Study of Neutron Wave Propagation

Neutron wave propagation in both non-multiplying and multiplying media is treated on the basis of the monochromatic telegrapher's equation. This approach makes it possible to take account of the aspects related to wave characteristics.

Using the Green function of the telegrapher's equation, the transient behavior of neutron waves propagating in a finite homogeneous medium is investigated. It is found that wave propagates in three different ways depending upon the size and nuclear properties of the medium. The asymptotic solutions of the telegraper's equation are next investigated. The dispersion relations of non-multiplying and multiplying media are derived.     

中子输运的随机理论

运用概率理论,考虑t时刻n个相空间点(r,u_iΩ_i)单位体元中分别出现N_i(i=1,2,…,n)个中子的概率P_N(r,t,uΩ),提出一个新的中子输运的随机理论,导出概率母函数Fn的非线性积分微分方程组。在某些近似下,n=1概率分布一阶矩方程恰好是中子平均数玻尔兹曼方程。将各向同性散射的单速中子随机理论应用于点堆模型。在一个超临界系统中,当t→∞时,出现有限个中子的概率为零,P_N=0(0＜N＜∞),即系统内或没有中子,或有无限多中子。给出了母函数的近似解,导出了母函数概率分布各阶矩的近似方程及解式。标准差公式表明,当初始中子数起伏ξ0较大,初始中子平均数N0不够多,或中子源强Q很弱时,对于0＜λ＜1的增殖系统,中子数的起伏很大,应予以重视。     

Neutron Wave Propagation in Heterogeneous Multiplying Medium

Neutron wave propagation is considered by multigroup diffusion theory in a heterogeneous multiplying system having periodic arrays of line fuel rods. The coupled equations are resolved into uncoupled equations by suitable linear transformation. By means of Green functions for a point or a line neutron source, the behavior of the neutron waves originating from a periodically varying neutron source are investigated in an infinite system.

In such a system, the neutron fluxes are represented by means of a Floquet solution relevant to the buckling B which depends on the angular frequency φ of the external neutron source. Neutron waves propagate analogously to classic waves spreading in accordance with Hüygens' & Fresnel's principles.     

改进准静态本征函数法求解反应堆时空中子动力学方程

本文提出一种改进准静态本征函数法并用于求解时空中子动力学方程。在改进准静态近似条件下,利用因子分解法将中子通量密度分解为幅函数和形状函数,进而采用本征函数法求解形状函数轴向和径向的近似解析解。在求解幅函数过程中,将幅函数化简为点堆模型下的形式进行求解,最终导出反应堆三维时空中子动力学方程的解析解。与其他解析法和数值解相比,改进准静态本征函数法的适用范围更广,计算速度更快。     

Bragg Cut-Off Effects on Neutron Wave Propagation in Graphite

Theoretical and experimental studies concerning the effects of Bragg cut-off on the neutron wave propagation in graphite are presented, as analyzed from the two following factors related to the Bragg cut-off.

First, the inordinately long mean free path of cold neutrons of energies below the Bragg cut-off causes in some cases serious problems due to the deep penetration and the strong leakage of the cold neutrons.

Secondly, the extremely low collision frequency of about 2,500 rad/sec for the cold neutrons produces a “singular frequency” at a point of even lower frequency, wherein the waves do not decay exponentially in the propagating direction.

The peculiar behavior of these cold neutrons severely restricts the conditions of experiment under which one can derive an exact value for the propagation parameter in graphite. In certain cases, however, one can obtain meaningful approximate information about wave propagation from experiments conducted beyond these limiting conditions, provided that certain precautions are taken in carrying out the experiment and in interpreting the results. Analysis to explain these phenomena and experiments to reveal the effects in graphite are described.     

Approximate Treatments of Neutron Wave Propagation Problems in Three-Dimensional System

Neutron wave propagation problems, espetially the Milne and the albedo problems, are investigated by making use of the Wiener-Hopf technique for treating the Boltzmann equation with an isotropic one-term degenerate kernel in a semi-infinite prism.

A pole of the solution of the integral-transformed Boltzmann equation corresponds to an eigenvalue—or physically, to a complex wave number varying on the two-dimensional complex plane with transverse buckling and wave oscillation frequency.

In the Milne problem, the solution ceases to exist when the imaginary part of the complex wave number exceeds σ_min. In the albedo problem, however, the solution always exists irrespectively of the oscillation frequency ω, and the discrete eigenvalue presents a continuous spectrum as soon as ω exceeds a critical frequency ω_c. Detailed forms of solutions are derived for the case of constant velocity, and complex eigenvalues are evaluated numerically.     

Solution of Multi-Dimensional Neutron Transport Equation by Finite Element Method

A scheme of treatment with the finite element Galerkin method is proposed for the approximation of solutions of multidimensional steady state neutron transport equations, and it is proved that the approximate solutions yielded by the treatment converge to the solutions of the transport equations under reasonable hypotheses. These approximate solutions are used also to show the existence of weak solutions of transport equations. Finally, based on the Galerkin method, the conditions for convergence are discussed for the multi-group approximation and for the method of spherical harmonics.     

小波尺度函数展开在连续能量中子输运方程计算中的应用

小波展开能够很好地拟合剧烈变化的函数,近年来已被应用于模拟中子角注量率随角度剧烈变化的问题,并取得了令人满意的结果.中子能谱在共振区具有剧烈震荡的特性,本文介绍了利用能群与小波尺度函数展开相耦合来离散连续能量中子输运方程中能量自变量的方法.对中子注量率在共振区关于能量用小波尺度函数进行拟合,而在快中子区和热中子区利用分群计算的方法.初步的数值结果表明,该方法使有效增殖系数计算精确,并能够得到中子注量率在共振区随能量的精细分布,对共振自屏蔽的精确计算具有重要意义.     

Neutron Wave Propagation Experiment in a Heterogeneous System of Plane Lattice

Measurements of neutron pulse propagation were carried out in heterogeneous systems of plane lattices constituted of alternate plane layers of graphite and hydrogeneous materials (either paraffin or meta-acrylite).

The effective diffusion parameters, diffusion length and diffusion coefficient for neutron propagation parallel and perpendicular to the layers of propagating medium were derived from measured complex relaxation lengths.

An elementary method of theoretical analysis is given, based on the P-1 equation with space dependent cross sections. Dispersion laws for parallel and perpendicular propagation are derived in the form of determinants of matrices. The complex inverse relaxation lengths for the experimental systems taken up have been calculated, which agree fairly well with the experimental results reported in terms of effective diffusion parameters.     

Analysis of Pulsed Neutron Experiment in Beryllium by the Method of Inelastic Scattering Expansion

Pulsed neutron experiments in beryllium have been analyzed by the method of inelastic scattering expansion. Considering the full range of energy, the present method is applied to the calculation of the time dependent spectra within a system as well as at its surface. For obtaining the decay curve of the neutron counting rate, the neutrons are classified according to the number of specified inelastic scatterings that scatter the neutrons over the Bragg cutoff energy. The pseudo mode is, as a principal result, shown to correspond to an eigen solution of the thermal bounded equation. The buckling dependence of the eigenvalue and the eigen spectrum of the thermal bounded equation are studied, and useful approximate expressions are derived. The calculated results are compared with the experiments by Fullwood et al. and other authors.     

非结构网格中子输运方程的球谐函数解法研究

从新的二阶自共扼角通量密度(SAAF：Self-Adjoint Angular Flux)中子输运方程出发．利用球谐函数对角度变量进行展开,导出了一组关于空间变量的偏微分方程组,中子通量密度的各个分量相互耦合,应用一定的迭代策略进行迭代求解。针对每一个方程,应用有限元方法对非结构网格进行离散求解。据此编写了二维球谐函数方法输运计算程序,对一系列基准题进行校算的数值结果表明,该方法具有较高的计算精度,克服了射线效应,并能用于非结构网格。     

中子输运计算界面流方法的微扰计算

以六角形几何中子积分输运计算界面流算法及其相对应的数学共扼方程计算为基础,利用微扰原理计算了当反应堆六角形组件中栅元核参数发生微量变化时系统反应性的变化。计算结果表明,本文所开发的基于六角形几何中子积分输运算法的微扰计算方法是正确的。     

中子输运方程的三角形节块SN方法研究

利用面积坐标思想,将任意三角形变换为正三角形,使用横向积分方法对正三角形节块进行处理.节块内横向积分通量、中子源的空间分布使用新的正交二次多项式近似;横向泄漏项的空间分布使用二阶多项式近似;中子通量和横向泄漏的角度通过离散纵坐标(SN)求积组离散.采用节块平衡有限差分方法建立稳定有效的迭代方案;编制了二维三角形节块SN输运计算程序(DNTR),对一系列基准题进行了验证.结果表明,本方法在同等计算精度下比细网差分程序(DOT4.2)快5～7倍,在同等计算精度和相同节块尺寸下比矩形离散节块输运方法(DNTM)快1～3倍,但DNTR程序可应用于非结构几何区域问题,具有DNTM等其它结构化节块SN程序无可比拟的优势.     

高保真中子输运计算的多级加速理论及应用

为解决高保真中子输运计算耗时严重的问题,本文提出了多级加速理论。其中,针对迭代求解过程,采用迭代加速的思路,即通过等价的低分辨率系统加速以减少迭代次数;针对瞬态求解,采用时间步加速方法,通过建立多级预估校正系统,实现在大时间步长下开展准确的高保真中子输运计算。最终引入不同分辨率系统的概念,将时间步加速方法与迭代加速方法整合形成一套完整的多级加速理论,并将其应用到精细化中子物理计算程序HNET中。采用典型瞬态基准题验证HNET程序加速效果。数值结果表明：多级加速理论能够在保证高保真中子输运计算精度的同时,极大地提升计算效率。     

强各向异性散射中子输运方程加速方法

采用修正的PN方程加速中子输运方程的SN数值求解过程.并将该加速方法应用到国际通用的ANISN程序中,对其进行了数值检验.数值计算结果表明,本文加速方法可以使迭代次数大量减少,特别是对于强各向异性散射问题具有明显的加速效果.