# Electron Phonon Interaction Potential

1. Aug 14, 2009

### vidur

The electron-lattice interaction potential is given by

$$V(r)=\sum_{i} Q_{i}\nabla V_{ei} \left( r- R_i\right)$$

where i is a summation over lattice sites, $$Q_i$$ is the lattice site displacement, and $$V_{ei}$$ is the coulombic interaction

Now According to Mahan's book Many particle physics, 2nd ed. pg 34 ,$$V_{ei}(r)$$ has a Fourier transform of the form

$$V_{ei}(r)=1/N \sum_{q}V_{ei}(q) e^{jqr}$$

I've had a hard time digesting this since this is a fourier summation of a periodic signal, and $$V_{ei}(r)$$ is not periodic, only a summation over all lattice sites is. Can anyone help me in pointing out what's wrong with my understanding

2. Aug 21, 2009

### olgranpappy

those 'q' are not reciprocal lattice vectors, then. they are the born van karman vectors that you can think of as being very densely spaced. I.e. we have forced V_{ei}(r) to be periodic but it is only periodic in the SYSTEM SIZE. which is not much of a constraint. get it?