Converting Between K-Space Sum and Integral for Macroscopic Solids"

[aaaa202]

How is it exactly i convert between a k-space sum an integral?
Assume that we have some macroscopic solid. Periodic boundary conditions leads to kx,ky,kz = 2π/L, so each k-space state fills a volume (2π/L)³ or has a density of V/(2π)³. To then count for instance the number of state with wavevector k<k0, what do you then do?
Intuitively I would multiply the volume of a cube of radius k0, but how does this translate into an integral exactly?

 

[andrien]

$∑_k=\frac{V}{(2\pi)^3}∫d^3k$

 

[DrDu]

You can also write the sum as an integral over a sum of delta functions.
For slowly varying test functions, the delta functions may then be replaced by their density $V/(2\pi)^3$.