Mathematical Definitions

Similar to the atomic model, $G(u)$ is initially defined as

\[G(h,k,l) = \sum_j e^{-i (x'_j (h+G_h) + y'_j (k+G_k) + z'_j (l+G_l))} \\\]

where $x'_j, y'_j, z'_j$ are atom positions and $h,k,l$ represent a distance away from some scattering vector $G_h, G_k, G_l$ in reciprocal space. However, $x'_j, y'_j, z'_j$ can be thought of as an addition of lattice spacings and displacement vectors, i.e. $x_j+ux_j, y_j+uy_j, z_j+uz_j$. Then, if $G_h,G_k,G_l$ are reciprocal lattice vectors, we find that $x \cdot G$ is an integer multiple of $2\pi$, so it does not affect the simulated electric field. We are then left with

\[G(h,k,l) = \sum_j e^{-i (x_j G_h + y_j G_k + uz_j G_l)} e^{-i (ux_j (h+G_h) + uy_j (k+G_k) + uz_j (l+G_l))} \\\]

Coarse graining to get a mesoscale model, we get

\[G(h,k,l) = \sum_j \rho_j e^{-i (x_j h + y_j k + uz_j l)} e^{-i (ux_j (h+G_h) + uy_j (k+G_k) + uz_j (l+G_l))} \\\]

Again, it is important that the $h,k,l$ value are integers and that they range from $-\frac{n}{2} \to \frac{n}{2}-1$, so both real space and reciprocal space positions must be scaled. The $x'_j,y'_j,z'_j$ positions should be shifted to lie between $0 \to 1$ and should be multiplied by $2\pi$ to capture the missing $2 \pi$ scaling in the Fourier transform exponent.

Usage

Calculating the loss function and its derivative for the mesoscale model is done in three steps. First, the BcdiCore.MesoState struct is created. Then, the atom positions are set by calling BcdiCore.setpts!. Finally, the loss function is calculated with BcdiCore.loss.

state = MesoState(lossType, scale, intens, G, h, k, l)
setpts!(state, x, y, z, rho, ux, uy, uz, getDeriv)
lossVal = loss(state, getDeriv, getLoss)

If the derivative is requested with the getDeriv variable, the results are stored in state.rhoDeriv, state.uxDeriv, state.uyDeriv, and state.uzDeriv.