The simulation of the neutronic behaviour (safety features, fuel evolution, radioprotection, etc.) of a nuclear reactor requires precise knowledge of the reaction rates, mainly elastic and inelastic scattering, fission, radiative capture, (n, a) and (n, xn) reactions, fission product yields, angular and energy distribution of the produced neutrons, and, of course, the decay time of the radioactive nuclei. The neutron energies in a reactor range from 0.01 eV to a few MeV.

The ADSR concept extends the neutron energy up to a few hundred MeV, and requires a precise knowledge of the interactions of charged particles with the nuclei of the spallation target (neutron emission, angular distribution, production of residues, etc.). These specific codes and data are presented in chapter 6.

At present, most of the codes used to simulate the neutronics of a reactor core use data libraries describing neutron-induced reactions from 10 peV to 20 MeV. The deterministic method and the Monte Carlo approach do not use the data bases in the same way. Both methods require, however, a huge number of data in order to perform precise calculations. The neutronic libraries provide evaluated values of the cross-sections at a given temperature.

There is at present no nuclear model able to predict exactly the neutron cross-sections in the energy range covered by reactor physics. Thus, most of the data available in the neutronic libraries are evaluated from experimental measurements. Different effects have to be taken into account, for instance background, impurities in the sample, etc. In general, the evaluation of neutronic data requires different types of measurement, which do not necessarily correspond to the same energy range. Thus, a compilation step is required, to combine different types of experimental data together, and produce a list of evaluated parameters of the cross-sections. This very impor­tant step consists of fitting the data with a nuclear model. One of the approaches that can be considered is the Bayesian one, which allows a compilation of different data sets. This statistical method is based on the equation

p(AB)^p(BA)p(A), (5.1)

where A is the quantity to be determined (a resonance parameter for example) and B is the set of data. p(A/B) gives the probability distribution of the evaluated parameter. p(B/A) is the probability that quantity B will be observed if the parameter is A (likelihood function). This quantity is deter­mined by the nuclear model chosen. Finally the last term p(A) represents what is known about A before the experimental measurement. This equation shows how the information contained in a new measurement can be added to the initial data base corresponding to p(A), in order to update the knowledge of the parameter A. A simplified example of this approach is detailed below.

Depending on the neutron energy, different models (p(B/A)) are used to extract the cross-section parameters from the measurements. The R-matrix theory [67, 68] is generally applied in the energy domain where the resonances can be resolved experimentally. This range of energy depends on the nucleus studied and on the quality of the measurements. It corresponds in general to thermal energies, up to a few tens of keV. This formalism considers only binary collisions: an ingoing wave function describes the two incident particles (the neutron and the nucleus); an outgoing wave function describes the emerging reaction products (the neutron and the excited nucleus, or fission fragments, etc.). The space is divided into, on one hand, the external area, where nuclear forces are negligible, and, on the other hand, the internal region, where nuclear forces dominate. The external region is handled by pure Coulombic wave functions of the particles. Different approximations can be used to describe the excited nucleus, the Breit and Wigner approach being preferred for an isolated resonance. For the different absorption reactions i, the cross-section is a Lorentzian:

where Гп is the neutronic width and Г = ^1 rt the total width.

In these equations,

• A = ЙД/2^ECM where ^ is the reduced mass and ECM the energy of the particle in the centre of mass system.

• gJ = (2J + 1) /2(2/ + 1) where I is the spin of the target nucleus, and J the spin of the compound nucleus level.

• E0 is the energy of the resonance being studied.

More precise models calculate the resonance profile in more complex cases. However, the different parameters (width, mean energy) have to be fitted on experimental measurements. We give here a simplified example of Bayes’ method, using the least-squares approximation. Let us consider a resonance of a given nucleus. We will assume that the partial widths of this resonance are perfectly known, and that we are trying to determine only its position in energy.

We consider that a first experimental measurement has been performed, and that we have prior knowledge of the resonance position, in the form of an estimated value e1 and an associated error oy. The prior distribution of E0, given e1 and o1, is then

p(E01"1,О) dE0 « exp ^- (E°2o.2"1 ) ) dE0. (5.4)

We suppose now that we perform a new experiment concerning the position of this resonance (for example a fission experiment), and that we obtain a data set fyig. We choose the Breit-Wigner description which leads to a theoretical observable fyth « oBW(Ei)g. Thus, the likelihood of obtaining the values f yi g, given the theoretical f ythg, is

P(yil/h df yig ~ exp (- X (Уі 2fly’ ) ) df Уіg (5.5)

where o2 = ((yi — yth)2) and dfyg = dyj dy2 • —dyN. Finally we obtain a new distribution of the energy position, given by the product of the like­lihood function by the prior distribution:

p(E„)iE„ = Ц-ДХ — X) dE«. (5.6)

This evaluation contains the knowledge taken from the two successive experimental data sets. The correlation between prior and new data has been neglected here, but could be taken into account.

This approach can be generalized to a vector of parameters X = {X1, X2,XN g (for example position, total and partial widths of a resonance). Prior knowledge of this vector is in the form of an estimated vector E and an associated covariance matrix A (with A = ((E — X)^(E — X))), which describes the uncertainties and correlations of these parameters. The probability distribution of X can be written as

p(XE, A) ~ exp(— 1 (E — X)fA—1 (E — X)) dX (5.7)

where dX is the volume element of the N-dimensional parameter space.

This evaluation step requires sophisticated computer programs. For example, the SAMMY [69] code has been developed at the Oak Ridge National Laboratory (ORNL), and allows one to parametrize the experi­mental data (essentially neutron time-of-flight data), using Bayes’ method, together with the multilevel R-matrix description. This code gives the best fit of the experimental data and the associated covariance matrix. Physics effects, such as the Doppler effect, are taken into account.

For heavy nuclei, the resonances can be separated for energies below a few tens of keV. Above this energy, the resonances are too close to be experi­mentally separated. In this domain, the evaluated cross-sections are built with resonance parameters that are chosen randomly. Individual partial widths are chosen following the Porter-Thomas [70][29] distribution:

n nx 2" 1

2Г(ЙІ T) e with x = г(а)/(Г(а)).^ The average values (T(q)) are extrapolated from the region of well-separated resonances or from nuclear model estimates. In the Porter-Thomas distribution, n is the number of degrees of freedom. For neutron elastic scattering widths, there is only one final state, so that n = 1. For gamma rays, there are many available levels for the primary gamma decays, n ~ 30-40. For fission, the relevant degrees of freedom are the Bohr and Wheeler transition states, and one finds typi­cally n ~ 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 [71] between next-neighbour levels with same spin and parity which reads P(S) =2 kS e—4KS with S = D/(D) and D the distance between two nearest neighbours.*

Families of resonances with different spins and/or parities are handled independently.

At higher energies, the resonance widths are larger than the spacing between each of them, and the cross-section varies smoothly with energy. In this energy domain, the optical model is generally used to evaluate the experimental measurements.

The evaluated cross-sections are usually found in nuclear data evaluated files like ENDF-B6, JEF 2.2, JENDL or BROND. These files contain the parameters of each resonance and sometimes the correlation matrix of these parameters, which allow sensitivity calculations. These files, as well as the experimental files ([30].EXFOR files in the CSIRS library), can be found on the National Nuclear Data Center (NNDC) site* at Brookhaven National Laboratory.

In general, an additional step is required and consists of decoding the data files, and producing a new one at the chosen temperature, which will be consistent with the code and the system studied. For deterministic codes, the reaction cross-section must be calculated and averaged on different energy groups, while Monte Carlo codes require, in general, continuous cross-section values obtained by a list of points and an interpola­tion procedure. For example, the program NJOY reads an ENDF format file, and writes a specific file for a Monte Carlo code such as MCNP, taking into account the Doppler broadening of the resonances.

The spallation source of the ADSR implies neutron energies of a few hundred MeV. Present nuclear data libraries contain neutronic cross-sections up to 20 MeV. Different experimental programmes aim to widen the neutronic cross-section available in the libraries up to 200 MeV.