Abstract

The study of the eigenvalues of the neutron transport operator yields an important insight into the physical features of the neutronic phenomena taking place in a nuclear reactor. Although the multiplication eigenvalue is the most popular because of its implication in the engineering design of multiplying structures, alternative interesting formulations are possible. In this paper the interest is focused on the multiplication, collision and time eigenvalues. The transport model is considered in the spherical harmonics approximation and the study is restricted to the one-dimensional plane geometry in the monokinetic case. The spectra of the different eigenvalues are investigated using a numerical code, validating its performance against the results available in the literature. The observation of the convergence trends allows to establish the performance of even- and odd-order approximations. It is shown that in general even-order approximations yield slightly less accurate results, nevertheless they appear to converge to the reference values.

The effect of the choice of the boundary conditions according to the methodologies proposed by either Mark or Marshak is also investigated. The analysis of all the results presented allows to characterize the convergence properties of the spherical harmonics approach to neutron transport.

The spectrum of the time eigenvalues retains a very rich physical meaning, as they are the actual time constants of the time-dependent solution of the transport problem. Therefore, in the last part of the paper the behavior of the pattern of the spectrum of the time eigenvalues when changing the scattering ratio and the order of the approximation is examined.

摘要

对中子输运算子的特征值的研究对核反应堆中发生的中子现象的物理特征产生了重要的认识。尽管由于乘积特征值在乘积结构的工程设计中具有影响，所以它是最受欢迎的，但也可以使用其他有趣的公式。在本文中，关注点集中在乘法，碰撞和时间特征值上。在球谐函数近似中考虑了输运模型，在单动情况下，研究仅限于一维平面几何。使用数字代码研究不同特征值的光谱，从而根据文献中的结果验证其性能。对收敛趋势的观察允许建立偶数和奇数阶逼近的性能。结果表明，一般而言，偶数阶近似的结果精度稍差，尽管如此，它们似乎收敛于参考值。

还研究了根据Mark或Marshak提出的方法选择边界条件的影响。对所有给出的结果进行分析，可以表征球谐方法在中子输运中的收敛特性。

时间特征值的频谱保留了非常丰富的物理意义，因为它们是运输问题的时间相关解的实际时间常数。因此，在本文的最后部分中，研究了当改变散射比和近似阶数时，时间特征值频谱图的行为。