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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:

\rho(r)=\frac{\rho_0}{1+e^{(r-R) / / t}}

where $\rho_0, R$ , and $t$ are positive constants, with $t \ll R$ . (a) Prove that $\rho=\rho_0 / 2$ at $r=R$ . (b) Show that the maximum value of $\rho$ occurs at $r=0$ and is very close to $\rho_0$ , given that $t \ll R$ . (c) Sketch $\rho$ as a function of $r$ . (d) Prove that as $r$ increases, the density falls from $90 \%$ to $10 \%$ of $\rho_0$ in a distance $\Delta r=4.4 t$ . (Thus $t$ characterizes the thickness of the surface region.)

(a) Substitute the binding-energy formula into the relation

m_{\mathrm{nuc}}=Z m_{\mathrm{p}}+N m_{\mathrm{n}}-\frac{B}{c^2}

to obtain the semiempirical mass formula. [To simplify matters, take $A$ to be odd, so that the pairing term in above is zero.] (b) Among any set of isobars, the nucleus with lowest mass is the most stable. To identify this nucleus, first write your mass formula in terms of the variables $A$ and $Z$ (that is, replace $N$ , wherever it appears, by $A-Z$ ). With $A$ fixed, differentiate $m$ with respect to $Z$ . The minimum mass is determined by the condition $\partial m / \partial Z=0$ . Show that this leads to a relation of the form

Z=\frac{A}{2} \frac{1+\alpha}{1+\beta A^{2 / 3}}

where $\alpha$ and $\beta$ are related to the coefficients as follows:

\alpha=\frac{\left(m_{\mathrm{n}}-m_{\mathrm{p}}\right) c^2}{4 a_{\mathrm{sym}}} \quad \text { and } \quad \beta=\frac{a_{\mathrm{Coul}}}{4 a_{\mathrm{sym}}}

(c) For $A=37,115,185$ , find the values of $Z$ that give the most stable nuclei and compare with the observed values from Appendix D.

In this problem you will calculate the electrostatic energy of a uniform sphere of charge $Q$ with radius $R$ . The electric potential $V(r)$ at any radius $r<R$ is given by the equation in the above problem, and the potential energy of a charge $q$ at radius $r$ is $q V(r)$ . (a) Write down the total charge contained between radius $r$ and $r+d r$ , and hence find the potential energy of that charge. (b) If you integrate your answer to (a) from $r=0$ to $r=R$ , you will get twice the total potential energy of the whole sphere since you will have counted the energy of any two charge elements twice. Show that the total Coulomb energy of the sphere is

U_{\text {Coul }}=\frac{3}{5} \frac{k Q^2}{R}

Question

Which of the following nuclei has a closed subshell for Z? Which for $N ?{ }^{10} \mathrm{~B},{ }^{11} \mathrm{~B},{ }^{17} \mathrm{O},{ }^{27} \mathrm{Al},{ }^{28} \mathrm{Si},{ }^{42} \mathrm{Ca},{ }^{48} \mathrm{Ca}$ .

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The nuclear magic numbers are $2$ , $8$ , $20$ , $28$ , $50$ , $82$ , and $126$ . The given nucleus has a closed subshell for its nucleon if the number of the particular nucleon is equal to these magic numbers. If it is not equal, then it has a subshell that is only partially occupied.