Low-temperature thermodynamics of Xe-doped fullerite C₆₀
Using a model of the fullerene C₆₀ molecule with carbon atoms uniformly distributed over its surface, the potential
 energy U(n) of a Xe atom in an octahedral void of C₆₀ is calculated. Within the framework of threedimensional
 harmonic oscillator, the lowest energy levels are estima...
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| Zitieren: | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ / M.S. Klochko, M.A. Strzhemechny // Физика низких температур. — 2015. — Т. 41, № 6. — С. 620-624. — Бібліогр.: 13 назв. — англ. |
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| author | Klochko, M.S. Strzhemechny, M.A. |
| author_facet | Klochko, M.S. Strzhemechny, M.A. |
| citation_txt | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ / M.S. Klochko, M.A. Strzhemechny // Физика низких температур. — 2015. — Т. 41, № 6. — С. 620-624. — Бібліогр.: 13 назв. — англ. |
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| container_title | Физика низких температур |
| description | Using a model of the fullerene C₆₀ molecule with carbon atoms uniformly distributed over its surface, the potential
energy U(n) of a Xe atom in an octahedral void of C₆₀ is calculated. Within the framework of threedimensional
harmonic oscillator, the lowest energy levels are estimated and the contribution of xenon impurity
atoms to the heat capacity of the Xe–C₆₀ system is determined. The contribution of Xe dopants to the total heat
capacity is shown to be essential compared to that of pure fullerite. Using the calculated energy spectrum we estimated
the contribution of Xe atoms to the thermal expansivity of C₆₀ with 37% of Xe. This contribution is in a
qualitative agreement with experimental findings. We estimated the Grüneisen parameter Г due to the anisotropic
part of U(n) to show that the negative part of Г is negligible due to the very small width of the five lower
oscillatory wave functions.
|
| first_indexed | 2025-12-07T18:06:53Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6, pp. 620–624
Low-temperature thermodynamics of Xe-doped fullerite C60
M.S. Klochko and M.A. Strzhemechny
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: klochko@ ilt.kharkov.ua
Received December 12, 2014, published online April 23, 2015
Using a model of the fullerene C60 molecule with carbon atoms uniformly distributed over its surface, the po-
tential energy U(n) of a Xe atom in an octahedral void of C60 is calculated. Within the framework of three-
dimensional harmonic oscillator, the lowest energy levels are estimated and the contribution of xenon impurity
atoms to the heat capacity of the Xe–C60 system is determined. The contribution of Xe dopants to the total heat
capacity is shown to be essential compared to that of pure fullerite. Using the calculated energy spectrum we es-
timated the contribution of Xe atoms to the thermal expansivity of C60 with 37% of Xe. This contribution is in a
qualitative agreement with experimental findings. We estimated the Grüneisen parameter Г due to the aniso-
tropic part of U(n) to show that the negative part of Г is negligible due to the very small width of the five lower
oscillatory wave functions.
PACS: 61.48.+c Fullerenes and fullerene-related materials;
61.72.Ji Point defects (vacancies, interstitials; color centers, etc.) and defect clusters;
65.40.De Thermal expansion; thermomechanical effects;
65.40.Ba Heat capacity.
Keywords: doped fullerite C60, low-temperature thermodynamics, rare gases.
1. Introduction
The thermal expansivities α measured [1] in doped C60
were negative within certain temperature intervals; the
authors related this effect with tunnel rotations of C60 mol-
ecules between the neighboring equilibrium states. How-
ever, even if negative warming was not documented, do-
pant-related expansivity as a function of temperature
showed maxima [2]. Dolbin et al. explained it as being due
to polyamorphic first-order transformations in orientational
glasses, also related with tunnelling motion of C60 mole-
cules. Both these interpretations were put to doubt, mostly
because, first, C60 has a huge rotational moment and, se-
cond, the corresponding rotational barriers are very high
and steep, so the maxima observed on the ( )Tα curves
could be easily explained as being due to impurities, espe-
cially, if they are molecules [3].
However, even in the case of an atomic (RG) impurity
such effect is still possible. In particular, it was shown [4]
that the excited states of a Xe atom in octahedral voids can
be represented as an oscillatory rotational modes. The re-
sulting spectrum resembled, with slight modifications, the
Devonshire spectra for molecular dopant species, that is,
the thermodynamic quantities of C60 with rare gas impuri-
ties in octahedral voids could include negative contribu-
tions to the respective Grüneisen parameter .Γ Such con-
tribution is usually associated with energy spectra specifi-
cally modified due to some tunnelling motion, which in the
case of rare gas atomic impurities were due to tunnelling in
the angular space.
Since rather detailed x-ray data are available [5] for the
same C60–Xe sample that was used in thermal expansion
measurements, it was evident that detailed thermodynamic
calculations are needed to have better insight into the low-
temperature thermodynamics of this system.
2. Interaction potential of Xe atom in O-void
Let us consider a Xe atom in an octahedral cavity. The
interaction between the Xe atom and any of the carbons
atom that constitute the cage of six fullerene molecules C60
(see Fig. 1) will be described by the Lennard–Jones (LJ)
potential with the parameters = 3.36σ Å and = 104ε K,
as reconstructed from rare gas scattering on graphite sur-
faces [6]. Any of the six C60 molecules will be represented
by a sphere of radius [7] 0 = 3.54r Å with carbon atoms
distributed homogeneously over the surface of every full-
erene sphere with the density 2
060/(4 ).rπ
Integration over the surfaces of the six nearest fullerene
molecules (the contribution of the farther molecules is ig-
© M.S. Klochko and M.A. Strzhemechny, 2015
Low-temperature thermodynamics of Xe-doped fullerite C60
nored) yields the following expression for the potential
energy of a Xe atom in the octahedral void
{ }
6 12
10 10
0 0=6
12( ) =
| | i i
ii
R R
r
− −
− +
εσ
Φ − −
+∑r
R r
{ }
6 6
4 4
0 0=6
30
| | i i
ii
R R
r
− −
− +
εσ
− −
+∑ R r
(1)
where 0 0= (| | )i iR r− + −R r and 0 0= (| | );i iR r+ + +R r r
is the radius-vector of the integration point; 0iR is the
radius-vector of the ith C60 molecule. Further analytical cal-
culations are impractical, so we need to make certain ap-
proximation when dealing with the numeric sums in Eq. (1).
Since the problem has a cubic (octahedral) symmetry,
we can represent the energy ( )Φ r in the form
even
( ) = ( ) ( )N N
N
U r IΦ ∑r n (2)
where = ,rr n ( )NI n are the cubic invariants of rank N.
Here we also remind that 2 ( ) = 0.I n
To seek the solution to the respective Schrödinger equa-
tion, as the first step we truncate the potential energy of
Eq. (2), leaving only two terms with = 0N and = 4,N
which are proportional to the respective invariants
0 ( ) = 1/ 4 ;I πn (3)
4 40 44 44
7 5( ) = ( ) ( ( ) ( )).
12 24
I Y Y Y+ +n n n n (4)
The respective energy functions ( )NU r can be calculated
numerically:
( ) = ( ) ( ) .N NU r r I dΦ∫ n n n (5)
As the second step we calculate 0 ( )U r :
0
( )( ) = ,
4
rU r dΦ
π∫
n n (6)
for the value 0 =a 14.04 Å, which is the = 0T value in
pure C60. This is a preliminary calculation in view of the
next steps explained below.
The third step consists in complying our estimates
with the structure data [5] on a 37% Xe-doped C60 sam-
ple. First we need to know how the depth of 0 ( )U R de-
pends on 0 = /2.R a As it follows from the 0 ( )U R
curves calculated as shown in Eq. (6) for different val-
ues of 0R (see also Fig. 2), the lowest value of the min-
imum is at (m n)
0 6.98 Å.iR This means that the octahe-
dral void would tend to shrink to values less even than
the value of (0)
0 = 7.02R Å in pure C60, whereas in the
above-mentioned doped sample (exp)
0 = 7 ..035 ÅR By in-
creasing σ we were able to shift (min)
0R to higher values
until it surpassed (exp)
0 .R Then we took into consideration
the fact that C60 resists elastically above (or constriction
below) the value (0)
0R in pure C60 at zero temperature. The
latter value is determined exclusively by the interactions of
C60 molecules between themselves. The corresponding
energy per octahedral site can be easily derived [13] to be
(0) 2
el 0 0 0( ) = 9 [ ] .U R Kr R R− (7)
Here K is the low-temperature bulk modulus known from
experiment [8,9] and 0R (which depends on )σ is the dis-
tance from the center of octa-void to the C60 molecule
which belongs to the octahedral cage.
For every given value of σ in a subsequent iteration
procedure we evaluated the resulting value of 0R with
account of the fact that the C60 lattice elastically resists
extension (or shrinking). This was done by solving the
following equilibrium equation
0 0 el 0
0
[ ( , ) ( )] = 0.d U R U R
dR
σ + (8)
Fig. 1. Geometry of the problem, showing an octahedral cage
with a xenon atom shifted from the void center.
Fig. 2. The potential well depth of (final)
0 ( )U r as a function of
the half of the lattice parameter a of fullerite C60, 0 = /2.R a
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 621
M.S. Klochko and M.A. Strzhemechny
If the solution fin
0R was not equal to (exp)
0 = 7 ,.035 ÅR we
varied σ until it was. In this way we determined the opti-
mum value of σ which was used to calculate the ( )rΦ as
in Eq. (1) and then, using Eq. (6), the final potential curve
(fin)
0 ( )U R to be utilized in further estimations.
We first perform the final fitting (cf. Fig. 3) of (final)
0 ( )U R
to a 2R function within the range 0. 5 Å0 2R≤ ≤ and ob-
tain the eigenvalue of the oscillation energy / =kω
(64.93 0.45) K,= ± where k is the Boltzmann constant.
For a 3D oscillator the eigenvalues for the nth excited
state are = ( 3/2)nE nω + and degeneracies =ng
(1/2)( 1)( 2).n n= + + Since we are going to take into ac-
count the five lowest oscillatory states, in Fig. 4 we show
the modulus of the eigenfunction with = 4n (the explicit
expressions for eigenfunctions can be found elsewhere
[4,10]); the ground-state wave function is roughly twice as
narrow. The span of this eigenfunction justifies our choice
for the width of the fitting range for the evaluation of .ω
One of our goals is to understand to what extent the ro-
tational tunnelling levels are split by the N = 4 potential
term (cf. Eqs. (2) and (5)). To that end, proceeding from
the general considerations that for sufficiently smooth func-
tions involved, as the actual case is, the small-argument as-
ymptote of ( )NU x must be ,Nx we obtained that
4
4 ( )U R BR with B = (7529 ± 33) KÅ–4. In our subsequent
numeric estimations we will use the following dependence:
4
4 ( )/ = ( / )U R Rω γ ρ (9)
where 1/2= ( / ) =Mρ ω 0.106 Å is the natural reference
distance in the quantum problem of harmonic oscillator
and 0.015γ is the dimensionless parameter which de-
termines the splitting of the energy levels due to the aniso-
tropic potential 4 ( , ).U R n
Now we evaluate the changes in the excitation spectrum
as well as the effects in low-temperature thermal expansion
due to angular tunnelling. From scratch we note that the
total span of the wave functions of the lowest five levels do
not exceed 0.25 Å. That, in its turn, means that at these dis-
tances the contribution of the correction term 4 4( ) ( )U r I n
must be inessential. Since the eigen-functions of the 3D
oscillator are well known [4,10], it is easy to evaluate the
corresponding splittings. Thus we found that the splitting
of, for example, the fourth excited level amounts to 1.4 K.
That means that the respective negative contribution for
the low-temperature thermal expansion of Xe-doped C60 is
very small compared to the classical contribution. Compar-
ing with a similar contribution due to a molecular dopant,
for instance, a nitrogen molecule with its bond length (ana-
log of diameter) of 1.1 Å, it is understandable that its radi-
us of 0.55 Å is sufficient to probe ranges closer to the oc-
tahedral cage molecules, rendering the splitting much
larger than in our case of the atomic Xe dopant.
3. Heat capacity
Now we calculate the Xe-related contribution to the low-
temperature heat capacity of the above-mentioned sample
with 37% of the Xe occupancy. To this end we use the fol-
lowing expression for the free energy per dopant particle
= ln exp ( / ),n n
n
F kT g E kT− −∑ (10)
where ng is the degeneracy of level n. In Fig. 5 we show
the experimental data [11] for pure C60 (solid triangles) as
well three theoretical curves as explained in the caption.
We should note that only the five lower energy levels were
taken into account in our estimates. The particular fraction
of 37% in octahedral voids was taken, because this fraction
was in a real sample in which the thermal expansion was
measured at low temperatures [8] (see also the next Sec-
tion). In Fig. 5 one can see that presence of xenon atoms
even at a level of 37% changes the ( )VC T dependence
significantly compared with that for pure C60. If all octa-
hedral voids were filled, there would have been a maxi-
mum at 15 K.T Fig. 3. Potential energy of a Xe atom inside an octahedral cage as
a function of the shift from cage center.
Fig. 4. The shape of the wave function of the n = 4 excited state.
622 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
Low-temperature thermodynamics of Xe-doped fullerite C60
4. Thermal expansion
Evaluation of the low-temperature linear thermal ex-
pansion coefficient α as a function of temperature was
carried out with the aid of the known relation [12]
2 2( ) = ( /3 ){ .T i i i i iT VkT E E Eα χ 〈 Γ 〉 − 〈 〉〉 γ 〉 (11)
Here the thermodynamic average is defined as
exp ( / )
=
exp ( / )
i i i
i
i
i i
i
f x E kT
x
f E kT
−
〈 〉
− 〉
∑
∑
(12)
where Tχ is the isothermal compressibility and
= ln / lni id E d VΓ − (13)
is the partial Grüneisen parameter.
As we noted above, the tunnelling-related contribution
to Γ is negligible, the remaining Γ associated with the Xe
impurities will be the same for all oscillatory levels, be-
cause the energy spacings between oscillatory levels of Xe
atoms depend on the volume of octahedral void but are the
same for all levels. We calculate the respective positive Γ
to be 6.89. In Fig. 6 we compare our estimates of the linear
expansivities as a function of temperature with the experi-
mental data.
5. Conclusions
In order to analyze the available experimental data con-
cerning the thermodynamic properties (heat capacity and
thermal expansivity) of C60 doped (partially) with xenon,
we calculated the potential energy ( )U R of a xenon atom
in an octahedral void of C60 making use of the LJ poten-
tial. The relevant LJ parameters used were close to those
which were derived from scattering of Ar atoms from
graphite surfaces [6], the range parameter σ being varied
to fit the actual lattice parameter [5] of fullerite C60 doped
with 37%Xe.
We calculated the temperature dependence of the con-
tribution due to the 37% Xe doping to the low-temperature
heat capacity of C60 and showed that this contribution is
quite appreciable and would change the net result. Unfor-
tunately, heat capacities of Xe-doped samples were not
measured.
We have also evaluated the contribution of 37%Xe to
the linear low-temperature thermal expansion ( )Tα and
compared it with the corresponding contribution measured
[1] on such a sample. Again, like ( ),C T ( )Tα has a char-
acteristic low and broad maximum close to 28 K. Our cal-
culations are in a reasonably good agreement with the
available experimental data, without resorting to an as-
sumption of tunnelling rotations of C60 molecules or phase
transitions between different realizations of the actual
orientational glass state.
We evaluated the splittings of the lowest four excited
vibrational level of a xenon atom in an octahedral cavity of
C60 lattice owing to the anisotropic part of the potential
energy of Xe in octahedral void. It was shown that the
splittings are small (about 0.14 K for the = 4n excited
oscillatory level of the Xe atom). This result is understand-
able, considering that the wave functions even of higher
excited states are spread to 0.3–0.4 Å where the contribu-
tion of the anisotropic part of the potential ( )U R is negli-
gible compared with its isotropic part.
The authors thank M.I. Bagatskii and V.V. Sumarokov
for providing original heat capacity data for pure fullerite
C60. Similar thanks are due to A.V. Dolbin for the detailed
experimental data concerning thermal expansivities of both
pure and Xe-doped fullerite C60.
Fig. 5. Heat capacity of pure and Xe-doped C60: solid triangles
show experimental data [11] for pure fullerite; theory: contribu-
tion of 37% of Xe in octahedral voids (empty circles); a sum of
VC of pure C60 and the contribution of 37% of Xe (empty
squares); a sum of VC of pure C60 and the contribution of 100%
of Xe (empty overturned triangles).
Fig. 6. Linear thermal expansivity of Xe-doped C60: experiment
[1] (empty squares) and theory (dashed curve).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 623
M.S. Klochko and M.A. Strzhemechny
1. A.N. Aleksandrovskii, A.S. Bakai, D. Cassidy, A.V. Dolbin,
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(2005) [Low Temp. Phys. 31, 429 (2005)].
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V.G. Manzhelii, G.E. Gadd, S. Moricca, D. Cassidy, and
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Phys. 35, 226 (2009)].
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Strzhemechny, D. Cassidy, G.E. Gadd, S. Moricca, B.
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624 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
1. Introduction
2. Interaction potential of Xe atom in O-void
3. Heat capacity
4. Thermal expansion
5. Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-127939 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:06:53Z |
| publishDate | 2015 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Klochko, M.S. Strzhemechny, M.A. 2017-12-31T13:57:02Z 2017-12-31T13:57:02Z 2015 Low-temperature thermodynamics of Xe-doped fullerite C₆₀ / M.S. Klochko, M.A. Strzhemechny // Физика низких температур. — 2015. — Т. 41, № 6. — С. 620-624. — Бібліогр.: 13 назв. — англ. 0132-6414 PACS: 61.48.+c, 61.72.Ji, 65.40.De, 65.40.Ba https://nasplib.isofts.kiev.ua/handle/123456789/127939 Using a model of the fullerene C₆₀ molecule with carbon atoms uniformly distributed over its surface, the potential
 energy U(n) of a Xe atom in an octahedral void of C₆₀ is calculated. Within the framework of threedimensional
 harmonic oscillator, the lowest energy levels are estimated and the contribution of xenon impurity
 atoms to the heat capacity of the Xe–C₆₀ system is determined. The contribution of Xe dopants to the total heat
 capacity is shown to be essential compared to that of pure fullerite. Using the calculated energy spectrum we estimated
 the contribution of Xe atoms to the thermal expansivity of C₆₀ with 37% of Xe. This contribution is in a
 qualitative agreement with experimental findings. We estimated the Grüneisen parameter Г due to the anisotropic
 part of U(n) to show that the negative part of Г is negligible due to the very small width of the five lower
 oscillatory wave functions. The authors thank M.I. Bagatskii and V.V. Sumarokov
 for providing original heat capacity data for pure fullerite
 C₆₀. Similar thanks are due to A.V. Dolbin for the detailed
 experimental data concerning thermal expansivities of both
 pure and Xe-doped fullerite C₆₀. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур 10th International Conference on Cryocrystals and Quantum Crystals Low-temperature thermodynamics of Xe-doped fullerite C₆₀ Article published earlier |
| spellingShingle | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ Klochko, M.S. Strzhemechny, M.A. 10th International Conference on Cryocrystals and Quantum Crystals |
| title | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ |
| title_full | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ |
| title_fullStr | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ |
| title_full_unstemmed | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ |
| title_short | Low-temperature thermodynamics of Xe-doped fullerite C₆₀ |
| title_sort | low-temperature thermodynamics of xe-doped fullerite c₆₀ |
| topic | 10th International Conference on Cryocrystals and Quantum Crystals |
| topic_facet | 10th International Conference on Cryocrystals and Quantum Crystals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/127939 |
| work_keys_str_mv | AT klochkoms lowtemperaturethermodynamicsofxedopedfulleritec60 AT strzhemechnyma lowtemperaturethermodynamicsofxedopedfulleritec60 |