Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects
The heat capacity of the interstitial solid solution (CH₄)₀.₄C₆₀ has been investigated in the temperature interval 1.4–120 K. The contribution of CH4 molecules to the heat capacity of the solution has been separated. The contributions of CH4 and CD4 molecules to the heat capacity of the solutions (C...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1195882017-06-08T03:04:16Z Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects Bagatskii, M.I. Manzhelii, V.G. Sumarokov, V.V. Dolbin, A.V. Barabashko, M.S. Sundqvist, B. Физические свойства криокристаллов The heat capacity of the interstitial solid solution (CH₄)₀.₄C₆₀ has been investigated in the temperature interval 1.4–120 K. The contribution of CH4 molecules to the heat capacity of the solution has been separated. The contributions of CH4 and CD4 molecules to the heat capacity of the solutions (CH₄)₀.₄C₆₀ and (CD₄)₀.₄C₆₀ have been compared. It is found that above 90 K the character of the rotational motion of CH4 and CD4 molecules changes from libration to hindered rotation. In the interval 14–35 K the heat capacities of CH₄ and CD₄ molecules are satisfactorily described by contributions of the translational and libration vibrations, as well as the tunnel rotation for the equilibrium distribution of the nuclear spin species. The isotope effect is due to mainly, the difference in the frequencies of local translational and libration vibrations of molecules CH₄ and CD₄. The contribution of the tunnel rotation of the CH₄ and CD₄ molecules to the heat capacity is dominant below 8 K. The isotopic effect is caused by the difference between both the conversion rates and the rotational spectra of the nuclear spin species of CH₄ and CD₄ molecules. The conversion rate of CH₄ molecules is several times lower than that of CD₄ ones. Weak features observed in the curves of temperature dependencies of the heat capacity of CH₄ and CD₄ molecules near 6 and 8 K, respectively, are most likely a manifestation of first-order polyamorphic phase transitions in the orientational glasses of these solutions. 2014 Article Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects / M.I. Bagatskii, V.G. Manzhelii, V.V. Sumarokov, A.V. Dolbin, M.S. Barabashko, B. Sundqvist // Физика низких температур. — 2014. — Т. 40, № 8. — С. 873-880. — Бібліогр.: 43 назв. — англ. 0132-6414 PACS 65.40.Ba, 65.80.–g, 66.35.+a, 81.05.ub http://dspace.nbuv.gov.ua/handle/123456789/119588 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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English |
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Физические свойства криокристаллов Физические свойства криокристаллов |
| spellingShingle |
Физические свойства криокристаллов Физические свойства криокристаллов Bagatskii, M.I. Manzhelii, V.G. Sumarokov, V.V. Dolbin, A.V. Barabashko, M.S. Sundqvist, B. Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects Физика низких температур |
| description |
The heat capacity of the interstitial solid solution (CH₄)₀.₄C₆₀ has been investigated in the temperature interval 1.4–120 K. The contribution of CH4 molecules to the heat capacity of the solution has been separated. The contributions of CH4 and CD4 molecules to the heat capacity of the solutions (CH₄)₀.₄C₆₀ and (CD₄)₀.₄C₆₀ have been compared. It is found that above 90 K the character of the rotational motion of CH4 and CD4 molecules changes from libration to hindered rotation. In the interval 14–35 K the heat capacities of CH₄ and CD₄ molecules are satisfactorily described by contributions of the translational and libration vibrations, as well as the tunnel rotation for the equilibrium distribution of the nuclear spin species. The isotope effect is due to mainly, the difference in the frequencies of local translational and libration vibrations of molecules CH₄ and CD₄. The contribution of the tunnel rotation of the CH₄ and CD₄ molecules to the heat capacity is dominant below 8 K. The isotopic effect is caused by the difference between both the conversion rates and the rotational spectra of the nuclear spin species of CH₄ and CD₄ molecules. The conversion rate of CH₄ molecules is several times lower than that of CD₄ ones. Weak features observed in the curves of temperature dependencies of the heat capacity of CH₄ and CD₄ molecules near 6 and 8 K, respectively, are most likely a manifestation of first-order polyamorphic phase transitions in the orientational glasses of these solutions. |
| format |
Article |
| author |
Bagatskii, M.I. Manzhelii, V.G. Sumarokov, V.V. Dolbin, A.V. Barabashko, M.S. Sundqvist, B. |
| author_facet |
Bagatskii, M.I. Manzhelii, V.G. Sumarokov, V.V. Dolbin, A.V. Barabashko, M.S. Sundqvist, B. |
| author_sort |
Bagatskii, M.I. |
| title |
Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects |
| title_short |
Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects |
| title_full |
Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects |
| title_fullStr |
Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects |
| title_full_unstemmed |
Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects |
| title_sort |
low-temperature dynamics of matrix isolated methane molecules in fullerite c₆₀. the heat capacity, isotope effects |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2014 |
| topic_facet |
Физические свойства криокристаллов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119588 |
| citation_txt |
Low-temperature dynamics of matrix isolated methane molecules in fullerite C₆₀. The heat capacity, isotope effects / M.I. Bagatskii, V.G. Manzhelii, V.V. Sumarokov, A.V. Dolbin, M.S. Barabashko, B. Sundqvist // Физика низких температур. — 2014. — Т. 40, № 8. — С. 873-880. — Бібліогр.: 43 назв. — англ. |
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Физика низких температур |
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2025-07-08T16:13:33Z |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8, pp. 873–880
Low-temperature dynamics of matrix isolated methane
molecules in fullerite C60. The heat capacity, isotope effects
M.I. Bagatskii, V.G. Manzhelii , V.V. Sumarokov, A.V. Dolbin, and M.S. Barabashko
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 January 9, 2014, revised February 24, 2014,
published online June 23, 2014
The heat capacity of the interstitial solid solution (CH4)0.4C60 has been investigated in the temperature inter-
val 1.4–120 K. The contribution of CH4 molecules to the heat capacity of the solution has been separated. The
contributions of CH4 and CD4 molecules to the heat capacity of the solutions (CH4)0.40C60 and (CD4)0.40C60
have been compared. It is found that above 90 K the character of the rotational motion of CH4 and CD4 mole-
cules changes from libration to hindered rotation. In the interval 14–35 K the heat capacities of CH4 and CD4
molecules are satisfactorily described by contributions of the translational and libration vibrations, as well as
the tunnel rotation for the equilibrium distribution of the nuclear spin species. The isotope effect is due to main-
ly, the difference in the frequencies of local translational and libration vibrations of molecules CH4 and CD4.
The contribution of the tunnel rotation of the CH4 and CD4 molecules to the heat capacity is dominant be-
low 8 K. The isotopic effect is caused by the difference between both the conversion rates and the rotational
spectra of the nuclear spin species of CH4 and CD4 molecules. The conversion rate of CH4 molecules is several
times lower than that of CD4 ones. Weak features observed in the curves of temperature dependencies of the heat
capacity of CH4 and CD4 molecules near 6 and 8 K, respectively, are most likely a manifestation of first-order
polyamorphic phase transitions in the orientational glasses of these solutions.
PACS: 65.40.Ba Heat capacity;
65.80.–g Thermal properties of small particles, nanocrystals, nanotubes, and other related systems;
66.35.+a Quantum tunneling of defects;
81.05.ub Fullerenes and related materials.
Keywords: heat capacity, fullerite C60, isotope effects, rotational dynamics, nuclear spin species.
Introduction
Investigations of the physical properties of solid solu-
tions (CH4)nC60 and (CD4)nC60 can provide valuable in-
formation about the isotopic effects and their role in the
low-temperature dynamics of isolated spherical rotators in
the octahedral cavities of the low-temperature phase Pa3
of the fullerite lattice.
The difference between the molecular masses of methane
and deuteromethane is relatively small (MCD4/MCH4 =
= 1.25) but the substances differ significantly in moments
of inertia (ICD4/ICH4 = 2), total nuclear spins of the A-, T-
and E- species of CH4 and CD4 molecules (SCH4 = 2, 1, 0
and SCD4 = 4, 2, 0) and magnetic moments of the light
atoms (µH/µD = 3.268). These differences can cause iso-
topic effects in the heat capacities CCH4(T) and CCD4(T) of
the admixed CH4 and CD4 molecules. Substitution of CD4
for CH4 in the solutions has a relatively slight effect on the
translational subsystem but changes dramatically the phys-
ical properties of the rotational subsystem of the CH4 and
CD4 molecules at helium temperatures at which the quan-
tum effects are dominant.
© M.I. Bagatskii, V.G. Manzhelii , V.V. Sumarokov, A.V. Dolbin, M.S. Barabashko, and B. Sundqvist, 2014
M.I. Bagatskii, V.G. Manzhelii , V.V. Sumarokov, A.V. Dolbin, M.S. Barabashko, and B. Sundqvist
The low-temperature dynamics of the solutions
(CH4)nC60 and (CD4)nC60 was investigated by the me-
thods of neutron [1–6] and x-ray [7,8] diffraction, NMR
spectroscopy of H [2,3] and 13C [9,10] atoms, dilatometry
[11,12] and adiabatic calorimetry [13].
Intercalation of fullerite with CH4 and CD4 molecules
(and other simple impurities [14–23]) leaves the structures
of the high-temperature and low-temperature phases of
fullerite unaltered. The lattice parameters of the high-tem-
perature phase of the solutions (CH4)0.91C60 and (CD4)0.87
C60 increase at T ≈ 300 K by 0.026 and 0.016 Å, respective-
ly [2,6], relative to that of pure C60 (a = 14.161 Å [24]).
This depresses the molecular interaction and smears the
orientational phase transitions to ~241 K in (CH4)0.91C60
and ~ 235 K in (CD4)0.87C60 [2,6] from ~ 260 K in pure
C60. Below these temperatures the third order axes of the
C60 molecules have the <111> orientations. There are two
orientational configurations of C60 molecules in which the
orientations of the rotation axes of C60 differ by 60°. These
are symmetrically nonequivalent configurations for C60 mo-
lecules known as pentagonal (p) and hexagonal (h) [6,25].
The energy difference between the p- and h-configurations
is ∆E ≈ 11 meV [6,26]. In the orientationally ordered phase
the C60 molecules execute fast p–h and h–p jumps. As the
temperature decreases, the fraction of the p-configurations
increases and the frequency of the jumps attenuates. Below
the temperature of glass transition (Tg ≈ 90 K) the p- and
h-configurations are frozen and the time of reorientation
increases to about an hour [6]. The intercalation of atoms
(molecules) to the fullerite C60 leads to a change in ∆E and
in the p- and h-fractions [6].
Investigations of the thermal expansion α(T) of the so-
lutions CH4–C60 and CD4–C60 show that at T = 4–6 K the
orientational glasses of these solutions undergo first-order
polyamorphic phase transitions evidenced as a hysteresis
and maxima in the temperature dependences of α(T)
[11,12]. After completing the dilatometric measurements,
an x-ray diffraction study of the CH4–C60 sample was per-
formed in the vicinity of the orientational phase transition,
T = 140–320 K [8].
The dynamics of the solutions (CH4)0.91C60, (CD4)0.87C60
and pure C60 was investigated at T = 1.5–40 K by Kwei
et al. [6] using the method of inelastic neutron scattering
in the energy range 0.1–100 meV. The energy differences
between the lower levels of the rotational spectra of
the nuclear spin A-, T- and E-species of CH4 molecules in
the octahedral cavities of C60 were measured (see Table 1
in Ref. 6). A nuclear spin conversion of CH4 molecules
was revealed and the time t1/2 ≈2.6 h of the conversion
between the ground states of the A- and T-species was
found. The authors of Ref. 6 analyzed comprehensively
their own results and the data [2,9,10] obtained by other
methods. It is found that the impurity molecules have a
minor effect on the motion of the molecules C60. The CH4
molecules execute rotational vibrations at helium tempera-
tures and a weakly hindered rotation at T ~ 120 K [6].
The heat capacity of the solution (CD4)0.40C60 in the
temperature interval 1.2–120 K was measured by Bagatskii
et al. [13] The results of analysis of the rotational-
vibrational states of the CD4 molecules in the octahedral
cavities in fullerite C60 [13] are in good agreement with the
inelastic neutron scattering [6] and NMR spectroscopic
[2,3] data on the dynamics of the molecular motion in the
solutions (CH4)0.92C60 and (CD4)0.88C60.
Our goal was to carry out a calorimetric investigation of
the isotopic effects in the low-temperature dynamics of
admixture molecules in the solid interstitial solutions
(CH4)xC60 and (CD4)xC60.
Experiment
The heat capacity C(T) of the interstitial solid solution
(CH4)0.40C60 was investigated under a constant pressure in
the interval T = 1.5–120 K. The difference between the
heat capacities at constant pressure and volume is negligi-
ble for both fullerite and the solution (CH4)0.40C60 below
120 K [24]. The heat capacity C(T) was obtained by sub-
tracting the addenda heat capacity Cad(T) from the total
heat capacity Cad+sol(T). The temperature dependence of
the Cad(T) (the heat capacity of an “empty” calorimeter
with the Apiezon vacuum grease) was measured in a sepa-
rate experiment [27,28]. The measurements were made by
the heat pulse technique using an adiabatic vacuum calo-
rimeter [27]. The variation of the sample temperature was
1i i iT T T+∆ = − , where Ti and Ti+1 are the temperatures of
the sample before switching on and after switching off
heating. It was 0.2–10% of Ti in a single measurement run.
The heat capacity corresponds to the average temperature
1( ) / 2i iT T T+= + . The heating time th was 1–10 min. The
time required to measure one value of heat capacity was
0.1–0.4 h. This was dictated mainly by the time te of equal-
izing the temperature in the sample after switching off
heating. The characteristic time tm of a single heat capacity
measurement was taken to be m h et t t= + . The tempera-
ture of the calorimeter was measured with a “CERNOX”
thermometer [27,28].
The sample was a cylinder ~8 mm high and 10 mm
in diameter. The samples were prepared from a high-puri-
ty (99.99%) C60 powder (SES, USA) with a grain size of
about 0.1 mm. The C60 powder was saturated with metha-
ne/deuteromethane under similar conditions: P ≈ 200 MPa
and T = 575 °C for 36 h. The procedure was performed at
the Australian Science and Technology Organization, Aus-
tralia. The CH4 or CD4 molecules filled about 50% of the
octahedral cavities in the samples according to thermal
gravimetric analysis (TGA). The CH4 and CD4–saturated
C60 powders were compacted at Umea University, Swe-
den. The technology of sample preparation and estimation
874 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8
Low-temperature dynamics of matrix isolated methane molecules in fullerite C60
of the methane (deuteromethane) concentration are de-
scribed in [29,30].
The concentration of admixture CD4 molecules n ≈
≈ 40% mol was measured in a separate experiment with a
low-temperature vacuum desorption gas analyzer [31] after
completing the measurement of heat capacity [13]. The
design and operation of the gas analyzer is detailed in [31].
The mass of the sample was m = (874.23 ± 0,05) mg.
The sample weighing and mounting in the calorimeter
along with hermetization of the vacuum chamber took sev-
eral hours. Then the vacuum chamber of the calorimeter
was washed several times with pure nitrogen gas and the
sample was held in the dynamic vacuum at room tempera-
ture for ~ 14 h. The residual N2 pressure in the vacuum
chamber was up to several mTorr. The calorimeter was
cooled through wires without using He as a heat exchang-
ing gas. The cooling from room temperature to ≈ 5 K took
about eight hours. The cooling of the calorimeter from 5 K
to about 1.4 K and the reaching a steady–state temperature
rate of 10–3–10–4 K/min took about 10 h. Several series of
measurement were performed. The measurement results
were only slightly dependent on the temperature prehistory
of the sample.
The measurement error in the heat capacity of (CH4)0.40C60
was ± 15% at T = 1.5 K, ± 4% at T = 2 K, and ± 2% at
T ≥ 4 K.
Results and discussion
The measured heat capacity C(T) of the solid solution
(CH4)0.40C60 normalized to unit mole of C60 is shown in
Fig. 1 (T = 1.4–120 K (a); T = 1.4–4 K (b)). The heat ca-
pacity Cf of pure fullerite C60 added on the figure to com-
parison has been measured previously in the same calorim-
eter [27,28]. The heat capacity of the solution increases
with temperature in the whole of temperature region. The
ratio C/Cf is about 9 at 1.5 K, 5 at 2 K, 2 at 4 K, 1.5 at
20 K, and decreases to 1.3 at 116 K.
Proceeding from the analysis of inelastic neutron scat-
tering data for the (CH4)0.92C60 and (CD4)0.88C60 solu-
tions a conclusion was drawn [6] that intercalation of
fullerite with both CH4 and CD4 molecules caused only a
slight decrease in the frequencies of the translational and
libration lattice modes of C60. Therefore, the increase in
C(T) against Cf(T) is mainly due to the rotational and trans-
lational motion of the CH4 molecules in the octahedral
cavities of the C60 lattice.
The dependence C(T) is analyzed assuming that doping
of C60 with CH4 molecules has a negligible small effect on
the lattice vibrations of the C60 molecules [6]. The heat
capacity component CCH4 was obtained by subtracting
Cf(T) of pure C60 [13,27] from the heat capacity C of the
solution (CH4)0.40C60 (CCH4(T) = C(T) – Cf(T)).
The temperature dependence CCH4(T) normalized to
the unit mole of CH4 in the solution is shown in Fig. 2
(T = 1.4–120 K (a); T = 1.4–18 K (b)) along with the cal-
culated heat capacity curves determined by local transla-
tional (Ctr, curve 2) and libration (Clib, curve 3) vibrations
and by tunnel rotation of the CH4 molecules in the poten-
tial field of the octahedral cavities of the C60 lattice for
the equilibrium (Crot,eq, curve 4) and “frozen” high–tempe-
rature (Crot,eq, curve 5 in Fig. 2(b)) distributions of the nu-
clear spin species of the CH4 molecules. The total heat
capacity Ccalc = Ctr + Clib + Crot,eq is described by curve 1.
The Ctr and Clib were calculated within the Einstein model.
The Ctr was calculated using the characteristic Einstein
temperature Θtr = 126.5 K found from inelastic neutron
scattering data [6]. The characteristic Einstein temperature
Θlib = 67 K used to calculate Clib was found from the con-
dition of the best fit to CCH4(T). The calculation of Crot,eq
and Crot,high is described below. It is seen in Fig. 2(a) that
experimental CCH4(T) and calculated Ccalc agree well in
the region T = 8–35 K.
As the temperature rises above 35 K, the curve CCH4(T)
goes upwards and reaches a maximum at glass-transition
temperature ≈ 90 K. On a further increase in the tempera-
ture the curve CCH4(T) descends. The fall of CCH4(T) at
(CH4)0.40C60 () and pure fullerite [27] () normalized to unit
mole of C60 in the temperature intervals 1.4–120 K (a) and 1.4–
4 K (b).
40 80 1200
40
80
120
C
, J
/(K
·m
ol
)
C
, J
/(K
·m
ol
)
T, K
(a)
1 2 3 4
0
0.5
1.0
1.5
T, K
(b)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8 875
M.I. Bagatskii, V.G. Manzhelii , V.V. Sumarokov, A.V. Dolbin, M.S. Barabashko, and B. Sundqvist
T > 90 K occurs because the rotational motion of the CH4
molecules changes from libration to hindered rotation.
At T = 35–120 K the calculated heat capacity Ccalc is
lower than CCH4(T). As for the solution (CD4)0.40(C60), we
assume that the discrepancy may be due to “additional”
rotational excitations of the CH4 molecules in the region
of a phase transition from a glass state to an orientationally
ordered phase. According to Ref. 6, the motion of CH4
molecules has a weak effect on the translational and rota-
tional lattice vibrations of the C60 molecules in the so-
lution; on the contrary, the motion of the C60 molecules
affects appreciably the local translational and rotational
vibrations of the CH4 molecules. The formations of an ori-
entation glass in the solution (CH4)0.40C60 changes the cha-
racter of the rotational motion of the C60 molecules, which
induces the “additional” rotational excitations of the CH4
molecules.
The contribution of the tunnel rotation of the CH4 mole-
cules to CCH4(T) dominates below 8 K (see Fig. 2(b)).
At T < 8 K the heat capacity CCH4(T) is determined by
the low-lying energy levels of the rotational spectra of
the nuclear spin A-, T- and E-species of the CH4 molecules
and by the correlation between the characteristic time τCH4
of CH4 conversion and the time tm of one measurement run
[32,33]. The low-energy parts of the rotational spectra of
the A-, T- and E-species of CH4 are as for CD4 dependent
on the symmetry and value of the crystal field. The rotators
CH4 and CD4 are therefore an effective probe of the crystal
environment in the octahedral cavities of the fullerite C60.
The A-species has the lowest–energy state. Therefore, un-
der the equilibrium condition at T = 0 K all the CH4 mole-
cules are in this state. At T >0 K the equilibrium distribu-
tion of the A-, T- and E- species is reached through state
conversion.
The energy differences between the low-lying energy
levels of the A-, T- and E- species of the CH4 molecule in
the octahedral cavity of the lattice in the solution
(CH4)0.92(C60) were found by the method of inelastic neu-
tron scattering [6] (see Table 1 in Ref. 6). It is found [6]
that (i) the rotational spectra are close in the p- and h-con-
figurations; (ii) the relaxation time of the occupancy of ro-
tational energy levels above the ground state of the A- and
T-species of CH4 molecules is rather short; (iii) the time of
conversion between the ground states of the A- and T-spe-
cies of CH4 molecules at T ≈ 4 K is t1/2 ≈ 2.6 h. A two-pa-
rameter model of a crystal potential field ( ) V w =
CH4 4 4 6 6[ ( ] ) ) (B V w V w= β +β in the octahedral cavity of C60
was proposed. Here the BCH4 = 7.538 K is the rotational
constant of the CH4 molecule, V4(w), V6(w) are basis func-
tions with symmetry Ā1А1 and β4, β6 are the dimension-
less parameters. Proceeding from the available experi-
mental data and assuming β4 = 2.1856β6 Kwei et. al.
calculated the energies of some low-lying levels in the ro-
tational spectra of the A-, T- and E-species of CH4 mole-
cules as a function of the potential field parameters (β4, β6)
(see Fig. 1 in Ref. 6).
Figure 3 illustrates our estimates (based on the data of
Table 1 in [6]) of the low-energy parts of the rotational
spectra of CH4 molecules in the octahedral cavity of
fullerite C60. The Crot,eq(T) (Fig. 2(a), curve 4) was calcu-
lated for the case of “fast” conversion (τCH4 << tm) when
the concentration distribution of the nuclear spin species of
CH4 at the test temperature can be considered as being in
equilibrium at any moments. This occurs when the conver-
sion is completed within the effective time tm of a single
measurement. Under such a condition the heat capacity can
Fig. 2. The contribution CCH4(T) () of the CH4 molecules to
the heat capacity C(T) of the solution at T = 1.4–120 K (a) and
1.4–18 K (b). The calculated molar heat capacities are determined
by local translational (Ctr, curve 2) and libration (Clib, curve 3) vib-
rations and tunnel rotation of the CH4 molecules for the equilibrium
(Crot,eq, curve 4) and “frozen” high temperature (Crot,high, curve 5 in
Fig. 2(b)) distributions of the nuclear spin species. The total heat
capacity Ccalc = Ctr + Clib + Crot,eq is described by curve 1.
0 20 40 60 80 100 120
20
40
60
80
3
4
2
1
(a)
T, K
T, K
0 2 4 6 8 10 12 14 16 18
2
4
6
8
10
3
5 4
2
1
(b)
C
C
H
4,
J/
(K
·m
ol
)
C
C
H
4,
J/
(K
·m
ol
)
Table 1. The Einstein temperatures of translational (ΘСН4,tr,
ΘСD4,tr) and librational (ΘСН4,lib, ΘСD4,lib) vibrations of CH4 and
CD4 [13] molecules in octahedral cavities of fullerite C60.
Sample Θtr, К Θlib, К
CH4 126.5 67
CD4 112.6 51
876 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8
Low-temperature dynamics of matrix isolated methane molecules in fullerite C60
be calculated using a unified rotational energy spectrum
including all species levels.
The Crot,high(T) was calculated (see Fig. 2(b), curve 5)
for the case when there is no conversion (τCH4 >> tex >> tm).
Under this condition the distribution of species remains con-
stant during the measurement time tm (several weeks) in the
whole temperature region and is equal to the high-tem-
perature “frozen” distribution xA:xB:xE = 5:9:2. Therefore,
rot,high ( ) ( )(5 /16) (9 /16 ( ) ( ),) (2 /16)A T EC T C T C C TT= + +
where the molar heat capacities CA,rot(T), CT,rot(T),
CE,rot(T) depend only on the transitions inside each species.
It is seen in Fig. 2(b) that the experimental heat capacity
CCH4(T) is inconsistent with the calculated curves
Crot,eq(T) and Crot,high(T). At τCH4 ∼ tm the heat capacity
CCH4(T) depends on the number of CH4 molecules which
convert during the time tm [32]. The characteristic time
τCH4(T) of the conversion in CH4 molecules can be de-
scribed as
CH4 ( ) / ln (1– )mТ t K ′τ = − , (1)
where CH4 rot,eq/( ) ( )K C T C T′ = is the fraction of CH4
molecules in the equilibrium distribution that converted
during the time tm.
The time of cooling of the calorimeter from 5 to 1.4 K
and achieving a steady-state rate of temperature was about
10 h. During this period the CH4 species came to a near-
equilibrium distribution over the sample. Most of the CH4
molecules were in the ground state of the A-species. In our
experiment τCH4 ≈ tm and CCH4(T) is determined by the
number of CH4 molecules which converted during the ef-
fective time tm of a single heat capacity measurement [32].
Above 8 K CCH4(T) corresponds to the equilibrium dis-
tribution of CH4 species. The relatively fast drop of the
theoretical Crot,eq(T) values in the region T = 8–12 K sug-
gests that the higher-lying energy levels of the molecules
CH4 (than for those shown in Fig. 3) are ignored in calcu-
lation of Crot,eq(T).
We estimated 4 rot,eq/( ) ( )CHK C T C T′ = and the aver-
age experimental times tm at T = 2–7.5 K. The temperature
dependence of the characteristic conversion time τCH4(T)
of the CH4 molecules in the octahedral cavities of C60 was
calculated using the ( )K T′ and tm(T) values according
Eq. (1). The dependences τCH4(T) and τCD4(T) [13] for the
solutions (CH4)0.40C60 and (CD4)0.40C60 are illustrated
in Fig. 4. The curves are in good qualitative agreement.
The τCH4(T) and τCD4(T) decrease with rising temperature.
At T = 4 K τCH4(T) is about 5.5 times lower than t1/2 and
about 4.7 times higher than τCD4(T) [13].
The isotopic effects in CCH4(T) and CCD4(T) are illus-
trated in Fig. 5 (T = 1.2–120 K (a); T = 1.2–18 K (b)). In
the interval T = 12–40 K the isotopic effect is mainly due
to the frequency difference between the local translational
and libration vibrations of CH4 and CD4 molecules. The
Einstein temperatures of the translational (ΘСН4,tr, ΘСD4,tr)
and libration (ΘСН4,lib, ΘСD4,lib) vibrations of CH4 and
CD4 molecules are given in Table 1.
In the region of formation of the orientational phase
transition from a glass state to an orientationally ordered
phase of fullerite C60 (∆T = 40–100 K) CCD4(T) and
CCH4(T) are influenced by “additional” excitations of CH4
and CD4 molecules (Fig. 5(a)). These excitations appear
because the motion of the C60 molecules in the lattice ful-
lerite and the motion of the impurity molecules are coupled.
At T = 80–120 K the curve CCH4(T) is higher than the
curve CCD4(T) (Fig. 5(a)). This may be due to an uncer-
tainty in the CH4 concentrations. The behavior of CCD4(T)
and CCH4(T) correlates with the effects of the impurity
molecules CH4 and CD4 on change of the lattice parameter
of C60. At T ∼ 300 K the lattice parameter of the solutions
(CН4)0.91С60 and (CD4)0.87С60 increases by 0,026 Å and
0,016 Å [6], respectively, relative to that of pure C60 (a =
= 14,161 Å) [24].
Fig. 3. The low-energy part of the rotational spectrum of free
CH4 molecules [34] and for CH4 molecules in the potential field
of the octahedral cavities in C60 [6] for the A-, T- and E- nuclear
spin species. J is the rotational quantum number and E is the en-
ergy (the degeneracy of the energy levels are shown in parenthe-
ses to the right).
0
40
80
120
92.26 (6)
7.08 (9)
ETA
0 (5)
15.08(9)
45.23 (25)
90.46 (77)
J = 2
J = 1
J = 0
J = 3
E,
K
Free rotor CH4
0 (5)
14.16 (9)11.72 (4)
20.20 (5)
92.26 (6)
Fig. 4. The characteristic conversion times τCH4(T) () and
τCD4(T) [13] (solid line) of CH4 and CD4 molecules in the octa-
hedral cavities of fullerite.
1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
T, K
τ,
h
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8 877
M.I. Bagatskii, V.G. Manzhelii , V.V. Sumarokov, A.V. Dolbin, M.S. Barabashko, and B. Sundqvist
The CCH4(T) and CCD4(T) exhibit a complex isotopic
effect below 14 K (Fig. 5(b)) when the contributions of the
translational and libration modes decrease exponentially
and quantum regularities scale up to a macroscopic level.
The dependences CCH4(T) below 8 K and CCD4(T) below
6 K are determined by the tunnel rotation of the CH4 and
CD4 molecules. The isotope effect is due to differences
between both the rates of conversion (see Fig. 4) and rota-
tional spectra of nuclear spin modifications of molecules
CH4 and CD4 (see Fig. 3 in this article and Fig. 4 in
Ref. 13). The rotational constant of CH4 molecules is twice
as large as that of CD4 molecules. Therefore, the distance
between the rotational tunnel energy levels of the CH4
species is also twice that in the case of CD4. The differ-
ences in the level degeneracy are due to the different total
nuclear spin species of the molecules. In the case of an
equilibrium distribution of CH4 and CD4 species the iso-
topic effect in CCH4(T) and CCD4(T) makes itself evident
in the differences between the curves describing Crot,eq
(Fig. 5(b), solid curve for CH4 and dashed curve for CD4).
Besides, below 5 K the conversion rate of CD4 molecules
is several times higher than that of CH4 molecules (Fig. 4).
The conversion of molecules deuteromethane has been first
found in the solid solutions (CD4)xKr [35,36]. The behav-
ior of the experimental temperature dependences τCH4(T)
and τCD4(T) [13], as in cases of solid solutions (CH4)xKr
[32,33], (CD4)xKr [35,36], agrees qualitatively with the
theoretical data on conversion in multiatomic molecules.
In contrast to diatomic molecules [37–40], other mech-
anisms of conversion are dominant in multiatomic mole-
cules at low temperatures. These are hybrid [41] and
“quantum relaxation” [42,43] mechanisms. According to
the theory of the hybrid conversion mechanism [41], the
highest conversion rate is determined by the intramolecular
magnetic interaction of protons and the intermolecular
noncentral interaction between the nearest molecules in the
lattice. The intramolecular interaction mixes the nuclear
spin states, while the intermolecular noncentral interaction
induces transitions between the rotational states transfer-
ring the conversion energy to the lattice. The hybrid mech-
anism is dominant at low temperatures [41]. The
noncentral interaction between CD4 and C60 molecules is
more than in case of CH4 and C60 molecules. This is be-
cause CD4 molecules have smaller amplitudes of zero-
point orientational vibrations and a larger effective electric
octupole moment in the ground state. Therefore, the prob-
ability of conversion energy transfer from the CD4 mole-
cules to the C60 lattice is higher than in the case of CH4
molecules. Hence, CD4 molecules have a higher conver-
sion rate.
The contribution of the “quantum relaxation” mecha-
nism of conversion increases with temperature [42,43].
The conversion rate is determined by the intramolecular
magnetic interaction of proton (deuteron) and the tunnel
exchange of the states of the species having equal energy
levels (no phonons are involved). The distance between the
lowest-lying rotational levels of the CD4 molecules is half
as large as that in case of CH4 molecules (see Fig. 3 in this
work and Fig. 4 in Ref. 13). Therefore, at equal tempera-
tures below 5 K the conversion rate of CD4 is higher than
that of CH4. In the regions T = 1.3–5.5 K and T = 1.4–8 K
the experimental heat capacities CCD4(T) and CCH4(T) are
dependent on the number of the molecules which convert-
ed during the effective time tm of a single run of heat ca-
pacity measurement [32]. The heat capacities CCD4(T)
above ≈ 5.5 К and CCH4(T) above ≈ 8 К correspond to
the equilibrium distribution of the molecular species.
It is interesting that at T = 5–6 K and T = 7–8 K the
scatter of the CCD4(T) and CCH4(T) data, respectively, is
considerably wider than at the other temperatures. This
may be due to the first-order phase transitions in the
orientational glasses of the CH4–C60 and CD4–C60 solu-
tions [11,12].
Fig. 5. The isotopic effects in CCH4(T) () and CCD4(T) [13] (+)
at T = 1.2–120 K (a) and T = 1.2–18 K (b). The solid and dashed
curves in Fig. 5(b) are the calculated heat capacities Crot,eq deter-
mined by tunnel rotation of CH4 and CD4 molecules, respective-
ly, in the case of the equilibrium distribution of the nuclear spin
species.Fig. 1. The heat capacities of the solid solution
20 40 60 80 100 1200
10
20
30
40
50
60
70
80
(a)
CD4
CD4 CD4
CH4
CH4
CH4
T, K
T, K
2 4 6 8 10 12 14 16 180
2
4
6
8
10
12
14
16
(b)
C
, J
/(K
·m
ol
)
C
, J
/(K
·m
ol
)
878 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8
Low-temperature dynamics of matrix isolated methane molecules in fullerite C60
Conclusions
The heat capacity of CH4 intercalated fullerite (CH4)0.4C60
has been first investigated in the interval 1.4–120 K. The
contribution CCH4(T) of the CH4 molecules isolated in
the octahedral cavities of the C60 lattice to the heat capaci-
ty of the solution has been separated. The contributions
of the CH4 and CD4 to the heat capacity of the solutions
(CH4)0.4C60 and (CD4)0.4C60 have been compared.
It has been found that above 90 K the rotational motion
of the CH4 and CD4 molecules changes from librational vi-
brations to a hindered rotation. The heat capacities CCH4(T)
in the range from 8 to 35 K and CCD4(T) in T = 14–35 K
are described well taking into account the contributions
of translational and librational vibrations, as well as the
tunnel rotation of the CH4 and CD4 molecules in the case
of an equilibrium distribution of the nuclear spin species.
The isotopic effect is caused mainly by the frequency dif-
ferences between the local translational and libration vibra-
tions of the CH4 and CD4 molecules.
The characteristic conversion times of the lowest-lying
levels of the CH4 species have been estimated at T < 7 K.
The isotopic effect in CCH4(T) and CCD4(T) is induced by
the differences between both the conversion rates and the
rotational spectra of the nuclear spin species of the CH4
and CD4 molecules.
The weak features observed in the curves CCH4(T) near
6 K and CCD4(T) near 8 K are most likely manifestations
of first-order phase transition in the orientational glasses of
the solutions [11,12].
The authors are indebted to A.I. Prokvatilov for a fruit-
ful discussion and to G.E. Gadd, S. Moricca, and D. Cassi-
dy for samples preparation.
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880 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8
Introduction
Experiment
Results and discussion
Conclusions
|