Problems of Nuclear Reactor Theory

The second class of problems is concerned with methods for obtaining or at least discussing the solutions of the reactor equations. The coding of these equations for calculating machines belongs to this class. There are, however, more subtle methods among which the transformation of the reactor equations into a variational principle has proved, so far, the most effective. Again, this transformation has been carried out so far only for the most simple problems and it is not known whether all problems of reactor theory can be reformulated as variational problems. 

Some remarks are made on a reformulation of diffusion theory which gives, at least in the cases considered so far, much more accurate results than the conventional theory. 

The assumptions of transport theory. As you well know, the theory which permits one to calculate the neutron densities or fluxes is an essentially statistical theory and is called transport theory. This theory goes back to the last century and Boltzmann s book on the kinetic theory of gases [1] can still be read to advantage. The fimdamental concept is the so-called neutron flux x,EySl t), This quantity gives for the time t the number of neutrons, multiplied with their speed, which satisfy the following conditions  

Naturally, O can be replaced by other, equivalent functions, such as EQ , etc. Similarly, the variables x, E, 2 can be replaced by equivalent variables. In particular, E and 2 can be replaced by the three components of the velocity. It may be proper, at this point, to summarize the basic assumptions of transport theory as used in reactor theory. 

1 The effect of neutron polarization on diffusion was recognized ten years ago by S. Borowitz and M. Hamermesh. See [3]. Actually, the situation is more complicated than represented in this article, or in the text. The reason is that quantum mechanics defines amplitudes rather than intensities for the two helicities and that these amplitudes are complex rather than real. However, one can define, from the two amplitudes, four real quantities, the so-called statistical matrix. One of the real quantities, Oo(JiP, E, 2, t) is the total flux of both polarizations the three other quantities O, Oy, Og, (which depend on the same variables) describe the state of polarization of the neutrons with energy E, velocity-direction SI, and position x. If these neutrons are impolarized = 0 if the state of polarization is complete, 

---

E.P. Wigner, Mathematical Problems of Nuclear Reactor Theory , Proceedings of Symposia in Applied Mathematics Vol. XI, Nuclear Reactor Theory, pp. 89-104, 1961. (American Mathematical Society). 

Nevertheless, very few research mathematicians have so far devoted serious effort to the mathematical problems of nuclear reactor theory. The present volume is intended to increase the number of such mathematicians, by indicating the great variety of interesting mathematical problems encountered in this fascinating field. As a by-product, it may help to put the design of future nuclear reactors on a more scientific basis. 

---