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Hello I am trying to find the density of states for the dispersion relation: $$E(k_x,k_y) =\cos(k_x a) -\cos(k_y a),$$ over an entire period, not simply around the minimum. For a crystal of length $L,$ I keep seeing the expression $$ \frac{L^2}{\pi^2} \int \delta(E_0 - E(k_x,k_y))~\mathrm dk_x\mathrm dk_y $$ but I am really struggling to compute this integral. I honestly don't even know where to start, or what the resulting density of states is respect to (i.e. energy or $k$-vector), or even why this is the density of states. Would anyone be able to give me a hint on how to begin or point me to a direction on where to begin? I have exhausted all resources I can find online.

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The way to calculate this integral is to change coordinates to $E$ and some independent variable. Because you know the dispersion, you can find a vector in k-space perpendicular to the curve at energy $E$ and use that. The resulting integral is then simply the length of the curve at $E_0$, since the delta function picks out this value of E, i.e. the density of states.

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The standard way to write the energy spectrum is $$E = -2t\big[\cos(kx a) + \cos(ky a)\big].$$

Your energy is the same if you shift the $x$ momentum or $y$ momentum by $\frac\pi a$. Anyway, I recall that the answer is $K_0\left(\frac EW\right)$ up to a numerical constant, where $K_0(z)$ is the modified Bessel function $H_0(iz)$. I'm in the process of learning how to get that exact result.

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  • $\begingroup$ Welcome on Physics SE :) Please note that the system accepts TeX-formulas - this makes for easier readable formulas :) $\endgroup$
    – Sanya
    Jun 15, 2017 at 9:04

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