The spatial and temporal distribution of thermal neutrons in a multiplying assembly following the introduction of a short burst of fast neutrons is investigated by means of an extension of the so-called “asymptotic reactor theory” to the time-dependent case. It is shown that the solution for an nth mode fast neutron source can be reduced to that for an nth mode thermal neutron source, so that only the latter need be considered. A formal solution to the time-dependent thermal diffusion equation with an nth mode thermal source is found for an arbitrary slowing-down kernel. The asymptotic behavior of the flux in the long-time limit is shown to be exponential, with a decay constant satisfying a generalized material buckling equation The asymptotic behavior following a burst of fast neutrons is also found to be exponential with the same time constant. In a continuous slowing-down model, all neutrons slow down in the same time implying that the time-dependent part of the time-dependent slowing-down kernel is a Dirac delta-function. In this case, an explicit expression for the flux following a burst can be derived from which the approach to the asymptotic behavior is clearly seen. The mean slowing-down time (t) is used to find an approximate expression for the asymptotic decay constant. To evaluate (t) for hydrogenous media, it is noted that the Laplace transform of the Boltzmann equation is identical with the time-independent Boltzmann equation if, in the latter,Σa (E) is replaced by Σa(E) + η/υ(E), where υ(E) is the neutron velocity and η the Laplace transform variable The resulting equation can then be solved by standard methods. The infinite medium B2 = 0) result of 0.92 µsec for the slowing-down time to 1.4 ev is in good agreement with the value 0.85 µsec obtained from Monte Carlo calculations. The validity and application of the method are discussed.
