K space sum to integral

1. Mar 31, 2014

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?

2. Apr 3, 2014

andrien

$∑_k=\frac{V}{(2\pi)^3}∫d^3k$

3. Apr 3, 2014

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 $V/(2\pi)^3$.

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