The wave vector in 1st B.Z in Bloch theory

[pallab]

TL;DR

why q=k-K

why the general wave vector q (in the proof of Bloch theorem in Ashcroft Mermin) is represented by k-K, where k is in the 1st BZ ? why not q=k+K ( usual vector form) what is special about k-K?

 

[dRic2]

What's the difference ? k and K can take both positive and negative values so all the possibilities are covered. You choose the definition which simplifies the math

 

[pallab]

no difference but I was thinking maybe k-K is the conventional one rather than k+K to use in this case or maybe there is "something special" with the k-K choice.

 

[dRic2]

No, there is nothing special. When you derive Bloch theorem just start by defining a new vector G = -K and then you get to define q as k+G but the whole proof remains the same. It's a matter of taste.