Calculating Phonon Dispersion from 2D Hamiltonian

[karlthecar1]

Hello,

I am constructing a 2d atomistic mass spring Hamiltonian with nearest neighbor bonds (harmonic potential) in an attempt of calculating the phonon dispersion. I solve the eigenvalue system and calculated the eigenvalues or phonon energy levels (y-values of dispersion). I am fine up to that point. So my question is how to correctly determine the wavevector (k) from the eigenvalue solution to plot omega vs. k/a. If I look at the mode shape (eigenvectors) for each eigenvalue can I determine the wavevector? If that is the case do I select a single atom within the lattice and look at the neighbors or do I have to do and average over all atoms? My next question is going to be how do you discern transverse from longitudinal modes. Any help would be great.

Thanks

 

[asheg]

I'm not sure how you implement the hamiltonian. one way is that you pick a lattice and impose periodic boundary condition on it. if you do so, you determine the k in your boundary condition and you will got several energy eigenvalues for each k you select.

If you make a hamiltonian for N*N atoms directly. then the first eigen value is for the longest phonon wavelength and shortest possible k and the higher ones (energy eigen values) related to higher k. But it would help us if you tell how you implement hamiltonian.

It's very difficult and irrational that you try to find each wave vector by looking at the eigenvectors directly. you can make an algorithm to find it. when you have N*N atoms you will have N*N k-vector space. just as an example, K_x = (U(i+1,j) - U(i-1,j))/(2a*(U(i,j))) which is ratio of derivative of displacement U to its amplitude.

U(x) = Ae^(ikx) => d(U(x))/dx / U(x) = ik