Integration of the reciprocal lattice

[mcodesmart]

I am studying Solid state physics from kittel and I am stuck at the following equation. I can see that the exponential term turns to the kroneckler delta, but I don't understand how the integral gives the volume of the specimen, Ω? What am I not seeing?

∫d³x f(x)e^iK.x = \sum a_G∫d³x e^i(K+G).x = Ω\suma_Gδ_k,-G

f(x) is the Fourier transform of the lattice, ie. the reciprocal lattice and he wants to prove that integration is not zero unless k is a vector in the reciprocal lattice G

 

[daveyrocket]

If k = -G, then the value in the exponential is zero, which means your integral reduces to the integral of just d^3x, which is going to result in the volume of the space over which the integration occurs.

 

[mcodesmart]

I see it now.

Is there any significance to the fact that k=-G? That is, k is in the opposite direction of the reciprocal lattice vector G