Thermodynamic Properties of the FCC Modification of Solid Fullerene C 60
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METHOD
The CUSF allows investigations to be made of thermodynamic properties of various materials whose interatomic or intermolecular forces are known. Its zeroth approximation can hold the main anharmonic terms. Usually, the terms of the fifth and higher orders remain small up to the melting temperatures. In this case, the zeroth approximation of CUSF gives the following expression for the Helmholtz free energy of a cubic strongly anharmonic crystal 253 Mat. Res. Soc. Symp. Proc. Vol. 359 01995 Materials Research Society
K0 5e Fo.F Ný-!-_ 0
120
2 24
X
__e (X.5)2_ 51
4
( - ÷5 13)
n [ ( 3 m12E93) 3/4D _ - 0 1l
S4K-- --
6X
6X
s-1,
Here NKo /2 is its static part, m the molecular mass,
X- K2 (-3P/2
(2)
&K4
K2 and K4 are isotropic parts of the second- and fourth-order force coefficients, p(X) is a solution
of the transcendental equation D-2, 5 (X*- 5 (X.)
(3- 3x
6X
(3)
D_1 , 5 (X' -ý)
where D, are the parabolic cylinder functions. In the case of pairwize forces with the intermolecular potential ( (r),
K .21- K
0, 0 V2_q ( ) I-) I .X; 1-0,1,2,..
(4)
where A. is the lattice matrix and n are integer-component vectors. CUSF enables one also to include many-body forces into the zeroth approximation " From (1) and (3) follows the equation of state under hydrostatic pressure poa[l
dKo
P36
dK 2
(3-()0 dK4
da
4K4
2 da 2K2 da
3v
(a is the nearest-neighbour distance and v (a) lattice part of the internal energy 0
._-N [K
E0.20[K
=
(913) E) 2
V / N the volume of the unit cell) and for the
(6)
Differentiating (1), (5) and (6) with respect to 0, v (or a) and components of the strain tensor, all thermodynamic functions for a cubic strongly anharmonic crystal are derived ,14,16 Higher anharmonicities (fifth and sixth) are taken into account using the perturbation theory 12, 13. CUSF allows quantum effects to be included as well 4. However, for C60 the de Boer parameter 17is
A
-
2rfh /aO --
3.1X10-3
(7)
254
where a is the point in which 1(o) = 0 and E is the depth of the potential well this, even at the SC - FCC transition temperature T, = 261.4 K, (A/ T)
2
. (EAlkT,)
2
18.
z 1.5x10-3
Because of (8)
and hence, quantum corrections to lattice properties of the FCC phase of C60 are negligible in all its diagram of state. INTERATOMIC INTERACTIONS IN SOLID C6 In general, the interaction between two fullerene molecules can be represented as the sum of the interactions between all the carbon atoms of one molecule and those of the other 18and also between effective charges of their electron bonds '9. Thus, intermolecular forces in fullerenes are noncentral, this provides the orientational order in the low-temperature phase. However, in the high-temperature modification as well as in the gas phase, the molecules rotate very rapidly, then they can be averaged over molecular orientations. Considering each C60 molecule as if it were a sphere with a surface consisting of a constant density of carbon atoms and using the (12-6) Lennard-Jones potential for C-C interactions, Girifalco has obtain
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