Variance Reduction Using $\delta$-scatter

Variance reduction techniques can be applied to the reference system's particles. We have applied survival biasing and Russian roulette [Lew93] to the reference system's particles and correlated the unperturbed and perturbed systems' particles to the reference system's particles. After determining the site of collision, the survival chance of the reference system particle is sampled from,

$$p_s^{ref} = \frac{\Sigma_s^{ref}+\delta^{ref}}{\Sigma_t^{ref}+\delta^{ref}},$$ (65)

where $\Sigma_s^{ref}$ is the macroscopic scattering cross section of the reference system. Next, the survival chance of the unperturbed and perturbed particles are sampled from,

$$p_s^{up} = \frac{\Sigma_s^{up}+\delta^{up}}{\Sigma_t^{ref}+\delta^{ref}},$$ (66)

and

$$p_s^{p} = \frac{\Sigma_s^{p}+\delta^{p}}{\Sigma_t^{ref}+\delta^{ref}},$$ (67)

respectively. The weights of the reference, unperturbed and perturbed particles are modified at the site of the j^th collision as follows,

W_j^ref = W_j-1^refp_s^ref,

W_j^up = W_j-1^upp_s^up,

W_j^p = W_j-1^pp_s^p.

After the weight adjustment, Russian roulette is applied to the reference system particles. In this procedure, one checks to determine if the reference system particle weight has fallen below some minimum value, in which case the reference system particle as well as the unperturbed and perturbed particles are terminated. In this research work we have simulated isotropically scattering systems. After each collision event the new angular distribution for a neutron is chosen isotropically.