Result estimators

The Monte Carlo transport simulation is run to obtain statistical estimates for integrals of the form:

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where $f$ is a response function that can be evaluated at an arbitrary position of the phase space, most typically a reaction cross section. These estimates are based on the collection of simulated events (collisions, track-lengths, surface crossings, etc.) that occur during the course of the simulated random walk.

The estimates can be divided into:

1. Analog estimates, based on recorded simulated physical events
2. Implicit estimators, based the expected frequency of events

Implicit estimators are derived from analog estimators, with the purpose of obtaining better statistics.

Analog estimators

Analog estimates are the most straightforward way to obtain physical results from the Monte Carlo transport simulation. Each particle history consists of a number of events containing relevant information on the transport process, which can be counted as-is, for example:

• Collisions
• Sampled reactions
• Crossed surfaces
• Neutrons emitted in fission

The integration domain is defined by separating the scores into different bins based on particle position, energy and time (and direction of motion). For example:

• Fission rate in a specific fuel pin -- count the number of simulated fission events in that fuel pin (integration over specific volume)
• Thermal neutron absorption in coolant -- count the number of neutrons absorbed in the coolant with energy in the thermal region (integration over specific volume and energy)
• Total fission rate as function of time -- count the number of fissions, and place the results in successive bins depending on the time of the event (integration over specific time)

These examples also illustrate the fact the results are always integrated over the variables.

The analog estimators used by Serpent 2 are listed below.

Implicit estimators

Implicit estimators are derived from analog estimators, but instead of scoring events that actually occured during the simulation, the estimates are based on the expected occurrance of the evens. The implicit estimators used in Serpent 2 are listed below.

Collision flux estimator (CFE)

Implicit estimators are best understood by considering the collision estimate of particle flux (CFE). When a collision occurs at a given position, the probability of sampling reaction $x$ is the ratio of the reaction cross section to material total:

$P_x = \frac{\Sigma_x}{\Sigma}$

The probability is the same whether the reaction was actually sampled or not, so counting $P_x$ as the result estimate means that the overall score reflects the statistically expected number of reactions $x$.

Since the total number of collisions is always greater than or equal to the number of sampled reactions, the implicit estimator gives better statistics. The overall score is given by the sum over all collisions:

$x_n = \sum_i s_i$

where the CFE is written as:

$s_i = \frac{f}{\Sigma}$

and $f$ is the response function and $\Sigma$ is the cross section that was used for sampling the path length. It should be noted that with delta-tracking this cross section is not necessarily the physical material total.