The uncertainties associated with the coupling between successive Monte Carlo (MC) steps and the integration of the (non-linear) differential equations were carefully studied by Brandan et al. [77]. They found that an integration step equal to one fifth of the shortest nuclide half-life was sufficient to reach convergence of the solution to the differential equations. They anticipate two types of error: statistical errors in the average cross­section determination after one MC step, and systematic errors due to the evolution of these cross-sections during the time of a single MC calculation.

The CPU time devoted to each MC step was fixed, thus fixing the total number of neutrons being followed and the statistical errors at all cycles, independent of the system multiplication. For instance, a CPU equal to 15 min (in a DEC Alpha 500/500 workstation) permits one to follow some 20 000 neutrons, which correspond to 400 source neutrons for a typical ks = 0.98. This way, the statistical errors in the calculated cross-sections were typically less than 10% in each cell. However, the total error after a series of MC steps may depend dramatically on the particular nucleus being studied. Tests were performed simulating the evolution of a 233U reactor over 5 years. In a first case, the differential equations were solved every 3 months of operation, i. e. the complete evolution included a total of 20 MC steps. These results were compared with a second case, with only one MC step for the 5-year solution.

Figure 5.8 illustrates the resulting time evolution of the number of 239Pu and 244Cm nuclei during the last year at the end of the evolution, for the two cases. The data points and thick curve show the results from the 20-step calculation. The thin solid curves show independent one-step results and the dashed curve represents their average. The dispersion of the one-step results reflects statistical fluctuations in the initial average cross-sections. These fluctuations, indicated in figure 5.8 by the (F) arrows, reach ~1% of the average for the production of 239Pu and ~10% for the production of 244Cm. Arrows (S) in figure 5.8 show the (systematic) discrepancies between the two procedures. The systematic error that could be made when treating the 5 year evolution with only one intermediate integration is of the order of 1% for 239Pu and of 20% for 244Cm. The difference

Time (y)

between these nuclei is due to the time evolution of their mean cross-sections, caused by the neutron spectral evolution during the operation. As figure 5.8 has shown, this variation cannot be properly taken into account by one-step calculations. All the results reported by Brandan et al. [77] have been obtained with 15 minute CPU Monte Carlo simulations per step and

20 MC steps per 5 year calculation; test calculations have shown that the systematic errors associated with this choice are negligible. Examples of particular cases where this detailed approach is absolutely required have been given by David et al. [79].