Normal coordinates for harmonic approximation (classical lattice vibration)
I am reading <PERSON> "Fundamentals of the Physcs of Solids" vol. 1. and i am very much stuck at this point (chapter 11.3.2 in the book):
In the harmonic approximation the potential energy of a lattice with a base can be written as:
$$U_{harm}= {1 \over 2} \sum \limits_{m, \mu, \alpha, n, \nu, \beta} \Phi^{\mu \nu}{\alpha,\beta}(m-n) u\alpha(m,\mu) u_\beta(n,\nu) $$
Here $m,n$ sum primitive cells, $\mu,\nu$ base atoms and $\alpha,\beta$ coordinates.
We want to express this in the normal coordinates of the Fourier transform of $\Phi^{\mu \nu}(m-n)$, called the dynamical matrix $D$ and the instantatious displacements $u_\alpha(m,\mu,t)$ expressed in the eigenvectors $e^{(\lambda)}$ of $D$ in the normal coordinates $Q$:
$$D^{\mu \nu}{\alpha,\beta}( \vec q)={1 \over \sqrt{M\mu M_\nu}}\sum_n \Phi^{\mu \nu}(m-n)e^{-i\vec q ( \vec R_m -\vec R_n)}$$
$$u_\alpha(m,\mu,t)={1 \over \sqrt{N M_\mu}} \sum \limits_{ \vec q , \lambda}e^{(\lambda)}{\mu,\alpha}(\vec q)Q\lambda(\vec q) e^{i \vec q \vec R_m} $$
Putting the $u_\alpha(m,\mu,t)$ in the equation for $U_{harm}$ gives the expression of my concern:
$$U_{harm}={1 \over 2} \sum \limits_{m, \mu, \alpha, n, \nu, \beta} \Phi^{\mu \nu}{\alpha,\beta}(m-n){1 \over \sqrt{N M\mu}} \sum \limits_{ \vec q , \lambda}e^{(\lambda)}{\mu,\alpha}(\vec q)Q\lambda(\vec q) e^{i \vec q \vec R_m} {1 \over \sqrt{N M_\nu}} \sum \limits_{ \vec q' , \lambda}e^{(\lambda')}{\nu,\alpha}(\vec q')Q{\lambda'}(\vec q') e^{i \vec q' \vec R_n} $$
The argument now goes that in the secons sum only summants with $\vec q' = \vec q $ contribute because $\Phi^{\mu \nu}(m-n)$ only depends on the difference in lattice vectors $\vec R_m$ i really don't get this and would very much appriciate an explanation of this arument or the fact tat only these summants contribute.
PS: as always any critique and questions that make it easyer for you to answer the question is apriciated. 818