= Collective nuclear model
= Bohr-Mottelson-Rainwater collective model
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{synonym}
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.<Elementary Modes of Excitation in the Nucleus by Ben Roy Mottelson>[pp. 236–237, coupled Hamiltonian and interpretation of low-energy excitation spectra]{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.<Background for the Spheroidal Nuclear Model Proposal by James Rainwater>[p. 264, Fig. 3 and discussion of the 1949 quadrupole compilation]{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>. Suppose
$$
V(q)=\frac12Cq^2-fq
=\frac C2\left(q-\frac fC\right)^2-\frac{f^2}{2C}.
$$
The minimum is $q_*=f/C$, with <energy> gain $f^2/(2C)$. For chosen values $C=20u$ and $f=4u$, where $u$ is an arbitrary positive <energy> unit, $q_*=0.20$ 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.
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