The Behavior of a Reactor at Prompt Critical when the Reactivity is a Linear Function of Time

A simple analytic formula is derived for the values of the prompt neutron density, the delayed neutron emitter densities, and the period in a reactor at the instant it has becòme prompt critical under the assumption that it has been brought to a prompt critical state from an arbitrary subprompt critical initial state by introducing reactivity at the constant rate of a dollars per second. For a fixed value of a, these formulas are asymptotic with respect to small values of the dimensionless parameter al/β, in which l is the mean lifetime of a neutron in the reactor and β is the fraction of fission neutrons which are delayed. For fast reactors, the quantity l/β is generally small, so that our formulas should be useful in estimating the power rise at prompt critical unless the rate of introduction of reactivity is quite large.