Electron Phonon Interaction Potential

[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

 

[olgranpappy]

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

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?