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?

 

[charng]

?

The electron-phonon interaction potential is a fundamental concept in solid state physics, and it plays a crucial role in understanding the behavior of electrons in materials. As you correctly stated, the potential is given by a sum over all lattice sites, where each term represents the interaction between the electron and the displacement of the lattice at that site. This potential is usually written as a function of the distance between the electron and the lattice site, r.

However, in order to fully understand the electron-phonon interaction, we need to consider the potential in momentum space as well. This is where the Fourier transform comes in. The Fourier transform allows us to express a function in terms of its frequency components, which in this case are represented by the wave vectors q. This is important because the electron-phonon interaction potential is not only a function of distance, but also of momentum.

In other words, the potential is not just dependent on the displacement of the lattice, but also on the momentum of the electron. This is why the summation over all lattice sites is not enough to fully describe the potential. The Fourier transform takes into account the periodicity of the lattice, and allows us to express the potential in terms of its frequency components.

So, to answer your question, there is nothing wrong with your understanding. The Fourier transform is necessary in order to fully describe the electron-phonon interaction potential, as it takes into account both the distance and momentum dependence of the potential. I would suggest further reading and studying of this concept in order to fully grasp its implications in solid state physics.