Modeling charge cloud dynamics in cross strip semiconductor detectors

When a $\gamma$-ray interacts in semiconductor detector, the resulting electron-hole charge clouds drift towards their respective electrodes for signal collection. These will expand over time due to both thermal diffusion and mutual electrostatic repulsion. Solutions profiles are well understood limiting cases accounting only repulsion, but general solution including effects can be solved numerically. Previous attempts model these have taken into account broadening of profile effects, simplified shape by assuming Gaussian distributions. However, detailed important impacts on sharing multi-electrode strip detectors. In this work, we derive an analytical approximation solution, that closely replicates width profiles. This simplifies modeling