Conditions in the ion source of a mass spectrometer can be adjusted to allow formation of multiply ionized atoms. (a) Rewrite the above equation for the case of an $n$ times ionized atom, and solve for $R$. (b) Taking the masses of ${ }^4 \mathrm{He},{ }^{12} \mathrm{C}$, and ${ }^{16} \mathrm{O}$ to be $4 \mathrm{u}, 12 \mathrm{u}$, and $16 \mathrm{u}$, respectively, write down expressions for the radii, $R$, for $\mathrm{He}^{+}, \mathrm{C}^{3+}$, and $\mathrm{O}^{4+}$. (c) Using the actual masses of the atoms, find the fractional differences in these three radii (He versus $\mathrm{C}$ and $\mathrm{C}$ versus $\mathrm{O}$ ). The tiny differences in these radii can be measured very accurately and allow precise comparisons of the three masses involved (one of which, ${ }^{12} \mathrm{C}$, defines the atomic mass scale).

Where does the methionine molecule in the abovefigure break to account for the two peaks at $74 u$ and $75 \mathrm{u}$ in its mass spectrum?

The spin-orbit energy is proportional to $\mathbf{S} \cdot \mathbf{L}$. To evaluate $\mathbf{S} \cdot \mathbf{L}$, note that since $\mathbf{J}=\mathbf{S}+\mathbf{L}$,

$$J^2=(\mathbf{S}+\mathbf{L}) \cdot(\mathbf{S}+\mathbf{L})=S^2+L^2+2 \mathbf{S} \cdot \mathbf{L}$$

You can solve this for $\mathbf{S} \cdot \mathbf{L}$ and then put in the values for $J^2, S^2$, and $L^2$. [For instance, $J^2=j(j+1) \hbar^2$ where $j=l \pm \frac{1}{2}$.] Show that if $j=l+\frac{1}{2}$ then $\mathbf{S} \cdot \mathbf{L}=$ $l^2 / 2$, and if $j=l-\frac{1}{2}$, then $\mathbf{S} \cdot \mathbf{L}=-(l+1) \hbar^2 / 2$. Use these answers to prove that the spin-orbit splitting increases with increasing $l$, as indicated in the above figure.

For each choice of $n$ and $l$ (with $l \neq 0$ ), a nucleon has two different possible energy levels with $j=l \pm \frac{1}{2}$. Prove that the sum of the degeneracies of these two levels is equal to the total degeneracy that the level $n l$ would have had in the absence of any $\mathbf{S} \cdot \mathbf{L}$ splitting.

(a) Use the energy levels of above figure to draw an occupancy diagram like the above figure. (b) for ${ }^{13} \mathrm{C}$. (b) What is $j_{\text {tot }}$ for the ground state? (c) The first excited state is formed by lifting one neutron from $1 p_{3 / 2}$ to $1 p_{1 / 2}$. Draw an occupancy diagram for this state, and explain what you would predict for $j_{\text {tot }}$ of this state. (Does your answer agree with the observed value, $j_{\text {tot }}=\frac{3}{2}$ ?) (d) Now predict $j_{\text {tot }}$ for the ground and first excited states of ${ }^{13} \mathrm{~N}$. (In this case the excitation involves a proton.)

In applying the shell model, we usually use the level ordering of above Figure, which is based on an assumed spherical potential well. For certain nonspherical nuclei, this assumption is incorrect and the ordering of the levels is different. Examples are ${ }^{19} \mathrm{~F}$ and ${ }^{121} \mathrm{Sb}$; check this by using the rule and above figure to predict the angular momenta of these two nuclei and comparing with the observed values in Appendix D.

(a) Use the levels shown in the above figure to draw an occupancy diagram like above Figure (b) for the ground state of ${ }^7 \mathrm{Be}$. (b) What is $j_{\text {tot }}$ ? (c) What would you predict for the first excited state? (Draw an occupancy diagram, and give $\left.j_{\text {tot }}\right)$

The ground state of ${ }^7 \mathrm{Li}$ has $j_{\text {tot }}=\frac{3}{2}$ and its first excited state has $j=\frac{1}{2}$. Explain these observations in terms of the shell model.

Use the shell model to predict $j_{\text {tot }}$ for the ground states of ${ }_{19}^{39} \mathrm{~K}_{20},{ }_{20}^{40} \mathrm{Ca}_{20}$, and ${ }_{21}^{41} \mathrm{Sc}_{20}$.

Apply Appendix D to calculate the chemical symbols and the natural percent abundances of the isotopes with (a) $Z=2, \quad A=3 ; \quad$ (b) $Z=6, \quad A=13$; (c) $Z=26, A=56$; (d) $Z=82, A=206$.Appendix D to find the chemical symbols and the half-lives of the four radioactive nuclei with (a) $Z=1, \quad A=3 ;$ (b) $Z=6, \quad A=14$; (c) $Z=19, A=40$; (d) $Z=94, A=244$.Appendix $\mathrm{D}$ to make a list of all the stable isotopes, isobars, and isotones of ${ }^{70} \mathrm{Ge}$. (b) Do the same for ${ }^{27} \mathrm{Al}$ and (c) ${ }^{88} \mathrm{Sr}$.

Use the levels shown in the above figure to predict the total angular momenta of the following nuclei: ${ }^{29} \mathrm{Si}$, ${ }^{33} \mathrm{~S},{ }^{37} \mathrm{Cl},{ }^{59} \mathrm{Co}$, and ${ }^{71} \mathrm{Ga}$.

Use the levels shown in the above figure to predict $j_{\text {tot }}$ for the ground states of the nine nuclei ${ }^{40} \mathrm{Ca},{ }^{41} \mathrm{Ca}$, ${ }^{48} \mathrm{Ca}$. Compare with the data in Appendix D.

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

(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.