Deformed Nuclei

Nuclear Deformation Parameters

The shape of a deformed nucleus is described by expanding the nuclear surface radius in terms of spherical harmonics. The general parameterization is:

The most important multipoles are the quadrupole ($\lambda=2$) and hexadecapole ($\lambda=4$) deformations. The quadrupole deformation parameters $\beta_2$ and$\gamma$ are defined through:

The parameter $\beta_2$ measures the magnitude of quadrupole deformation, while$\gamma$ specifies the shape type:

• - $\gamma = 0^\circ$: Prolate (cigar-shaped), axially symmetric
• - $\gamma = 60^\circ$: Oblate (disc-shaped), axially symmetric
• - $0^\circ < \gamma < 60^\circ$: Triaxial deformation

The hexadecapole deformation $\beta_4$ adds a higher-order correction to the shape:

Typical values for well-deformed rare-earth nuclei are $\beta_2 \approx 0.2\text{--}0.35$ and$|\beta_4| \lesssim 0.1$. The volume-conservation condition ensures that the nuclear volume remains constant under deformation: $R_0 = r_0 A^{1/3}$.

Quadrupole Deformation & Moments

The intrinsic (body-frame) electric quadrupole moment is related to the deformation parameter by:

The spectroscopic (lab-frame) quadrupole moment for a state with spin $J$ in the ground-state rotational band is related to $Q_0$ by the projection factor:

where $K$ is the projection of $J$ onto the symmetry axis. For the ground-state band of an even-even nucleus ($K=0$):

Note that $Q_{\text{spec}}(0^+) = 0$ by symmetry, and for $J=2$:$Q_{\text{spec}}(2^+) = \frac{2}{7}Q_0$. A positive $Q_0$ corresponds to prolate shape, while negative indicates oblate.

The Nilsson Model (Deformed Shell Model)

The Nilsson model describes single-particle motion in a deformed potential. The Hamiltonian for an axially symmetric deformed harmonic oscillator is:

The oscillator frequencies are related to the deformation parameter $\delta$:

Each Nilsson level is labeled by the asymptotic quantum numbers $[Nn_z\Lambda]\Omega^\pi$, where $N$ is the total oscillator quantum number, $n_z$ counts quanta along the symmetry axis, $\Lambda$ is the orbital angular momentum projection, and$\Omega = \Lambda + \Sigma$ is the total angular momentum projection ($\Sigma = \pm\frac{1}{2}$).

The Nilsson diagram shows single-particle energies as functions of the deformation parameter. It successfully predicts ground-state spins and parities of odd-A deformed nuclei by filling Nilsson levels up to the Fermi surface.

Rotational Bands

A deformed nucleus can rotate collectively about an axis perpendicular to the symmetry axis. The rotational energy spectrum for an axially symmetric rotor is:

For $K=0$ bands (ground-state band of even-even nuclei), only even spins appear:$J^\pi = 0^+, 2^+, 4^+, 6^+, \ldots$ The characteristic energy ratios serve as diagnostics for the rotational character:

For odd-A nuclei, the rotational band is built on a Nilsson orbital with quantum number$K = \Omega$, and the band contains spins $J = K, K+1, K+2, \ldots$ (for $K \neq 1/2$). When $K = 1/2$, a decoupling parameter $a$ modifies the energies:

Moment of Inertia

The nuclear moment of inertia lies between two extremes. The rigid-body value for a uniformly charged ellipsoid is:

The irrotational-flow moment of inertia for a superfluid nucleus is:

Experimentally, nuclear moments of inertia are typically 30-50% of the rigid-body value, indicating that nuclei behave as superfluids with significant pairing correlations. The Inglis cranking formula gives:

where the sum runs over occupied (i) and unoccupied (j) single-particle states. Pairing correlations reduce the moment of inertia below the rigid-body value by creating an energy gap in the single-particle spectrum.

E2 Transition Strengths & B(E2) Values

The reduced transition probability for E2 transitions within a rotational band is directly related to the intrinsic quadrupole moment:

For the important $0^+ \to 2^+$ ground-state transition:

Deformed nuclei have very large B(E2) values, typically 100-300 Weisskopf units (W.u.), reflecting their collective nature. The Weisskopf single-particle estimate is:

The ratio $B(E2)/B_W(E2)$ measures the collectivity: values $\gg 1$ indicate coherent contributions from many nucleons.

Superdeformation & Hyperdeformation

At very high angular momentum, nuclei can be trapped in secondary minima of the potential energy surface with very large deformations:

Superdeformed bands are stabilized by shell effects at large deformation, where new magic numbers appear (e.g., N or Z = 60, 80 at 2:1 axis ratio). The moment of inertia for superdeformed bands is close to the rigid-body value:

The transition energies in a superdeformed band follow a nearly linear pattern:$E_\gamma(J \to J-2) = \frac{\hbar^2}{\mathscr{I}}(2J-1)$, giving equally spaced gamma rays, which is the experimental signature used to identify superdeformed bands.

Ground-State Correlations

The nuclear ground state contains important correlations beyond the mean-field description. Pairing correlations are described by the BCS theory adapted to nuclei:

where $v_k^2$ is the occupation probability and $u_k^2 + v_k^2 = 1$. The pairing gap $\Delta$ is determined self-consistently:

where $E_k = \sqrt{(\epsilon_k - \lambda)^2 + \Delta^2}$ is the quasiparticle energy and$G$ is the pairing strength. Empirically, $\Delta \approx 12/\sqrt{A}$ MeV. Quadrupole correlations in the ground state lead to zero-point fluctuations in the deformation, which are especially important near shape-transitional nuclei.

Python Simulation: Nilsson Diagram & Rotational Bands

Simplified Nilsson diagram showing single-particle level splitting with deformation, alongside rotational band energies for different deformation parameters.

Fortran Implementation: Quadrupole Properties

Computes intrinsic quadrupole moments, B(E2) values, and moment of inertia ratios for a selection of well-deformed nuclei. Also examines superdeformation and hyperdeformation.