Windowed multipole representation of R -matrix cross sections

Abstract

Nuclear cross sections are basic inputs to any nuclear computation. Campaigns of experiments are fitted with the parametric R-matrix model of quantum nuclear interactions, and the resulting cross sections are documented-both pointwise and as resonance parameters (with uncertainties)-in standard evaluated nuclear data libraries (ENDF, JEFF, BROND, JENDL, CENDL, TENDL): these constitute our common knowledge of fundamental low-energy nuclear cross sections. In the past decade, a collaborative effort has been deployed to establish a new nuclear cross-section library format-the Windowed Multipole Library-with the goal of considerably reducing the computational cost of cross-section calculations in nuclear transport simulations. This paper lays the theoretical foundations underpinning these efforts. From general R-matrix scattering theory, we derive the windowed multipole representation of nuclear cross sections. Though physically and mathematically equivalent to R-matrix cross sections, the windowed multipole representation is particularly well suited for subsequent temperature treatment of angle-integrated cross sections, in particular Doppler broadening, which is the averaging of cross sections over the thermal motion of the target atoms. Doppler broadening is of critical importance in neutron transport applications, as it ensures the stability of many nuclear reactors (negative thermal reactivity). Yet, Doppler broadening of nuclear cross sections has been a considerable bottleneck for nuclear transport computations, often requiring memory-costly pretabulations. We show that the windowed multipole representation can perform accurate Doppler broadening analytically (up to the first reaction threshold), from which we derive cross-section temperature derivatives to any order-all computable on the fly (without precalculations stored in memory). Furthermore, we here establish a way of converting the R-matrix resonance parameters uncertainty (covariance matrices) into windowed multipole parameters uncertainty. We show that generating stochastic nuclear cross sections by sampling from the resulting windowed multipole covariance matrix can reproduce the cross-section uncertainty in the original nuclear data file. The windowed multipole representation is therefore a novel nuclear physics formalism able to generate Doppler broadened stochastic nuclear cross sections on the fly, unlocking breakthrough computational gains for nuclear computations. Through this foundational paper, we hope to make the windowed multipole representation accessible, reproducible, and usable for the nuclear physics community, as well as provide the theoretical basis for future research on expanding its capabilities.

Shadow poles in the alternative parametrization of R -matrix theory

“…All the other R-matrix expressions linking these operators to the scattering matrix (19), and thereby the cross Sec. (22) remain unchanged in the Reich-Moore formalism.…”

“…thus preserving the scattering matrix (19) and ultimately the cross section (22), as long as we choose more (or equal) solutions to (59) than there are levels: N S N λ . Note that our generalization (63) does not make the Kapur-Peierls operator Hermitian, since the generalized alternative level A R.M.…”

“…To validate theorems 2 and 3, we first create a verisimilar fictitious single-channel 134 Xe isotope in R-matrix formalism (instead of MLBW) by setting all the capture widths (explicit γ or eliminated capture) to zero and treating the resulting purely scattering system with Rmatrix equations, i.e., (19) and (17). We then convert these Wigner-Eisenbud R-matrix parameters into alternative parameters by solving the generalized eigenvalue system (34) TABLE V. 134 Xe alternative parameters for spin-parity group J π = 1/2 (−) .…”

“…Finally, to validate theorem 4, as well as the entire generalization to the Reich-Moore formalism we establish in Sec. IV A, we construct the corresponding cross sections using the 134 Xe resonance parameters from ENDF/B-VIII.0 with the exact R-matrix and Reich-Moore equations, i.e., (19) and (17), to compute (22). The resulting point wise crosssection values are plotted in Fig.…”

“…This means that in order to convert nuclear data libraries to alternative parameters, the nuclear physics community must first agree on how to continue the Rmatrix operators to complex wave numbers. We argue in favor of analytical continuation in our follow-up article [18], and use this to establish the windowed multipole representation or R-matrix cross sections in the third and last article [19] of what constitutes our xenon trilogy on pole parametrizations of R-matrix theory (see Supplemental Material [20]).…”