Material Specification and Cross Section Processing Overview¶
Introduction by M. L. Williams and B. T. Rearden
XSProc (Cross Section Processing) provides material input and multigroup (MG) cross section preparation for most SCALE sequences. XSProc allows users to specify problem materials using easily remembered and easily recognizable keywords associated with mixtures, elements, nuclides, and fissile solutions provided in the SCALE Standard Composition Library. For MG calculations, XSProc provides cross section temperature correction and resonance self-shielding as well as energy group collapse and spatial homogenization for systems that can be represented in celldata input as infinite media, finite 1D systems, or repeating structures of 1D systems, such as uniform arrays of fuel units. Improved resonance self-shielding treatment for nonuniform lattices can be achieved through the use of the MCDancoff (Monte Carlo Dancoff) code that generates Dancoff factors for generalized 3D geometries for subsequent use in XSProc. Cross sections are generated on a microscopic and/or macroscopic basis as needed. Although XSProc is most often used as part of an integrated sequence, it can be run without subsequent calculations to generate problem-dependent MG data for use in other tools.
This chapter provides detailed descriptions of the methods and modules used for self-shielding. Self-shielding calculations are effectively a problem-specific extension of the processing procedures used to create the SCALE cross section libraries. SCALE includes continuous energy (CE) and several MG (MG) cross section libraries described in the chapter on SCALE Cross Section Libraries. The AMPX nuclear data processing system [wiarda_ampx_2015] was used to convert evaluated data from ENDF/B into CE cross sections, which were then averaged into problem-independent MG data at a reference temperature of 300K, weighted with a generic energy spectrum (see the SCALE Cross Section Libraries chapter). After being transformed in probability distributions by AMPX, the CE data require no further modifications for application to a specific problem except for possible interpolation to the required temperatures. However, in MG calculations, reaction rates depend strongly on the problem-specific energy distribution of the flux, which implies that the problem-independent MG data on the library should be modified into problem-dependent values representative of the actual flux spectrum rather than the library generic spectrum. The neutron energy spectrum is especially sensitive to the concentrations and heterogeneous arrangement of resonance absorbers, which may dramatically reduce the flux at the resonance peaks of a nuclide, thus reducing its own reaction rate —a phenomenon known as self-shielding. In general, the higher the concentration of a resonance nuclide and the more the interaction between heterogeneous lumps (e.g. fuel pins), the greater the degree of self-shielding for the nuclide.
Reference [williams_resonance_2011] gives a general description of the SCALE self-shielding methods. The individual computational modules perform distinct functions within the overall all self-shielding methodology of XSProc. More theoretical details about individual computational modules are given in 7-2 through 7-7. XSProc provides capabilities for two different types of self-shielding methods, which are summarized below.
The Bondarenko approach [ilich_bondarenko_group_1964] uses MG cross sections pre-computed over a range of self-shielding conditions, varying from negligibly (infinitely dilute) to highly self-shielded. Based on the following approximations [stammler_methods_1983] it can be shown that the degree of self-shielding in both homogeneous and heterogeneous systems depends only on a single parameter called the background cross section, “sigma0,” and on the Doppler broadening temperature:
neglect of resonance interference effects,
intermediate resonance approximation, and
During the SCALE MG library processing with AMPX, self-shielded cross sections are computed using a CE flux calculated at several background cross section values and temperatures. These are used to calculate ratios of the shielded to unshielded cross sections, called “Bondarenko factors” (a.k.a. shielding factors or f-factors). As described in the SCALE Cross Section Libraries chapter, Bondarenko factors are tabulated on the SCALE libraries as a function of sigma0 values and Doppler temperatures for all energy groups of each nuclide.
Bondarenko factors are multiplicative correction factors that convert the generic unshielded data into problem-dependent self-shielded values. The BONAMI computational module performs self-shielding calculations with the Bondarenko method by using the input concentrations and unit cell geometry to calculate a sigma0 value for each nuclide and then interpolating the appropriate MG shielding factors from the tabulated library values.
Self-shielding calculations with BONAMI are fast and are always performed for all SCALE MG sequences. However, due to the approximations (a)–(c) listed in the previous section, a more rigorous method is also provided which can replace the BONAMI results over a specified energy range, usually encompassing the resolved resonance ranges of important absorber nuclides. This approach is designated as the CENTRM/PMC method, named after the two main computational modules, although several additional modules are also used. CENTRM/PMC eliminates the main approximations of the BONAMI approach by performing detailed neutron transport calculations with a combination of MG and CE cross sections for the actual problem-dependent compositions and unit cell descriptions [williams_computation_1995]. This provides a problem-dependent pointwise (PW) flux spectrum for averaging MG cross sections, which reflects resonance cross-interference effects, an accurate slowing down treatment, and geometry-specific transport calculations using several available options. Shielded MG cross sections processed with CENTRM/PMC are usually more accurate than BONAMI, so it is the default for most SCALE MG sequences. However, depending on the selected transport option, CENTRM/PMC may run considerably longer than BONAMI alone.
The CENTRM/PMC methodology first executes BONAMI, which provides shielded cross sections outside the specified range of the PW flux calculation. Then the computational module CRAWDAD reads CE cross section files and bound thermal scatter kernels and interpolates the data to the desired temperatures for CENTRM. Using a combination of shielded MG data from BONAMI and CE data from CRAWDAD, CENTRM calculates PW flux spectra by solving the deterministic neutron transport equation for all unit cells described in the input. CENTRM calculations cover the energy interval 10-5 eV to 2 × 107 eV spanned by the SCALE MG libraries. This energy range is subdivided into three sections: (a) upper MG range: E>demax, (b) PW range: demin<E<demax, and (c) lower MG range: E<demin, where demin and demax are the boundaries of the PW range, which can be defined by user input. The default values are demin=10-3 eV and demax=2 × 104. The values encompass the resolved resonance ranges of essentially all actinide and fission product nuclides. MG transport calculations are performed in the upper and lower ranges, which are coupled to the PW transport calculation by the scattering sources.
Several methods are available for the CENTRM transport solutions within each energy range, and the default methods can be changed through parameters in the XSProc input. The discrete Sn method is default for homogeneous media and for arbitrary one dimensional (1D) slab, spherical, and cylindrical geometries with general boundary conditions. A unit cell model is used for self-shielding arrays of spherical or cylindrical fuel regions. For the common case of a square-pitch lattice with cylindrical fuel pins, the default transport solver is the 2D method of characteristics (MoC). The CENTRM MoC solution exactly models the outer rectangular cell surface using a reflected boundary condition. CENTRM also has an option for discrete Sn calculations using a 1D Wigner-Seitz cell with a white outer boundary condition. The 1D cell model is always used for spherical fuel arrays (e.g., pebbles), and can also be selected as a faster alternative than MoC for cylindrical fuel lattices. Finally, a two-region collision probability method can be used for any type of array. The two-region solver executes very fast but is usually more approximate than the MoC and Sn methods.
After CENTRM computes the average PW flux for each material zone, PMC uses the spectra to process the CE cross sections into problem-specific MG values for each material zone. A typical energy grid for the flux solution consists of 50,000–90,000 points, providing good resolution of the spectral fine-structure caused by resonance self-shielding. PMC has several options for processing the MG data, such as correcting for resonance absorption effects on the elastic removal. Shielded cross sections from PMC may also be used to perform an optional MG eigenvalue calculation with the XSDRNPM Sn module for cell-averaging and/or group collapsing of the MG values.
A variation of the standard CENTRM/PMC method is used to perform self-shielding for doubly heterogeneous cells in which cylindrical or spherical fuel elements, composed of small spherical fuel particles dispersed in a moderator material, are distributed in an array configuration. Self-shielding of this type of system requires multiple CENTRM/PMC passes, effectively representing the two levels of heterogeneity [goluoglu_modeling_2005]. First-level CENTRM calculations are performed for each type of fuel particle using a spherical unit cell to represent the array of multi-layered fuel particles distributed in the moderator matrix. Space-dependent CE fluxes from these calculations are used in the CHOPS module to compute CE disadvantage factors (fuel-average flux divided by cell-average flux) for generating cell-averaged, CE cross sections representative of the homogenized fuel compact. The spatially averaged CE cross sections are used in a second-level CENTRM transport calculation corresponding to a 1D unit cell model for the array of fuel elements, with homogenized number densities for the fuel compact. The CE flux spectrum from this calculation is used in PMC to process the final MG, problem-dependent cross sections. This entire procedure is transparent to the user and has been automated in XSProc. Reference 2 provides more details about the SCALE treatment for doubly heterogeneous fuel.
Treatment of Non-Uniform Lattice Effects
For self-shielding of lattice configurations, both the BONAMI and CENTRM/PMC approaches assume that the fuel is arranged in an infinite, uniform array of identical cells. For most pins in an actual lattice, the uniform-array approximation is satisfactory; however, self-shielding of some cells may be affected by boundary effects along the edge of the array or by the presence of water holes or control rods. These effects can be treated by incorporating a nonuniform Dancoff factor into the self-shielding calculations for the affected cells. The SCALE module MCDancoff performs a simplified one-group Monte Carlo calculation to compute Dancoff factors for arbitrary absorber mixtures within a complex (nonuniform) 3D array. The input for MCDancoff is described in 7-7. This module must be run as a standalone executable prior to the self-shielding calculations for a given sequence, and the computed Dancoff factors must be entered as XSProc input. The input Dancoff factor is used directly in defining the background cross section for BONAMI calculations. In the CENTRM/PMC methodology, the input Dancoff factor is used in CENTRM to calculate a Dancoff-equivalent unit cell, which defines a uniform lattice pitch that produces the same Dancoff value as the nonuniform lattice. The CENTRM transport calculation then proceeds as usual using 2D MoC or 1D Sn for the unit cell.