Electronic States SO-Coupling and Crystal Symmetry

The ideal crystalline solid is an infinite array of identical primitive cells so that the crystal is invariant under lattice translations 

The translation symmetry group is cyclic and has one-dimensional irreducible representation. It is simple to show that the solutions, V ( ) to 

The solutions are thus plane waves modulated by the functions fc(r) which have the periodicity of the lattice, and the states are labelled by the wavevector k. With p = —hiV it is seen that Uk(r) satisfies 

When spin-orbit coupling is added (eq. (1)), the relation (5) is replaced by 

The Hamiltonian is invariant under lattice translations, if V r) is invariant, even with inclusion of the SO term. The eigenfunctions will be of the Bloch form, but they will in general not correspond to pure spin states a or the spin functions which diagonalize (the z-axis is taken as quantization axis). Often one labels the Bloch function by arrows f i, 

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