# Biases and Statistical Errors in Monte Carlo Burnup Calculations: An Unbiased Stochastic Scheme to Solve Boltzmann/Bateman Coupled Equations

Nuclear Science and Engineering / Volume 167 / Number 2 / February 2011 / Pages 165-170

Technical Paper / dx.doi.org/10.13182/NSE09-100

External linking scripts between Monte Carlo transport codes and burnup codes, and complete integration of burnup capability into Monte Carlo transport codes, have been or are currently being developed. Monte Carlo-linked burnup methodologies may serve as an excellent benchmark for new deterministic burnup codes used for advanced systems; however, there are some instances where deterministic methodologies break down (i.e., heavily angularly biased systems containing exotic materials without proper group structure) and Monte Carlo burnup may serve as an actual design tool. Therefore, researchers are also developing these capabilities in order to examine complex, three-dimensional exotic material systems that do not contain benchmark data.

Providing a reference scheme implies being able to associate statistical errors to any neutronic value of interest like keff, reaction rates, fluxes, etc. Usually in Monte Carlo, standard deviations are associated with a particular value by performing different independent and identical simulations (also referred to as “cycles,” “batches,” or “replicas”), but this is only valid if the calculation itself is not biased. And, as will be shown in this paper, there is a bias in the methodology that consists of coupling transport and depletion codes because Bateman equations are not linear functions of the fluxes or of the reaction rates (those quantities being always measured with an uncertainty). Therefore, we have to quantify and correct this bias. This will be achieved by deriving an unbiased minimum variance estimator of a matrix exponential function of a normal mean. The result is then used to propose a reference scheme to solve Boltzmann/Bateman coupled equations, thanks to Monte Carlo transport codes. Numerical tests will be performed with an ad hoc Monte Carlo code on a very simple depletion case and will be compared to the theoretical results obtained with the reference scheme. Finally, the statistical error propagation during the time-dependent depletion process will be discussed. It will be proven that the nonlinearity of Bateman equations induces a bias in the coupling. The paper provides a formula to correct it, which is verified in a simple example.