Collective nuclear model

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.

A collective oscillation of the quadrupole shape coordinates of an atomic nucleus. Near spherical equilibrium, the ideal harmonic description has five components, corresponding to the five magnetic substates of angular momentum two. Each quantum carries $I^{\pi }=2^{+}$; it is an excitation of the many-nucleon system rather than an extra constituent. Its frequency is $\omega$, and its energy spacing $\hslash \omega$ follows the quantum harmonic oscillator.

One quantum produces a $2^{+}$ level at $\hslash \omega$. Two identical quadrupole quanta can couple to $I=0,2,4$ in a symmetric state, giving a degenerate $0^{+},2^{+},4^{+}$ multiplet at $2\hslash \omega$ in the ideal model. Odd total $I$ would be antisymmetric under their exchange. Thus the first $4^{+}$ to first $2^{+}$ excitation ratio is $2$, rather than the nuclear rotational band value $10/3$. Real nuclear quadrupole vibrations can be anharmonic and their multiplets split: the ratio is a diagnostic of a limiting model, not a universal classification rule.

There is a useful state-counting check. Two identical quanta distributed among five substates span $5(5+1)/2=15$ symmetric states. The proposed multiplet contains $(2\cdot 0+1)+(2\cdot 2+1)+(2\cdot 4+1)=1+5+9=15$. This verifies the angular-state bookkeeping independently of the chosen energy scale.

Excitation of an atomic nucleus by the electric field of a passing charged projectile. For suitable collision energies and separations, the electromagnetic interaction can populate collective states without requiring contact through the nuclear force. Electric quadrupole transition strengths connect excitation yields with collective charge motion; inferring a shape additionally assumes a nuclear model. Contributions from the nuclear force, competing transitions and the collision kinematics must be controlled before identifying a yield with a single quadrupole strength.

In the straight-trajectory dipole model, retain the Coulomb-coupling convention of $V(t)=Z{e}^2(bx+vty)/(b^2+v^2{t}^2{)}^{3/2}$. In SI units the coupling denoted $e^2$ here is $e^2/(4\pi {ϵ}_0)$. First-order time-dependent perturbation theory expresses a transition through the Fourier transform of this pulse at the transition angular frequency.

For positive $\omega$ put $\beta =\omega b/v$. The two Cartesian pulse transforms arewhere $K_{\nu }$ is a Modified Bessel function of the second kind. One may derive these from the Fourier transform of $(1+s^2{)}^{-1/2}$, differentiating in its scale and integrating its spatial derivative by parts. For $\beta \ll 1$, $\beta K_1(\beta )\to 1$ while $\beta K_0(\beta )\to 0$, so the transverse impulse gives the probability in the title. If the $x$ matrix element vanishes, the smaller longitudinal term can be the leading nonzero contribution.

The conditions include $b$ large compared with the atomic extent for the electric-dipole approximation, a small total transition probability for first-order theory, and a collision time $b/v$ short compared with $1/|\omega |$. Large $\beta$ instead gives exponentially suppressed excitation in this model through the large-argument behavior of $K_{\nu }$.