Free Electron Gas in Crystals with Unequal Dimensions

	If we consider a crystal with dimensions Lx, Ly, Lz, it has the volume V = Lx· Ly· Lz.
		All we have to do is to replace the periodic boundary conditions ψ(x + L) = ψ(x) by:
		ψ(x + Lx, y, z)  =  ψ(x , y + Ly, z)  =  ψ (x, y, z + Lz)  =  ψ(x, y, z)
	This leads to simple expressions for the allowed wave vectors k:
		kx  =  0,    ± 2π Lx ,    ± 4π Lx ,    ...               ky    0,    ± 2π Ly ,    ± 4π Ly ,    ...               kz   0,    ± 2π Lz ,    ± 4π Lz ,    ..
		The pre-exponential factor, which was (1/L)3/2, now changes to (1/V) 1/2.
	Since all relevant quantities are usually expressed as densities, i.e. divided by V, and the quantization of k is usually given up in favor of a continuous range of k's, we may just as well stick to the more simple description of a crystal with equal sides - the results are the same.

With frame

2.1.1 Essentials of the Free Electron Gas

© H. Föll (Semiconductors - Script)