Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II
The heat capacity CP of solid Kr–n CH₄ solutions with the CH₄ concentrations n = 0.82, 0.86, 0.90 as well as solutions with n = 0.90, 0.95 doped with 0.002 O₂ impurity has been investigated under equilibrium vapor pressure over the interval 1–24 K. The (T,n)-phase diagram was refined and the regio...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1217742017-06-17T03:02:55Z Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II Bagatskii, M.I. Mashchenko, D.A. Dudkin, V.V. Classical Cryocrystals The heat capacity CP of solid Kr–n CH₄ solutions with the CH₄ concentrations n = 0.82, 0.86, 0.90 as well as solutions with n = 0.90, 0.95 doped with 0.002 O₂ impurity has been investigated under equilibrium vapor pressure over the interval 1–24 K. The (T,n)-phase diagram was refined and the region of two-phase states was determined for Kr–n CH₄ solid solutions. The contribution of the rotational subsystem, Crot, to the heat capacity of the solutions has been separated. Analysis of Crot(T) at T < 3 K made it possible to estimate the effective conversion times τ and the energy gaps E₁ and E₂ between the tunnel levels of the A-, T- and A-, E-nuclear-spin species of CH₄ molecules in the orientationally ordered subsystem, and to determine the effective energy gaps E₁ between the lowest levels of the A- and T- species. The relations τ(n) and E₁(n) stem from changes of the effective potential field caused as the replacement of CH₄ molecules by Kr atoms at sites of the ordered sublattices. The effective gaps EL between a group of tunnel levels of the ground-state libration state and the nearest group of excited levels of the libration state of the ordered CH₄ molecules in the solutions with n = 0.90 (EL = 52 K) and 0.95 (EL = 55 K) has been estimated. 2007 Article Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II / M.I. Bagatskii, D.A. Mashchenko, V.V. Dudkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 728-734. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 65.60.–i; 64.60.Cn; 65.40.–b; 33.15.Mt http://dspace.nbuv.gov.ua/handle/123456789/121774 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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Classical Cryocrystals Classical Cryocrystals |
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Classical Cryocrystals Classical Cryocrystals Bagatskii, M.I. Mashchenko, D.A. Dudkin, V.V. Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II Физика низких температур |
| description |
The heat capacity CP of solid Kr–n CH₄ solutions with the CH₄ concentrations n = 0.82, 0.86, 0.90 as
well as solutions with n = 0.90, 0.95 doped with 0.002 O₂ impurity has been investigated under equilibrium
vapor pressure over the interval 1–24 K. The (T,n)-phase diagram was refined and the region of two-phase
states was determined for Kr–n CH₄ solid solutions. The contribution of the rotational subsystem, Crot, to
the heat capacity of the solutions has been separated. Analysis of Crot(T) at T < 3 K made it possible to estimate
the effective conversion times τ and the energy gaps E₁ and E₂ between the tunnel levels of the A-, T- and A-, E-nuclear-spin species of CH₄ molecules in the orientationally ordered subsystem, and to determine
the effective energy gaps E₁ between the lowest levels of the A- and T- species. The relations τ(n) and E₁(n)
stem from changes of the effective potential field caused as the replacement of CH₄ molecules by Kr atoms
at sites of the ordered sublattices. The effective gaps EL between a group of tunnel levels of the ground-state
libration state and the nearest group of excited levels of the libration state of the ordered CH₄ molecules in
the solutions with n = 0.90 (EL = 52 K) and 0.95 (EL = 55 K) has been estimated. |
| format |
Article |
| author |
Bagatskii, M.I. Mashchenko, D.A. Dudkin, V.V. |
| author_facet |
Bagatskii, M.I. Mashchenko, D.A. Dudkin, V.V. |
| author_sort |
Bagatskii, M.I. |
| title |
Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II |
| title_short |
Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II |
| title_full |
Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II |
| title_fullStr |
Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II |
| title_full_unstemmed |
Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II |
| title_sort |
phase transitions in solid kr–ch₄ solutions and rotational excitations in phase ii |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
Classical Cryocrystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121774 |
| citation_txt |
Phase transitions in solid Kr–CH₄ solutions and rotational excitations in phase II / M.I. Bagatskii, D.A. Mashchenko, V.V. Dudkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 728-734. — Бібліогр.: 34 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT bagatskiimi phasetransitionsinsolidkrch4solutionsandrotationalexcitationsinphaseii AT mashchenkoda phasetransitionsinsolidkrch4solutionsandrotationalexcitationsinphaseii AT dudkinvv phasetransitionsinsolidkrch4solutionsandrotationalexcitationsinphaseii |
| first_indexed |
2025-07-08T20:30:11Z |
| last_indexed |
2025-07-08T20:30:11Z |
| _version_ |
1837112085272592384 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 728–734
Phase transitions in solid Kr–CH4 solutions and rotational
excitations in phase II
M.I. Bagatskii, D.A. Mashchenko, and V.V. Dudkin
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
Received October 20, 2006
The heat capacity CP of solid Kr–n CH4 solutions with the CH4 concentrations n = 0.82, 0.86, 0.90 as
well as solutions with n = 0.90, 0.95 doped with 0.002 O2 impurity has been investigated under equilibrium
vapor pressure over the interval 1–24 K. The (T,n)-phase diagram was refined and the region of two-phase
states was determined for Kr–n CH4 solid solutions. The contribution of the rotational subsystem, Crot, to
the heat capacity of the solutions has been separated. Analysis of Crot(T) at T < 3 K made it possible to esti-
mate the effective conversion times � and the energy gaps Å1 and Å2 between the tunnel levels of the A-, T-
and A-, E-nuclear-spin species of CH4 molecules in the orientationally ordered subsystem, and to determine
the effective energy gaps Å1 between the lowest levels of the A- and T- species. The relations �(n) and Å1(n)
stem from changes of the effective potential field caused as the replacement of CH4 molecules by Kr atoms
at sites of the ordered sublattices. The effective gaps EL between a group of tunnel levels of the ground-state
libration state and the nearest group of excited levels of the libration state of the ordered CH4 molecules in
the solutions with n = 0.90 (EL = 52 K) and 0.95 (EL = 55 K) has been estimated.
PACS: 65.60.–i General studies of phase transitions;
64.60.Cn Order-disorder transformations; statistical mechanics of model systems;
65.40.–b Thermal properties of crystalline solids;
33.15.Mt Rotation, vibration, and vibration-rotation constants.
Keywords: low-temperature specific heat, phase transitions, nuclear-spin species, conversion.
Introduction
Solid solutions of simple substances which are charac-
terized by multipole (ÑÍ4, CD4) and central (Ar, Kr) in-
teractions are popular objects for investigating a number
of topical problems of physics of solids: phase transi-
tions, dynamics of ensembles of rotors with different
quantum parameters in orientational ordered and disor-
dered sublattices of crystals, quantum effects in the rota-
tional motion of molecules in crystals [1,2]. Different
states of the orientational subsystem can be obtained eas-
ily by varying the concentration of components and the
temperature.
Under equilibrium vapor pressure, solid CH4 exists
in two phases: disordered phase I with a FCC structure
(T > 20.4 K) and partially orientationally-ordered anti-
ferrorotational phase II with a FCC structure of the mo-
lecular centers of mass (T < 20.4 K) [1]. 75% of molecules
at lattice sites D2d-symmetry feel a strong potential field
and form a long-range orientational order. In 25% of the
sites (Oh-symmetry) according to the symmetry condition
the octopole–octopole interaction (molecular field) can-
cel out and the molecules see only a cubic crystall field.
The molecules at these sites having 12 nearest neighbors
(ordered molecules) execute a weakly hindered motion.
For simplicity, the discussion below will be concerned
with orientationally ordered/disordered ÑÍ4 molecules.
The low-energy parts of the rotational spectra of or-
dered and disordered molecules calculated in [3] are
shown in Fig 1. We recall that the ÑÍ4 molecule can be in
three A-, T- and E-nuclear-spin states with the total nu-
clear spin S = 2, 1, 0, respectively. The lowest energy state
is A, therefore at T = 0 K at equilibrium all ÑÍ4 molecules
are in this state. At T < 3 K the behavior of the heat capac-
ity Crot of the rotational subsystem is determined (due to
conversion) by the transitions between the tunnel levels
of the A-, T- and E-species of ordered molecules and be-
tween the lowest levels of the A- and T-species of disor-
© M.I. Bagatskii, D.A. Mashchenko, and V.V. Dudkin, 2007
dered molecules. Thus, analysis of Crot(T) can furnish in-
formation about the effective conversion times � and the
energy gaps Å1 and Å2 between the tunnel levels of the A-,
T- and A-, E-species of ordered molecules, and determine
the effective energy gaps Å1 between the lowest levels of
the A- and T-species of disordered molecules.
Kr atoms and ÑÍ4 molecules have close Lennard–
Jones parameters [1,2]. The Êr impurity is therefore a
very suitable non-active component that can suppress the
noncentral interaction in phase II of methane. The intro-
duction of Êr into the ÑÍ4 lattice leaves the FCC struc-
ture of the centers of masses unaltered and the lattice
dilatation is rather small. No long-range orientationally
order is formed in solutions with Êr concentrations above
0.20 [2].
The phase diagram of solid Kr–n CH4 solutions and
the dynamics of the rotational subsystem in phase II at liq-
uid helium temperatures were investigated by different
methods, such as calorimetry [4,5], NMR [6], inelastic
neutron scattering [7], permittivity measurement [8]. The
heat capacity CP of solutions with n > 0.80 was measured
in 1936 [4] (n = 0.8440, 0.9255, 0.9630) in the tempera-
ture region �T = 12–25 K and in 1971 [5] (n = 0.9352) at
�T = 2.5–16 K. The discrepancy among the gaps E1 be-
tween the lowest levels of the rotators and among the
characteristic conversion times � obtained by different
method [9–11] exceed the experimental errors. The re-
gion of the two-phase states of solid Kr–n CH4 solutions
was not identified in the (T,n)-phase diagram.
In this study we performed a detailed calorimetric in-
vestigation of solid Kr–n CH4 solutions with n > 0.80 in a
wide temperature interval T = 1–25 K. We also investi-
gated how the relatively small quantity of the paramag-
netic O2 impurity influences the heat capacity of the solu-
tions. The basic goal of the study was to obtain
information about the phase transitions and diagram of
solid Kr–n CH4 solutions, as well as about the character-
istic conversion times and the low-energy part of the rota-
tional spectrum of the rotators in phase II solutions. It was
expected in particular that results of this study and
[12–14] along with the available literature data would
permit us to obtain a complete concentration dependence
(0 � n � 1) of the conversion rate and E1 for the solutions.
Experiment
The heat capacities at equilibrium vapor pressure CÐ
of solid Kr–n CH4 solutions with the CH4 concentration
n = 0.8240, 0.8600, 0.9000, and solutions with n =
= 0.8980, 0.9500, doped with 0.002 of O2 impurity were
measured in the interval T = 1–25 Ê. The measurements
were performed by pulse heating using an adiabatic va-
cuum calorimeter [15]. The heating time th was 2–6 min.
The effective time tm of one heat capacity measurement
was tm = th + te, where te is the time needed to achieve a
steady time dependence of temperature operation of the
calorimeter since the moment of switching off the heat-
ing. The te was 50–10 min. The purity of the gases used
was: CH4 (99.94%) contained 0.04% N2, � 0.01% O2, and
Ar; Kr (99.79%) contained 0.2% Xe, 0.01% N2. The solid
solutions were prepared in the calorimeter at T � 75 K by
condensing gas mixtures into the solid phase. This tech-
nology ensured homogeneous solutions. Before measure-
ments the calorimeter was cooled from T � 1.3 K to
T � 0.5 K during 6 hours and kept at this temperature
for � 18 hours. Because of the conversion, the majority
of CÍ4 molecules change during this period went to the
ground state of the A-species. The error of the heat capac-
ity measurement was 4% at 1 K, 1% at 2 K and no more
than 0.5% at T > 4 K.
At T < 14 K the rotational heat capacity Crot can be
written as Crot = CP – Ctr – �Ctr, where Ctr is the trans-
lational component of pure CÍ4, �Ctr is the change in the
translational heat capacity due to heavy Kr impurities
in the CH4 lattice (the contribution of quasi-local vibra-
tions). Ctr was calculated by the Jacobian matrix method
[16] and the characteristic temperature � = 140 K. �Ctr
was calculated by the Jacobian matrix method [16,17]
disregarding the changes in the force constants for the
mass ratios mKr/mCH4
= 5.
In the solutions containing 0.002 impurity O2, Crot
was separated by subtracting the relatively small ro-
tational contribution of the O2 molecules. The calorimet-
Phase transitions in solid Kr–CH4 solutions and rotational excitations in phase II
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 729
0
1
2
3
70
75
80
0 (5)
12.7 (9)
30.8 (12)
42.0 (9)
61.3 (18)
67.0 (4)
83.9 (18)
A
T
ET
T
AT
E
T
0 (5)
1.9 (9)
2.8 (2)
A
T
E
70.3 (3)
71.5 (4)
72.0 (3)
74.1 (6)
75.9 (3)
78.7 (3)
79.9 (10)
T
E
T
T
T
T
A
0
10
20
30
40
50
60
70
80
90
Oh D
2d
E
,
K
Fig. 1. Rotational energy spectrum of CH4 molecules at the
sites of lattices with Oh (disordered molecules) and D2d (or-
dered molecules) symmetries in phase II of methane [3]. E is
the energy (level degeneracies are in brackets, A-, T- and E-
are nuclear-spin species).
ric results obtained on the Kr–O2 solutions were used
[18–20].
Results and discussion
The experimental ÑÐ of solid Kr-n CH4 solutions (n =
= 0.824, 0.86, 0.90) and solutions doped with 0.002 O2
(n = 0.90, 0.95) are given in Fig. 2,a,b. It is seen that
first-order phase transitions occur in solutions with
n > 0.80 as the temperature drops below T = 20 K. The
transitions are accompanied by partial orientational or-
dering.
Qualitative discussion of results
In phase II the temperature dependences of the heat ca-
pacity CP(T) suggests that the phase transitions in the in-
vestigated solid solutions (Fig. 2), like in pure CH4, CD4
[1,21], CF4 [22], CCl4 [23], are mixed (partially smooth)
second-order-first phase transitions [24,25]. The distur-
bance of the long-range order (as in second-order phase
transitions) occurs in a wide interval below Òtr. We inves-
tigated in greater detail the behavior of the heat capacity
CP(T) in the solution with n = 0.95 CH4. Two series of
measurement were performed. After the first series the
calorimeter was cooled from 24 to 17 K. The other series
was made with smaller temperature increments �T ~ 0.1 K
during a single heat capacity measurement. Results of
both series are in good agreement. The curve CP (T) ex-
hibits jumps at temperatures ~0.1 K below and ~1.5 K
above Òtr. These temperatures are close to the tempera-
tures at which the low-temperature phase II transforms to
two phases and then the two-phase region changes into
the high-temperature phase I. It is taken below that Òtr
corresponds to the temperatures at which the low-tempe-
rature single-phase region changes into the two-phase
region (Fig. 2,a, solid lines). At Òtr the second-order
phase transitions transform into the first-order phase
transitions. It is assumed that the transitions from the
two-phase region to the high-temperature single-phase
region are completed at T2 at which the sign of the deriva-
tive CP(T) changes (Fig. 2,a, dashed lines).
The interval Òtr–T2 corresponds to two-phase states
(Figs. 2,a and 3). At all temperatures of measurement, ex-
cept the interval Òtr–T2, the CP values are independent of
the temperature prehistory of the sample. In the interval
Òtr–T2 , the CP values are reproducible if the sample was
cooled before measurement to the equilibrium tempera-
ture at T < Òtr. When 0.002 O2 is introduced into the
Kr–0.90 CH4 solution, Òtr decreases by � 0.3 K and
CP,max increases by 5%. At the first-order phase transi-
tions, the long-range orientational order is completely
destroyed. Therefore, at T > T2, rotation of CH4 mole-
cules is correlated and hindered. As the temperature is
730 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
M.I. Bagatskii, D.A. Mashchenko, and V.V. Dudkin
0 2 4 6 8 11 12
1
2
3
4
5
6
7
8
T, K
T, K
12 14 16 18 20 22 24
0
20
40
60
C
,
J·
m
o
l
·K
P
–
1
–
1
C
,
J·
m
o
l
·K
P
–
1
–
1
a
b
Fig. 2. Heat capacity CP(T) of solid (CH4)nKr1–n solutions.
The solid and dashed curves show (from right to left) the tem-
peratures of the onset Òtr and the end T2 of the first-order
phase transition in solutions with n: 0.82 (�), 0.86 (�), 0.90
(�), 0.90 (with 0.002 O2) (�), 0.95 (with 0.002 O2) (+). Òtr is
taken to be equal to the temperature of the corresponding
CPmax. Ò2 is taken as equal to the temperature were the corre-
sponding derivative CP(T) changes sign.
0.70 0.75 0.80 0.85 0.90 0.95 1.00
12
16
20
T
,
K
PHASE I PHASE II
n (CH )4
Fig. 3. The low temperature region of the phases phase dia-
gram of solid Kr–CH4 solutions. Experimental results: tem-
peratures of the onset Òtr and the end Ò2 (see Fig. 2,a) of the
first-order phase transition (this study) (�, �); Òtr [4] (heat ca-
pacity) (�); Òtr [6] (NMR) ( ); Òtr [8] (dielectric permittivity)
(
); Òtr= 20.48 Ê of ÑÍ4 [21] (heat capacity) (�).
rised, the degree of correlation decreases and the hin-
dered motion of the molecules changes to a diffusive
(close to free) rotation. It is commonly accepted that
Òtr corresponding to the highest CP(T) is the temperature
of the phase transitions
The low temperature part of the phase diagram of solid
Kr1–n(CH4)n solutions is shown in Fig. 3. Our Ttr agree
well with calorimetric [4], NMR [6], dielectric [8] and in-
elastic neutron scattering (INS) [7] data.
The CP(T) dependence at T < 8 K (see Fig. 2,b) is de-
termined by the heat capacity of the rotational subsystem
Crot(Ò). Below 3 K the contribution of Crot(Ò) to CP(T) is
over 0.97. At T < 3 K, the solution with n = 0.90 and the
same solution doped with 0.002 O2 have different magni-
tudes of the heat capacities and dependenses CP(T). The
distinctions are due to the paramagnetic O2 impurity
which enhances the conversion in the ordered sublattices
and thus increases the contribution of the tunnel excita-
tions to the heat capacities CP(T). It was found for the
first time in [12,13] that a hybrid conversion mechanism
was dominant at T < 3 K [26]. According to [26], the con-
version rate depend weakly on temperature at T < 3 K
[9–13]. As the temperature goes above 3 K, the conver-
sion rate increases rapidly [9–11] and at T > 6 K the distri-
bution of nuclear-spin species of CH4 molecules in the so-
lutions with n = 0.90 comes to equilibrium within the time
tm of one heat capacity measurement. It is evident that the
heat capacities CP(T) of the solutions with and without O2
approach each other as the temperature increases, and at
T > 6 K they coincide (see Fig. 2,b). Thus, the experimen-
tal CP(T) values of the solutions with n = 0.90 are in equi-
librium at T > 6 K. So are the CP(T) of the solutions with
n < 0.90 (T > 6 K) because the conversion rate increases
when the Kr concentration increases [9–11]. In the solu-
tions with n = 0.95 the distribution of nuclear-spin species
becomes equilibrium at high temperatures.
Quantitative analysis results
Now we analyze in more detail the experimental re-
sults in the temperature region below 3 K. We recall that
before measurement, the samples were kept at T � 0.5 K
for about 24 hours. During this time the majority of the
CH4 molecules in the orientationally ordered and disor-
dered sublattices go, due to the conversion to the ground
A-state. Therefore, at T < 3 K the rotational heat capaci-
ties of solutions CR(T) = Crot(T)/(n,R), normalized to the
CH4 concentration n and the universal gas constant R, are
determined by the changes in the occupancy during the
time tm of one measurement (i) between the tunnel levels
of the libration ground state of the A-, T- and A-, E-species
of the ordered molecules with the energy gaps E1 and E2
(the structure of the tunnel levels E1/E2 =2/3 is as for or-
dered molecules in pure CH4 [3]), and (ii) between the
lowest levels of the A- and T-species of the disordered
molecules with the energy gaps E1.
Model
The quantitative analysis of the experimental depend-
ences CR(T,n) at temperatures below 3 K was performed
using a simple model.
It is assumed that:
— three fourths of the molecules are in the same strong
effective potential field. They form a long-range orien-
tational order and execute small librations about the or-
dering axes and tunnel rotation;
— one fourth of the molecules are in the same compar-
atively weak effective field. They are orientationally dis-
ordered and execute hindered motion;
— the low energy parts of the spectra of the ordered
and disordered molecules in phase II of CH4 [3] (see
Fig. 1) and in ÑÍ4–Kr solutions are qualitatively similar.
The normalized experimental heat capacity CR,exp
per mole at Ò < 3 K was written as a sum of the contribu-
tions from the molecules in the orientationally ordered
CR,ord = 3
4
�K C Rord ord,eq, and orientationally disordered
CR,dis = 1
4
�K C Rdis dis,eq, sublattices:
C K C K CR R R,exp , , �
�3
4
1
4ord ord,eq dis dis,eq . (1)
The ratio
� K C CR Rord(dis) ord(dis) ord(dis),eq, ,/ (2)
is the fraction of CÍ4 molecules of the equilibrium distri-
bution which in real experiment moved from the tunnel
level of the A-species to the tunnel level of the Ò- and
Å-species in the ordered sublattices (from the lowest level
of the A-species to the lowest level of the Ò-species in the
disordered sublattices) during the time tm of one measure-
ment. The normalized heat capacities CR,ord (dis),eq for
the equilibrium distribution in the orientationally ordered
(disordered) sublattices were calculated as
CR,ord(dis),eq(T) = T �2(<E2> – <E>2) , (3)
where
� � �
� �E Z E g E kTi
i
i i
2 1 2 exp ( )
and
� � �
� �E Z E g E kTi
i
i i
1 exp ( )
are the mean rotor energies squares and energies, respec-
tively,
Z g E kTi
i
i �
� exp ( )
Phase transitions in solid Kr–CH4 solutions and rotational excitations in phase II
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 731
is the statistical sum, Ei and gi are the energy and degene-
racy of level ³. The CR,ord,eq(T) was calculated for three
tunnel levels i = 0, 1, 2 of the effective single spectrum for
the molecules in the ordered sublattice. The CR,dis,eq(T)
was calculated for two lowest levels i = 0, 1 of the effec-
tive single spectrum of the molecules in the disordered
sublattice (Fig. 1).
Table 1. Parameter characterizing experimental CR(Ò,n) depend-
ences of solid (CÍ4)nKr1–n solutions at T � 3 K.
n, %
Ordered molecules Disordered molecules
�K � , h Å1, K Å2, K �K �, h Å1, K
82 0.43 1.5 6.5 9.8 � 1 � 0.2* 8.5
86 0.19 3.9 6.0 9.0 � 1 � 0.2* 9.4
90 0.13 6.2 3.6 5.4 � 1 � 0.2* 11.4
90** 0.64 0.96 3.6 5.4 � 1 � 0.2* 11.4
95** 0.62 1.01 3 4.5 0.8 0.6 12.6
N o t e s: * is the estimated highest value of �. ** for samples with
0.002 Î2 impurity.
Table 1 presents the effective parameters E1, E2, �K
and � (the characteristic conversion times � considered
below) for CH4 molecules in the orientationally ordered
and disordered sublattices. We obtained them from the
condition of an optimal description for experimental heat
capacities CR,exp of the investigated solutions at T < 3 K.
The lines in Fig. 4 show the heat capacities CR(T) calcu-
lated from Eqs. (1)–(3) using the parameters of Table 1. It
is seen that the calculated values describe the experiment
fairly well. As we can see in Table 1, the addition of 0.002
O2 impurity to Êr–0.90 ÑÍ4 solution raises �K to 0.64.
Our results on Å1,CL are shown in Fig. 5 together with
other authors’ data obtained from heat capacities [12–14],
inelastic neutron scattering Å1,INS [27–32], and nuclear
magnetic susceptibility Å1,NMR [9,10] for 0 < n < 1. Note
that the calorimetric investigation was performed using
the same adiabatic calorimeter [15]. The effective Å1,CL
in the solutions with n < 0.1 and n > 0.9 in the
orientationally disordered sublattices are in good agree-
ment with Å1,INS [28–30,32]. In the region 0.9 > n > 0.8,
Å1,CL are significantly lower than Å1,INS [31,32]. Figure 5
shows that in the orientationally ordered sublattice Å1,CL
is larger than Å1,INS even at small Kr concentrations. This
difference increases smoothly with Kr concentrations.
For the Kr concentration 0.2, Å1,CL of the disordered and
ordered sublattices move closer to each other. The differ-
ences between Å1,CL and Å1,INS can be explained as fol-
lows. The Å1,INS corresponds to the highest intensities in
the inelastic neutron scattering spectra [31,32]. Both lines
describing the tunnel rotation of the molecules and the
nearly free rotation of the molecules exhibit specific
asymmetric smearing towards higher and lower energies
[32]. The smearing becomes pronounced as the Kr con-
centration increases. Thus, Å1,INS for orientationally or-
dered and disordered molecules shifts towards the effec-
tive Å1,CL values obtained in this study.
We estimated the effective gap EL between the group
of tunnel levels of the ground-state libration state and the
nearest group of levels of the excited libration state in the
732 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
M.I. Bagatskii, D.A. Mashchenko, and V.V. Dudkin
1.0 1.5 2.0 2.5 3.0
0
0.1
0.2
0.3
0.4
0.5
T, K
C
/(
n
R
)
ro
t
Fig. 4. Rotational heat capacity CR = Crot/(nR) of solid
(CH4)nKr1–n solutions, normalized to the CH4 concentration n
and to the universal gas constant R, with n: 0.82 (�), 0.86
(�), 0.90 (�), 0.90 (with 0.002 O2) (�), 0.95 (with 0.002 O2)
(+).The curves were calculated using Eqs. (1)–(3) and parame-
ters of Table 1.
0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
12
14
phase II
phase I
E
,
K
1
n (CH )4
Fig. 5. Concentration dependence of the effective energies
E1 of disordered and ordered CH4 molecules in phases I and II
of solid (CÍ4)nKr1–n solutions. Experimental results: this work
(�), heat capacity [12–14] (�), heat capacity [21] (
), INS
[28] ( ), INS [29] (�), INS [30] (�), INS [31] (
), INS [32]
(�), INS [27] (�), NMR [9,10] (�).
solution with n = 0.90 and 0.95 CH4. The upper levels
were substituted by a single effective level with the de-
generacy equal to a sum of the degeneracies of the levels
[3] (see Fig. 1). The normalized heat capacity CR,exp at
T = 12 K was represented in terms of Eq. (1) taking the
level EL into account and using the gaps for the tunnel le-
vels Å1 = 3.6 K, Å2 = 5.4 K (n = 0.90) and Å1 = 3 K,
Å2 = 4.5 K (n = 0.95). Since at T > 7 K the nuclear-spin
species come to a practically equilibrium distribution in
the ordered and disordered sublattices during tm, we ob-
tain � � K Kord dis 1. Assuming that at T = 12 K the norma-
lized rotational heat capacity of the disordered sublattice
is equal to the ultimate high-temperature CR,dis,eq = 3/2,
we obtain EL = 52 K (n = 0.90) and EL = 55 K (n = 0.95)
(EL �70 K in pure CH4 [3]).
Proceeding from [13] and using �K ord and �K dis for the
ordered and disordered molecules (and times tm), we
could estimate the characteristic times � of the conversion
between the lowest tunnel states of the A-, T-, E-species of
ordered molecules and the lowest states of the A- and
T-species of disordered molecules at T < 3 K.
The following expression was obtained in [13]:
�ord dis ord dis( ) ( )/ ln ( ) � � �t Km 1 , (4)
The � values calculated at T < 3 K from Eq. (4) are
given in Table 1. Our own (this work and [12–14]) and lit-
erature data on � in the region 1 > n > 0 of the Kr–n CH4
solutions are shown in Fig. 6. It is seen that in the ordered
sublattices the conversion of CH4 molecules slows down
significantly as the CH4 concentration increases. Our ef-
fective � values (in solutions free of O2) in the ordered
sublattices are in good agreement with literature data
[9–11]. In the disordered sublattices the rate of CH4
conversion is much higher than that in the ordered sub-
lattices. Our results agree qualitatively with other experi-
ments [9,10]. At shorter times � in the region of conver-
sion time � < tm, the error in � calculated from Eq. (4)
increases and at 4� � tm only the upper estimate of � can
be obtained from the condition 4� = tm. This condition
is met in disordered sublattices of the solutions with
0.8 � n � 0.9. In phase I of the Kr–n CH4 solutions with
n > 0.60, this condition is obeyed if the equilibrium distri-
bution of the CH4 species is established in these solution
during the characteristic times tm of the experiment [13].
The work was supported by the Ukraine Ministery
of Education and Science (Project «New quantum and anhar-
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