In Figure 6.4, the states at the critical points X and L, in the [100] and [111] directions, respectively, correspond to different symmetries from the T_d symmetry of the crystal as a whole. Determine what these symmetries are, by explicitly determining which rotations are no longer included as operations that map the system, including the k-vector, into itself. Are the splittings of some of the Γ-point degeneracies seen in the band diagram in the [111] direction consistent with your assignment of the symmetry in that direction? To do this requires access to a set of symmetry tables, which can be found at the website associated with this book, (www.cambridge.org/snoke).

eV 4 2 L56 -12 17. L56 -10 L L A 1 r r X Xay V X Xy Xo X U.K M r