Forward $\delta$-scatter

As with the single perturbation case, a $\delta$-scatter cross section is added to the total cross sections of the reference, unperturbed and perturbed systems. The $\delta$-scatter cross section for the reference system ($\delta^{ref}$) is chosen, depending upon the problem, such that,

$$\delta^{ref} \ge \mid \Sigma_t^{ref} - max(\Sigma_t^{p_i}) \mid\;\;\;; if\; max(\Sigma_t^{p_i}) \gt \Sigma_t^{up},$$ (92)

or,

$$\delta^{ref} \ge \mid \Sigma_t^{ref} - \Sigma_t^{up} \mid\;\;\;; if\;\Sigma_t^{up} \gt max(\Sigma_t^{p_i}).$$ (93)

The $\delta$-scatter cross section for the unperturbed system ($\delta^{up}$) and perturbed systems ($\delta^{p_i}$, i = 1,2,3,...,N) are chosen as,

$$\delta^{up} = \Sigma_t^{ref} + \delta^{ref} - \Sigma_t^{up},$$ (94)

and,  

$$\delta^{p_i} = \Sigma_t^{ref} + \delta^{ref} - \Sigma_t^{p_i}.$$ (95)

The conditions imposed by equation (4.5) and (4.6) ensure that all the $\delta$-scatter cross sections are nonnegative. In our test problems the above four equations are used to determine forward $\delta$-scatters for the reference, unperturbed and all perturbed systems. The distance to collision d in the reference system is sampled from ${({\Sigma_t}^{ref}+\delta^{ref})}{exp(-(\Sigma_t^{ref}+\delta^{ref})d)}$, and the modified biasing factors for the unperturbed and all perturbed systems become,

$$WF^{up} = {{{({\Sigma_t}^{up}+\delta^{up})}{exp(-(\Sigma_t^{... ...ef}+\delta^{ref})}{exp(-(\Sigma_t^{ref}+\delta^{ref})d)}}} = 1,$$ (96)

and,

$$WF^{p_i} = {{({\Sigma_t}^{p_i}+\delta^{p_i})}{exp(-(\Sigma_t... ...ref}+\delta^{ref})}{exp(-(\Sigma_t^{ref}+\delta^{ref})d)}} = 1.$$ (97)

This avoids large fluctuations in WF^up and WF^p_i.