The Coulomb potential in SMASH includes the electric and magnetic field. For simplicity we assume magnetostatics such that the fields can be directly calculated as

\[ \mathbf{E}(\mathbf{r}) = -\boldsymbol{\nabla} \phi(\mathbf{r}) = -\boldsymbol{\nabla}\int\frac{\rho(\mathbf{r}')} {|\mathbf{r}-\mathbf{r}'|} dV' = \int\frac{\rho(\mathbf{r}')(\mathbf{r}-\mathbf{r}')} {|\mathbf{r}-\mathbf{r}'|^3}dV' \]

and

\[ \mathbf{B}(\mathbf{r}) = \boldsymbol{\nabla}\times\mathbf{A}(\mathbf{r}) = \boldsymbol{\nabla}\times \int\frac{\mathbf{j}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}dV' = \int\mathbf{j}(\mathbf{r}')\times \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3}dV'\;. \]

These integrals are solved numerically on the SMASH lattice, where the discretized equations read

\[ \mathbf{E}(\mathbf{r}_j) = \sum_{i\neq j} \frac{\rho(\mathbf{r}_i)(\mathbf{r}_j-\mathbf{r}_i)} {|\mathbf{r}_j-\mathbf{r}_i|^3}\Delta V \]

and

\[ \mathbf{B}(\mathbf{r}_j) = \sum_{i\neq j}\mathbf{j}(\mathbf{r}_i)\times \frac{\mathbf{r}_j-\mathbf{r}_i} {|\mathbf{r}_j-\mathbf{r}_i|^3} \Delta V \]

with the lattice cell volume \( \Delta V \). For efficiency the integration volume is cut at \( R_\mathrm{cut} \), which is taken from the configuration. Note that in the final equations the summand for \(i=j\) drops out because the contribution from that cell to the integral vanishes if one assumes the current and density to be constant in the cell.

R_Cut — double, required

The radius value in fm at which the integration volume is cut.