Neutron density distribution

Anti-protonic atoms. Recently neutron density distributions in a series of nuclei were deduced from anti-protonic atoms [30], The basic method determines the ratio of neutron and proton distributions at large differences by means of a measurement of the annihilation products which indicates whether the antiproton was captured on a neutron or a proton. In the analysis two assumptions are made. First a best fit value for the ratio I / of the imaginary parts of the free space pp and pn scattering lengths equal to unity is adopted. Secondly in order to reduce the density ratio at the annihilation side to a a ratio of rms radii a two-parameter Fermi distribution is assumed. The model dependence introduced by these assumptions is difficult to judge. Since a large number of nuclei have been measured one may argue that the value of Rj is fixed empirically. 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

Dimensionless flux distributions may be obtained by multiplying the neutron density distributions by the neutron speed ratio v/vo ... 

D. Vretenar, G. Lalazissis, P. Ring, Neutron density distributions for atomic parity nonconservation experiments, Phys. Rev. C 62 (2000) 045502. 

The flux (j)yjj is the first-generation flux distribution developed in the perturbed reactor starting from the fission-neutron density distribution in the unperturbed reactor. The calculation of 4)pQ takes into account all the interactions the fission-neutron can undergo up to the first fission event. Thus, p, accounts for perturbations in all nuclear parameters except in the fission cross section. Whereas p (and (/> [,) includes effects of single collision events in the perturbed system, 0pp includes effects of multiple collisions. Consequently we expect that, for many problems, Pid[0fd> ] will be more accurate than Pid[0fl> 0 ]-... 

Next we consider the problem of extracting model independent nuclear structure information from analyses of medium energy proton-nucleus elastic scattering data. Spedfically, we have in mind the ground state neutron density distributions. Studies of this type are plentiful in the literature and will not be reviewed here. The reader may refer to refs. [Ba 87, Ra79, Ra 81aj. 

In the preceding discussion it is apparent that the NRDD model systematically underestimates the overall magnitude and slope of the ditferential cross section data and it does not reproduce the angular positions of the diffractive minima. The fits can be recovered empirically by either adjusting the effective interaction [Ke 89b] or by varying the neutron density distribution, particularly in the surface region. Here we discuss the latter approadi, since it provides a simple measure of the absolute accuraqr of the theoretical model and its energy dependence. 

She third defect mentioned in the previous section for the thermal flux distribution vas the omission of the contribution of the slowlng-dowi neutrons to the thermal spectrum. It must be recognised that the thermal and epithermal flux distributions overlap, it Is here vhere It, 1s most useful to coofblne the two distributions and again introduce the Westcott fhix formulation. Westcott defines a neutron-density distribution per unit speed interval as follows ... 

An understanding of the properties and behavior of nuclear chain reactors is achieved through a study of the neutron population which supports the chain. Information about the neutron population is conveniently expressed in terms of the neutron-density-distribution function. 

The detailed features of the chain reaction are determined by the various nuclear processes which can occur between the free neutrons and the materials of the reactor system. As in chemical chain reactions, the rates of the reactions involved in the chain are directly dependent upon the density of the chain carrier, in the present case the neutrons. Thus in order to determine the various properties of a reactor, such as the power-production rate and the radiation-shielding requirements, it is necessary to obtain the fission reaction rate throughout the system and, therefore, the neutron-density distribution. In fact, all the basic nuclear and engineering features of a reactor may be traced back ultimately to a knowledge of these distribution functions. 

The subject of reactor analysis is the study of the analytical methods and models used to obtain neutron-density-distribution functions. Since these functions are intimately related to various neutron-induced nuclear reactions, a knowledge of at least the basic concepts of nuclear physics is essential to a thorough understanding of reactor analysis. 

Much of the attention in reactor analysis is devoted to the calculation of the fractions/a, / , etc., defined above. A knowledge of these quantities forms the basis for determining the neutron-density distributions in the reactor. We will later derive suitable procedures for computing these quantities, not only as a function of the spatial coordinates, but also in terms of other independent variables of interest, such as the neutron energies. 

Frequently coupled with this problem is the determination of the optimum nuclear configuration which yields a minimum fuel mass. Reasonable estimates for preliminary studies can be made with relatively little effort, and many crude analytical models are available for this purpose. Accurate estimates require more elegant methods or the use of critical experiments. Although precise mass figures per se are only infrequently required in modern practice, this information is usually available in every reactor study as the by-product of solutions to more essential problems involving neutron-density distributions. As a practical matter, relatively large discrepancies in mass estimates can be readily accommodated with the increased availability of high-enrichment fuel samples. 

Another class of time-dependent problems of concern to the reactor physicist are questions on fuel burnup, poison production and burnup, breeding ratio, and the like. These problems differ from those on reactor stability in that they involve time scales measured in hours (or years) in contrast to stability problems which are concerned with fractions of a second. Reactor-analysis problems, such as the determination of critical mass and neutron-density distributions, are based on the steady-state operating condition of the reactor. The day-to-day operation of the reactor at steady state involves, however, long-time changes in the fuel concentration. Except in the case of circulating-fuel reactors, the fuel is introduced into the reactor according to some predetermined cycle. As the fuel is consumed, some gradual adjustments can be made by means... 

In this treatment we examine the nuclear reactor with the aid of an infinite- medium model which describes the neutron-density distributions in terms of the kinetic energy of the neutrons. This model is the immediate extension of the one-velocity approximation introduced in Chap. 3. In the present analysis we discard the one-velocity restriction and attempt to describe the trajectory of the neutron in energy space as it is slowed down by elastic-scattering collisions with the nuclei of the medium. In... 

These conditions are perfectly general and must be met, at least in approximate form, by whatever function is specified for the neutron-density distribution. The neutron flux defined by the elementary diffusion theory cannot meet all these conditions. The function 0(r) is much too coarse a description of the neutron population in that it gives the track length per unit volume per unit time of all neutrons of a single speed. Conditions such as (2) above require, however, a much more detailed statement, and so must be replaced by less exacting requirements. Before inquiring into the nature of these approximations, let us first examine the implications of the three conditions as stated. 

Figure A5.6 Normalized thermal neutrons density distribution along cell of the operating channel (1) experimental curve and (2) design curve.

Distribution deformation near the end of operating period was explained by the nonuniform fuel bumup. The results proved a possibility of elementary diffusion theory application for determining neutron density distributions and showed the impact of the arrangement of the superheated steam channels on power distribution. 

These values are to be subtracted from the experimental total cross sections to obtain the absorption cross sections of gold and indium, and the average scattering cross sections of aluminum and iron. The experimental absorption cross sections will be corrected for hardening by use of Fig. 25.2 and connected to the cross section at the most probable velocity of the neutron-density distribution. The experimental measurements for scattering cross section will yield an average value over the neutron-density distribution. 

A plot of the neutron-density distribution will be made and compared with the expected Maxwellian distribution. The distribution as seen from the corrected counts in each channel versus velocity will be of the form... 

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