Measurements of Neutron Slowing Down Time in Graphite by Capture Gamma-Ray Detection

The neutron slowing down time in graphite was measured by means of capture γ-ray detection. Bursts of fast neutrons produced by D-T reaction were injected into a graphite pile. Detection was by Nal(Tl) scintillator covered by metallic foil of either In, Cd or Gd, serving as resonance absorber. The slowing down time was determined from the maximum point of the reaction rate registered by this counter and presented in the form of time spectrum. The values thus determined were 19.2±0.6, 39±3, 43.0±0.7, 46.1±1.0, 47.5±0.8, 55±2 and 35±6 μs, for 1.58, 0.48, 0.44, 0.40, 0.36, 0.30 and 0.17eV respectively. These values agree well with calculated based on crystal model. The time response of the resonance detector also was in reasonable agreement with calculation. The effect of the medium being finite and that of detector position on the observed slowing down time were estimated. From this experiment, it became clear that the process of the neutron slowing down shows some definite influence of the crystal effect of graphite and is explained pretty well by Williams' model.     

Measurement of Neutron Slowing Down Time in Graphite

A series of pulsed neutron experiments was performed to investigate the chemical binding effect on the neutron slowing down time. Bursts of D-T neutrons of 1 μsec width were generated in a hexagonal prism with 240 cm high with 120 cm flanks, made of reactor grade graphite with a density of 1.54. The slowing down neutrons were detected by bare as well as energy-selective filter-covered BF₃ counters, and analyzed with a 256 channel time analyzer. Slowing down times in graphite were determined by interpreting the increment of the difference of events between the two counters to be due to the contribution made by the fraction of the slowing down neutrons at the time of measurement that was below the cut-off energy of the filter. The results of measurements showed good agreement with calculation based on the crystal model.     

Effective Parameter Method in Time-Dependent Neutron Slowing Down Problem

Examining properties common to various analytical solutions obtained by many authors, a new method is proposed for solving the time-energy distribution of neutrons in an infinite homogeneous medium. The essence of this method is to express the solution in terms of an effective parameter [Etilde](t, u) which may always be approximated by a simple rational fraction of time. Coefficients in this rational fraction are determined by using the time moments.

The present method gives a variety of practically correct closed-form solutions relevant to the set of moments used. Applications of the present method to several cases show that a practically correct solution can be obtained with use made of only the first several time moments.     

Effective Parameter Method in Space-Dependent Neutron Slowing Down Problem

A solution of the space-dependent neutron slowing down equation is derived in terms of two effective parameters Σ(z,u) and ξ(z, u) which are shown to characterize the spatial and lethargy distributions, respectively. The expression thus obtained for the solution is quite simple, and permits easy understanding of the neutron transport phenomena, especially at large distances. The solution consists of two components—one representing the first few collisions between neutrons and atoms, the other corresponding to further collisions between them. The spatial distribution of the latter at large distances is mainly governed by that of the former.

Thus, the problem reduces mainly to the evaluation of these two parameters. The parameters are proved, from both qualitative and quantitative considerations, to be smooth functions of the spatial variable. They can be approximated in terms of simple rational fractions of the spatial variable all over the space of interest. Hence, the moment-matching procedure can very effectively be applied to the determination of the coefficients included in the rational fractions.     

Some New Applications of Neutron Slowing Down Time Spectrometry

A lead neutron slowing-down-time spectrometer LESP (LEad Standard Pile or LEad neutron SPectrometer) with a number of novel features was devised. It was applied to measurements on the effective sensitivity of counters and on the effective absorption and total cross sections of reactor materials. The basic principle of this method centers around the correlation that exists between the average neutron energy and the neutron slowing-down-time after fast neutrons are pulsed into a block of heavy medium with small neutron absorption such as a large lead assembly. This is analogous to the well established time-of-flight spectrometry.

The proposed method of spectrometry should be suitable for measurement of such effects as the resonance self and mutual shielding and geometrical heterogeneity. As trial experiment, the effective efficiency of a ²³⁵U fission chamber was measured and found to correspond fairly well with BNL-325 data. As another example, blocks of heavy resonance material such as natural uranium, antimony and tungsten were placed in the LESP for producing the standard neutron spectrum fields of these materials, and the neutron spectra and effective cross sections measured on these materials were compared with calculation.

It is concluded that these new applications of the method are quite practical for measurements of such properties as the effective efficiency of neutron counters and group averaged cross sections.     

Dependence of Effective Resonance Integrals on Moderator Slowing Down Properties

Effective resonance integrals calculated by means of the usual approximations do not have the dependence on moderator slowing down properties which is shown by “exact” calculations.

In this paper a modified effective resonance integral is defined in such a way as to fit the exponential form of the resonance escape probability. Application of a generalised “intermediate-resonance” approximation to this modified effective resonance integral shows a dependence on the slowing down properties of the moderating nuclei in good agreement with “exact” calculations.     

An Improvement to the G.G. Approximation in Neutron Slowing Down through the Method of Generalized Function

The neutron slowing down problem in an infinite homogeneous medium is treated within the G.G. approximation through the theory of generalized function (g.f.). As test function for defining the g.f.'s, the source importance for the slowing down is chosen. In place of the Taylor expansion of the collision term of slowing down equation in the G.G. method we expand the adjoint collision term of the importance equation. Solutions obtained with this method clearly reproduce the Placzek wiggles, which do not appear in corresponding solutions by the orthodox G.G. method using the same order of approximation, and our solutions are in very good agreement with the exact Placzek function.     

Contribution of Heavier Nucleus on Neutron Slowing Down in Hydrogenous Mixture, (II)

The consistent method of solving the coupled slowing down equations developed in the previous paper was extended to a study of the second spacial moment of neutron slowing down in a hydrogenous mixture. The duonucleic system constituted by a mixture of hydrogen and non-hydrogen nuclei was treated analytically by application of the G.G.A. and Fermi approximations to the latter nucleus.

Applying the G.G.A. approximation, an interference effect of scattering due to the respective nuclei appeared in the quantities concerning the first and the last flight of neutrons, and it is shown that, when the mass of the non-hydrogen nucleus becomes infinitely large, our general expression coincides with the result obtained by Volkin.

Numerical calculations were performed to compare the different methods proposed in the past.     

Measurement of Neutron Slowing Down Time in Light Water,Ice, Paraffin and Santowax

Measurements of the neutron slowing down times in light water, ice, paraffin and santowax have been made by pulsed neutron technique. Bursts of D-T neutrons of 0.1_μsee width were generated in moderator cubes of 40×40×40 cm³. The slowing down neutrons were detected by bare as well as energy-selective filter-covered BF₃ counters, and analyzed with a 256-channel time analyzer. Slowing down times in the moderator were determined by interpreting the increment of the difference of events between the two counters as attributable to the fraction of the neutrons slowing down at the time of measurement below the cut-off energy of the filter.

The measured slowing down times below 0.63 and 0.43 eV agreed well with theoretical values on the 0°K free gas model. On the other hand, the measured values below 0.20 eV were found to be appreciably greater than the theoretical, which would appear to indicate that he effect of thermal agitation and chemical binding come into play at this range of energy.     

Contribution of Heavier Nucleus on Neutron Slowing Down in Hydrogenous Mixture, (I)

The problem of neutron slowing down in hydrogenous mixtures was studied from an analytical approach. The coupled slowing down equations derived from P ₁-diffusion, using the G.G. or Fermi approximations, were solved consistently, and detailed expressions for special cases are discussed.

It is shown that, in a system comprising two kinds of nuclei, the neutron energy spectrum has a transient term near a source, representing an interference effect between the neutron moderation by the two different kinds of nuclei. The consequence of the G.G. approximation is limited mainly to the transient region so long as absorption and leakage are negligible. When the mass of the non-hydrogenous nucleus becomes infinitely large, our general expression for the neutron energy spectrum derived from the G.G. approximation coincides with the result from the simple S.G. method.     

Measurements of Neutron Thermalization Time Constant of Light Water by Pulsed Neutron Method

The thermalization parameters of light water have been measured using a pulsed neutron source and non-l/v absorbers (Cd, Gd and Sm). The pulsed neutron source utilizing the D-D reaction was placed at the center of an effectively infinite system of water poisoned with non-1/v; absorbers. The time dependent capture γ-rays from these absorbers were detected by a Nal(Tl) scintillation counter. The neutron thermalization time constant of light water was determined to be 5.3 ± 0.3,μsec from the difference of the decaying reaction rates of the γ-rays, expressed as the sum of two asymptotical exponential components. The resulting value was larger than that of Möller's experiment (4.1 ± 0.4μsec), but agreed fairly well with the theoretical value of 5.6μsec, calculated by Hoshino based on the Nelkin model. A supplementary experiment was performed using thick neutronic filters of Cd and Ag.     

子群法与特征线法结合的中子共振计算

传统的中子共振自屏计算方法采用了有理近似,局限于处理简单的共振模型,在处理复杂燃料栅元/组件时会引入较大误差。为提高复杂情况下共振计算的精度,将子群法共振模型与特征线方法结合,推导了子群法-特征线法方程。基于WIMSD格式的69群数据库,编制了可用于任意二维几何中子共振计算的SGMOC程序。通过数值验证表明,该程序计算结果与MCNP程序计算结果吻合良好,具有较高的计算精度与几何通用性。     

Intermediate Neutron Resonance Absorption with Interference Scattering in Heterogeneous Systems

A new method for intermediate treatment of resonance absorption developed for homogeneous systems has been applied to heterogeneous systems. In the present method, the interference between potential and resonance scattering is taken into account. Also taken into account is the temperature dependence of the parameters used in the intermediate treatment.

By making use of the Wigner type of simple rational approximation for the collision probability, heterogeneous systems are treated by means of the intermediate-resonance absorption, and an analytic expression is given for calculating the resonance integral. Discussions are given on the assumption of the spatially flat flux, from the intermediate resonance approach.

Calculations of the resonance integral have been performed for typical resonances of ²³⁶U in square lattices of UO₂ fuel rods and moderator. The results for the resonance integral obtained with the IR approximation are compared with the exact results. Comparisons are also made for the Doppler coefficient of the resonance integral. The results obtained with the present method will be shown to be infairly good agreement with those obtained from exact numerical approach.     

Equivalence of Taylor Series Approximation with Transcendental Equation of Neutron Slowing Down Theory under Constant Cross Sections

Abstract

Neutron moderation in an infinite homogeneous medium with constant cross sections, which is traditionally treated by Taylor series approximation methods, bears a close correspondence with the exact transcendental equation approach, as has been demonstrated by reducing the exact result to approximate forms. In this paper we give an alternative approach in which the convergence of the approximations to the exact result is derived by allowing the order of approximation to increase indefinitely. The approach is based on the fact that the usual approximations do not give a self consistent definition of the effective mass parameter. The present result can hence be considered as the slowing down counterpart of Davison's treatment of the one speed equation in the P_l approximation.     

Generalized Exact Solution of the Slowing Down Equation

The slowing down equation for an infinite homogeneous monoatomic medium is solved exactly. The solution is obtained in analytical form. The calculated collision density is compared with the one obtained by Teichmann and Stefanovi'c respectively. For the special case of hydrogen, the present solution reduces to Bethe's solution.     

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.