Question

How to evaluate the following Fourier calculation?

Answer 1

Let k⃗ k→ and r⃗ r→ represent coordinates in Fourier space and real space of a crystal. If there is no translational symmetry in the real space, is it possible to evaluate the following Fourier calculation?

Let u(r⃗ )u(r→) be the displacement of the particle at r⃗ r→and u(r⃗ ′)u(r→′) be the the displacement of the particle at r′→r′→ In the Fourier representation,

u(r⃗ )=∑kv(k⃗ )exp(ik⃗ ⋅r⃗ )u(r→)=∑kv(k→)exp⁡(ik→⋅r→)

apart from the normalizing constant NN.

What is the product of u(r⃗ )u(r→) and u(r⃗ ′)u(r→′),

u(r⃗ )u(r⃗ ′)=∑k⃗ ,k′→v(k⃗ )v(k′→)exp[i(k⃗ ⋅r⃗ +k⃗ ′⋅r⃗ ′)]u(r→)u(r→′)=∑k→,k′→v(k→)v(k′→)exp⁡[i(k→⋅r→+k→′⋅r→′)]

if there is no translational symmetry? How do I evaluate this product? Because r⃗ ′r→′ is not equal to r⃗ +R⃗ r→+R→, as there is there is no translation vector in this system.