The Crystallographic Modeling of C 60 Orientations in a Cubic Lattice
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VENIAMIN SH. SHEKHTMAN, RUBEN A. DILANYAN, OKSANA G. RYBCHENKO Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow district, 142432, Russia.
ABSTRACT The symmetry conditions which rule the matching of a fulleren molecule's main axis in a translational lattice are considered. The full set of orientational states for the structure of fullerene are obtained. The different levels of correlations between symmetry axes of icosahedral molecule and cubic lattice are considered. This result used for analysis of orientational ordering models, including modulation structure and discrete orientational glass.
INTRODUCTION The study of fullerene compounds puts forward a number of novel interesting problems in the spheres of crystallochemistry and material science. The orientational characteristics of molecules discussed in a set of works [1-4] require, in particular, special attention. One of the key problems here, to our opinion, is the analysis of unique topological situations which are associated with the coordination of a high-symmetry configuration of atoms with a translational lattice. The given work considers the creation of the cubic Bravais lattices from the atomic designs belonged to the icosahedral system. The authors proceed from the fact that consistent application of basic notions for the modern theory of crystals (quasicrystals) is quite appropriate to the systematic study of orientational ordering of C6 0 molecules. MATCHING OF SYMMETRY AXES We should begin with the statement that the icosahedral (Y) and cubic (C) point groups refer to the rank system of a complete group of the ball's symmetry. The the groups m3 and 23 are the subgroups of the both holohedral Y- and C- groups. According to the Curie principle the truncation may be put down as m35r-m3m=m3, (I) which, in particular, preassigns a point symmetry for the structure of cubic C60 . The geometrical analysis of(1) fully reproduces the picture of orientations of the icosahedral molecule in a cubic unit cell. It is significant that the Y-group consists often 3-fold and fifteen 2-fold axes, and the Cgroup - of 4 3-fold and 3 2-fold axes. Bearing this in mind the fulfillment of(1) coordinates the axes of the molecule's symmetry with those of the unit cell's symmetry: (3-fold axes)y 11(11 )C (2-fold axes)y I1 ( 00 1 )C (2) 181
Mat. Res. Soc. Symp. Proc. Vol. 359 01995 Materials Research Society
A 5-fold axes is not associated with any rational direction and according to [5]: (5-fold axis)y II(ITO), where T = (1+45)/2. Thus four of ten 3-fold axes and three of fifteen 2-fold ones coincide with corresponding cubic axes. Further on, the observation of (2) permits us to describe the indexes of directions and the interorientation of all the symmetry axes of a molecule's in the coordinate system of a cubic lattice. 3-FOLD AXES (Ti), Table 1. One often 3-fold axes (TI 0 ) coincides with the direction [11 I]C and is not presented in the Table. The rest nine axes can be divided into two groups. The six axes (T1 :T6 ) form the angle
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