International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 2.2, p. 299
Section 2.2.8. Bloch functions^{a}Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria |
We can provide a physical interpretation for a Bloch function by the following considerations. By combining the group-theoretical concepts based on the translational symmetry with the free-electron model, we can rewrite a Bloch function [see (2.2.4.18)] in the form where denotes the plane wave (ignoring normalization) in Dirac's ket notation (2.2.5.3). The additional superscript j denotes the band index associated with (see Section 2.2.6.2). The two factors can be interpreted most easily for the two limiting cases, namely:
According to another theorem, the mean velocity of an electron in a Bloch state with wavevector and energy is given by If the energy is independent of , its derivative with respect to vanishes and thus the corresponding velocity. This situation corresponds to the genuinely isolated atomic levels (with band width zero) and electrons that are tied to individual atoms. If, however, there is any nonzero overlap in the atomic wavefunctions, then will not be constant throughout the zone.
In the general case, different notations are used to characterize band states. Sometimes it is more appropriate to label an energy band by the atomic level from which it originates, especially for narrow bands. In other cases (with a large band width) the free-electron behaviour may be dominant and thus the corresponding free-electron notation is more appropriate.