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.

中文翻译：

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R 矩阵横截面的窗口多极表示

核截面是任何核计算的基本输入。实验活动符合参数化$\mathrm{电阻}$- 量子核相互作用的矩阵模型，以及由此产生的横截面在标准评估的核数据库（ENDF、JEFF、BROND、JENDL、CENDL、TENDL）中记录——逐点和作为共振参数（具有不确定性）：这些构成了我们的基本低能核截面的常识。在过去的十年中，已经部署了一项合作努力来建立一种新的核横截面库格式——窗口多极库——目标是大大降低核输运模拟中横截面计算的计算成本。本文奠定了支撑这些努力的理论基础。从一般$\mathrm{电阻}$-矩阵散射理论，我们推导出核横截面的加窗多极表示。虽然物理上和数学上等价于$\mathrm{电阻}$矩阵横截面，加窗多极表示特别适用于角度积分横截面的后续温度处理，尤其是多普勒展宽，这是对目标原子热运动的横截面的平均。多普勒展宽在中子传输应用中至关重要，因为它确保了许多核反应堆的稳定性（负热反应）。然而，核横截面的多普勒展宽一直是核输运计算的一个相当大的瓶颈，通常需要占用大量内存的预制表。我们表明，加窗多极表示可以在分析上执行准确的多普勒展宽（达到第一反应阈值），从中我们可以导出任意阶次的横截面温度导数——所有这些都可以即时计算（无需预先计算存储在内存中）。此外，我们在这里建立了一种转换$\mathrm{电阻}$-矩阵共振参数不确定性（协方差矩阵）到窗口多极参数不确定性。我们表明，通过从生成的窗口多极协方差矩阵中采样来生成随机核横截面可以重现原始核数据文件中的横截面不确定性。因此，窗口多极表示是一种新颖的核物理形式，能够动态生成多普勒加宽的随机核截面，为核计算解锁突破性的计算增益。通过这篇基础论文，我们希望使窗口多极表示对核物理界易于访问、可重复和使用，并为未来扩展其能力的研究提供理论基础。