Monte Carlo methods

Monte Carlo calculations follow the history of individual neutrons. The most used codes are MORSE[42] and MCNP[44]. The CERN group has writen its own code, MC2[45], which is, however, not in the public domain. The physics involved is basically the same in all these codes. Neutrons are propagated on straight paths in a medium, until they escape the medium or suffer a nuclear interaction which occurs with probability $\Sigma_{T}^{(i)}(E)$ characteristic of medium i. If the neutron exits the medium without interaction, it is, then, followed on the same trajectory, but with the new medium cross-sections. If the neutron interacts,

1.

firstly the struck nucleus (l) is chosen randomly with a weight proportional to the partial macroscopic total cross-section of this nucleus $p_{(l)}=\frac{n_{(l)}\sigma_{T,(l)}(E)}{\Sigma_{T}(E)}$

2.

secondly the type of interaction $\alpha$ is chosen randomly according to its partial weight $\frac{\sigma_{\alpha,(l)}}{\sigma_{T,(l)}}$

The cross-sections are evaluated from experimental data. They are, usually found in nuclear data evaluated files like ENDF-B6, JEF 2.2, JENDL or BROND. These files, as well as the experimental files (*.EXFOR files in the CSIRS library), can be found on the National Nuclear Data Center(NNDC) site^4.16 at Brookhaven National Laboratory. It is important to note that all resonances appearing on the evaluated files have not, necessarily, been experimentally observed.

In the resonance region the evaluation process proceeds in the following way:

1.

Extract the resonances parameters from the experimental data. These are the resonance energy E_R, the resonance width $\Gamma_{T}$ and the partial widths: neutron $\Gamma_{n}$, gamma $\Gamma_{\gamma}$ and fission $\Gamma_{f}$. In the evaluation process the widths are corrected for the broadening due to the thermal Doppler effect, and for the experimental broadening.

2.

Compute the average values of the widths $<\Gamma^{(\alpha)}>$, of the level spacing <D> and of the strength functions $<\frac{\Gamma^{(\alpha)} }{D}>$.

3.

Reconstruct the cross-sections with the corrected resonances parameters, in the region where the experimental data show well separated resonances.

4.

In regions where resonances are not well separated on the experimental data, simulated cross-sections are built with resonances parameters chosen randomly. Individual partial widths are chosen following the Porter and Thomas[46]^4.17 distribution which reads $P_{n}(x)=\frac{n}{2\Gamma(\frac{n}{2})}\left( \frac{nx}{2}\right) ^{\frac{n}{2}-1}e^{-\frac{n}{2}x}$ with $x=\frac{\Gamma^{(\alpha)}} {<\Gamma^{(\alpha)}>}$^4.18. The average values $<\Gamma^{(\alpha)}>$ are extrapolated from the region of well separated resoances or from nuclear model estimates. In the Porter and Thomas distribution n is the number of degrees of freedom. For neutron elastic widths, there only one final state, so that n=1. For gamma rays, there are many available levels for the primary gamma decays, $n\simeq$30-40. For fission, the relevant degrees of freedom are the Bohr and Wheeler transition states, and one finds, typically $n\simeq$3-4. Note that large values of n correspond to small fluctuations around the average. Resonance energies are chosen according to the Wigner interval distribution[47] between next neighbor levels with same spin and parity which reads: $P(S)=\frac{\pi}{2}Se^{-\frac{\pi}{4}S^{2}}$with $S=\frac{D}{<D>}$ and D the distance between two nearest neighbors^4.19. Families of resonances with different spins and(or) parities are treated independently.

In the continuum region, where experimental cross-sections are not available, the optical model is used to obtain cross-sections. This approach is limited to energies below 20 MeV. Efforts are presently being made to extend the optical model calculations and experimental data between 20 and 100 MeV[48]

Monte Carlo methods allow exact treatment of the most complicated geometries, the only limitation being the statistics. However for reactors close to criticality or, even more, for super critical reactors a special difficulty comes from the fact that more and more chains become infinitely long. To overcome this difficulty one stops the calculation after a fixed time t_step, or number of generations n_step, and resume it at that point with a limited sample of the results. The time over which the calculation step is carried out should be long as compared to the generation time, but small, as compared to the evolution time: $\tau_{D}<<t_{step}<\frac{\tau_{D}}{k-1}$ or $1<<n_{step}<\frac{1}{k-1}$. This condition is not, always, easily fulfilled, when the system becomes very super cricital.

The influence of very long multiplication chains on the accuracy of Monte Carlo simulations have been recently discussed by the CERN group[49]. $M=\frac{1}{1-k}$ being the total number of neutrons originating from one initial neutron, these authors give the number N of cascades to be generated to obtain a relative error $\varepsilon$ on M : $N=2.56\frac{M}{\varepsilon^{2}}\left( \frac{\nu}{2.55}\right)$ with $\nu$ the neutron number per fission. Equivalently, the precision for N cascades is $\varepsilon=1.6\sqrt{\frac{M}{N}\frac{\nu}{2.55}}$