D.79 Shell model quadrupole moment

The result for one proton is readily available in literature and messy to derive yourself. If you want to give it a try anyway, one way is the following. Note that in spherical coordinates

\begin{displaymath}
3z^2 - r^2 = 2 r^2 -3 r^2\sin^2\theta
\end{displaymath}

and the first term produces $2\langle{r}^2\rangle$ simply by the definition of expectation value. The problem is to get rid of the $\sin^2\theta$ in the second expectation value.

To do so, use chapter 12.8, 2. That shows that the second term is essentially $3\langle{r}^2\rangle$ modified by factors of the form

\begin{displaymath}
\langle Y_l^l \vert sin^2\theta Y_l^l\rangle
\quad\mbox{and}\quad
\langle Y_l^{l-1} \vert sin^2\theta Y_l^{l-1}\rangle
\end{displaymath}

where the integration is over the unit sphere. If you use the representation of the spherical harmonics as given in {D.65}, you can relate these inner products to the unit inner products

\begin{displaymath}
\langle Y_{l+1}^{l+1} \vert Y_{l+1}^{l+1}\rangle
\quad\mbox{and}\quad
\langle Y_{l+1}^l \vert Y_{l+1}^l\rangle
\end{displaymath}

Have fun.

The expression for the quadrupole moment if there are an odd number $i$ $\raisebox{-.5pt}{$\geqslant$}$ 3 of protons in the shell would seem to be a very messy exercise. Some text books suggest that the odd-particle shell model implies that the one-proton value applies for any odd number of protons in the shell. However, it is clear from the state with a single hole that this is untrue. The cited result that the quadrupole moment varies linearly with the odd number of protons in the shell comes directly from Krane, [30, p. 129]. No derivation or reference is given. In fact, the restriction to an odd number of protons is not even stated. If you have a reference or a simple derivation, let me know and I will add it here.