Does the energy conserve in scattering of electron from periodic potential?

Question

Consider the electrons in a weakly perturbed periodic potential. It's turn out that any plane-wave state $\mathbf{k}$ can scatter into another plane-wave state $\mathbf{k}'$ only if these two plane waves are separated by a reciprocal lattice vector.

$$\mathbf{k}'-\mathbf{k}=\mathbf{G}$$

In the wave scattering process, the energy is also conserved. Does this also true for electrons?

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Edit: Consider for instance, one dimensional case, We know that $\epsilon(k)\sim k^2$, if the energy are equal then one needed $k=\pm k'$. Taking $G=-2\pi n/a$, We have $$k'=-k=\frac{n\pi}{a}$$ It shows that energy is not conserved for arbitrary scattering.

Answer 1

If there is a weakly periodic potential with a periodicity of a $a$ such that $G = 2\pi/a$, then the relationship $\epsilon(k)\sim k^2$ does not hold for all values of $k$ of the electron. The electron also has to share the periodicity of this potential, i.e., $\epsilon(k) = k^2$ for $-\pi/a \leq k < \pi/a$ and $\epsilon(k) = \epsilon(k+G)$. The electron state cannot be a simple plane wave because of this restriction. Instead, one has to write its wave function as $\psi(x) = e^{ikx} u(x)$ where $u(x)$ has the same periodicity as the periodic potential.