10.3 The k-Space and the E vs. k Diagram

Returning now to the simpler one-dimensional case again, because the crystal potential is periodic in x with a period, or a primitive translation, a,

As a periodic function in x , it can be put in the form of a spatial Fourier series:

(10.10)

where

(10.11)

The V _n are the spatial Fourier coefficients. Integral multiples of a are the lattice vectors in the direct physical space. By analogy, integral multiples of 2 ?/a, or G _n, are the lattice vectors in a reciprocal lattice k-space. This is much like expanding a time-varying electrical signal ?( t + T )= ?( t) with a period T in a Fourier series:

If the potential in the crystal is zero everywhere, or V _cr( x) = 0, then the normalized solution (10.7a) of the Schr dinger equation (10.7) is simply:

(10.12)

The dispersion curve (or E vs. k curve) of the corresponding de Broglie wave is that of a free particle and is shown as the solid curve in Figure 10.3(a).

Introducing the periodic potential (10.10) as a perturbation, the corresponding eigen function and eigen value of the Schr dinger equation become, respectively, ? _{E ( k )}( x) and E( k):

(10.13)

This equation can be solved by the perturbation technique, as outlined in Section 9.1, if V _cr is...