A simplified kinetic model describes the population of prompt neutrons in a compact uranium assembly by the factor keff, defined as the average number of neutrons in one generation divided by the number in the preceding generation.
During an experiment, an initially sub-critical assembly (keff=0.95) is rapidly modified by inserting additional uranium rings, after which keff rises above unity.
(a) Derive an expression for the net number of neutrons Nn present in the nth generation in terms of keff and the initial neutron number N0.
(b) Show that the condition for criticality can be written as keff=1 and explain, in terms of conservation principles, why values of keff less than, equal to, or greater than unity correspond to different reactor states.
(c) Immediately after the additional rings are inserted, measurements indicate that the neutron population triples after four generations. Estimate the new value of keff and state whether the assembly is prompt-critical. Show all reasoning.
(d) To return the assembly to a safe state, engineers propose coating the inner surface of the cavity with a thin boron-rich paint. Develop a qualitative argument—including interaction cross-sections and neutron balance—that explains how such a coating can reduce keff below unity without altering the assembly’s mass or geometry.