Reference System

We have chosen to define the reference system by,

$${\Sigma_x^{ref}} = {1 \over (N+1)}{\left(\Sigma_x^{up} + {\sum\limits _{i=1}^N}\Sigma_x^{p_i}\right)}\;\;\;; i=1,2,3,...,N,$$ (89)

where x denotes a different cross section type. It is possible to choose a different reference system; the optimum choice may vary from one problem to another. For all the numerical results shown in this chapter we have used equation (4.2) to determine the reference system. The Monte Carlo particle tracking is done in this reference system. The distance to collision d is sampled from $\Sigma_t^{ref}exp(-\Sigma_t^{ref}d)$. The adjusting weight factors for the unperturbed and all perturbed systems are then given respectively by,

$$WF^{up} = {{{\Sigma_t}^{up}}\over{{\Sigma_t}^{ref}}} exp{({\Sigma_t}^{ref}-{\Sigma_t}^{up})}d,$$ (90)

and

$$WF^{p_i} = {{{\Sigma_t}^{p_i}}\over{{\Sigma_t}^{ref}}} exp{({\Sigma_t}^{ref}-{\Sigma_t}^{p_i})}d\;\;\;;i=1,2,3,...,N.$$ (91)