= Intrinsic nuclear quadrupole moment
{title2=$Q_0$}
The axial <electric quadrupole moment> of an intrinsic <nuclear charge distribution>, with the origin at its center and the $z$ axis along its symmetry axis:
$$
Q_0=\int \rho_{\mathrm{ch}}(\mathbf r)(2z^2-x^2-y^2)\,d^3r.
$$
This convention includes <electric charge> in $Q_0$; a tabulated moment divided by the <elementary charge> $e$ instead has units of area. For the uniform <spheroid>, mapping a uniform unit ball to $(ax,ay,cz)$ gives $\langle x^2\rangle=a^2/5$ and $\langle z^2\rangle=c^2/5$, so
$$
Q_0=\frac25Ze(c^2-a^2),\qquad
\frac{Q_0}{ZeR^2}=\frac25\left(\eta^{4/3}-\eta^{-2/3}\right).
$$
The formula explains the signs in the figure. It measures an anisotropy of the <nuclear charge distribution>, rather than counting individual <nucleons> on a surface. An illustrative choice $\eta=\sqrt8$ gives $a=R/\sqrt2$, $c=2R$ and $Q_0=7ZeR^2/5$, while preserving $a^2c=R^3$ exactly.
<James Rainwater>'s lecture contrasts measured laboratory <electric quadrupole moments> with intrinsic moments inferred through <Coulomb excitation>, including <even-even nuclei>. Extracting intrinsic shape therefore uses a nuclear model, rather than equating every measured moment to $Q_0$.<Background for the Spheroidal Nuclear Model Proposal by James Rainwater>[p. 266, Fig. 4 and preceding Coulomb-excitation discussion]{ref}
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