Plotter

Commands for adding atoms to the scene and animating them.

Mathematics

The key equation is: [1]

\[\mathbf{u}(jl,t) = \sum_{\mathbf{q},\nu} \mathbf{U}(j,\mathbf{q},\nu) \exp(i[\mathbf{q} \mathbf{r}(jl) - \omega(\mathbf{q},\nu)t])\]

Where \(\nu\) is the mode identity, \(\omega\) is frequency, \(\mathbf{U}\) is the displacement vector, and \(\mathbf{u}\) is the displacement of atom \(j\) in unit cell \(l\). We can break this down to a per-mode displacement and so the up-to-date position of atom \(j\) in cell \(l\) in a given mode visualisation

\[\mathbf{r^\prime}(jl,t,\nu) = \mathbf{r}(jl) + \mathbf{U}(j,\mathbf{q},\nu) \exp(i[\mathbf{q r}(jl) - \omega (\mathbf{k},\nu) t])\]

Our unit of time should be such that a full cycle elapses over the desired number of frames.

A full cycle usually lasts \(2\pi/\omega\), so let \(t = \frac{2\pi f}{\omega N}\); \(-\omega t\) becomes \(-\omega \frac{2 \pi f}{\omega N} = 2 \pi f / N\) where \(f\) is the frame number.

\[\mathbf{r^\prime}(jl,t,\nu) = \mathbf{r}(jl) + \mathbf{U}(j,\mathbf{q},\nu) \exp(i[\mathbf{q r}(jl) - 2 \pi f/N])\]

The arrows for static images are defined as the vectors from the initial (average) positions to one quarter of the vibrational period (i.e. max displacement)

[1]	Dove, Introduction to Lattice Dynamics (1993) Eqn 6.18

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