It was recently demonstrated that an uncertainty decrease and non-zero correlation terms between different nuclear data reactions can be obtained when using integral information such as criticality benchmarks [ 1 ] see Refs. Such approach can be useful to lower calculated uncertainties on integral quantities based on nuclear data covariance matrices, without artificially decreasing cross section uncertainties below reasonable and unjustified values.

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This is appropriate when the propagation of uncertainties from differential data to large-scale systems indicates an apparent discrepancies between uncertainties on measured integral data neutron multiplication factor, boron concentration, isotopic contents and the calculated ones. In this reference, the correlation terms between reactions for a specific isotope and the decrease of differential uncertainties were calculated using a simple Bayesian Monte Carlo method. In the present work, the same method is applied 1 to obtain correlation terms this time between different isotopes, and 2 to decrease the uncertainties for important reactions, using again criticality-safety benchmarks.

The approach and the equations used in the present work are the same as in [ 1 ]. In the following, the case of the U and U isotopes will be considered and the Bayesian update will be performed using a specific criticality benchmark with high sensitivity to these isotopes: the intermediate metal fast number 7 benchmark, or imf7 also known as Bigten [ 5 ]. First the method will be recalled in simple terms, then the application with the imf7 benchmark will be presented. The updated benchmark value, cross sections, correlations and uncertainties will be compared to the prior values, thus demonstrating the results for the differential quantities.

This is of interest in the context of nuclear data evaluations, where both nominal values and covariance matrices can reflect the present results.

## Neutron cross section

The basic principles of the method were already presented in [ 1 ]. We will outline here the major equations. The Bayesian updates of the prior information is obtained using a Monte Carlo process: random nuclear data are produced following specific probability density functions pdf. Such pdf were obtained as follows: starting from uniform distributions, comparisons between calculations and differential measurements from EXFOR were performed.

Following the description of reference [ 6 ] and as presented below for integral data , weights are derived from such comparisons and pdf of TALYS model parameters are updated. The next step is to sample from these specific parameter pdf to produce random nuclear data;. As a prior for the nuclear data, the random U and U cross sections and emitted particles and spectra are obtained from the TENDL library [ 7 ].

The sampling between these two isotopes is performed in independent manner, so that no correlation between U and U can exist other than from the model themselves. The prior correlation matrices for U and U are simply obtained from the n random files, using the conventional covariance and standard deviation formula. The n random ACE files are then used in n MCNP6 simulations [ 15 ], leading to n values of calculated k eff , i with i varying from 1 to n.

The weights are then assigned to the corresponding U and U nuclear data files for both isotopes together which lead to k eff,i. Considering n random files for each isotopes, there is n 2 possible combinations; in the following, we will consider only n combinations such as 1,1 , 2,2 ,… i,i. Examples for the weights of the random U and U nuclear data are presented in Figure 1. In this example, one iteration i corresponds to the use of one specific random file for U and another one for U. Due to this large range of weights, a large number of random files is necessary to obtain meaningful results.

As quantities in these equations average cross sections, standard deviations and correlation factors come from a Monte Carlo process, one has to check their convergence as a function of the iteration number, as presented in Figure 2. One can see that in both cases considering or not weights w i , the final correlation values are different, and the difference is outside the standard errors defined as for the non weighted case.

### So, what am I looking at here?

As it can be seen on this figure, the non weighted running correlation evolves smoothly with the increasing number of samples, while the weighted running correlation exhibits large jumps for low iteration i where high weight samples are added to the calculation as seen in [ 16 ] showing same kind of behavior. In the following, more details will be given on the imf7 benchmark together with the results regarding the prior and posterior information for the uranium isotopes.

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Calculated weights w i for the random cases considered in this work. The number on the right are the percent of weights within the space defined by the arrows.

The weight comes from the imf7 benchmark. The gray band is the standard error on the correlation factors without weights. The work presented in [ 1 ] was limited to the single Pu isotope, since it was applied to integral experiments from the PMF subtype Plutonium Metal Fast of the ICSBEP collection [ 5 ], for which only Pu nuclear data dominate the benchmark calculation result.

The imf7 benchmark intermediate enrichment uranium metallic fast number 7 , also known as Bigten, is a highly enriched uranium core, surrounded by a massive natural uranium reflector. Bigten is a cylindrical assembly with a core composed entirely of fissionable material in metal form. This imf7 configuration has long been known by evaluators to be sensitive to nuclear data for both U and U isotopes.

This double dependency is so strong that mixing nuclear data for U from one source e.

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Some examples are presented in Table 2 by repeating the benchmark calculation with different nuclear data evaluations for U and U. As observed, if both uranium isotopes come from the same library, the calculated k eff is close to the experimental value. On the other hand, a mixture of the library of origin leads to very different calculated k eff. These cases can be interpreted as the effective presence of correlated isotopes in current evaluated nuclear data libraries.

This spectrum is averaged for the whole benchmark. Average neutron energy in keV causing fission or capture in the two main zones of the imf7 benchmark. Comparison of k eff calculation for imf7 by mixing the sources of the evaluations for U and U. In all cases, the probability tables are included. By extending the methodology described in reference [ 1 ], such cross-isotopes correlations can be rigorously quantified.

All combinations of neutron incident energy, observables cross sections, prompt fission neutron spectra, nubar, etc. Correlation matrices for a selection of cross sections, nubar and pfns in the case of U and U. Top: correlation without taking into account the imf7 benchmark; bottom: same, but taking into account imf7. See text for details.

In each sub-block, the cross sections are presented as a function of the incident neutron energy the lower-left part corresponds to the lower neutron energy range, whereas the higher-right part corresponds to the higher neutron energy. Four blocks are separated by two red lines, each block represents the correlation and cross-correlation for these isotopes: bottom-left: U- U, bottom-right: U- U, top-left: U- U and top-right: U- U.

As it can be seen, cross-isotopes correlations between isotopes are zero, since model parameters for both isotopes were independently sampled in this study. The lower panel shows the full U- U correlation matrix for the TMC samples of U and U, weighted according to equation 2 , where k exp is the experimental value of the imf7 benchmark, and k eff,i that derived from the U and U sampled files, indexed by i.

Obviously, that lower panel exhibits cross-isotopes correlations contrary to the upper one, and it also exhibits correlations between different types of observables like those discussed in [ 1 ].

Although the TMC treatment allows the constructions of covariance matrices between all the nuclear data observables, the matrices shown in Figure 4 are restricted to the observables which are expected to have a strong influence of k eff ; hence the n,p , n,2n , and other cross sections are not shown in this figure. The color coding of the amplitude of the correlation in Figure 4 reflects four levels of correlations: zero or very low white , low lighter blue or red , moderately strong intermediate blue or red , and very strong darker blue or red , with red identifying positive correlations, and blue negative ones.

The correlations between observables from different isotopes in the off-diagonal blocks sit in the low range. The U or U sub-matrices display some stronger correlations, mostly along the diagonal, but also for observables derived from the optical model potential total, non elastic and elastic cross sections , highlighting the role played by that model in inducing correlations in nuclear data.

As expected, similarly to the conclusions of references [ 1 , 16 ], a weak negative correlation for the posterior is observed see Fig.

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## Book Evaluated Fast Neutron Corss Sections Of U

The correlation matrix between the U capture and fission cross sections Fig. Although the crosses materializing the mean energies leading to fission and capture reactions in the core and blanket regions of the assembly both sit in the weak correlation region of the map close to the negligible correlations zone white , there are regions of stronger correlation, both positive and negative, nearby. That complex structure of the U capture and fission correlation might result from the interplay between U in the core region fast spectrum and the blanket region slower neutronic spectrum.

Again, this can be understood in order to compensate for the loss of neutrons caused from a specific inelastic cross section for instance n,inl by another one for instance n,inl 2. A very prevalent weak anti-correlation can also be observed between the fission cross section of U and the total elastic cross section of U presented in an enlarged format in Fig. They are anti-correlated since a weaker fission cross section of U can be compensated by a more efficient neutron reflector U n,el , which reflects leaking neutrons back into the U core for another attempt to fission U.

Correlation sub-matrix between the of U and the fission cross section of U. The red cross indicates the average energy of the neutron causing fission events Tab. As in Figure 4 : correlation sub-matrix between the fission and capture cross sections of U.

The red and black crosses indicate the average energy of the neutron causing fission and capture events in the core and blanket regions, respectively. As in Figure 4 : correlation sub-matrix between the fission cross section of U and the capture cross section of U. The cross indicate the average energy of the neutron causing U fission and U capture events.

As in Figure 4 : correlation sub-matrix between the fission cross section of U and the elastic cross section of U. Comparison between the posterior weighted , prior unweighted and the IAEA standard U n,f cross section and evaluated uncertainties the lines denotes the cross sections whereas the bands are the uncertainties. The weighting of TMC samples according to equations 1 and 2 not only introduces correlations between observables, but it also leads to modifications of the central values of nuclear data as well as a reduction of the variances of the various nuclear data observables.

Such updated cross sections and variances are presented in Figures 9 and 10 and for all considered quantities. The changes in the posterior cross sections are to some extent depending on the prior uncertainties. If the prior uncertainties are small, the changes will also be small.

Therefore the changes presented in Figure 10 can be different for different prior. In the case of U, that reduction brings the variance in the same order of magnitude as that of the existing experimental differential data.