Energy bands

Schwarz, K.

doi:10.1107/97809553602060000639

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 298

Section 2.2.6.2. Energy bands

K. Schwarz^a ^*

^a Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.6.2. Energy bands

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Each irreducible representation of the space group, labelled by $[{\bf k}]$ , denotes an energy $[E^{j}({\bf k})]$ , where $[{\bf k}]$ varies quasi-continuously over the BZ and the superscript j numbers the band states. The quantization of $[{\bf k}]$ according to (2.2.4.13) and (2.2.4.15) can be done in arbitrary fine steps by choosing corresponding periodic boundary conditions (see Section 2.2.4.2). Since $[{\bf k}]$ and $[{\bf k+K}]$ belong to the same Bloch state, the energy is periodic in reciprocal space: $[E^{j}({\bf k})=E^{j}({\bf k+K}).\eqno(2.2.6.6)]$ Therefore it is sufficient to consider $[{\bf k}]$ vectors within the first BZ. For a given $[{\bf k}]$ , two bands will not have the same energy unless there is a multidimensional small representation in the group of $[{\bf k}]$ or the bands belong to different irreducible representations and thus can have an accidental degeneracy. Consequently, this can not occur for a general $[{\bf k}]$ vector (without symmetry).

References

International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 298