The multiplication factor (K) is defined as the dominant eigenvalue of the neutron transport equation. The eigenvalue K can also be expressed as the ratio of the number of neutrons in one generation to the number in the previous generation in a system containing fissionable material and in the absence of any external source. Monte Carlo eigenvalue simulation starts with hundreds or thousands of neutrons and follows these neutrons through many generations [Goa59, Lie68, Mih67, Whi66]. These neutron generations are referred to as batches, cycles, or stages. Within each batch, a random walk simulation, from birth to death, of all the neutrons is done. The simulation of a neutron lifetime from birth by fission, to death by capture or leakage, is called a ``history''. Appropriate probability density functions, representing physical processes such as fission, scattering, transmission, leakage etc., are sampled to determine the different states of a random walk simulation. If fission occurs, the site of the fission and the number of neutrons produced at that site are stored for the next batch. After the simulation of all the histories for a particular batch is completed, the fission production is normalized to maintain the original number of starting fission neutrons. Most codes require that the starting number of fission neutrons be approximately the same for all generations. This prevents population extinction if K < 1 and memory overflow if K > 1, and it makes the coding simpler.
One problem associated with Monte Carlo eigenvalue calculations is that the spatial source distribution for the first batch is not known a priori [Men68]. The usual procedure for handling this situation is to start with some arbitrary source distribution for the first batch, and then for each subsequent batch, select source sites based on the fission neutron production distribution obtained from the previous batch. This iterative procedure requires that a number of initial batches be discarded to eliminate the effect of the arbitrary starting distribution. It is important that enough initial batches be discarded so that the fundamental mode source distribution is reached before batch eigenvalue estimates are accumulated [Whi71].
In a Monte Carlo eigenvalue calculation there is a tradeoff between the number of batches versus the number of neutrons per batch. It is difficult to determine what combinations of batches and histories per batch will provide the most accurate result for a fixed number of neutrons (i.e., neutrons per batch times the number of batches). If a very large number of neutrons per batch is used for an eigenvalue calculation, then the variance in the eigenvalue estimator for each batch will be small. However, the fundamental mode distribution of fission neutrons may not be reached due to the relatively fewer number of batches used. The other alternative is to follow a small number of neutron histories for many batches. This will allow the neutron source distribution to reach the fundamental mode, but a large variance will arise in the fission production estimator for each batch.
Bias is another important issue involved with Monte Carlo eigenvalue calculations [Bow83, Eno90, Gel74, Gel91, Zol83]. It is known that the calculation of K using Monte Carlo batches produces biased results. This is due to the need to generate and maintain the fundamental mode eigenfunction, which is usually achieved by a batch-to-batch settling process involving some kind of normalization of the neutron population at the end of each batch. With some experience in Monte Carlo eigenvalue computation and using the relationship [Bri86, Gel94] between bias and standard deviation as guidance, the effect of bias can be made insignificant for most cases. The estimators for the variance of the eigenvalue over the batches can also be biased [Gel81, Gel90a, Mac73, Moo76, Gas75], especially for systems with high dominance ratios, due to the correlations between the neutron histories from one batch to another.