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K space sum to integral 
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#1
Mar3114, 05:09 PM

P: 1,005

How is it exactly i convert between a kspace sum an integral?
Assume that we have some macroscopic solid. Periodic boundary conditions leads to kx,ky,kz = 2π/L, so each kspace 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
Apr314, 08:10 AM

P: 1,020

##∑_k=\frac{V}{(2\pi)^3}∫d^3k##



#3
Apr314, 02:50 PM

Sci Advisor
P: 3,596

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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