update time based animation

[minhsueh]

Summary

This PR builds on the related CTK PR

Major changes:

• Implemented time-based phonon animation
• Add velocity

[esoteric-ephemera]

Looks good! Couple of questions just to make sure I'm following:

• Where do the phases get set?
• With this approach, are we only sending the eigendisplacements + eigenvalues to the user and then having the browser set up the animation?

Plus some math notes for later reference:

Eigen-modes (the time-dependent displacements) are:

$$\vec x = \vec u \exp[ i(\vec q \cdot \vec R - \omega t)],$$

where $\vec u$ are the eigendisplacements (the eigenvectors of the dynamical matrix), $\vec q$ are vectors in reciprocal space ($\Gamma$ would be $\vec q$ = (0,0,0)), $\omega$ is the frequency of the mode / eigenvalue, and $\vec R$ are the positions of the atoms in real space (the cartesian coords).

With

$$\alpha = \vec q \cdot \vec r - \omega t,$$

and

$$\vec u = \vec u' + i \vec u '',$$

the real and imaginary parts of the displacement are:

$$\vec x = ( \vec u' \cos \alpha - \vec u'' \sin \alpha ) + i [ \vec u'' \cos \alpha + \vec u' \sin \alpha]$$

We need only plot the real part of $\vec x$, the term in parenthesis, which is what's done here.

[minhsueh]

@esoteric-ephemera
Thanks for the review and the math note!

• The omega ($ω$), phase ($\vec{q}$ * $\vec{r}$), eigenVectors ($\vec{u}$), and amplitude ($A$) via crystal toolkit
  • The final formula is: $\vec{x} = A \vec{u} exp[i(\vec{q}\cdot\vec{r}+ωt)]$
• Yes, that is exactly how it works!
  • Advantages:
    • Avoid large memory consumption for frame-based animation
    • No glitch when the bonds rotate
    • Velocity can be adjusted by the elapsed time $t$ with the velocity coefficient