Dynamical matrix of amorphous system

[squirmingToad]

Dear Alamode users & developers.

I wish to ask how Alamode creates a dynamical matrix (with a wave vector q) for an amorphous system.

I calculated the phonon states of an amorphous system at different k-points. In this case, Alamode identified a supercell as a primitive cell, as expected because it does not have symmetries.

What I am not sure of is, since the supercell is the primitive cell, how exponential components necessary to build a dynamical matrix are defined.

$$ D_{\alpha\beta}(k_{1}k_{2}; \bold{q}) = \sum_{l_{2}} \Phi_{\alpha\beta}(k_{1}l_{1}; k_{2}l_{2}) \exp[i\bold{q}\cdot(R(l_{2})-R(l_{1}))] $$

For example, for crystalline systems, a difference between lattice points; R(l2)-R(l1) is used, but in amorphous systems, it only has a single lattice point (supercell = primitive cell), those expression cannot be applied.

I found a related part on the source code : line 586, dynamical.cpp

for (i = 0; i < 3; ++i) { vec[i] = system->xr_s[atm2_s][i] + xshift_s[icell][i] - system->xr_s[system->map_p2s[atm2_p][0]][i]; }

but difficult to understand, since I am not sure what these variables mean.

Thank you for your help,
Youhwan Jo

[ttadano]

In your case, the lattice index $\ell_1$ of the first atom $\kappa_1$ is always 0, namely $R(\ell_1)=0$), as you described. For each $\kappa_1$, an image (or images) of atom $\kappa_2$ having the minimum distance $d = |R(\ell_2 \kappa_2) - R(0 \kappa_1)|$ is searched following the minimum image convention. The found index (or indices) $\ell_2$ is used for the phase factor in dynamical matrix.

[squirmingToad]

Thank you Dr. Tadano. Yes I finally understood, in the case of the supercell, I should consider periodic boundary conditions to modulate a phase difference.