Neutron transport equation

Different perturbation theory expressions for reactivity are obtained from different formulations of the neutron transport equation. 

G. H. Pimbley, Solution of an initial value problem for the multi-velocity neutron transport equation with a slab geometry, J. Math. Mech. vol. 8 (1959) pp. 837-866. 

H. R. McCready and D. B. Vollenweider, An IBM-704 Program lor the Solution ol the Neutron Transport Equation in Filly Concentric Cylindrical Annuli by the Weil Method, Program L, APEX-468. 

The discrete ordinates (DO) approximation is also a multiflux model. The discrete ordinates approximation was originally suggested by Chandrasekhar [19] for astrophysical applications, and a detailed derivation of the related equations was discussed by several researchers for application to neutron transport problems [33, 57-61], During the last two decades the method has been applied to various heat transfer problems [62-81]. 

Wigner by this time (1941) had worked out many of the most familiar results from neutron transport theory (Papers 3 and 4) for example, he showed that the distribution of monoenergetic neutrons around a point source diminished asymptotically as (l/r)e, where k was the solution of the transcendental equation... 

Although the elementary laws of the interaction between neutrons and the medium of the reactor can be calculated only on the basis of quantum mechanical theories, the wave nature of the neutrons can be disregarded and classical mechanics forms the basis of the transport equations. This is evident already from the simultaneous speciflcation of energy and position in the flux. There is no reason to doubt this assumption the only case in which the wave nature of the neutrons plays a macroscopic role is the diffraction in crystalline media. Even this can be taken into account within the framework of classical transport equations by the use of anisotropic cross sections. 

In the moderator the neutron density should satisfy (17) with the i determined from (15). The expression for Ki represents a higher order of approximation to the true solution of the transport equation than is justified by the simple slowing down picture of the previous section. We therefore shall use the simplified, first-order expression for ki (not involving the 2/5 aa/o- correction) in our comparison of the two calculations. In this approximation /cf = 3iVi crti<7i i/ri, and we have in the moderator. 

IV. Spatially dependent velocity distributions. When the spectrum is independent of position, the central problem is the determination of the energy-transfer cross sections. The calculation of the spectrum once these cross sections are known is a straightforward procedure. The cross-section aspect of the problem is both more difficult from a physics point of view, and more time consuming from the point of view of machine computation. This situation is reversed when we come to consider the spatial dependence of the slow neutron spectrum. The cross sections needed are the same ones that already have been computed for the infinite medium spectrum problem. The transport equation must now be solved in at least two variables, and in a form for which the existing approximate techniques are not very well adapted. The focus of the problem therefore shifts to the development of appropriate techniques for solution of the transport equation when the energy and position variables are coupled in such a way that neutrons can both gain and lose energy in a collision. 

Analytical formulation. Call /(r, co. A) the fiux of photons or neutrons at the point r = (x,y,z), traveling in a direction co = (wxyOjy.coz) = (0,cp) with an energy represented by a suitable parameter A (wavelength for photons or lethargy for neutrons). This flux obeys the transport equation... 

The simplest example in which the singular solutions of (4) form a continuous spectrum, but can be easily overlooked, concerns one of the simplest problems of transport theory the diffusion of monoenergetic neutrons in plane geometry. In this case, t, as well as y, z, cease to be variables because the flux is assumed to be independent of them. Furthermore, since the flux is zero except for a single value of E, this is not a significant variable either. Finally, the double variable 2 can be replaced by the direction cosine /jl = 0.x of the velocity with respect to the x axis because the flux does not depend on the azimuthal angle either. If the further simplification is made that the scattering is spherically symmetric and if the total cross section is measured in appropriate units, the transport equation assumes the form... 

A counterexample. I shall, however, begin with a simple example. This may be visualized physically as the case of neutrons going unimpeded around a circular track with constant velocity (case S = 2 = 2/ = 0). The transport equations then reduce, in suitable units, to... 

Introduction. In this paper we discuss time-dependent neutron transport in a medium. Our interest centers on problems in which the neutrons interact with the medium without affecting it, and in which the neutrons do not interact with each other. Equations describing such processes are not too difficult to write down [1]. In the classical formulation they are linear, although certain new approaches to the theory have produced non-linear versions [2 3 4]. In any event, the structure of these equations usually makes it impossible to find explicit solutions, and extremely difficult to determine rigorously what properties the solutions possess. From the mathematician s viewpoint, even the existence of the solution is often not obvious. 

When this information is available, one has a well-defined stochastic initial-value problem, which one might, in principle, hope to solve by the transport equation. Since the neutron population is here assumed to have spherical symmetry, this is an integro-differential equation in four independent variables. However, a stepwise numerical integration of it is a formidable task, even for modem computing machines. 

The Monte Carlo method is in a certain sense complementary to the numerical methods based on the transport equation, in which the neutron population is treated as a continuously distributed infinity of particles while the phase space is represented by a usually rather coarse network of discrete points in the Monte Carlo method the phase space is treated as a continuum while only a finite number of neutrons are treated, as discrete particles. 

Invariant imbedding and neutron transport theory. II Functional equations,... 

The transport equation. The transport equation is linear in the neutron flux N and of first order in the variables t, R, and Q. More precisely, if N is the number of neutrons flowing in the direction ii at position R, per unit time and unit area, and if v is the neutron velocity (assumed constant), then the equation is given by ... 

The convenience of this second case has been noticed by several authors (6-8). The adjoint equation provides an alternative exact representation (dual) for the characteristic. Then, if several values of S are being studied, the effect on N may be computed by quadrature involving only one solution of the adjoint equation. This alternative exact procedure therefore avoids recomputing the equation in N for every S. Thus, Pendlebury (9) Studies the multiplication in a subcritical system as an external source of neutrons impinges at different places on the surface and in doing so has only to solve the (adjoint) transport equation once. 

The 40-group BPG code solves the transport equation for neutron flux, current, and slowing down density as a... 

A series of critical experiments with annular cylinders of uranium metal has been performed at the (WNL Critical Eiqperiments Facility to determine the adequacy of the Sq method of solving the transport equation for this geometry. The uranium has a density of 18.76 g/cm and is ouriched to 93.2% in the U-235 isotope. UnreOected annuli with outside diameters as large as 15 in. and inside diameters as smaU as 7 in. have been assembled to delayed critical, and the prompt-neutron decay constant has been measured by the Rossi-a technique. The resulting values of the decay cmistant, together with the measured multiplication cemstant of the various assemblies, are shown in Table 1. 

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