Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆
The isochoric thermal conductivity of solid C₆H₆ is described within a model in which the heat is transferred by phonons and above the phonon mobility edge by “diffusive” modes migrating randomly from site to site. The mobility edge ω₀ is found from the condition, that the phonon mean-free path re...
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Інститут фізики конденсованих систем НАН України
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Pursky, O.I. Konstantinov, V.A. 2017-06-14T11:21:13Z 2017-06-14T11:21:13Z 2006 Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ / O.I. Pursky, V.A. Konstantinov // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 709–717. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 66.70+f, 63.20.Ls DOI:10.5488/CMP.9.4.709 https://nasplib.isofts.kiev.ua/handle/123456789/121451 The isochoric thermal conductivity of solid C₆H₆ is described within a model in which the heat is transferred by phonons and above the phonon mobility edge by “diffusive” modes migrating randomly from site to site. The mobility edge ω₀ is found from the condition, that the phonon mean-free path restricted by the examined mechanisms of scattering cannot become smaller than half the wavelength. The contributions of phononphonon, one and two-phonon scattering to the total thermal resistance of solid C₆H₆ are calculated under the assumption of additive contribution of different scattering mechanisms. Significant deviations from the dependence Λ ∝1/T are explained by thermal conductivity approaching its lower limit. Iзохорна теплопровiднiсть твердого C₆H₆ описується в рамках моделi, в якiй тепло переноситься фононами, а вище вiд граничної рухливостi – “дифузними” модами, що мiгрують випадковим чином з вузла на вузол. Границя фононної рухливостi ω₀ знаходиться iз умови, що довжина вiльного пробiгу фононiв, котра визначається розглянутими механiзмами розсiяння фононiв, не може стати меншою половини довжини хвилi. Внески фонон-фононного, одно та дво-фононного розсiяння в повний тепловий опiр твердого C₆H₆ розраховано в припущеннi адитивностi внескiв рiзних механiзмiв фононного розсiяння. Значнi вiдхилення вiд залежностi Λ ∝1/T пояснюються наближенням теплопровiдностi до її нижньої границi. This study was supported by the Ukrainian Ministry of Education and Science, Project No. 121–97. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ Фононне розсiяння та перенесення тепла “дифузними” модами в твердому C₆H₆ Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ |
| spellingShingle |
Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ Pursky, O.I. Konstantinov, V.A. |
| title_short |
Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ |
| title_full |
Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ |
| title_fullStr |
Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ |
| title_full_unstemmed |
Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ |
| title_sort |
phonon scattering and heat transfer by “diffusive” modes in solid c₆h₆ |
| author |
Pursky, O.I. Konstantinov, V.A. |
| author_facet |
Pursky, O.I. Konstantinov, V.A. |
| publishDate |
2006 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Фононне розсiяння та перенесення тепла “дифузними” модами в твердому C₆H₆ |
| description |
The isochoric thermal conductivity of solid C₆H₆ is described within a model in which the heat is transferred
by phonons and above the phonon mobility edge by “diffusive” modes migrating randomly from site to site.
The mobility edge ω₀ is found from the condition, that the phonon mean-free path restricted by the examined
mechanisms of scattering cannot become smaller than half the wavelength. The contributions of phononphonon,
one and two-phonon scattering to the total thermal resistance of solid C₆H₆ are calculated under
the assumption of additive contribution of different scattering mechanisms. Significant deviations from the
dependence Λ ∝1/T are explained by thermal conductivity approaching its lower limit.
Iзохорна теплопровiднiсть твердого C₆H₆ описується в рамках моделi, в якiй тепло переноситься
фононами, а вище вiд граничної рухливостi – “дифузними” модами, що мiгрують випадковим чином з вузла на вузол. Границя фононної рухливостi ω₀ знаходиться iз умови, що довжина вiльного
пробiгу фононiв, котра визначається розглянутими механiзмами розсiяння фононiв, не може стати
меншою половини довжини хвилi. Внески фонон-фононного, одно та дво-фононного розсiяння в
повний тепловий опiр твердого C₆H₆ розраховано в припущеннi адитивностi внескiв рiзних механiзмiв фононного розсiяння. Значнi вiдхилення вiд залежностi Λ ∝1/T пояснюються наближенням
теплопровiдностi до її нижньої границi.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121451 |
| citation_txt |
Phonon scattering and heat transfer by “diffusive” modes in solid C₆H₆ / O.I. Pursky, V.A. Konstantinov // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 709–717. — Бібліогр.: 25 назв. — англ. |
| work_keys_str_mv |
AT purskyoi phononscatteringandheattransferbydiffusivemodesinsolidc6h6 AT konstantinovva phononscatteringandheattransferbydiffusivemodesinsolidc6h6 AT purskyoi fononnerozsiânnâtaperenesennâtepladifuznimimodamivtverdomuc6h6 AT konstantinovva fononnerozsiânnâtaperenesennâtepladifuznimimodamivtverdomuc6h6 |
| first_indexed |
2025-11-25T23:28:34Z |
| last_indexed |
2025-11-25T23:28:34Z |
| _version_ |
1850583696593125376 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 4(48), pp. 709–717
Phonon scattering and heat transfer by “diffusive”
modes in solid C 6H6
∗
O.I.Pursky†1, V.A.Konstantinov2
1 Taras Shevchenko National University of Kyiv, Faculty of Physics, 6 Glushkova Ave., Kyiv 03022, Ukraine
2 Institute for Low Temperature Physics and Engineering of the National Academy of Science of Ukraine,
47 Lenin Ave., Kharkov 61103, Ukraine
Received July 15, 2005, in final form November 19, 2005
The isochoric thermal conductivity of solid C6H6 is described within a model in which the heat is transferred
by phonons and above the phonon mobility edge by “diffusive” modes migrating randomly from site to site.
The mobility edge ω0 is found from the condition, that the phonon mean-free path restricted by the examined
mechanisms of scattering cannot become smaller than half the wavelength. The contributions of phonon-
phonon, one and two-phonon scattering to the total thermal resistance of solid C6H6 are calculated under
the assumption of additive contribution of different scattering mechanisms. Significant deviations from the
dependence Λ ∝1/T are explained by thermal conductivity approaching its lower limit.
Key words: molecular crystals, heat transfer, phonons, librons
PACS: 66.70+f, 63.20.Ls
1. Introduction
At low temperatures, well below the Debye temperature of solids (ΘD), heat transport in sim-
ple molecular crystals is adequately described by basic theoretical models [1]. However, in high
temperature region (T> ΘD) experimental and theoretical understanding of thermal conductivity
of molecular crystals remains incomplete. For example, at T> ΘD, the classical theoretical mod-
els of heat transfer [1] predicted that thermal conductivity should be inversely proportional to
temperature (Λ ∝1/T ), whereas experimental investigations of thermal conductivity of molecular
crystals show considerable deviations from the above dependence [2]. Essentially all the basic con-
cepts of heat transfer were created based on the studies of the simplest crystalline structure, i.e.,
atomic crystals. Therefore, the features typical of molecular crystals were not taken into account
therein. One of the features that can affect the temperature dependence of thermal conductivity
is the translation-rotation coupling. In the present study we wish to emphasize that thermal con-
ductivity approaching its lower limit turns out to be another factor capable of determining the
temperature dependence of thermal conductivity.
The purpose of this paper was to study the basic features of heat transfer in molecular crystals at
T> ΘD. Our previous measurements have revealed a considerable deviation of the isochoric thermal
conductivity of solid C6H6 from the dependence 1/T [2]. The effect was explained qualitatively
but we did not provide a quantitative interpretation. The present work continues investigating the
observed effect on the solid C6H6. In the current study we analysed the temperature dependence of
isochoric thermal conductivity of C6H6 using the model, which assumes that the heat is transferred
by low-frequency phonons and above the mobility edge by “diffusive” modes, and taking into
account phonon-phonon and phonon-rotation scattering.
∗The paper submitted to the Proceedings of the conference “Statistical physics 2005: Modern problems and new
applications” (August 28–30, 2005, Lviv, Ukraine).
†E-mail: pursky o@ukr.net
c© O.I.Pursky, V.A.Konstantinov 709
O.I.Pursky, V.A.Konstantinov
2. The object
Solid benzene under the pressure of its own saturated vapor has only one crystallographic
modification: it has the orthorhombic spatial symmetry Pbca (D15
2h) with four molecules per unit
cell [3,4]. Benzene melts at 278.5 K and the melting-caused change in the entropy is ∆Sf/R =4.22
[5], which is much higher that Timmermans criterion for orientationally disordered phases. Here R
is universal gas constant. The high-temperature magnitude of the Debye temperature of C6H6 is
120 K [6]. In the interval 90–120 K the second NMR moment of C6H6 drops considerably as a result
of the molecule reorientations in the plane of the ring around the sixfold axis [7]. The activation
energy of reorientational motion estimated from the spin-lattice relaxation time is 0.88 kJ/mole.
The frequency of molecular reorientations at 85 K is 104 s−1. On a further rise of the temperature
it increases considerably, reaching 1011s−1 near melting temperature. The basic frequency of the
benzene molecule oscillations about the sixfold axis at 273 K is 1.05 ·1012 s−1 [8].
At present, the thermal conductivity of solid C6H6 has been experimentally studied in the
temperature range from 80 K to the melting temperature [2]. The isobaric thermal conductivity of
solid C6H6 was also measured under the pressure above 100 MPa [9].
3. Model
The calculation was performed based on the Debye’s expression for thermal conductivity [1]
using the approach of Roufosse and Klemens [10] who used the idea of a lower limit for the phonon
mean free path:
Λ =
kB
2π2υ2
ωD
∫
0
l(ω)ω2dω, (1)
where υ is the polarization-averaged sound velocity, (ωD = (6π2)1/3υ/a) is the Debye frequency,
a3 is the volume per atom (molecule), and l (ω) is the phonon mean-free path.
Heat transfer in molecular crystals at T > ΘD is determined mainly by phonon-phonon and
phonon-rotational interactions. We assumed that the phonon-rotation relaxation time is determined
by the one and two-phonon scattering processes [11]. Then, l (ω) – the combined phonon mean-free
path determined by all of the examined mechanisms of scattering can be written as:
lΣ (ω) =
(
∑
i
li (ω)
−1
)−1
. (2)
To explain the behavior of the thermal conductivity in solid CH4 and CD4, the authors of [11]
used the analogy between molecular and spin systems. In a number of magnetic crystals the thermal
conductivity was observed to increase above the magnetic phase transition. The reason for these
anomalies is the scattering of phonons by critical fluctuations of the short-range magnetic order
above the Neel point. In molecular crystals an increase of the isochoric thermal conductivity with
increasing temperature is due to the weakening of phonon scattering by fluctuations of the short-
range orientational order. By analogy, using the equations for one and two-phonon relaxation times
[12], the phonon mean free path of each of the examined scattering mechanisms can be expressed as:
lu (ω) = υ
/
ATω2, (3)
lI (ω) = ρυ5
/
B2ΛrotTω2, (4)
lII (ω) = πρ2υ8
/
C2kBCrotT
2ω4, (5)
A =
18π3
√
2
kBγ2
ma2ω3
D
, (6)
where the Grüneisen parameter γ = − (∂ ln ΘD/∂ lnV )T , lu (ω) is the phonon mean-free path de-
termined by U -processes, lI (ω) and lII (ω) are the phonon mean-free paths for one and two-phonon
710
Heat transfer in solid C6H6
scattering, respectively, m is the average atomic (molecular) weight, B and C are the constants of
non-central intermolecular interactions, Λrot is thermal conductivity of the orientational subsys-
tem, Crot is rotational heat capacity per unit volume. In the first approximation: B = C2 [11]. The
coefficient B can be found from the pressure dependence of melting temperature:
B = −
(
1
χT
)
∂ (lnTf )
∂P
, (7)
where χT is the isothermal compressibility, Tf is temperature of the phase transition, P is pressure.
Thermal conductivity Λrot can be found from the well-known gas-kinetic expression:
Λrot =
1
3
Crota
2τ−1, (8)
where τ is the characteristic time of orientational excitation transfer from one lattice site to an-
other. This time was estimated from the relation τ ≈ ~/∆E [13], where non-central part ∆E of
the intermolecular interaction was calculated based on the expression given in [14] relating it to
the phase transition temperature of C6H6.
By substituting (3)–(5) in (2), the combined phonon mean-free path can be expressed as follows:
lΣ (ω) =
(
ATω2
υ
+
B2ΛrotTω2
ρυ5
+
C2kBCrotT
2ω4
πρ2υ8
)−1
. (9)
Expression (9) is not applicable if l (ω) becomes of the order of or smaller than half the phonon
wavelength: λ/2 = πυ/ω. A similar situation was considered previously for the case of U -processes
alone [15]. Let us assume that in the general case:
l (ω) =
{
lΣ (ω) , 0 6 ω 6 ω0,
απυ/ω = α λ/2, ω0 < ω 6 ωD,
(10)
where α is the numerical coefficient of the order of unity. The frequency ω0 can be found from the
condition:
(
ATω2
0
υ
+
B2ΛrotTω2
0
ρυ5
+
C2kBCrotT
2ω4
0
πρ2υ8
)−1
=
απυ
ω0
. (11)
It is equal to
ω0 = −
u
(
−η +
√
u3 + η2
)
1/3
+
(
−η +
√
u3 + η2
)
1/3
, (12)
where the parameters u and η are:
u =
πρ2υ7
3C2kBCrotT
(
A +
B2Λrot
ρυ4
)
; η = −
ρ2υ7
2αC2kBCrotT 2
. (13)
The condition (11) is the well-known Ioffe-Regel criterion which implies localization. Therefore,
we can assume that the excitations whose frequencies are above the phonon mobility edge ω0
are “localized” or “diffusive”. Since completely localized modes do not contribute to the thermal
conductivity, we supposed that the localization is weak and the excitations are capable of hopping
from site to site diffusively, as was suggested by Cahill and Pohl [16].
If ω0 > ωD the mean free path of all modes exceeds λ/2 and the thermal conductivity is
determined exceptionally by the processes of phonon scattering. At ω0 6 ωD the integral of thermal
conductivity (1) is subdivided into two parts describing the contributions to the heat transfer from
the low-frequency phonons and high-frequency “diffusive” modes:
Λ = Λph + Λdif . (14)
711
O.I.Pursky, V.A.Konstantinov
In the high-temperature limit (T > ΘD) these contributions are:
Λph =
kB
2π2υ2
ω0
∫
0
ω dω
C2kBCrotT
2ω3
πρ2υ8
+
ATω
υ
+
B2ΛrotTω
ρυ5
, (15)
Λdif =
αkB
4πυ
(
ω2
D − ω2
0
)
. (16)
In the case of orientationally ordered phases equation (15) gives the well-known dependence
Λ ∝1/T at ω0 > ωD.
Λph =
kBωD
2π2υAT
. (17)
4. Results and discussion
As was mentioned above, at temperature close to or above the Debye temperature T > ΘD
the thermal conductivity of pure crystals should be inversely proportional to temperature [1]. To
obey the law 1/T, the volume of the crystals should remain invariable, because the modes would
otherwise change and so would the temperature dependence of the thermal conductivity [9,17].
Nevertheless, isochoric studies of thermal conductivity of molecular crystals [2,15,18] show a
considerable deviation from this law. These discrepancies may be due to the thermal conductivity
approaching its lower limit. The concept of the lower limit of thermal conductivity is based on the
following: the mean free paths of the oscillatory modes participating in heat transfer are essentially
limited but it cannot become smaller than half the phonon wavelength λ/2, and the site-to-site
heat transport proceeds as a diffusive process [10]. In this case the lower limit of the lattice thermal
conductivity Λmin can be written as [16]:
Λmin =
(π
6
)
1
3
kBn
2
3
∑
i
υi
(
T
Θi
)2
Θi/T
∫
0
x3ex
(ex − 1)
2 dx
. (18)
The summation is over three (one longitudinal and two transverse) sound modes with the
sound velocities υi. Θi is the Debye cut-off frequency for each polarization in Kelvin’s: (Θi =
υi (~/kB)
(
6π2n
)1/3
), n = 1/a3 is the number of atoms per unit volume.
Figure 1. Calculated temperature dependence of the isobaric and isochoric
(Vmol=70.5 cm3/mole) sound velocities of solid C6H6. υ
P
, υ
P
` , υ
P
t and υ
V
, υ
V
` , υ
V
t are
mean, longitudinal, and transversal phonon velocities for isobaric and isochoric conditions,
respectively.
The calculated values of Λmin were as a rule considerably smaller than the experimental ones
[2,15,18]. The most obvious reason for this difference is that the site to site transfer of the rota-
tional energy was not taken into account. In molecular crystals the heat is transferred by mixed
712
Heat transfer in solid C6H6
translation-rotation modes, whose heat capacity is saturated in proportion to the total molecular
degree of freedom. Taking into account these features of molecular crystals, the lower limit of the
thermal conductivity can be represented as [15]:
Λ∗
min =
1
2
(π
6
)
1
3
(
1 +
z
3
)
kBn
2
3 (υ` + 2υt) , (19)
where υ` and υt are the longitudinal and transversal sound velocities, respectively, z is the number
of rotational degree of freedom.
To our knowledge, no experimental data are available on the sound velocity of solid C6H6. In
this respect, it was calculated using the method described in [19]. The necessary data were taken
from [20,21]. Figure 1 shows the calculated sound velocity of solid C6H6. The isochoric speed of
sound corresponds to the molar volume Vmol=70.5 cm3/mole.
The heat capacity of molecular crystal can be written as a sum of contributions from the
translational Ctr and rotational Crot subsystems and a contribution due to the intramolecular
vibrations of the atoms Cin:
C
V
= Ctr + Crot + Cin . (20)
To separate the partial contributions to the heat capacity, we used the method described in [22].
The heat capacity at constant volume Cv is difficult to measure. In practice, it is recalculated,
when data on thermal expansion β [20] and Grüneisen parameter γ [6] are available, from the
values of the heat capacity at atmospheric pressure Cp [23] by using the known thermodynamic
relation:
C
V
=
Cp
(1 + γβT )
, (21)
Ctr was calculated in the Debye approximation using the characteristic temperature ΘD =120 K
obtained from the expression
ΘD = υ (~/kB)
(
6π2n
)
1
3
and is close to 3R. Cin was calculated in the Einstein approximation using the intramolecular vibra-
tional frequencies νi [6]. Crot was determined as Crot = Cv −Ctr−Cin. Our results for components
of the heat capacity are shown in figure 2. The heat capacity of librational subsystem Crot con-
stantly increases with the temperature increase, and at premelting temperatures approaching the
values that are considerably larger than 3/2R, which is characteristic of a free three-dimensional
rotator.
Figure 2. Temperature dependence of the heat capacity contributions in solid C6H6: Cp show
the data [23] on the heat capacity at constant pressure, while Cin, Ctr, and Crot correspond
to the calculated curves for the heat capacity at constant molar volume Vmol=70.5 cm3/mole
and for the intramolecular, translational, and rotational components of the heat capacity Cv,
respectively.
To correctly compare the experimental results of thermal conductivity with the theory it is
necessary to use data at constant density to exclude the effect of thermal expansion. The iso-
choric thermal conductivity of C6H6 (Vmol=70.5 cm3/mole) according to [2] is shown in figure 3
713
O.I.Pursky, V.A.Konstantinov
(squares). The behavior of isochoric thermal conductivity Λv of solid benzene is inconsistent with
the theoretical prediction. Satisfactory agreement with the classical law 1/T is observed only below
180 K (Λv ∼T−0.8). At T >180 K, Λv is practically constant and even starts to grow slightly at
premelting temperatures.
Figure 3. Isochoric thermal conductivity Λv of solid C6H6 (squares) [2]. The solid line is the
fitting curve for isochoric thermal conductivity. Λph and Λdif are contributions of phonons and
“diffusive” modes to heat transfer, respectively. The lower limits of the thermal conductivity
Λmin and Λ∗
min calculated according to equations (18), (19).
The computer fitting of the thermal conductivity using equations (12)–(16) was performed
using the least square method, varying the coefficients A,B,C, and α. The parameters of the
Debye model for thermal conductivity used in the fitting (a, υ), and the fitted values A,B,C, and
α are listed in table 1.
Table 1. Parameters of the Debye model of thermal conductivity used in the fitting, and other
quantities which were used for calculation.
Vmol, cm3/mole a, 10−8cm υ, m/s γ α A, 10−17s/K B C
70.5 4.9 2275 3.3 4.4 7.6 6.8 2.6
The fitting results for isochoric thermal conductivity are shown in figure 3 (solid line). The same
figure shows the contributions (dot-and-dash lines) to the heat transfer from the low-frequency
phonons Λph and the high-frequency “diffusive” modes Λdif (calculated by equations (15), (16)).
The dotted line shows the lower limit of thermal conductivity Λ∗
min (19) calculated taking into
account the possibility of site to site rotational energy transfer. The dashed line in the lower part
of the figure is the lower limit of thermal conductivity Λmin (18) calculated according to Cahill and
Pohl within the framework of the Einstein model for the diffusive transfer of heat directly from
atom to atom [16].
It is seen (figure 3) that the “diffusive” bahavior of oscillatory modes appears prior to 70 K. As
temperature rises, the amount of heat transferred by the “diffusive” modes increases, and at 110 K
it becomes equal to the heat transferred by the low-frequency phonons. Above 150 K most of the
heat is transported by the “diffusive” modes. As demonstrated in figure 3, our theoretical findings
for thermal conductivity agree well with experimental data cited in [2]. The discussion of the lower
limit of thermal conductivity of molecular crystals brings up inevitable question: should the site-
to-site transport of the rotational energy of the molecules be taken into account? The minimum
values of the experimental thermal conductivity Λv is 1.2 times higher than Λ∗
min calculated by
equation (19), and 2.5 times higher than Λmin calculated by equation (18). The above correlation
between the Λmin and Λ∗
min suggests a positive answer.
In high temperature region of existence of molecular crystals the contribution of librons Λrot
to the heat transfer is assumed to be relatively small [2,15,24]. The calculation of Λrot according
to (8) gives the values 1.42·10−3W/m·K at T=80 K and 1.5·10−2 W/m·K at T=278 K. These
714
Heat transfer in solid C6H6
results prove the above assumption. At the same time, the role of librons in scattering processes is
important [2,24,25].
To answer questions as to basic peculiarities of phonon-rotational coupling in the solid C6H6,
we have undertaken to separate the phonon-phonon and phonon-rotational contributions to the
total thermal resistance. A number of molecular crystals have several solid phases, which substan-
tially differ in the character of orientational ordering. If the non-central forces are strong and the
temperature is low, there is a long-range orientational order in the location of molecular axes. The
molecules perform small vibrations around the selected axes (librations), so that the motion of
the neighbouring molecules is correlated and the collective orientation excitations (librons) propa-
gate in the crystal. In the first approximation, the translation-orientation interaction in molecular
crystals leads to an additional contribution to the thermal resistance W=1/Λ [17].
Figure 4. Contributions of the phonon-phonon scattering Wpp, and one-phonon Wpr1 mecha-
nisms of scattering to the total thermal resistance of solid C6H6. Square symbols indicate total
thermal resistance W=1/Λph. The solid line shows the sum of thermal resistances Wpp, Wpr1,
and Wpr2.
If the noncentral forces are relatively weak, and the temperature is high enough, the molecules
can have a considerable orientational freedom. In this case a number of orientations are accessible to
the molecule, which can pass from one orientation to another. In individual cases the limit of such
reorientational motion can be a continuous rotation. The additional phonon scattering (compared
to the phonon-phonon one) may originate in molecular crystals due to the collective rotational
motion of molecules. As a rule, the unfreezing of molecular rotation is accompanied by an increase
of the isochoric thermal conductivity due to the weakening of phonon-rotational scattering [2,15].
We assume that the contributions of different scattering mechanisms to the thermal resistance are
additive [1]:
∑
i
Wi = Wpp + Wpr1 + Wpr2, (22)
where Wpp is the phonon-phonon thermal resistance, Wpr1 and Wpr2 are thermal resistances
determined by one and two-phonon mechanisms of phonon-rotational scattering, respectively. Using
(1) and (3)–(5) we have:
Wpp = 2π2υAT
kB
ω0
∫
0
dω
−1
, (23)
Wpr1 = 2π2B2ΛrotT
kBυ3ρ
ω0
∫
0
dω
−1
, (24)
Wpr2 = 2πC2T 2Crot
υ6ρ2
∣
∣
∣
∣
∣
∣
ω0
∫
0
dω
ω2
∣
∣
∣
∣
∣
∣
−1
. (25)
715
O.I.Pursky, V.A.Konstantinov
Figure 4 shows the calculated results for thermal resistance C6H6. The total thermal resistance
W=1/Λph is shown with black squares. The solid curve is the sum
∑
i
Wi of the thermal resistances,
calculated by equations (23)–(25). The phonon-phonon component of the thermal resistance Wpp
increases with temperature. As temperature rises the thermal resistance Wpr1 due to the one-
phonon scattering at rotational excitations of molecules also increases and becomes substantial
after 120 K. The two-phonon component Wpr2 of the total thermal resistance is practically zero
(it does not exceed 5·10−5m·K/W). From the aforesaid, it is clear that phonon-rotation scattering
cannot be a reason for an increase of isochoric thermal conductivity of C6H6.
5. Conclusions
It is shown that the isochoric thermal conductivity of C6H6 can be described in a model, where
the heat is transferred by phonons and above the phonon mobility edge by “diffusive” modes
migrating randomly from site to site and taking into account the transfer of rotational energy
between lattice sites. The total thermal resistance in molecular crystals is determined by phonon-
phonon and phonon-rotation scattering mechanisms. In turn, the phonon-rotation relaxation time
is determined by the one and two-phonon scattering. The contributions of phonon-phonon, one and
two-phonon scattering to the total thermal resistance of solid C6H6 are calculated in supposition
of additive contribution of different scattering mechanisms.
Based on these studies, it seems justified to conclude that the main reason for the essential
deviations of the isochoric thermal conductivity of C6H6 from the dependence 1/T is the additional
heat transfer by “diffusive” modes. The latter is caused by the thermal conductivity approaching
its lower limit.
Acknowledgements
This study was supported by the Ukrainian Ministry of Education and Science,
Project No. 121–97.
References
1. Berman R. Thermal Conduction in Solids. Clarendon Press, Oxford, 1976.
2. Pursky O.I., Zholonko N.N., Konstantinov V.A., Fiz. Nizk. Temp., 29, 1021 (Low Temp. Phys., 2003,
29, 771).
3. Cox E.G., Cruickshank D.W.J., Smith J.A.S., Proc. R. Soc., 1958, 247, 1.
4. Bacon G.E., Curry N.A., Wilson S.A., Proc. R. Soc., 1964, 279, 98.
5. Ubbelohde A.R. Melting And Crystal Structure. Clarendon Press, Oxford, 1965.
6. Kitaigorodskii A.I. Molecular Crystals. Nauka, Moscow, 1971 (in Russian).
7. Andrew E.R., Eades R.G., Proc. Roy. Soc., 1953, A218, 537.
8. Parsonage N.G., Staveley L.A.K. Disorder in Crystals. Clarendon Press, Oxford, 1978.
9. Ross R.G., Andersson P., Bäckström G., Mol. Phys., 1979, 38, 377.
10. Roufosse M.C., Klemens P.G., J. Geophys. Res., 1994, 79, 703.
11. Krupskii I.N., Koloskova L.A., Manzhelii V.G., J. Low Temp. Phys., 1974, 14, 403.
12. Kawasaki K., Prog. Theor. Phys., 1963, 29, 801.
13. Mori H., Kawasaki K., Progr. Theor. Phys., 1962, 27, 529.
14. James H.M., Keenan T.A., J. Chem. Phys., 1959, 31, 12.
15. Konstantinov V.A., Fiz. Nizk. Temp., 2003, 29, 567 (Low Temp. Phys., 2003, 29, 422).
16. Cahill D.G., Watson S.K., Pohl R.O., Phys. Rev., 1992, B46, 6131.
17. Manzhelii V.G., Freiman Y.A. Physics of cryocrystals, Woodbury, AIP Press, New York, 1997.
18. Pursky O.I., Zholonko N.N., Konstantinov V.A., Fiz. Nizk. Temp., 2000, 26, 380 (Low Temp. Phys.,
2000, 26, 278).
19. Pursky O.I., Zholonko N.N., Fiz. Tverd. Tela, 2004, 46, 1949 (Phys. Solid State, 2004, 46, 2015).
20. Pursky O.I., Zholonko N.N., Ukr. J. Phys., 2004, 49, 1105.
716
Heat transfer in solid C6H6
21. Heseltine C.W., Elliot D.W., Wilson O.B., J. Chem. Phys., 1964, 40, 2584.
22. Isakina A.P., Prokhvatilov A.I., Fiz. Nizk. Temp., 1993, 19, 201 (Low Temp. Phys., 1993, 19, 142).
23. Kikoin I.K. Tables of Physical Data. Atomizdat, Moskow, 1976 (in Russian).
24. Konstantinov V.A., Manzhelii V.G., Smirnov S.A., Phys. St. Sol. (b), 1991, B163, 369.
25. Konstantinov V.A., Manzhelii V.G., Reviakin V.P., Smirnov S.A., Physica B, 1999 262, 421.
Фононне розсiяння та перенесення тепла “ дифузними”
модами в твердому C6H6
О.I.Пурський1, В.А.Константiнов2
1 Нацiональний унiверситет iм. Т.Шевченка, фiзичний факультет, вул. Глушкова 6, 03022 Київ, Україна
2 Фiзико-технiчний iнститут низьких температур НАН України, 61103 Харкiв, пр. Ленiна, 47
Отримано 15 липня 2005 р., в остаточному виглядi – 19 листопада 2005 р.
Iзохорна теплопровiднiсть твердого C6H6 описується в рамках моделi, в якiй тепло переноситься
фононами, а вище вiд граничної рухливостi – “дифузними” модами, що мiгрують випадковим чи-
ном з вузла на вузол. Границя фононної рухливостi ω0 знаходиться iз умови, що довжина вiльного
пробiгу фононiв, котра визначається розглянутими механiзмами розсiяння фононiв, не може стати
меншою половини довжини хвилi. Внески фонон-фононного, одно та дво-фононного розсiяння в
повний тепловий опiр твердого C6H6 розраховано в припущеннi адитивностi внескiв рiзних меха-
нiзмiв фононного розсiяння. Значнi вiдхилення вiд залежностi Λ ∝1/T пояснюються наближенням
теплопровiдностi до її нижньої границi.
Ключовi слова: молекулярнi кристали, перенос тепла, фонони, лiброни
PACS: 66.70+f, 63.20.Ls
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