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Physics 418, Homework Assignment
Due in Class, Wednesday, February 4, 2004

Isaac Asimov in his novel The Gods Themselves describes a universe where the most stable nuclide with A=186 is not $^{186}_{74}$ W but rather $^{186}_{94}$ Pu. This is claimed to be a consequence of the ratio of the strengths of the strong and electromagnetic interactions being different from that in our universe. Assume that only the electromagnetic coupling constant, $\alpha$ , differs and that both the strong interaction and the nucleon masses are unchanged. How large must $\alpha$ be in order that $^{186}_{82}$ Pu, $^{186}_{88}$ Pu and $^{186}_{94}$ Pu are stable?
In the Liquid Drop Model, the binding energy for nuclei is:

$\begin{displaymath} \rm B(Z,A) = a_v A - a_s A^{2/3} - a_c \frac{Z^2}{A^{1/3}} - a_a \frac{(A - 2Z)^2}{A} + \frac{\delta}{A^{1/2}} \end{displaymath}$

The Coulomb term in the semi-empirical mass formula (aka. liquid drop model) is $a_c Z^2/ A^{\frac{1}{3}}$ . Try to estimate the value of with simple electrostatics. First, show that the potential energy due to electrostatic forces on a uniformly charged shpere of total charge and radius is $3Q^2/(20\pi\epsilon_0 R)$ . Estimate by using $R = 1.25 \times A^{\frac{1}{3}}$ , the values for given in lecture, and the binding energy (not mass!) of $^{184}_{74}{\rm W}$ . For the binding energy, use the value given in: http://atom.kaeri.re.kr/ton/.
The -decay of a Pu ( = 127 years) nuclide into a long-lived U ( = 3.5 x 10 years) daughter nucleus releases 5.49 MeV kinetic energy. The heat so produced can be converted into useful electricity by radio-thermal generators (RTG's). The Voyager 2 space probe, launched 20-Aug-1977, flew past four planets, including Saturn, which it reached on 26-Aug-1981. Saturn's separation from the Sun is 9.5 AU (1 AU = Earth-Sun distance).
- How much plutonium would an RTG on Voyager 2 have to carry in order to deliver at least 400 W electrical power when the probe flies past Saturn? Assume a power conversion efficiency of 6.2%.
- How much electric power would then be available at Neptune, 30.1 AU from the Sun, on 24-Aug-1989?
- For comparison: the largest-ever ``solar paddles'' used in the space laboratory Skylab would have produced 10.5 kW from an area of 730 m, 1 AU from the Sun. What area of solar cells would Voyager 2 have needed if those were its power source?
Griffiths: Problem 1-3.
Fraunhofer diffraction by a circular disk with diameter D produces a ring-shaped diffraction pattern with the first minimun at an angle $\theta = 1.22 \lambda/D$ . Calculate the angular separation of the diffraction minima of $\alpha$ particles with energy E $_{kin}$ = 130 MeV scattered of a $^{59}$ Co nucleus. The nucleus should be considered as an impenetrable disk.

Thomas J. & 2004-01-28