[en] This paper introduces nonstatistical probability tables that take into account the actual neutron slowing down. The authors' goal is to enable the use of probability tables for any reactor physics application. Toward this purpose, they introduce cross-probability tables that contain the information lost by the simple probability tables. This method has the advantage of being general, without specific treatment for the wide or the intermediate resonances. The authors have shown that the method can work, but its accuracy needs improvement

[en] To compute the self shielding coefficient, it is necessary to know the point-wise cross-sections. In the unresolved resonance region, we do not know the parameters of each level but only the average parameters. Therefore we simulate the point-wise cross-section by random sampling of the energy levels and resonance parameters with respect to the Wigner law and the X² distributions, and by computing the cross-section in the same way as in the resolved regions. The result of this statistical calculation obviously depends on the initial parameters but also on the method of sampling, on the formalism which is used to compute the cross-section or on the weighting neutron flux. In this paper, we will survey the main phenomena which can induce discrepancies in self shielding computations. Results are given for typical dilutions which occur in nuclear reactors. 8 refs

[en] ''Probability tables'', established to describe neutron cross-sections in a given energy group, can be introduced directly into the multigroup system of equations: this is the principle of the subgroup or multiband method. We proposed to decompose the flux on a complete eigenflux basis. For this purpose we consider the eigenvalue problem for one-dimensional geometries. Our aim is to establish space probability tables which will describe the eigenflux as accurately as required. So far we succeed in the cases of the infinite slab and the sphere. These probability tables are the tables required by the GAUSS quadrature and are well suited to any integral calculation. We outline the application of this method to the treatment of the transport equation. 9 refs

[en] The idea of describing neutron cross-section fluctuations by sets of discrete values, called ''probability tables'', was formulated some 15 years ago. We propose to define the probability tables from moments by equating the moments of the actual cross-section distribution in a given energy range to the moments of the table. This definition introduces PADE approximants, orthogonal polynomials and GAUSS quadrature. This mathematical basis applies very well to the total cross-section. Some difficulties appear when partial cross-sections are taken into account, linked to the ambiguity of the definition of multivariate PADE approximants. Nevertheless we propose solutions and choices which appear to be satisfactory. Comparisons are made with other definitions of probability tables and an example of the calculation of a mixture of nuclei is given. 18 refs

[en] 1 - Description of problem or function: In the computation of Doppler- broadened resonance cross sections, use is made of the symmetric and anti-symmetric line shape functions. These functions usually denoted as Psi and Phi (Psi and Chi in Anglo-Saxon formalism) are defined in terms of the real and imaginary parts of the error function for complex arguments. They are the product of the convolution of a Gaussian function with the symmetric and anti-symmetric Breit-Wigner functions, respectively. FPSPH and DFPSPH compute these functions. 2 - Method of solution: For (1+x²) > 20 Beta², the calculation is based upon the asymptotic expansion: Psi+(i*Phi) = 1/(1-ix)*(1-t+3t²-3.5t³+3.5+7t⁴---), with: t = 1/(2z²); z = (1-ix)/Beta. The half-plane (Beta,x) is split in several parts, and use is made of PADE approximants. For 1 + x² < 20 Beta², the calculation is based upon the relation with the erf function: Psi + i*Phi = SQRT(Pi)/Beta*(e^(z2))*(1-erf(z)) (z = (1-ix)/Beta, and erf(z) being calculated from its analytic expansion: erf(z) = 2/SQRT(Pi)*z*e^(-z2)*(1+z²/3+z⁴/(3*5) + z⁶/(3*5*7)+---). PADE approximants are used to compute the expansion and e^z2

[en] The resonance self-shielding in the statistical energy range is generally treated by generating ladders of resonances, based upon the average value of parameters. From this statistical approach results dispersion inherent to Monte-Carlo methods. Several methods have been proposed to reduce this dispersion, but they introduce a bias. We propose a method -regularisation of the partial width sampling- which is basically exempt of bias. For resonance energies the most exact method -eigenvalues of a random matrix- greatly reduces the dispersion, and has to be recommended. We compare our results to few other calculations - in particular to values obtained by analytical methods, which appear to be too much approximate for medium and low dilutions. The possible accuracy is limited by the knowledge of data, and cannot be improved by calculations. The only way to improve this accuracy is to provide more information -in the circumstances to provide the parameters of the greatest resonances, even if they are few, the others being statistically treated

[en] Full text of publication follows: The exact treatment of the R-matrix formalism implies the inversion of a high order matrix: either the channel matrix R, or the level matrix A. There is a perfect agreement between the two inversion processes. Accepting a statistical assumption, the Reich-Moore method, based on a channel elimination method for the R matrix, provides very good agreement, except in the resonance dips. Based on an expansion of the A matrix , two multilevel approximations can be generated: -) the well known MLBW approach, which computes that the total cross-section as the sum of the partial cross-sections, -) and another used at CEA for many years named BWMN (Breit-Wigner Multi-Niveaux) which computes the scattering cross-section as the difference between the total cross-section calculated from an expansion of the A matrix. The BWMN approximation, at second order, gives negative values of the cross-section when the approximation is not valid, but provides exact average cross-sections; on the contrary, the MLBW approximation never gives negatives values, but systematically overestimates the total and scattering cross-sections. An extension to fourth order of the BWMN approach is examined. The Endf-102 format specification states: 2.3.1, Formats: 'Only the SLBW formalism for unresolved resonance parameters is allowed.' This statement is not appropriate and cannot be justified nowadays. The external resonances may introduce a bias of the total and scattering cross-sections if an asymmetry occurs (in the energy range, above and below the calculated energy rang). It appears difficult to correct such an asymmetry when using the Reich-Moore method, while this correction is possible with the multi-levels approximation is used. (author)

[en] Full text of publication follows: The common use is to accept the integer values which are given in the evaluations for ν, the number of degrees of freedom of the resonance parameter distributions; and ν=0 for the radiation widths, i.e. a constant value. We dispute these choices. For the neutron widths there are experimental results which state to ν>1, and this affects sensibly the self-shielding; assuming ν=2 for a fission where there are 2 fission channels is generally unright, and this has a small effect on the self-shielding. Accepting ν=0 for the radiation widths is also unright, and there are cases, such as ²³²Th, which may require several channels. Taking into account a possible (Γ_n,Γ_γ) correlation should also be deepened. These improvements imply modification of the evaluation format; including the Endf strange specification that the average neutron width has to be multiplied by the number of degrees of freedom, which provides errors (the case of latest ²³⁹Pu evaluations). The usual choice for the energy interpolation between average parameter (modint=2) is also inexact (case of ²³⁹Pu), and we propose to use modint=l, and give an example for ²³²Th. Also we should clearly distinguish the local fluctuations of average parameters from the true nuclear average, and not treat them in the same way. (author)

[en] Full text of publication follows: It is well known that ν, the number of prompt neutrons emitted by fission of Pu²³⁹ varies significantly in the resolved resonance range. It has been demonstrated that these fluctuations could be described by allocating specific ν values to the two spin states 0⁺ and 1⁺. Furthermore, a different ν value to the (n,γf) process of the 1+ spin state can be attributed. From these facts, it may result a variation of ν_eff (used in multigroup calculations or in the unresolved resonance range) when the self-shielding process is taken into account. Calculations in both resolved and unresolved energy ranges have been performed to determine the importance of this effect. (author)

[en] The straightforward method for calculation of resonance self-shielding is to generate one or several resonance ladders, and to process them as resolved resonances. The main drawback of Monte Carlo methods used to generate the ladders, is the difficulty of reducing the dispersion of data and results. Several methods are examined, and it is shown how one (a regularized sampling method) improves the accuracy. Analytical methods to compute the effective cross-section have recently appeared: they are basically exempt from dispersion, but are inevitably approximate. The accuracy of the most sophisticated one is checked. There is a neutron energy range which is improperly considered as statistical. An examination is presented of what happens when it is treated as statistical, and how it is possible to improve the accuracy of calculations in this range. To illustrate the results calculations have been performed in a simple case: nucleus ²³⁸U, at 300 K, between 4250 and 4750 eV. (author)