Computer Algebra Recipes

236 CHAPTER 5. LINEAR PDE MODELS. PART 1
5.2.4 Hugo and the Atomic Bomb
If the radiance of a thousand suns were to burst forth at once in the
sky, that would be like the splendor of the Mighty One.
Bhagavad Gita, a philosophical dialogue that is a sacred Hindu text, found in
the Mahabharata, one of the ancient Sanskrit epics (250 BC{250 AD)
Hugo, who was formerly a scientist in country X, has emigrated to the New
World in search of a better life. Unfortunately, he has been forced to temporarily
drive taxis until he can ¯nd a job more suitable to his educational training.
While waiting for his next fare, he tries to keep his mind sharp by carrying out
model calculations on his laptop computer. At this particular moment, Hugo is
working on the problem of the growth of the neutron density in a nuclear chain
reaction. Let's eavesdrop on what Hugo is doing.
Hugo knows that if uranium nuclei are bombarded with neutrons, a given
nucleus may absorb a neutron, resulting in the splitting of the uranium nucleus
into two parts with the release of substantial energy as well as two or three of
the neutrons that were already present in the nucleus. This splitting process
is called nuclear ¯ssion andisthe¯rststepinachain reaction. Whether the
reaction will keep on going depends on how many of the released neutrons are
available to initiate another ¯ssion process. The factor by which the number of
neutrons increases between one step and the next in the chain reaction is called
the multiplication factor. In a nuclear reactor, the multiplication factor is kept
at unity (called the critical condition) by using boron or cadmium control rods
to \soak up" excess neutrons. In this case, the chain reaction proceeds at a
constant rate with a steady output of energy. If the multiplication factor is
greater than unity (the supercritical condition), the chain reaction leads to a
geometrically increasing number of ¯ssions in a very short time interval with the
accompanying release of an enormous amount of energy. A nuclear explosion
takes place | the basis of the atomic bomb.
Hugo decides that he can learn more about the underlying role that the
neutrons play in the chain reaction by modeling the time evolution of some
speci¯ed initial neutron distribution inside a mass of ¯ssionable material. For
calculational purposes, he takes the mass to be cylindrical in shape with a radius
r = a, the lower face of the cylinder at z = 0 and the upper face at z = h.
For simplicity, Hugo takes the neutron density N (number of neutrons per unit
volume) to be independent of the angular coordinate μ, i.e., N = N(r; z; t). In
the absence of any production of neutrons by ¯ssion, the neutron density would
obey the linear di®usion equation. To account for the production of neutrons
by ¯ssion, Hugo adds a neutron source term ¯N,where¯is a positive rate
constant, to the di®usion equation, viz.,
@N
@t = d r2N + ¯N: (5.10)
In order to use the cylindrical polar form of the Laplacian, he calls up the
VectorCalculus package.