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A description of an atomic nucleus in terms of coordinated motion of many nucleons, especially changes in shape and orientation. The nuclear shell model organizes individual nucleon states in a nuclear mean field; the collective nuclear model organizes low-lying energy levels by nuclear deformation, nuclear rotational bands and nuclear quadrupole vibrations. These are complementary organizations of the same nuclear many-body problem. A deformed nuclear mean field can support individual-particle states while its orientation supplies a slow collective coordinate.

In his 1975 lecture, Ben Roy Mottelson described coupling individual-particle motion to nuclear shape coordinates, and distinguished observed rotational spectra from vibrations about spherical equilibrium. This connects the two descriptions without treating the collective coordinates as extra constituent particles.^[ref]

James Rainwater's 1975 account identifies unexpectedly large electric quadrupole moments as a motivation for a deformable nuclear mean field. The challenge concerned their magnitude, even where spherical-shell trends had the correct sign.^[ref]

A simple original potential energy example makes the coupling tangible. Let $q$ be a dimensionless signed shape coordinate for a chosen intrinsic configuration, $C>0$ its restoring stiffness, and $f$ a configuration-dependent coupling, both with units of energy. SupposeThe minimum is $q_{\ast }=f/C$, with energy gain $f^2/(2C)$. For chosen values $C=20u$ and $f=4u$, where $u$ is an arbitrary positive energy unit, $q_{\ast }=0.20$ and the gain is $0.40u$. This illustrates how a configuration can favor deformation when the core can respond. It is not a fitted nuclear surface or a claim that every configuration chooses the same shape; competing configurations and nuclear pairing can favor a spherical minimum.

The predictive attraction is economy: one collective scale relates several energy levels, while electric quadrupole transitions independently test their common structure. Agreement in both is much more restrictive than assigning a separate free parameter to every level. Nuclear magnetic moments test magnetic structure and angular momentum assignments; they are distinct observables from the electric quadrupole moments used to infer charge deformation.

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]