By Hartman, E.F.; Sandia National Laboratories.; United States. Dept. of Energy.; United States. Dept. of Energy. Office of Scientific and Technical Information
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22) is the number of electrons which lie in an element dr of real space and in dk of wavenumber space (or in an element dr, dk of phase space) at time t, such that / = 1 if all states are occupied and / = 0 if all are empty. 11. g. Ashcroft & Mermin 1976, p. 43). Returning now to the Boltzmann equation and following the discussion above, we assume a semiclassical description of the electron gas whereby the 'particles' are wavepackets of well-defined position r and crystal momentum ftk, but allow also for the Pauli exclusion principle.
Furthermore, since v(k) is normal to the constant energy surface in kspace, it will not in general be parallel to hk. For this reason (although somewhat misleadingly) hk is called the crystal momentum of the electron. If the wavepacket contains a spread Ak in wavenumber, it must extend over a region ~ 1/Ak in real space. It is required that Ak be small compared with the Brillouin zone to obtain the necessary energy resolution and so the wavepacket must extend over many unit cells. This means that any fields and temperature gradients must be essentially constant over such regions.
4. The degree of order is given by the square of the amplitude of this wave. More complicated structures can be built up from a superposition of different ordering waves. For example, a ternary ABC2 Heusler alloy can be generated from a (100) wave of wavelength a0 and a ( i i i ) wave of wavelength la^ljz (de Fontaine 1979). e. 20) where fa is a non-zero wavevector defined in the first Brillouin zone of the disordered alloy, c the mean alloy composition and A(($j,t) the amplitude of the fij composition wave which may evolve with time.