The low-temperature heat capacity of fullerite C₆₀
The heat capacity at constant pressure of fullerite C₆₀ has been investigated using an adiabatic calorimeter in a temperature range from 1.2 to 120 K. Our results and literature data have been analyzed in a temperature interval from 0.2 to 300 K. The contributions of the intramolecular and lattice...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1279652018-01-01T03:03:35Z The low-temperature heat capacity of fullerite C₆₀ Bagatskii, M.I. Sumarokov, V.V. Barabashko, M.S. Dolbin, A.V. Sundqvist, B. Наноструктуры при низких температурах The heat capacity at constant pressure of fullerite C₆₀ has been investigated using an adiabatic calorimeter in a temperature range from 1.2 to 120 K. Our results and literature data have been analyzed in a temperature interval from 0.2 to 300 K. The contributions of the intramolecular and lattice vibrations into the heat capacity of C₆₀ have been separated. The contribution of the intramolecular vibration becomes significant above 50 K. Below 2.3 K the experimental temperature dependence of the heat capacity of C60 is described by the linear and cubic terms. The limiting Debye temperature at T → 0 K has been estimated (Θ0 = 84.4 K). In the interval from 1.2 to 30 K the experimental curve of the heat capacity of C₆₀ describes the contributions of rotational tunnel levels, translational vibrations (in the Debye model with Θ0 = 84.4 K), and librations (in the Einstein model with ΘE,lib = 32.5 K). It is shown that the experimental temperature dependences of heat capacity and thermal expansion are proportional in the region from 5 to 60 K. The contribution of the cooperative processes of orientational disordering becomes appreciable above 180 K. In the high-temperature phase the lattice heat capacity at constant volume is close to 4.5 R, which corresponds to the high-temperature limit of translational vibrations (3 R) and the near-free rotational motion of C60 molecules (1.5 R). 2015 Article The low-temperature heat capacity of fullerite C₆₀ / М.I. Bagatskii, V.V. Sumarokov, M.S. Barabashko, A.V. Dolbin, B. Sundqvist// Физика низких температур. — 2015. — Т. 41, № 8. — С. 812–819. — Бібліогр.: 54 назв. — англ. 0132-6414 PACS: 65.40.Ba, 65.80.–g, 81.05.ub http://dspace.nbuv.gov.ua/handle/123456789/127965 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| topic |
Наноструктуры при низких температурах Наноструктуры при низких температурах |
| spellingShingle |
Наноструктуры при низких температурах Наноструктуры при низких температурах Bagatskii, M.I. Sumarokov, V.V. Barabashko, M.S. Dolbin, A.V. Sundqvist, B. The low-temperature heat capacity of fullerite C₆₀ Физика низких температур |
| description |
The heat capacity at constant pressure of fullerite C₆₀ has been investigated using an adiabatic calorimeter in
a temperature range from 1.2 to 120 K. Our results and literature data have been analyzed in a temperature interval
from 0.2 to 300 K. The contributions of the intramolecular and lattice vibrations into the heat capacity of C₆₀
have been separated. The contribution of the intramolecular vibration becomes significant above 50 K. Below
2.3 K the experimental temperature dependence of the heat capacity of C60 is described by the linear and cubic
terms. The limiting Debye temperature at T → 0 K has been estimated (Θ0 = 84.4 K). In the interval from 1.2 to
30 K the experimental curve of the heat capacity of C₆₀ describes the contributions of rotational tunnel levels,
translational vibrations (in the Debye model with Θ0 = 84.4 K), and librations (in the Einstein model with
ΘE,lib = 32.5 K). It is shown that the experimental temperature dependences of heat capacity and thermal expansion
are proportional in the region from 5 to 60 K. The contribution of the cooperative processes of orientational
disordering becomes appreciable above 180 K. In the high-temperature phase the lattice heat capacity at constant
volume is close to 4.5 R, which corresponds to the high-temperature limit of translational vibrations (3 R) and
the near-free rotational motion of C60 molecules (1.5 R). |
| format |
Article |
| author |
Bagatskii, M.I. Sumarokov, V.V. Barabashko, M.S. Dolbin, A.V. Sundqvist, B. |
| author_facet |
Bagatskii, M.I. Sumarokov, V.V. Barabashko, M.S. Dolbin, A.V. Sundqvist, B. |
| author_sort |
Bagatskii, M.I. |
| title |
The low-temperature heat capacity of fullerite C₆₀ |
| title_short |
The low-temperature heat capacity of fullerite C₆₀ |
| title_full |
The low-temperature heat capacity of fullerite C₆₀ |
| title_fullStr |
The low-temperature heat capacity of fullerite C₆₀ |
| title_full_unstemmed |
The low-temperature heat capacity of fullerite C₆₀ |
| title_sort |
low-temperature heat capacity of fullerite c₆₀ |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2015 |
| topic_facet |
Наноструктуры при низких температурах |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/127965 |
| citation_txt |
The low-temperature heat capacity of fullerite C₆₀ / М.I. Bagatskii, V.V. Sumarokov, M.S. Barabashko, A.V. Dolbin, B. Sundqvist// Физика низких температур. — 2015. — Т. 41, № 8. — С. 812–819. — Бібліогр.: 54 назв. — англ. |
| series |
Физика низких температур |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8, pp. 812–819
The low-temperature heat capacity of fullerite C60
М.I. Bagatskii, V.V. Sumarokov, M.S. Barabashko, and A.V. Dolbin
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: bagatskii@ilt.kharkov.ua
B. Sundqvist
Department of Physics, Umea University, SE-901 87 Umea, Sweden
Received April 8, 2015, published online June 25, 2015
The heat capacity at constant pressure of fullerite C60 has been investigated using an adiabatic calorimeter in
a temperature range from 1.2 to 120 K. Our results and literature data have been analyzed in a temperature inter-
val from 0.2 to 300 K. The contributions of the intramolecular and lattice vibrations into the heat capacity of C60
have been separated. The contribution of the intramolecular vibration becomes significant above 50 K. Below
2.3 K the experimental temperature dependence of the heat capacity of C60 is described by the linear and cubic
terms. The limiting Debye temperature at T → 0 K has been estimated (Θ0 = 84.4 K). In the interval from 1.2 to
30 K the experimental curve of the heat capacity of C60 describes the contributions of rotational tunnel levels,
translational vibrations (in the Debye model with Θ0 = 84.4 K), and librations (in the Einstein model with
ΘE,lib = 32.5 K). It is shown that the experimental temperature dependences of heat capacity and thermal expan-
sion are proportional in the region from 5 to 60 K. The contribution of the cooperative processes of orientational
disordering becomes appreciable above 180 K. In the high-temperature phase the lattice heat capacity at constant
volume is close to 4.5 R, which corresponds to the high-temperature limit of translational vibrations (3 R) and
the near-free rotational motion of C60 molecules (1.5 R).
PACS: 65.40.Ba Heat capacity;
65.80.–g Thermal properties of small particles, nanocrystals, nanotubes and other related systems;
81.05.ub Fullerenes and related materials.
Keywords: heat capacity, fullerite C60, lattice dynamics.
Introduction
Since the discovery of the fullerite molecule C60 [1],
the low-temperature physical properties of fullerite C60
have been investigated by various methods: inelastic neu-
tron scattering [2,3], infrared and Raman spectroscopy [4,5],
x-ray [3,6,7], neutron [8] and electron [9,10] diffraction,
NMR [11], dilatometry [12–15] and calorimetry [16–25].
It has found that fullerite C60 is a molecular crystal in
which the molecules are bonded by the van der Waals
forces and its physical properties are largely determined by
the dynamics of the rotational motion of the C60 mole-
cules. At Tc ≈ 260 K fullerite undergoes an orientational
phase transition from a high-temperature face-centered
cubic (FCC) lattice to a low-temperature simple cubic (SC)
one [7,8]. The high-temperature phase has no long-range
orientational order and the rotational motion of the mo-
lecules is slightly hindered. In the low-temperature phase
the centers of gravity of the molecules remain in FCC sites
and molecules form four SC sublattices having different
orientations of the axes of three-fold symmetry along the
body diagonals of the cube (<111> directions). In the orien-
tationally ordered phase by the rotation of the molecules
about the <111> axes the molecules can be in six potential
wells with global and local minima, which correspond to
the pentagonal (p) and hexagonal (h) configurations, res-
pectively. In the p-configuration one of the five-fold axes
of the molecule is directed toward the middle of one of the
double bonds of the neighboring molecule. In the h-confi-
guration the three-fold axis of the molecule is directed to-
ward the center of the double bond of the neighboring
molecule [8,26]. The barrier between the wells is about
© М.I. Bagatskii, V.V. Sumarokov, M.S. Barabashko, A.V. Dolbin, and B. Sundqvist, 2015
The low-temperature heat capacity of fullerite C60
2900 K and the energy difference between the minima in
the potential wells of the p- and h-configurations is ≈ 120 K.
Below Tc the C60 molecules either perform orientational
vibrations (librations) in the potential wells or execute re-
tarded rotation hopping between the nearest potential
wells. The phase transition is cooperative in nature. The mo-
lecule concentrations np and nh in the p- and h-configura-
tions are dependent on temperature. In the low-temperature
phase np ≈ 63% near Tc ≈ 260 K (see Fig. 10 in [2]). At
lowering temperature np increase and the frequency of
hopping decreases. At the temperature of glass formation
Tg (80–90 K) the reorientational motion of the molecules is
frozen and fullerite changes to the state of orientational
glass with np ≈ 83% (see Fig. 10 [2]). This occurs because
the energy of the molecules is no longer sufficient to over-
come the potential barrier between the p- and h-configura-
tions [2,8,27]. The presence of a high (≈ 2900 K) barrier
between the p- and h-configurations places the emphasis
on the influence of the temperature prehistory on the phys-
ical properties of C60 crystals, which entails the hysteretic
phenomena in the regions of phase transition.
The heat capacity at constant pressure Cp(T) of C60
was investigated by the adiabatic method in Refs. 16, 17
(at a temperature range 11–300 K), [18] (13–300 K), [19]
(5–340 K), [20] (6–350 K), [21] (1.2–30 K) and by thermal
relaxation method in [22,23] (1.4–20 K), [24] (4–300 K)
and [25] (0.2–190 K). In interval from 4 to 160 K the dis-
crepancy between the data in [16–23] is within 25%. The data
in [24] are 50–100% over those in [16–23]. Olson et al. [25]
investigated a solid ~15% C70–C60 mixture. Therefore
their Cp(T) differs significantly in value and behavior from
the results in [16–24]. The systematic discrepancy between
both the curves Cp(T) and the Tc — values taken on heat-
ing the samples in adiabatic calorimeters [16–21] are due
to the variations in the purity [16,17,28] and perfection
[20] of the samples. The systematic scatter of the data on
the thermodynamic properties of C60 is also caused by the
influence of the temperature prehistory of the samples and
by thermocycling [29–31]. Grivei et al. [24] observed a
large hysteresis of Cp(T) in the region from 160 to 286 K.
The heat capacity of C60 was investigated below 4 K
[21–23,25]. Beyermann et al. [22,23] performed two series
of Cp(T) measurements. In series 2 the samples were pre-
annealed in vacuum at 430 K. Above 4 K the data discre-
pancy between series 1 and 2 was within 16%. At T < 2 K
the results of series 1 were an order of magnitude higher
than in series 2. The distinctions between the Cp(T) data
in series 1 [22] and 2 [23] were attributed [23] mainly to
the different concentrations of the solvent impurity in the
C60 samples. In the interval from 4 to 20 K the results of
series 2 [23] are systematically 5–12% higher than the data
in [21]. As temperature decreases, the data distinctions be-
tween [21] and [23] increase and at 1.4 K the Cp in [23] is
about five times higher than in [21].
Below 2.2 K [23] and 0.7 K [25] the data are described
by the linear and cubic terms, respectively:
3
1 3( )pC T A T A T= + . (1)
According to [23], the linear term in Cp(T) is more sen-
sitive to the solvent impurity. It is determined by the con-
tribution of the tunnel levels in the orientational glass
phase of fullerite C60 [22,25]. The contribution described
by the cubic term is fully dependent on the density of states
of acoustic phonons. The limiting Debye temperature
Θ0 = 80 K at T → 0 was estimated using A3 [25]. Beyer-
mann et al. [23] and Nemes et al. [32] analyzed the expe-
rimental results on Cp(T). By varying four parameters A1,
Θ0, ΘE,tr and ΘE,lib they calculated the contributions of
tunnel levels (A1), acoustic phonons within the Debye mo-
del with the temperature Θ0 and optical translational and
librations within the Einstein model with characteristic
temperatures ΘE,tr and ΘE,lib, respectively. They obtained
Θ0 = 37 K [23] and Θ0 = 32 K [32]. From the analysis of
Cp(T) above 50 K Θ0 ≈ 50 K [16,17] and Θ0 ≈ 60 K [33].
Houen et al. [34] calculated Θ0 ≈ 100 K from the low-
temperature Young modulus of single-crystalline C60.
Θ0 ≈ 54 K was derived from the analysis of temperature
dependence of the thermal expansion coefficient of poly-
crystalline C60 and its solutions with Ne and Ar [14]. In [7]
the ultrasonic velocities of polycrystalline C60 were ana-
lyzed and extrapolated to low temperatures, which yielded
Θ0 ≈ 55.4 K. Shebanovs et al. [35] obtained the Debye
temperature ΘD(280 K) ≈ 53.9 K by analyzing the x-ray
diffraction data for single-crystalline C60 at 280 K.
ΘD(300 K) ≈ 66 K was obtained from the ultrasonic veloci-
ties measured in a C60 single crystal at 300 K [36].
Mikhalchenko [37] analyzed literature data on Θ0 of fullerite
C60. Author [37] calculated Θ0 = 77.12 K from the harmonic
elastic constants cijkl of a C60 single crystal at 0 K.
The large scatter of Θ0-data is a good motivation to
continue low-temperature investigation of the physical pro-
perties of fullerite C60. Θ0 is a constant which characterizes
the properties of a crystal, such as heat capacity, electric
and thermal conductivities, x-ray spectra intensity, elastic
features. Θ0 is also a characteristic parameter separating
the high temperature region (T >> Θ0), where the lattice
vibrations can be described within the classical theory, and
the low-temperature region (T << Θ0), where the quantum
mechanical effects become significant [38].
Note that the data on the heat capacity of C60 are essen-
tial for analyzing the heat capacities of new C60-containing
nanomaterials, for example, elementary atomic and mole-
cular gaseous substances that form interstitial solutions in
the octahedral voids of fullerite C60 [39–41].
Goal of this study was to investigate the low-tempera-
ture dynamics of fullerene C60 by the calorimetric method.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8 813
М.I. Bagatskii, V.V. Sumarokov, M.S. Barabashko, A.V. Dolbin, and B. Sundqvist
Experiment
The heat capacity at constant pressure Cp(T) of fullerite
C60 was investigated in a temperature interval from 1.2 to
120 K in an adiabatic calorimeter [21]. Two experiments
were made. In the first experiment the heat capacity of the
sample was measured from 1.2 to 30 K [21]. In the second
experiment the sample was first heated to ~ 320 K and held
in the dynamic vacuum (≈ 1·10–3 Torr) for about 48 hours.
The calorimeter was cooled through wires without using
helium as an heat exchange gas. Then measurements were
made in the interval from 1.2 to 120 K. Cp(T) of fullerite
C60 was obtained by subtracting addenda Cad(T) (the heat
capacity of an empty calorimeter with the Apiezon grease)
from the total heat capacity of the calorimeter with the
sample. Cad(T) was measured in a special experiment.
The sample was a cylinder about 6 mm high and 10 mm
in diameter. It was prepared at Umea University (Sweden)
by compacting a C60 powder under pressure about 1 kbar.
The characteristic sizes of the C60 crystallites varied within
0.1 to 0.3 mm. The C60 purity was 99.99%. The masses of
the C60 sample and the Apiezon grease were mf = (586.48 ±
± 0.05) mg and mA = (0.45 ± 0.05) mg, respectively. The in-
formation about the C60 sample and the calorimeter is de-
tailed elsewhere [14,21].
The contribution of the sample to the total heat capacity
of the calorimeter with the sample was 45% below 2 K,
about 70% in the interval from 4 to 20 K, 45% at 30 K and
23% at 120 K. The random experimental error in the spe-
cific heat of fullerite C60 was ±40% at 1.3 K, ±30% at 2 K,
±5% at 4 K and ±1.5% in the interval from 30 to 120 K.
Results and discussion
The experimental results on the specific heat Cp(T) of
fullerite C60 and the literature data [16–25] are illustrated in
Fig. 1 for the temperature regions 0.2–300 K (a), 0.2–120 K (b)
and 0.2–50 K (c) in the lg-lg scale representation. Our re-
sults obtained in experiments 1 [21] and 2 coincide. Ac-
cording to [7], the difference Cp(T)–Cv(T) is 0.18 mJ/(g·K)
at 120 K, 1.17 mJ/(g·K) at 160 K and 4 mJ/(g·K) at 290 K,
which makes about 0.1%, 0.4% and 0.6% of Cp(T), respec-
tively. The difference Cp(T)–Cv(T) is negligible below 160 K.
It is seen that two experimental curves [24,25] are sig-
nificantly distinct from the others. There is a giant hystere-
sis in temperature range from 160 to 286 K [24]. The curve
Cp(T) measured on heating the calorimeter has a maximum
at 286 K and a minimum at 255 K. The curve Cp(T) taken
on cooling the calorimeter has two maxima at 157 and 252 K
and two minima at 202 and 266 K [24] (see Fig. 1(a)). For
our opinion the distinctions between [25] and [16–24] is
the effect of the impurity ~15% C70 in the sample. Note
that above 40 K the contribution of the impurity C70 is
higher even at low concentrations because the C70 mole-
cule has 30 intramolecular frequencies more than the C60
molecule.
Fig. 1. (Color online) The temperature dependence of the specific
heat of C60 in temperature range 0.2–300 K (a), 0.2–120 K (b),
and 0.2–50 K (c), lg-lg scale. Experimental results: (○) — this
work and [21], (+) — [16,17], (∆) — [18], (×) — [19], () — [20],
(∇) — [23], (■) — [24], (●) — [25]. The contribution of intra-
molecular vibrations Cin (solid line) to the heat capacity of C60
was calculated within the Einstein model using the vibration fre-
quencies of the carbon atoms in the C60 molecule [42].
814 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8
The low-temperature heat capacity of fullerite C60
It is seen (Figs. 1(a),(b)) that our results agree well with
the data of Atake et al. [16,17] (in interval 11–120 K) and
Beyermann et al. [23] (4–20 K). At T < 4 K the distinction
between our results and [23] increases. Near 1.5 K the data
of [23] are five times higher than our results. Near 1.4 K
our Cp values are close to those in [25] (see Fig. 1(c)). In
the interval from 8 to 40 K the data of [18–20] are on the
average 8% higher than our results. Note that in [16–20]
the samples were hermetically sealed in calorimeter ves-
sels, and the sample-calorimeter thermal contact was im-
proved by feeding helium gas to the calorimeter at room
temperatures. At this temperature the He atoms can occupy
the octahedral voids in the C60 crystal [43,44]. Saturation
of C60 with He takes several hours [44]. The presence of
helium in the calorimetric vessels and the samples can lead
to higher systematic errors. In our experiments the calori-
metric cell was cooled through wires, without using helium
as an heat exchange gas. Above 40 K the distinctions in
the data of [16–25] increase. This may be caused by the
presence of the C70 impurity (this work, Fig. 1(a) and [28])
and the solvent (see Fig. 2 in [17]) or disturbance of the
equilibrium concentrations np and nh at Tg as well as by
helium gas.
The heat capacity of a C60 single crystal with a mini-
mum of structural defects was measured in a temperature
range from 6 to 350 K by Miyazaki et al. [20]. They ob-
served the sharpest maximum in the curve Cp(T) and high-
est phase transition temperature Tc = 262.1 K. It was found
that the transition from the orientational glass state to a par-
tially orientationally-ordered phase occurred in a range
from 80 to 90 K. The difference between the heat capaci-
ties above and below Tg ≈ 84.6 K is ΔC = 3.6 J/(K mol).
The corresponding data in [18] and [19] are ΔC =
= 7 J/(K·mol) at Tg ≈ 86.8 K and ΔC = 4.5 J/(K·mol) at
Tg ≈ 86 K, respectively. Below Tg the C60 concentrations
in the p- and h-configurations are frozen (np ≈ 83% [2]).
Above Tg the concentration np(T) decreases when the tem-
perature rises (see Fig. 10 in [2]). The decrease/increase in
np is attended with heat absorption/release in the crystal
[18,19]. The data on heat capacity [18] and thermal con-
ductivity [45] show that the characteristic time τ of p–h
relaxation increases from ~103 s at 90 K to ~104 s at 80 K.
Our calorimeter was cooled from 150 to 80 K rather fast
(2.5 hours). Therefore the concentration np frozen in the
sample below 80 K can correspond to the equilibrium con-
centration at Tf which is higher than Tg. As a result, the
heat capacities measured at T < Tf can be lower than in the
case of the equilibrium concentration np(Tf).
In the region of the glass phase transition the scatter of
data is several times higher than at T < 70 K and T > 90 K
(see Fig. 1(b)). This may be attributed to the influence of
the temperature prehistory of the sample.
The results obtained and the literature data in tempera-
ture range from 0.2 to 300 K [16–25] were analyzed as-
suming an additive contribution of translational, rotational
and intramolecular degrees of freedom to the heat capacity
of fullerite. Accoding to the group-theory analysis [46],
174 intramolecular vibrations may be grouped into 46 fun-
damental modes having characteristic symmetries: Ag (two
modes), Au (one mode), T1g (three modes), T1u (four modes),
T2u (five modes), Gg (six modes), Hg (eight modes), Hu
(seven modes). The contribution of the intramolecular vib-
rations Cin(T) was calculated using the Einstein model and
the data on the vibration frequencies of the carbon atoms in
a C60 molecule [42]. Cin(T) is illustrated in Fig. 1 (solid
line). The contribution Cin(T) becomes appreciable above
50 K. Cin/Cp ≈ 0.5 at ≈ 100 K (the low-temperature phase)
and Cin/Cp > 0.9 at 270 K (the high-temperature phase)
(see Fig. 1(a)). The lattice heat capacity Cp,lat(T) = Cp(T) –
– Cin(T) of fullerite C60 is illustrated in Fig. 2. It is seen
that Cp,lat is weakly dependent on temperature in the inter-
val from 60 to 150 K. The contribution of the cooperative
processes of orientational disordering to the lattice heat ca-
pacity of C60 is appreciable above 180 K. The peak in the
curve Cp,lat(T) at Tc ≈ 260 K is due to the orientational
order-disorder phase transition. At 290 K the CV,lat value is
close to 4.5 R (R is a gas constant). CV,lat was obtained from
the Cp,lat data [20] by subtracting the correction Cp–CV ≈
≈ 3.0 J/(K·mol) [7]. This behavior of the heat capacity is
consistent with the data in [2,48] which suggest that the
rotation of C60 molecules is nearly free in the high-tem-
perature phase.
At low temperatures the experimental Cp,lat(T) was ana-
lyzed taking into account the contributions of the rotational
tunnel levels in orientational glass (Ctun), translational (CD)
and libration (CE,lib) vibrational modes. Below 2.3 K our
temperature dependence Cp,lat(T) is described by the expres-
sion 3
,lat ( ) 0.01 0.00322pC T T T= + (J/(K·mol)) (see Fig. 3).
The linear term describes the contribution of the rotational
tunnel levels in orientational C60 glass [15,22,23,25,48,49].
This term is sensitive to even low impurity concentrations
[23,25]. The impurities randomly distributed in the crystal
generate random deformation fields leading to higher
stochastization of tunnel levels [50,51].
Fig. 2. (Color online) The lattice specific heat Cp,lat(T) =
= Cp(T) – Cin(T) of fullerite C60. Experimental results: (○) — this
work and [21], (+) — [16,17], (∆) — [18], (×) — [19], () —
[20], (∇) — [23].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8 815
М.I. Bagatskii, V.V. Sumarokov, M.S. Barabashko, A.V. Dolbin, and B. Sundqvist
The contribution described by the cubic term
3(0.00322T J/(K·mol) at T ≤ 2.3 K) is made by acoustic
translational vibrations. The limiting Debye temperature at
T → 0 K (Θ0 = 84.4 K) was found using the coefficient
at T3. This value agrees with literature Θ0 = 80 K obtained
for solid solution 15% C70 in C60 in experiment [25]. Cal-
culation of Debye temperature Θ0 = 77.12 K was made
in [37] by using the harmonic elastic constants cijkl of
a C60 single crystal at 0 K. Literature data for Debye tem-
perature are given in the Table 1. This scatter (unusual for
solids) in the quantities Θ0 for the fullerite C60 is most
likely associated not only with the limitations of the Debye
law T3 (for the C60 the corresponding range is extremely
narrow, T ≤ 2 K) but also with the procedures used for cal-
culation.
The contribution CD of translational vibration modes
above 2.3 K was calculated within the Debye model with
Θ0 = 84.4 K. The contribution of libration modes becomes
significant above 2.3 K. The contribution of optical libra-
tion modes CE,lib was calculated within the Einstein model.
The best fitting between the experimental Cp,lat and calcu-
lated ,lat 1 0 ,lib ,lib( ) ( , ) ( , )V D E EC T A T C T C T= + Θ + Θ curves
in temperature interval from 1.2 to 30 K is observed for
A1 = 0.01 J/K2mol, Θ0 = 84.4 K and ΘE,lib = 32.5 K.
The lattice heat capacities Cp,lat and CV,lat of C60 are com-
pared in Fig. 4. It is seen that the contributions of the
acoustic translational modes and the tunnel levels become
dominant below 3 K. The contribution of the librations
prevails in the interval from 4 to 20 K.
The densities of states of lattice vibrations (arbitrary
units) are illustrated schematically in Fig. 5. The density of
states (solid line) was obtained in experiment on inelastic
neutron scattering [2]. The dotted and solid lines show the
densities of states of the translational modes (Debye model)
and optical libration modes (Einstein model) obtained from
the analysis of our Cp,lat(T) data. It is seen in Fig. 5 that the
first libration (L) maximum in the density of states curve is
close to the Einstein temperature ΘE,lib = 32.5 K.
The behavior of the temperature dependences of the heat
capacity Cp(T) and the linear thermal expansion α(T) [14]
Table 1. Debye temperature
Authors Year Θ0, К Property Temperature range, K
T. Atake et al. 1992 50* Heat capacity 10–300
W.P. Beyermann et al. [23] 1992 37 Heat capacity 1.4–20
J.R. Olson et al. [25] 1993 80 Heat capacity 0.2–190
N. A. Aksenova et al. [7] 1999 55.2* Sound velocity 300
N. P. Kobelev et al. [36] 1998 66 Sound velocity 300
A.N. Aleksandrovskii et al. [14] 2003 54** Heat expansion 6–12
S. Hoen et al. [34] 1992 100 Young’s modulus 80–120
M. N. Magomedov et al. 2005 58.75 Calculation 0
V. P. Mikhal’chenko [37] 2010 77.12 Calculation from the harmonic elastic constants 0
This paper 2015 84.4 Heat capacity 1–120
C o m m e n t s : * — From analysis of high-temperature data
** — From analysis of temperature dependencies of heat expansion of C60 and C60 with Ne and Ar admixtures.
Fig. 4. (Color online) The lattice specific heat Cp,lat(T) =
= Cp(T) – Cin(T) of C60. Experiment: (1) — this work and [21].
Calculation: (3) dot line — contribution of tunnel levels and trans-
lational vibrations of the lattice (A1T+CD), (4) dash line — contri-
bution of librations (CE,lib) and (2) solid line — the total heat ca-
pacity. ,lat 1 0 ,lib ,lib( ) ( , ) ( , )V D E EC T A T C T C T= + Θ + Θ .
Fig. 3. (Color online) The low-temperature heat capacity of C60
in the coordinates C/T vs T2.
816 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8
The low-temperature heat capacity of fullerite C60
of C60 are compared in Fig. 6 in a range from 1 to 20 K.
The linear thermal expansion was scaled along the ordinate
axis to match the curves Cp(T) and α(T). The measure-
ments of Cp(T) and α(T) were made on the same C60 sample.
Two series of α(T) measurement were made [14]. The re-
sults are illustrated in Fig. 6 (first — dashed line, second —
solid line). The second series was made after series 1 and
annealing the sample in vacuum (1 mTorr) at 450°C for
72 hours. It is seen that the temperature behavior of Cp(T)
and α(T) measured on the non-annealed and annealed
samples is similar in interval from 5 to 20 K. In this tem-
perature region the contribution of the orientational molec-
ular vibrations (librations) to Cp,lat(T) is dominant and
therefore α(T) is mainly determined by the anharmonicity
of the orientational vibrations. Below 5 K the curves Cp(T)
and α(T) are close only for the annealed sample. The val-
ues of α(T) of the no annealed sample are negative at liquid
helium temperatures. We attribute this to the influence of
the impurities and defects present in the sample. The linear
thermal expansion α(T) is very sensitive to these factors in
C60 below 5 K [14].
The data on α(T) of a C60 single crystal [52] and Cp(T)
of our C60 powder sample are compare in the region from
5 to 120 K in Fig. 7. α(T) and Cp(T) exhibit similar tem-
perature dependences in the interval from 5 to 63 K. Above
60 K the curve Cp(T) goes upwards due to the increasing con-
tribution of the intramolecular vibrations (see Figs. 1(a), (b)
and Fig. 2). The ratio 3α/Cp determines the temperature
dependence of the thermodynamic Grüneisen parameter
3( )
V T
VT
C
α
γ =
χ
, (2)
because the relation between the molar volume V(T) and
the isothermal compressibility χT(T) is only slightly tem-
perature dependent at low temperatures. The correction
Cp–CV is negligible at 60 K (0.2 J/(K·mol) [7]). As seen in
Figs. 6 and 7, γ(T) of fullerite C60 is only slightly depend-
ent on temperature in the interval from 5 to 60 K. Accord-
ing to [53] γ(T) ≈ 3. The behavior of α(T) and γ(T) below
5 K is discussed in [14,48,49].
In the interval from 70 to 100 K the curve α(T) taken on
a C60 single crystal [52] has a feature – a sharp rise and a
jump of α(T) at Tg ≈ 86 K (see Fig. 7). This feature of α(T)
is also observed in x-ray and neutron diffraction studies
(see Fig. 1 in [7]). Figure 7 suggests that the ratio α/Cp and
hence γ(T) have a jump in the region of the glass phase
transition. Jumps of γ(T) during orientational phase transi-
tions (Tg ≈ 86 K, Tc ≈ 260 K) were observed in photoaco-
ustic experiments [54]. The anomalous values of γ(T) (and
other physical properties) are essentially lower during the
glass phase transition than in the case of an order-disorder
phase transition.
Fig. 5. (Color online) The density of states of the lattice vibra-
tions measured on the C60 powder by the method of inelastic
neutron scattering (solid curve). L and T refer to libration and
translation features, respectively The densities of state derived
from the analysis of our Cp,lat(T) data are shown for the acoustic
mode in the Debye model (straight dashed line), the optical libra-
tion mode in the Einstein model (straight solid line).
Fig. 6. (Color online) The comparison of the experimental curves
Cp(T) ((○) — our data) and the linear thermal expansion α(T)
(dashed line — series 1, solid line — series 2) [14] of fulle-
rite C60.
Fig. 7. (Color online) The comparison of the experimental curves
Cp(T) ((○) — our data) and α(T) (solid line — [52]) of fullerite C60.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8 817
М.I. Bagatskii, V.V. Sumarokov, M.S. Barabashko, A.V. Dolbin, and B. Sundqvist
Conclusion
The heat capacity of fullerite C60 at constant pressure
has been investigated in the interval from 1.2 to 120 K
using an adiabatic calorimeter. Our results and literature
data were analyzed in the temperature interval from 0.2 to
300 K. The contributions of intramolecular and lattice
vibrations to the heat capacity C60 have been separated.
The contribution of intramolecular vibrations becomes
essential above 50 K, being 50% at 100 K and over 90%
above 270 K.
The contribution of acoustic modes to the heat capacity
of C60 is dominant below 2.3 K. The Debye temperature at
T → 0 K has estimated (Θ0 = 84.4 K). The scatter of val-
ues of Θ0 in literature data is caused mainly due from cal-
culations involving the physical properties of C60 above
4 K as well as from the analysis procedures.
In the interval from 1.2 to 40 K the experimental curve
of the heat capacity of the C60 lattice is caused the contri-
butions of the rotational tunnel levels, translational vibra-
tions (within the Debye model with Θ0 = 84.4 K) and lib-
rations (within the Einstein model with ΘE,lib = 32.5 K).
ΘE,lib agrees well with the libration maximum in the curve
of the density of states estimated by the method of inelastic
neutron scattering.
It is found that the experimental temperature depend-
ences of heat capacity and thermal expansion are propor-
tional in the region from 5 to 60 K. This suggests that the
Grüneisen coefficient γ(T) of fullerite C60 is only slightly
dependent on temperature in this interval.
The features of the temperature dependences of the heat
capacity of C60 in the regions of the heat capacity of C60 in
the regions of the orientational glass Tg and order-disorder
Tc phase transitions are determined by the potential field
structure and the presence of p- and h-molecule configura-
tions. In the orientationally ordered phase by the rotation of
the molecules about the <111> axes the molecules can be in
six potential wells with global and local minima, which cor-
respond to the pentagonal (p) and hexagonal (h) configura-
tions, respectively. The presence of a high (≈ 2900 K) bar-
rier between the p- and h-configurations places the emphasis
on the influence of the temperature prehistory on the phys-
ical properties of C60 crystals, which entails the hysteretic
phenomena in the regions of phase transition.
In the high-temperature phase the specific heat of the lat-
tice at constant volume is close to 4.5 R, which corre-
sponds to the high-temperature limit for the translational
lattice vibrations (3 R) and the near-free rotational motion
of C60 molecules (1.5 R).
Acknowledgments
The authors are indebted to A.I. Prokhvatilov, A.I. Kriv-
chikov and S.B. Feodosyev for a fruitful discussion.
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Introduction
Experiment
Results and discussion
Conclusion
Acknowledgments
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