 Remember the free-e- model needs a correction for ionic potential.

 In Chap 3, we have shown that ions in crystals form a periodic lattice.

 Now let’s see the effect of such periodic ionic potential.

In free-e- model : $\left[ -\frac{\hbar^2}{2m}\nabla^2 + U(\vec r) \right] \psi(\vec r)= \varepsilon \psi(\vec r)$

Independent e- approximation (neglect e--e- interactions) still holds, so:

  • $U(\vec r)$ is for ionic potential only (NO repulsive interaction from other e- )

  • $\psi (\vec r)$ is a wave function for a single e- (so we can still use the single-e- equation)

Periodic Ionic Potential: $U(\vec r)=U(\vec r+\vec R)$, where $\vec R$ is a lattice vector.

Besides this periodic ionic potential, we still have the periodic boundary condition: $\psi(\vec r)= \psi (\vec r+\vec L)$

Combing periodic ionic potential and periodic boundary condition:
$$
\left[ -\frac{\hbar^2}{2m}\nabla^2 + U(\vec r) \right] \psi(\vec r)= \varepsilon \psi(\vec r) \
with \ U(\vec r)=U(\vec r+\vec R)\neq0 \ and \ \psi(\vec r)= \psi (\vec r+\vec L)
$$
Lattice vector $\vec {R}= n_x\vec {a}+n_y\vec {b}+n_z\vec {c}$ , L—size of the crystal.