Infinite-medium Criticality Problem

Each of the original no neutrons will have a different history, but, on the average, a certain fraction of them will disappear as a result of processes (3) and (4). Only those absorbed in nuclei have a chance to produce additional neutrons a certain fraction of the absorbing nuclei 

The ratio of the neutron population in two successive generations, such as no and ni, is a useful number for describing the characteristics of the chain reaction. This ratio is called the multiplication constant of the reactor and is formally defined as 

As noted above, the system is at steady state when k = In this case, each generation reproduces itself, and the neutron population remains fixed. If fc 1, the subsequent generations of neutrons increase in population if /c 1, the subsequent generations decrease in population. The case k = 1 corresponds exactly to the condition given by Eq. (3,13), i.e., the criticality condition. This is easily demonstrated for the infinite-medium reactor model defined by the assumptions (3.7). 

In our treatment of the infinite homogeneous reactor model, we argued that the neutron density was a spatially invariant function. Thus, if we determine the neutron-balance conditions in any one volume element of this infinite system, these conditions will apply to all such elements throughout the space. Let us write, then, the quantities involved in our definition of the multiplication constant in terms of the nuclear events which occur in a unit volume of the reactor. It is convenient here to write Eq. (3.16) in the form 

According to the definition of the collision density, the neutron production per unit volume per unit time is given by vl /nv. These neutrons are the offspring of the previous generation, of which there must have been exactly Sa/w produced per unit volume per unit time. That Sony is indeed the number of neutrons introduced follows from the fact that all neutrons must eventually be absorbed somewhere. Although very few of the neutrons introduced into the unit volume are actually absorbed there, every one of the neutrons which leave this volume element to be 

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As an example of the application of these concepts, consider the analysis of the slowing-down problem in an infinite, homogeneous multiplying medium which is at steady state, i.e., critical. Let the distribution of fission neutrons be given by the normalized spectrum 3(w) defined in Eq. (4.152), and let 2a(w), S/(m), S (r), and 2,(w) denote the nuclear constants of the medium. The total collision density for this system is given by... 

A multi-dimensional generalization of the Stefan problem which includes surface tension is studied as a mathematical model for growth in a metastable medium. Planar solutions are shown to exist for all time only if the data is sufficiently small, otherwise the velocity of the front becomes infinite in finite time. Planar fronts which exist for all time are shown to be morphologically unstable without surface tension and to be stable with respect to perturbations of short wavelength when surface tension is included. Above a critical value of the surface tension, planar fronts are completely stable. The mathematical techniques used are a combination of soft analysis based on the maximum principle and functional analytic/integral equation type hard estimates. 

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