Optimal Reflectors to Achieve Criticality

Within the framework of a simplified one-dimensional model, the following problem is considered. Given a segment of some fissionable material with a length less than its natural critical length, construct, if possible, reflectors that provide albedos sufficient to make the segment critical (or achieve a prescribed degree of supercriticality) and do this in an optimal way (i.e. with minimum weight or cost). It is shown via asymptotic solutions to the one-dimensional Boltzmann equations that the appropriate left and right albedos lie on a segment of a hyperbola. For any pair of these albedos and for a wide class of optimization criteria, the optimal reflectors can be designed using the technique of dynamic programming. The solution to the problem is then found by a simple minimization along an arc of the hyperbola which relates the left and right albedos. Numerical examples are provided to illustrate the method when the optimization criterion is minimum weight.