# Density of states in 1D semiconductor

I'm given the dispersion relation for the energy band of my semiconductor:

$$E(k)=\alpha+\beta\cos(ka)$$

where $$a, \alpha, \beta$$ are know parameters and I must obtain the density of states from that.

However, I'm not given a box size to have the k's quantized,so in order to manage that, a periodic boundary condition should be applied. But it confuses me a bit that I'm imposing a periodicity in the k space instead of the real space, and I'm not really sure what this L (box length) should be in order to have the $$\Delta k=\frac{2\pi}{L}$$ spacing between k-states.

What I tried so far:

$$n(|k|)=4\int_{0}^{|k|}\frac{L}{2\pi}dk$$

Using my dispersion relation:

$$dE=-\beta a\sin(ka)dk \leftrightarrow dk=-\frac{dE}{\beta a\sin(\cos^{-1}((E-\alpha)/\beta)})$$

then my DOS is $$g(E)=-\frac{2L}{\pi\beta a\sin(\cos^{-1}((E-\alpha)/\beta)})$$

since

$$n(E)=\int g(E)dE$$, at 0 T

If my calculations are correct how should I obtain an expression for a generic L?