Speaker
Description
Abstract
The Blackshaw-Murray[1] elastic kernel represents the effect of neutron up-scattering caused by thermal motion of target nuclei and resonance elastic scattering on the multigroup scattering matrices. A first implementation of this model was proposed by Ouisloumen and Sanchez and made available in proprietary cross-sections libraries of the PARAGON and APOLLO lattice codes.[2] Later, the
same technique was reimplemented as module RESK in the NECP-Atlas cross-section generating code by J. Xu, T. Zu and L. Cao.[3,4] Here, we propose an Open-Source implementation of the same module, renamed RESKR in NJOY-2012 and NJOY-2016.[5]
Introduction
The new RESKR module of NJOY generates Doppler-broadened resonance elastic scattering kernels (RESKs) in PENDF format starting from piecewise linear cross sections also in PENDF format. The input cross sections are recovered from RECONR and from a previous BROADR run.
The exact Doppler broadened energy transfer kernel produces a strong up-scattering effect in the resolved resonance energy range, above the upper limit where the $S(\alpha,\beta)$ free gas model of the THERMR module is set. The RESKR module is used to pursue Doppler broadening of the elastic kernel for resonant nuclides above the thermal domain considered by THERMR.
The RESKR module implements the Blackshaw-Murray kernel according
to the Ouisloumen and Sanchez formulas. Firstly, the resonance elastic scattering kernel (RESK) formulations for anisotropic scattering up to any Legendre order is adopted to represent the exact Doppler broadened energy transfer kernels. A semi-analytical integration method is applied to perform the RESK calculations. Combining with the RESK calculations, a linearization algorithm is proposed to generate the RESK interpolation tables. These interpolation tables are Legendre moments $\ell$ of the elastic scattering
kernels of the form $P_\ell(E \rightarrow E')$. They are written in the
output PENDF file as a new MF6 MT300 reaction type.
The main entry point is subroutine RESKR exported by module reskm. The coding logic of module reskm is similar to the one used by module thermm.
An unassigned reaction type number MT300 in ENDF-6 format is assigned to store and output the interpolation table of the RESK data and the MF3 MT300 and MF6 MT300 reaction types are defined. The incident energy grid of reaction MF3 MT300 is a subset of energy grid in MF3 MT2 selected between lower (elo) and upper (ehi) incident energy boundaries for the RESK calculation. The incident energy grid of reaction MF6 MT300 is a coarser grid set to reduce computing cost.
At each reconstructed incident energy, the moments of energy transfer kernels for the different orders are linearized simultaneously on a single unionized grid by the conventional interval-halving techniques of Cullen[6]. It ensures that all orders of the moments are reconstructed smoothly. Meanwhile, the moments of energy transfer kernels for the different orders are interpolated simultaneously once the interpolation interval is found.
A data structure in ENDF-6 format is defined as described in Table 1 where HEAD, TAB2, TAB1 and LIST are the standard types of records; ZA and AWR are the standard material charge and mass parameters; NL is the maximum Legendre order number of this table; EP is the secondary energy; P0 indicates the 0th moment of energy transfer kernel and P1 represents the 1st moment of energy transfer kernel. The TAB1 structure with embedded LIST structures is repeated for all NE incident energies. Each TAB1 structure contains NL$-1$ embedded LIST structures containing kernel values for $P_1$, $P_2$ and higher Legendre moments.
Table 1: MF6 MT300 reaction type
[MAT, 6, MT / ZA, AWR, NL, 0, 0, 0 ] HEAD
[MAT, 6, MT / T, 0.0, 0, 0, NR, NE / (INT) ] TAB2
[MAT, 6, MT / 0.0, E1, 0, 0, NRP, NEP / EP / P0(E1->EP) ] TAB1
[MAT, 6, MT / 0.0, 0.0, 0, 0, NW, 0 / P1(E1->EP) ] LIST
Repeat the LIST structure for all NL-1 orders
Repeat the TAB1 structure for all NE incident energies
After the processing, the data will be output into derived files which are the point-ENDF (PENDF) files. The reusable PENDF files can be used for generating the different multi-group cross sections and scattering matrices faced with the different requirements of dilutions and energy group structures.
References
[1] G. L. Blackshaw and R. L. Murray, ``Scattering functions for low-energy neutron collisions in a Maxwellian monoatomic gas," Nucl. Sci. Eng., 27, 520--532 (1967).
[2] M. Ouisloumen and R. Sanchez, ``A Model for Neutron Scattering of Heavy Isotopes that Acccounts for Thermal Agitation Effects," Nucl. Sci. Eng., 107, 189--200 (1991).
[3] T. Zu, J. Xu, Y. Tang, H. Bi, F. Zhao, L. Cao and H. Wu, ``NECP-Atlas: A new nuclear data processing code,'' Ann nucl. Energ, 123, 153--161 (2019).
[4] J. Xu, T. Zu and L. Cao, ``Development and verification of resonance elastic scattering kernel processing module in nuclear data processing code NECP Atlas,'' Progress nucl. Energy, 110, 301--310 (2019).
[5] R. E. MacFarlane and A. C. Kahler, ``Methods for Processing ENDF/B-VII with NJOY," Nuclear Data Sheets, 111, 2739 (2010).
[6] D. E. Cullen, ``Program SIGMA1 (Version 74-1),'' Lawrence Livermore Laboratory report UCID-16426 (January 1974).