Intrinsic nuclear quadrupole moment ($Q_0$)

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: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 $⟨x^2⟩=a^2/5$ and $⟨z^2⟩=c^2/5$, soThe 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 $\eta =\sqrt{8}$ gives $a=R/\sqrt{2}$, $c=2R$ and $Q_0=7Ze{R}^2/5$, while preserving $a^{2}c=R^3$ 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 $Q_0$.^[ref]