Windowed multipole representation of $R$-matrix cross sections

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.

Figure 1

Windowed multipole representation of $R$-matrix cross sections: ${}^{238}\mathrm{U}$ total cross-section (minus potential scattering) meromorphic continuation into the complex $z$ plane, for $z=\pm \sqrt{E}$ in eV. This surface's crest and thalweg line along the real axis is the $R$-matrix cross section above the zero threshold. (a) ${}^{238}\mathrm{U}$ first resonances (three $s$ waves and four $p$ waves). (b) ${}^{238}\mathrm{U}$ windowed multipole cross-section surface. (c) ${}^{238}\mathrm{U}$ first $s$-wave resonance peak. Negative z in (b) are on the shadow branch $\left\{E,-\right\}$ of mapping (2). (c) shows the resonance peaks are the saddle points between the complex conjugate poles. The black circle in (c) represents the contour integrals around the poles of the complex cross section which enable both conversion to windowed multipole covariances (Theorem t2) and analytic Doppler broadening (Theorem t3).

Figure 2

Xenon ${}^{134}\mathrm{Xe}$ Reich-Moore cross sections for spin-parity group $J^{\pi }=1/2^{(-)}$ $p$-wave resonances: the cross sections are generated using the multipole parameters from Table 1 in the multipole representation total cross section (114), as well as the reaction cross section (117) and interference one (119) to compute the scattering cross section as (31), while the capture cross section is the difference between the total and the scattering. All cross sections are identical to those computed using the Reich-Moore approximation $R$-matrix equations with the ENDF/B-VIII.0 resonance parameters. (a) First p-wave resonance and (b) Second p-wave resonance.

Figure 3

$R$-matrix cross-section uncertainty, computed either from the ENDF/B-VIII resonance parameters covariance $\mathbb{V}\mathrm{ar}\left(\mathrm{\Gamma }\right)$ (Table 2 in the Appendix), or from the multipoles covariance $\mathbb{V}\mathrm{ar}\left(\mathrm{\Pi }\right)$, as converted through (129), for both the stochastic cross sections (132) and (133) and the sensitivities approach (130) and (131). (a) ${}^{238}\mathrm{U}$ first capture resonance, parameters sampled from ENDF/B-VIII uncertainty. 30 samples shown here and (b) Cross section histogram at resonance energy $E_{\lambda }=6.67428\mathrm{eV}$.

Figure 4

$R$-matrix cross-section uncertainty, computed either from the enlarged ENDF/B-VIII resonance parameters covariance $\mathbb{V}\mathrm{ar}\left(\mathrm{\Gamma }\right)$ (Table 2 in the Appendix), or from the multipoles covariance $\mathbb{V}\mathrm{ar}\left(\mathrm{\Pi }\right)$, as converted through (129), for both the stochastic cross sections (132) and (133) and the sensitivities approach (130) and (131). (a) ${}^{238}\mathrm{U}$ first capture resonance, parameters sampled from an enlarged ENDF/B-VIII uncertainty, 30 samples shown here and (b) Cross section histogram at resonance energy $E_{\lambda }=6.67428\mathrm{eV}$.

Figure 5

Multipole sensitivities to $R$-matrix parameters $\left(\frac{\partial \mathrm{\Pi }}{\partial \mathrm{\Gamma }}\right)$. Trajectories of pole $p$ as resonance parameters $\left\{\mathrm{\Gamma }\right\}$ vary, using the SLBW approximation of the first resonance of ${}^{238}\mathrm{U}$ (the Appendix). The blue points show how the pole changes as $E_{\lambda }$ is varied with equal spacing within three standard deviations of the enlarged covariance matrix, while the green points result from equally spaced variations of ${\mathrm{\Gamma }}_{\gamma }$ within their uncertainty range (three standard deviations of enlarged covariance matrix). The Jacobians $\left(\frac{\partial \mathrm{\Pi }}{\partial \mathrm{\Gamma }}\right)$ from system (128) are the tangents of these trajectories from the mean pole $p$ (red reference point) and are shown in solid lines. Complex pole $p$ units are $\sqrt{\mathrm{eV}}$.

Figure 6

Accuracy of different Doppler-broadening methods. Using the SLBW resonance description given in the Appendix, the cross section (A1) is reconstructed at $T=0$ K. For each temperature $\{300,{10}^5,{10}^7\}$ K, the cross section is broadened using four different methods: (i) numerical integration of the Solbrig kernel (136); (ii) using the ${\psi }_T/{\chi }_T$ approximation (143) for SLBW Doppler broadening (144); (iii) conversion (A5) of the resonance parameters $\left\{\mathrm{\Gamma }\right\}$ to multipoles $\left\{\mathrm{\Pi }\right\}$ and analytic Doppler broadening of windowed multipole representation (A4) from Theorem t3 Eq. (147); and (iv) formulation of the parameters in ENDF format and processing using njoy [91]. For each temperature, the right column shows the absolute relative error for methods (ii), (iii), and (iv) to the direct integration of the Solbrig kernel (i). Note: njoy was run with a tolerance parameter of ${10}^{-2}$ as higher accuracy required a prohibitively long computation time. (a) T = 300 K, (b) $\mathrm{T}={10}^5\mathrm{K}$, and (c) $\mathrm{T}={10}^7\mathrm{K}$.