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K space sum to integral

by aaaa202
Tags: integral, space
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aaaa202
#1
Mar31-14, 05:09 PM
P: 1,005
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?
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andrien
#2
Apr3-14, 08:10 AM
P: 1,020
##∑_k=\frac{V}{(2\pi)^3}∫d^3k##
DrDu
#3
Apr3-14, 02:50 PM
Sci Advisor
P: 3,593
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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