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x,x,x; x, \ + x, \—x\ \ - x,x, \+x x,x,x; ¿ — rl+x,x; x,\-x,\+x; \+x,x, \-x x,y,z\ ix,\+y,\-z-, \-x,y,\+z z,x,y; j + z, ~z,\+x,\-y\ ¿-z,x,\+y y,z,x; \+y,\~z,x; y, $ + z,\-x; \-y,z,\+x x,y,z; x,\-y,\+z\ ±+x,y,{-z z,x,y; \-z,\+x,y\ z,\-x,\+y\ \+z,x,\-y y,z,x; \-y,\+z,x-, y,\-z,\+x\ ¡+y,z,\-x

"Origin at center (3).

Fig. 7.4. Two views of C60 in Pa3 symmetry showing the 10 inequivalent carbon sites. The dotted lines on the right indicate cross-sectional cuts normal to the threefold crystal symmetry axis, which is shown as the dark arrow in the figure. By adding mirror planes appropriate to the m3 site symmetry, it is easy to see how the 10 inequivalent sites in this figure go into the 3 sites shown in Fig. 7.3[7.52].

Fig. 7.4. Two views of C60 in Pa3 symmetry showing the 10 inequivalent carbon sites. The dotted lines on the right indicate cross-sectional cuts normal to the threefold crystal symmetry axis, which is shown as the dark arrow in the figure. By adding mirror planes appropriate to the m3 site symmetry, it is easy to see how the 10 inequivalent sites in this figure go into the 3 sites shown in Fig. 7.3[7.52].

representations contained in are used to obtain symmetry properties for the electronic and vibrational dispersion relations in the solid state.

The presence of four atoms per unit cell affects the optical lattice mode symmetries. Using for the equivalence transformation (Ag + Tg) for the four C60 molecules per unit cell in the Pa3 phase, the zone-folded vibrations for a mode with symmetry T at the zone center transforms as Y®(Ag + Tg). We use Table 4.15 to relate the normal modes T of the Ih icosahedral group to groups of lower symmetry (such as Th). We use Table 4.16 to obtain the irreducible representations after taking the indicated direct products. In this way the direct product Y <g> (Ag + Tg) is evaluated and the results are given in Table 7.6 for various symmetries at the zone center, including S6(3), Th(m3), and the space group 7® with four C60 molecules per unit cell. For example, the optically silent fourfold degenerate Gg modes in the molecule

Table 7.8

Xafor symmetry sites for the simple cubic space group (or Pah) [7.49],

Table 7.9

Character table for the point group S6(3).

Table 7.9

Character table for the point group S6(3).

m

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