Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

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.

•• (Section 16.2) Same as Problem 1, but make plots for 12C and 244Pu, drawing both curves on the same plot.

Problem 1

•• (Section 16.2) Same as Problem 2, but in part (c) use graph-plotting software to plot ρ(r) for 208Pb from r = 0 to r = 15 fm, given the following parameters: ρ0 = 3.25 × 1017 kg/m3, t = 0.55 fm, and R = R0A1/3, where R0 = 1.07 fm.

Problem 2

•• The density of mass inside a nucleus is shown in Fig. 16.2. There is no simple theory that predicts the exact shape shown, but it is found that the shape can be approximated by the following mathematical form, known as the Fermi function:

where ρ0, R, and t are positive constants, with t ≪ R. (a) Prove that ρ = ρ0/2 at r = R. (b) Show that the maximum value of ρ occurs at r = 0 and is very close to ρ0, given that t ≪ R. (c) Sketch ρ as a function of r. (d) Prove that as r increases, the density falls from 90% to 10% of ρ0 in a distance Δr = 4.4t. (Thus t characterizes the thickness of the surface region.)