Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
••• To get a rough feeling for critical sizes, consider the following: A stream of 1-MeV neutrons impinges normally on a slab of 235U, 1 cm thick. (1 MeV is the average energy of neutrons ejected in fission.) (a) Given that the fission cross section at 1 MeV is 1.25 barns, find the fraction, N/Ninc, of incident neutrons that induce a fission as they pass through the slab. [Hint: In Eq. (17.44) we wrote the number of reactions as N = σnincNtar, where ninc is the incident flux of projectiles (number/area) and Ntar is the total number of target particles. Here it is more convenient to use the equivalent expression
N = σNincntar
where Ninc is the total number of neutrons and ntar is the number of 235U atoms per unit area of the slab. To find ntar, you need to know that the density of uranium is about 19 g/cm3.] (b) On average, the number of neutrons produced in a fission is 5/2; therefore, in a self-sustaining reaction at least 2/5 of the neutrons produced must induce further fission. Use your answer to estimate how many centimeters of slab are needed to give N/Ninc = 2/5. This answer gives the order of magnitude of the critical radius of a solid sphere of 235U. (c) Using this estimate of the critical radius, find the approximate value of the critical mass of a sphere of 235U.
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