# PHYS 598 hw7 - 1 Critical Mass An infinite slab of fissile...

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Physics 498/MMA Handout 7 Fall 2007 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Critical Mass : An infinite slab of fissile material has thickness L . The neutron density n ( r ) in the material obeys the equation ∂n ∂t = D 2 n + λn + μ, where n is zero at the surface of the slab at x = 0 , L . Here D is the neutron diffusion constant, the term λn describes the creation of new neutrons by induced fission, and μ is the rate of production per unit volume of neutrons by spontaneous fission. Assume that n depends only on x . a) Expand n as a series n ( x, t ) = X m a m ( t ) ϕ m ( x ) where the ϕ m are a complete set of functions you think suitable for solving the problem. b) Find an explicit expression for the coefficients a m ( t ) in terms of their intial values a m (0). c) Determine the critical thickness, L crit , above which the slab will explode. d) Assuming that L < L crit , find the equilibrium distribution n eq ( x ) of neutrons in the slab. (You may either sum your series expansion to get an explicit closed-form answer,

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• Fall '07
• Stone
• Mass, Neutron, one year, Prof. M. Stone, ESB University of Illinois

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