A Markov chain algorithm incorporating correlated sampling

A particle of energy, $\varepsilon$, and charge, Q, is sampled. MC radiation transport is performed, h times. Energy, $E_{\mbox{DEP}}$, is deposited in the medium, at position ${\bf x}=(x,z,y)$, and ionisation in air, I, is calculated via the stopping power ratio of air to water, as well as dose, D_i, for a range of oblique angles, $\theta_i$. Intercepts with the surface are computed for the range of incident angles by rotation, R_i, of the position, ${\bf x}$, and direction, $\bf u$. If the intercept with any surface, t_i, is less than the path-length, s of the particle it exits in the $i^{\mbox{th}}$ geometry. When the particle history has been completed, the contribution is accepted and added to the previous estimate, X, if, $\chi^2$ is less than the variance, $\sigma^2$.