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1850575739741536256 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 778–782
The peculiarities of heat transfer in CO2 and N2O solids
at low temperatures
V.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
P. Stachowiak and A. Je¿owski
V. Trzebiatowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences
P.O. Box 1410, 50-950 Wroclaw, Poland
E-mail: p.stachowiak@int.pan.wroc.pl
a.jezowski@int.pan.wroc.pl
The thermal conductivities of CO2 and N2O solids have been investigated in the low-temperature range
1–40 K. The thermal conductivities of CO2 and N2O are large compared with those of simple molecular crys-
tals such as N2, CO, or O2 in the whole investigated temperature range. Analysis of the experimental data by
the Callaway method shows that relatively large size of crystalline grains, low density of dislocations and
weak phonon–phonon interaction might be the reasons for the good thermal conduction in these crystals at
temperatures near the maxima. A comparison between calculated values of the intensity of normal phonon
scattering processes and experiment gives evidence that in N2O there is an additional (in comparison with
CO2) giant scattering of phonons. This scattering is described in the frameworks of soft potential model by
the resonance phonon scattering on tunnel states and low-energy vibratons.
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
CO2 and N2O cryocrystals have much in common.
They have close molecular and crystal parameters (mo-
lecular mass, spacing between nearest neighbors, ze-
ro-oscillation energies [1,2]).
Under equilibrium vapor pressure, CO2 and N2O have
the Pa3 crystal structure below their triple-point tempera-
tures 216.57 and 182.35 K [1,2], respectively. The mole-
cular axes in these crystals are oriented along the body
diagonals of the cubic unit cell. The main distinction be-
tween solid CO2 and N2O is due to that the N2O (N–N–O)
molecule, unlike the CO2 (O–C–O) one, is asymmetric.
N2O molecules are head-tail disoriented [3]. The residual
entropy (difference between spectroscopic and calorimet-
ric entropies) �S /Rres ln . .2 1 04 0 17� � (Ref. 4) indicates
that this disorder persists at low temperatures.
The experimental investigation of the low-temperature
thermal conductivity of CO2 and N2 crystals [5–7] reveals
a very strong phonon scattering in the N2O solid (in com-
parison with CO2). By fitting a model to experimental
curves, the authors of Ref. 7 found the expression for the re-
laxation time, allowing then to describe effectively the con-
tribution of this extra (as compared to CO2) phonon scatte-
ring to the thermal conductivity, as a function of the
temperature and phonon frequency: � a
�1 ~ ��1 5 2/ T . It was
assumed that the additional thermal resistance was caused
by phonon scattering on the head-tail disorder of N2O
molecules. The orientation of the N2O axes along the
cube body diagonal is practically frozen since the times of
the 180° reorientation of the molecules are very long at
low temperatures. These N2O molecules, localized at their
sites, can have one of two stable orientations that differ
by 180° and are separated by a barrier. They form some-
© V.V. Sumarokov, P. Stachowiak, and A. Je�owski, 2007
thing like a two-level orientational subsystem. It is simi-
lar to an orientational glass, which influences the temper-
ature behavior of the thermal conductivity. The typical
properties of glasses usually manifest themselves below
1 K. The low-temperature properties are commonly de-
scribed within a two-level system model [8]. In the
low-temperature region (T < 1 K), the heat capacity is
proportional to temperature, and the thermal conductivity
is proportional to the temperature squared. The major
contribution to the thermal conductivity of glasses in this
temperature region comes from resonance scattering of
phonons at two-level systems. In a wider temperature in-
terval the properties of glasses can be described effi-
ciently by the model of soft atomic potentials (SPM) [e.g.,
see Ref. 9 and references therein]. The role of the N-pro-
cesses on heat transfer was not cleared in Ref. 7.
This study concerns the role of normal processes in
heat transfer of CO2 and N2O cryocrystals.
Experiment
The thermal conductivity of CO2 and N2O crystals was
measured using the steady-state flow method.
Since we meant to compare the thermal conductivities
of these crystals, special attention was concentrated on
the quality of the samples. A special technique was deve-
loped, which allowed us to prepare perfect crystalline
samples of CO2 and N2O. The samples were grown and
investigated in a cylindrical glass ampoule 36 mm high
with an inner diameter of 4.2 mm.
The crystals were grown directly from the gas phase.
The condensation temperature was about 173 and 162 K
for CO2 and N2O, respectively. The growth rate was
about 1.5 mm/h. The quality of the crystals is sensitive to
the annealing and cooling conditions. Special effort was
therefore made to exclude their effect upon the thermal
conductivity in these crystals. Thus, the CO2 sample and
the last N2O sample were cooled at a rate of ~ 0.1 K/h
down to 100 K, ~ 0.2 K/h in the temperature range
100–70 K, and about 0.5 K/h below 70 K.
The sample growth could be monitored through spe-
cial windows in the cryostat with the cold shields open.
The crystalline CO2 and N2O samples were transparent,
without visible defects after cooling to helium tempera-
tures. The gases employed for sample sgrowth had the
natural isotopic composition. The impurity content was
not higher than 0.001%.
The sample temperature and the temperature gradient
over the sample were measured with germanium resis-
tance thermometers placed at a 12 mm distance between
them.
Other details of experiment are described elsewhere
[5–7,10].
Results and discussion
The thermal conductivity of the N2O and CO2 crystals
was measured in the interval 1–40 K. The experimental
results [5–7] are shown in Fig. 1. It is interesting that (i)
the maxima differ considerably in shape, (ii) the thermal
conductivity coefficients of both crystals are very high at
the maxima exceeding greatly the corresponding coeffi-
cients of simple molecular cryocrystals N2 (Ref. 11), CO
(Ref. 12) and O2 (Ref. 13) whose thermal conductivity at
the maximum is about 200 mW/(cm�K). The high value of
the thermal conductivity at the maximum is due to the high
degree of perfection of the N2O and CO2 samples and to
the sound-velocity relationship in these crystals. The maxi-
mum in CO2 is a sharply pointed peak. In the case of N2O
the maximum is broad as if it was cut off by additional
phonon scattering. It was assumed [7] that the additional
thermal resistance is caused by the phonon scattering
at N2O molecules, which are head-tail disordered.
Phenomenologically, the relaxation time of the extra (as
compared to the CO2 crystal) scattering was estimated [7].
The authors did not allow for the contribution of normal
processes to the thermal conductivity of these crystals.
Let us consider the effect of N-processes on the heat
transfer in N2O and CO2 crystals and the heat transfer fea-
tures related to the orientational N2O subsystem. It is well
known that N-processes play a special role in thermal
conduction. Although they do not contribute to the ther-
The peculiarities of heat transfer in CO2 and N2O solids at low temperatures
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 779
1 10
10
1
10
2
10
3
10
4
CO2
N O2
T, K
Fig. 1. The temperature dependence of the low-temperature
thermal conductivities of CO2 and N2O solids. Experiment:
N2O (�, �), CO2 (�). Solid curves: the best fitting.
mal resistence [14] directly, they participate in the energy
redistribution in the phonon subsystem and thus affect
thermal conduction.
There is a number of theoretical methods [14–16] that
take account of N-processes. The Callaway method is
most suitable for our purpose. The experimental results are
described using the Callaway relaxation method [16] within
the Debye model. The Callaway expression for the thermal
conductivity of a dielectric crystal can be written as
�
1 2, (1)
where
�1
3
0
� �GT f x dxc
/T�
( ) , (2)
�
�
�
� �2
3
0
2
0
�
�
�
��
�
�
�
��
� �GT f x dx f x dxc
N
/T
c
N R
/T� �
( ) ( )
�
�
��
�
�
�
��
�1
.
(3)
Here
G
k B�
4
2 32� v�
, f x
x x
x
( )
( )
,�
�
4
21
e
e
x /k TB� �� is the dimensionless variable, kB is the
Boltzmann constant, � is the Planck constant, � is the
phonon frequency. � is the characteristic Debye tempera-
ture, �R, �N are the relaxation times for «resistive» and
normal interaction processes, �c is the combined phonon
relaxation time, v v v�
� � �[( ) ]l t
//3 3 1 32 3 is the sound ve-
locity averaged over the longitudinal vl and transverse vt
polarizations [17].
The low-temperature results were analyzed disregard-
ing either librons or phonon scattering on librations be-
cause the lowest-energy excitation level of librations is
about105 K (CO2) and 100 K (N2O) [2].
Assuming that different types of scattering are inde-
pendent, the relaxation times �c and �R can be written as
� � �c R N
� � ��
1 1 1, (4)
� �R i
� �� �1 1. (5)
Here � i
�1 (i = b, p, d, u) denotes the relaxation times for
different phonon-scattering mechanisms. The tempera-
ture and frequency dependence of the relaxation times
[14] for phonon scattering at grain boundaries, 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 u ua x T a /T� � �1
1
2 5
2exp( ). (6)
The role of normal processes for low-energy phonons
was investigated by Herring [18]. He found that the relax-
ation time of normal processes involving acoustic phon-
ons in the high-symmetry (cubic) crystals at low tempera-
tures can be written as
� �N T x T� � �1 2 3 2 5. (7)
Later, rather elaborate expressions (e.g., see Ref. 19)
were obtained for relaxation times, which took into ac-
count the phonon polarization in different processes of
phonon decay/production (l�l+t, l�t+t, etc.) as a func-
tion of temperature and frequency. The thermal conduc-
tivity asymptotes obtained (both for low and high temper-
atures) have extremely limited areas of applicability, well
beyond the temperature range of experiment. Usually, the
polarization-averaged relaxation rate for N-processes is
obtained from the analysis of thermal conductivity expe-
rimental results for cryocrystals. The inverse relaxation
time � �N T�1( , ) is therefore found using expressions with
fitting parameters [11,14,18–21]. However, comparison
with low-temperature experiment is most often made us-
ing Eq. (7) and neglecting the contribution of transverse
phonons (e.g., see Refs. 18, 21). To analyze our experi-
mental results, we used Eq. (7) and other expressions
[11,19,20] as trial functions.
Out results were approximated using the procedure
from [11]. The parameters aj (j = b, p, d, 1u, 2u, N)
were estimated through minimizing the functional
� �i ci ei ei/[( ) ] , 2 where êci and êei are the calculated
and experimental thermal conductivity coefficients, re-
spectively, at the i-th point.
The calculation was performed using the following va-
lues [2]: � = 141 K, vl = 2676.3 m/s, vt = 1513.1 m/s for
N2O, and � = 151.8 K, vl = 2806.4 m/s, vt = 1605.7 m/s
for CO2.
The fitting to the experimental temperature depend-
ence of the thermal conductivity of crystalline CO2 was
best described by Eq. (1) with
� R . . x T . xT� � �
�
1 5 4 4 41 07 10 1 9 8 4 10
�781 23 82 5x T . Texp( / );
� N . x T� � �1 3 2 51 62 10 . (8)
The relaxation times providing the best description of
the fitting for N2O are
� � � � � �R b p d u a
� � � � � ��
1 1 1 1 1 1, (9)
and � N
�1 is given by Eq. (7).
The term � a
�1 is introduced into Eq. (9) to take into ac-
count the extra (compared to CO2) phonon scattering. We
assume that the extra scattering is caused by the low-fre-
quency excitations related to the orientational subsystem
780 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
V.V. Sumarokov, P. Stachowiak, and A. Je�owski
of N2O. The relaxation rate � a
�1 of the acoustic phonons
can be written within the SPM [9] model as
� a c xT
x
c xT c xT� �
1
1 2
4
3
3
2
tanh ( ) . (10)
The first term describes the resonance phonon scattering
on the tunnel states of two-level systems. The other terms
concern scattering on low-energy soft quasiharmonic os-
cillations.
Applying the approximation procedure, we could esti-
mate the scattering intensities:
� R . . xT . xT� � �
�
1 5 4 34 47 10 5 77 1 06 10( )
� �
�
1 63 10 10 39 2 7 10
2
3 2 5 5. x T . /T . xT
x
exp( ) tanh
�43 3 22 104 4 3( ) .xT xT , (11)
� N x T� � �1 3 2 52 7 10. . (12)
Figure 1 shows the approximation curves (solid lines)
that provide the best description of the experimental re-
sults on the thermal conductivity of N2O and CO2. It is
seen that below 20 K the description is quite good.
The coefficients in the summands (which describe
phonon scattering on grain boundaries, stress fields of
dislocations, and in U-processes) of Eqs. (8) and (11) tes-
tify that in both CO2 and N2O solids indicate large size of
crystalline grains, low density of dislocations, and weak
phonon–phonon interaction. These might be reasons for
the good thermal conduction in these crystals at tempera-
tures near the maximum.
We assume that the differences in the low-temperature
thermal conductivity of these crystals are due to this
phonon scattering � a
�1, Eq. (10). If we add this contribu-
tion � a
�1 of Eq. (10) to the fitting procedure CO2, the re-
sulting curve will describe well the temperature depen-
dence of the thermal conductivity of N2O. Indeed, Fig. 2
illustrates the experimental temperature dependences of
the thermal conductivity of the CO2 and N2O crystals. It
also shows hypothetical curves 1 (dashed line) and 2 (solid
line). Curve 1 was obtained as follows. The thermal con-
ductivity ê was calculated from Eq. (1) for CO2 using the
relaxation times given in Eq. (8). In this procedure the
term � a
�1 of Eq. (10) with the coefficients ci obtained in
Eq. (11) by approximating the curve for N2O was intro-
duced into the term � R
�1 of Eq. (8) for CO2. It is seen that
hypothetical curve 1 provides a qualitatively adequate de-
scription of the thermal conductivity of N2O below the
maximum. Curve 1 can be transformed into curve 2 by
taking into account the difference of the parameters of the
U-processes in N2O. Hypothetical curve 2 describes qual-
itatively the temperature dependence of N2O. The expres-
sion � a
�1, Eq. (10), thus permits us to describe effectively
the extra (as compared to CO2) phonon scattering in the
N2O crystal. It is a very surprising result. The phonon
scattering on low-energy excitations, associated with the
frozen disordered orientations of N2O molecules, have a
glass-like character in the crystalline N2O.
The intensity of N-processes can be expressed in terms
of the characteristics of the crystal [21]. To estimate the
intensity of normal phonon scattering, we used for � N
�1 the
expression [19–21]
�
�
�N
D
b
T� �
�
��
�
�
��
�
�
�
�
�
1
2 3
�
, b
R
M
k
t
�
�
��
�
�
��
16
735 3
3 2
0
3 5
� � �
�
�
v
,
(13)
where R0 is the distance between nearest neighbors, M is
the molecular mass, � is the Grüneisen constant. Equation
(13) can be rewritten as
� N NA x T� �1 2 5 (14)
with
A .
V
N /
� �1 27 10 9
2
2 3 5
�
� �
, (15)
where � and V are the molar mass and volume, respec-
tively. According to Eq. (15), the intensity of normal pro-
cesses AN is determined by the physical parameters of the
substance. The intensities of obtained from experiment
The peculiarities of heat transfer in CO2 and N2O solids at low temperatures
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 781
1 10
2
1
10
1
10
2
10
3
10
4
CO2
N O2
T, K
Fig. 2. Thermal conductivity of CO2 and N2O solids. Experi-
ment: N2O (�,�), CO2 (�). Calculation: curves 1 and 2. The
explanation is in the text.
and calculated from Eq. (15) are given in Table 1. We see
quite a good agreement.
Table 1. The intensities of normal processes, resulting from thermal
conductivity experiment and calculations according to Eq. (15).
�,
10
–3
kg/mole
�
V,
10
–6
m
3
/mole
�, K
AN, 10
3
s
–1
K
–5
experiment calculation
CO2 44 2.13 25.796 151.8 1.62 1.86
N2O 44 2.16 27.018 141.0 2.70 2.69
The dependence on the Debye temperature in the coef-
ficient AN can be separated:
A A VN N
/* � �� �� �2 2 3 5� . (16)
Figure 3 illustrates the normal process intensities in loga-
rithmic coordinates (A A VN N
/* � �� �2 2 3 in reduced coor-
dinates) for CO2 and N2O as a function of the Debye tem-
perature. It also shows literature data for the hydrogens
[21,22] and Ne [23]. It is seen that the reduced intensity of
normal processes AN
* is inversely proportional to � 5 for
CO2, N2O and Ne crystals. This power law holds for the
hydrogens [21,22] to within a constant factor (2 2).
So, comparison between calculated and experimental
values of the intensity of normal phonon scattering in
CO2 and N2O shows that they are determined by the phy-
sical parameters of the substance.
Conclusions
Thermal conductivities of the CO2 and N2O crystals
have been measured in the temperature interval 1–40 K. A
technique has been developed that permitted us to grow
perfect samples of solid CO2 and N2O. It is found that
both CO2 and N2O have very high thermal conductivities
at the maxima but differ surprisingly widely from each
other. The experimental temperature dependence of the
thermal conductivity is described using the Callaway re-
laxation method within the Debye model. Analysis of the
experimental results shows that relatively large size of
crystalline grains, low density of dislocations and weak
phonon–phonon interaction might be reasons for the good
thermal conduction in these crystals at temperatures near
the thermal conductivity maximum. It is shown that the
normal process intensities in these cryocrystals are deter-
mined by the physical parameters of the substances. In
N2O solid there is a very large extra (as compared to CO2)
contribution to the phonon scattering at temperatures near
the thermal conductivity maximum. This scattering is de-
scribed within the model of soft potentials by the reso-
nance phonon scattering at the tunnel states and low-ener-
gy quasiharmonic oscillations.
The authors gratefully thank Yu.A. Freiman, B.Ya. Go-
rodilov and M.A. Strzhemechny for fruitful discussion.
1. Cryocrystals, B.I. Verkin and A.F. Prichot'ko (eds.), Nau-
kova Dumka, Kiev (1983) [in Russian].
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60 100 140 180
D2
H2
Ne
AN
*
10
–1
10
–2
10
–3
10
0
CO2
N O2
Fig. 3. Dependence of reduced intensity of normal processes for
molecular cryocrystals CO2 and N2O on Debye temperature.
The data for Ne, H2, D2 are taken from the of Refs. 21–23.
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