# 4.4: Application - electron in a crystalline solid


The above is the 1D statement of Bloch’s Theorem, the basis of study of electrons in solids. If we imagine applying an electric field $$(\mathcal{E})$$ in the $$x$$-direction, then the rate at which work is done is:

$−e\mathcal{E} v_g = \frac{dE}{dt} = \frac{dE}{dk}\frac{dk}{dt} \nonumber$

Using the expression for $$v_g$$ we find that the rate of change of $$\hbar k$$ is proportional to the external force, rather like Newton’s second law.

$−e\mathcal{E} = F = \hbar \frac{dk}{dt} \nonumber$

If we now consider acceleration:

$a = \frac{dv_g}{dt} = \frac{dv_g}{dk} \frac{dk}{dt} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2} F \nonumber$

we find a quantity $$\hbar^2 / \frac{d^2E}{dk^2}$$ which is known as the effective mass, relating external force to acceleration in a solid, and allowing us to avoid further consideration of the effect of the lattice.

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