Calculation of Azimuthally Dependent Thermal-Neutron Albedos for Slab by Method of Invariant Embedding

The method of invariant embedding has been applied to the calculation of differential thermal-neutron albedos for a semi-infinite ordinary concrete slab. The calculations have been performed in both cases of isotropic and anisotropic scattering in the laboratory system.

The calculated albedo data are compared with those obtained by the experiments and the semi-empirical formula fitted the detailed data obtained by Monte Carlo method. The calculated results assuming isotropic scattering are in good overall agreement with the values obtained by Monte Carlo and S_N methods, but there are some errors for azimuthally anisotropic scattering when azimuthal angle becomes large.

In this method, much less computing times within given accuracy are required for azimuthally isotropic scattering, but it is pronounced that the necessary computing times are heavily dependent on N in DP _(N/2)-1 (ξ)T_N(μ) quadrature sets when the azimuthally anisotropic scattering is considered.

It is found that, except for large N for the case of azimuthally anisotropic scattering, the calculation of differential albedo data by using invariant embedding method is much faster than those by using the Monte Carlo and the discrete ordinates methods.     

Albedo Analytical Method for Multi-Scattered Neutron Flux Calculation in Cavity

A simple formula which describes multi-scattered neutron flux in a spherical cavity was derived based on the albedo concept. The formura treats a neutron source which has an arbitrary energy-angle distribution and is placed at any point in the cavity.

The derived formula was applied to the estimation of neutron fluxes in two cavities, i. e. a spherical concrete cell with a 14-MeV neutron source at the center and the “YAYOI” reactor cavity with a pencil beam of reactor neutrons. The results of the analytical formula agreed very well with the reference data in the both problems. It was concluded that the formula is applicable to estimate the neutron fluxes in a spherical cell except for special cases that tangential source neutrons are incident to the cavity wall.     

Unbiasedness Evaluation of Statistical Error in Staged Monte Carlo Calculation

Fuel rod failure behavior has been studied under a reactivity initiated accident condition in Nuclear Safety Research Reactor (NSRR), JAERI. In the studies, inetallurgical observations showed that the incipient fuel rod failure mode was oxygen-induced embrittlement of the cladding independent of the test conditions such as fuel designs and cooling environments except for pressurized and waterlogged fuels. Development of the oxidation layers and embrittlement of β-Zry were quantitatively evaluated through the metallurgical examinations. A diffusion equation of oxygen was solved under a finite system with moving boundary conditions to obtain the oxygen concentration and evaluate the cladding embrittlement. The calculation showed that the wall thinning due to the cladding melt is needed for the complete embrittlement because the wall thinning enhances the oxygen concentration in the β-Zry, which well explain the experimental results. Therefore the failure threshold energy is determined by the cladding melting temperature. The failure threshold derived from this study is expected to be applicable to predicting the fuel rod failure behavior in computer analyses and also useful to evaluate the failure threshold energy for the new types of fuel rod.     

Considerations on Fast Neutron Albedo for Iron,Concrete and Iron-Covered Concrete Slab

Albedo data were calculated by ANISN for iron-covered concrete slab as well as for iron and concrete single layer slabs. Neutron is allowed to be incident on a slab with each of 14 energy groups ranging from 10 MeV to 1 keV and reflected with energy between and including the incident group and the lowest group. Neutron direction is described with 8 discrete angles for incidence and reflection respectively. Discussion was made for the dependency on slab thickness and on angle and energy of incidence and reflection. As a result, albedo data calculated by ANISN showed good agreement with the other similar data and consequently they were concluded to be sufficiently valid for use in shield design calculation.     

西安脉冲堆中子照相屏蔽体改建设计计算

文章采用蒙特卡罗程序耦合抽样技巧，对西安脉冲堆中子照相孔道外的屏蔽体进行改建设计计算，确定了中子照相孔道口出射束流参数和多种材料组合的屏蔽参数。实验测试结果表明，理论计算可靠，达到了改建指标。     

Neutronics of Polyethylene Thermal Moderator of Wing and Slab Geometries on Pulsed Neutron Source

Pulse width and intensity of thermal neutrons emitted from a polyethylene moderator on a pulsed neutron source are compared in two geometries of a moderator and target, namely, slab (radial type) and wing geometries (tangential type).

The pulse width for a 10cm thick moderator of the wing geometry, at which thickness saturated beam intensity is achieved, corresponds nearly to that of a 4 cm thick moderator of the slab geometry, although the maximum beam intensity for the wing geometry is some-what less than that of slab geometry. Analysis of figures of merit indicates that the wing geometry can be adopted in scattering experiments which require neutron beams that are not contaminated much by other radiations, even if it causes slight deterioration of neutronic characteristics in comparison with the slab geometry.     

Solutions of Monoenergetic Time Dependent Neutron Transport Equation in Slab Geometry

Approximate solutions to the one-velocity time dependent neutron transport equation are derived by means of the eigenfunction expansion method of Case for a slab geometry. A finite set of discrete time eigenvalues is determined by a simple formula. This formula, which is derived for the case of large slab thickness, is found to be valid also for fairly thin slabs. Each time eigenvalue generates a pair of spatial modes. The expansion coefficients to these modes are examined by numerical evaluation for various slab thicknesses. Higher spatial modes are found to play an important role in the transient time evolution of neutrons.     

Assessment of Neutron Cross Sections for Titanium with Neutron Spectrum in Titanium Pile

The measurement of angular neutron spectrum in a quasi-spherical pile of Ti was carried out by the linac time-of-flight method for the assessment of neutron cross sections for Ti in the energy range from a few keV to a few MeV. The measured spectrum in the pile is generally in good agreement with the calculated one from ENDF/B-IV (MAT = 1,286 for Ti) except in the energy range from about 60 keV to a few 100 keV, where the calculation gives considerably lower neutron flux than the measurement.

In order to investigate the cause of this discrepancy between the measured and calculated spectra, the total cross sections for Ti were measured by the transmission method. The results give larger values of total cross sections for Ti by about 30% than ENDF/B-IV below 200 keV, and smaller values by about 10% above 200 keV. These results were ensured at 55 and 147 keV by the measurement using a Si-filtered neutron beam. The calculation based on the measured cross sections shows better agreement with the measured spectrum than that based on ENDF/B-IV. The discrepancy is still observed around 100 keV.

The sensitivity analysis shows the importance of cross sections above 1 MeV and elastic cross sections in the resonance energy region to solve the disagreement between the measured spectrum and the calculated one.     

New Efficient Albedo Monte Carlo Method for Neutron Streaming Analyses

An efficient albedo Monte Carlo method newly developed has been studied by analyzing two types of experiments on neutron streaming. The method is characterized by employing the energy-angle dependent doubly differential albedos for slab, which can be calculated in a short computer time with a one-dimensional transport theory, such as the Sn method and more efficient invariant imbedding method. This paper describes the features of the present albedo Monte Carlo method, including fundamental formulas. In the analyses of the neutron streaming experiments, the calculated results agreed with the measured data within a factor of 2 for a benchmark experiment at the YAYOI reactor and within a factor of 3 for an SNR sodium duct mock-up experiment.

It is concluded that the present albedo Monte Carlo method is practical and applicable to the reactor shielding analysis concerning radiation streaming.     

Development of Discrete Ordinates SN Code in Three-Dimensional (X,Y, Z) Geometry for Shielding Design

A discrete ordinates transport code ENSEMBLE in (X, Y, Z) geometry has been developed for the purpose of shielding calculations in three-dimensional geometry. The code has some superior features, compared with THREETRAN which is the only code of the same kind so far developed. That is, the code can treat higher order anisotropic scattering and employs a coarse mesh rebalancing method. Moreover it has a negative flux fix-up routine using a variable weight diamond difference equation scheme and has a ray-effect fix-up option using a fictitious source based on S_N→P_N-1 conversion technique. Formulations for these advanced features in three-dimensional space have been derived.

As the demonstration of the capabilities of the code, several numerical analyses and an analysis of an annular duct streaming experiment in JRR-4 at Japan Atomic Energy Research Institute, have been performed.

As a result of these analyses, confirmation has been obtained for the prospect of applicability of ENSEMBLE to practical shielding design.     

Diffusion Calculation Model for Streaming Effects in Low Density Channels in LMFBRs

A study is made of the use of modified diffusion theory to calculate the negative reactivity worth of recently proposed flow activated reactor shutdown devices, in which sodium is voided from purely sodium filled channels located at the boundary between the core and the radial blanket in a Liquid Metal cooled Fast Breeder Reactor (LMFBR). Three-dimensional diffusion theory calculations using various definitions of modified diffusion coefficients in the channels are compared with each other and with three-dimensional transport theory computations. While normal diffusion coefficient is found to be inadequate for these reactivity worth calculations, the various modified diffusion coefficients appear adequate for the considered case. Modelling method of the small thickness of steel wall and interstitial sodium layer of the modules are also studied.     

适用于任意几何的特征线边界条件处理方法

边界条件处理是特征线方法（MOC）向任意三维几何拓展时遇到的难点之一。本文提出一种边界条件处理方法,既保留循环特征线中首尾相连的特性,又能像插值方法一样适用于任意几何。首先推导了平源近似下的特征线方程,提出了一种将源项和边界角通量分离处理的内迭代解法。然后证明了该解法具有唯一解,并类似于循环特征线方法给出解的构造方法。最后借助数值积分和权重插值给出迭代计算流程。采用Takeda算例、单铀球水腔模型和C5G7算例进行验证计算,k_eff的最大计算误差分别为21、319和138.8 pcm,表明方法可靠。该方法可应用于任意几何,且不需存储边界通量和进行边界迭代。     

Analysis of Neutron and Secondary Gamma-Ray Transport in Air-Over-Ground Geometry by PALLAS-2DCY-FC Code

Comparisons are made between PALLAS calculated with meaured neutron and γ-ray doses above the ground in an air-ground medium for HENRE accelerator and BREN reactor, yielding good agreements except for a 8.23-m height of HENRE source, in which the calculation overestimates the neutron dose by a factor of 1.5 due to the use of a rough angular quadrature set. Disregard of the ground results in a decrease by a factor of 2 in both neutron and γ-ray doses compared with those for the presence of the ground, while for an infinite-air medium both these doses increase with distances from the source, which indicates that the ground should not be. ignored in the neutron and secondary p-ray transport calculation. For assumed neutron skyshine calculations disregard of the ground results in a decrease in the neutron dose by a factor of 1.5 and 1.7 and also in the secondary γ-ray dose by a factor of 1.35 and 3~5 respectively for a 14-MeV source and a fission source. In addition to the importance of inclusion of the ground in neutron skyshine calculations, an additional essential factor is the secondary γ-ray production due to neutron inelastic scattering interaction with nitrogen for the 14-MeV source, or the one due to neutron capture interactions in both ground and air for the fission source.     

Boundary Condition Processing Method for MOC Calculation in Arbitrary Geometry

Boundary condition processing is one of the difficulties encountered in the application of method of characteristics (MOC) to arbitrary three-dimensional geometry. In this paper, a boundary condition processing method was proposed, which not only preserved the track continuity as cyclic track method, but also could be applied to arbitrary geometry as the interpolation method. The MOC equation was derived under the flat source approximation and an internal iterative method was proposed in which the source term and the boundary angular flux were processed separately. It was proved that the equation had a unique solution which could be constructed similarly to the cyclic track method. The iterative calculation flow was given by numerical integration and weight interpolation. Takeda benchmark, single uranium sphere model with water cavity and C5G7 benchmark were calculated to test the accuracy. The maximum error of k_eff is 21, 319 and 138.8 pcm respectively, which shows that the method is reliable. This method can be applied to arbitrary geometry without storing boundary fluxes and performing boundary iteration.     

Discrete Eigenvalues of Neutron Transport Equation with Quadratically Anisotropic Scattering

The general properties of discrete eigenvalues of the one-speed neutron transport equation with quadratically anisotropic scattering are studied analytically. We investigate the asymptotic behaviors of boundary curves of the complex discrete eigenvalues carefully and plot the distribution of the discrete eigenvalues in parameter space. Our predictions agree with numerical calculation     

Vectorization of Continuous Energy Monte Carlo Method for Neutron Transport Calculation

The vectorization method was studied to achieve a high efficiency for the precise physics model used in the continuous energy Monte Carlo method. The collision analysis task was reconstructed on the basis of the event based algorithm, and the stack-driven zone-selection method was applied to the vectorization of random walk simulation. These methods were installed into the vectorized continuous energy MVP code for general purpose uses. Performance of the present method was evaluated by comparison with conventional scalar codes VIM and MCNP for two typical problems. The MVP code achieved a vectorization ratio of more than 95% and a computation speed faster by a factor of 8–22 on the FACOM VP-2600 vector supercomputer compared with the conventional scalar codes.     

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。     

Gamma Ray Albedo Data Generated by the Invariant Embedding Method

The energy-and-angle dependent doubly differential γ-ray albedos for homogeneous semi-infinite medium have been calculated for water, ordinary concrete, soil, heavy concrete, iron, tin and lead. The procedure of calculation employed in the present paper is the invariant embedding method which is being developed to solve neutral particle transport problems in homogeneous one dimensional medium. The calculated γ-ray albedos are stored in the data base. One can easily obtain from this data base the various kinds of albedos (number, dose, energy, etc.) with a simple interface program. The accuracy of the calculated γ-ray albedos is ascertained by comparing with the Monte Carlo calculations (MCNP4A and EGS4).     

Coarse-Mesh Three-Dimensional Transport Calculation Method

A coarse-mesh 3-D (X-Y-Z, Hexagonal-Z) discrete ordinates transport calculation method has been developed. This method employs an weighted diamond difference approximation, the weight in which is a function of neutron direction and scalar flux, and can be easily incorpolated into conventional discrete ordinates transport codes. Results obtained in four-group S₄P₀ calculations on simple fast reactors indicate that, though the computer time of this method has increased by 30–50% compared with that of the conventional finite difference method using the same mesh, the errors of k_eff and the power distribution are reduced remarkably.