Interacting Boson Model

s and d Bosons

The Interacting Boson Model (IBM), developed by Arima and Iachello, describes low-lying collective states of even-even nuclei in terms of bosons. Each boson represents a correlated pair of valence nucleons (protons or neutrons). The model uses two types of bosons:

• s-boson: Angular momentum $\ell = 0$, representing a $J=0^+$ nucleon pair (1 component)
• d-boson: Angular momentum $\ell = 2$, representing a $J=2^+$ nucleon pair (5 components: $m = -2, -1, 0, 1, 2$)

The total number of bosons $N$ is conserved and equals half the number of valence nucleon pairs (counting from the nearest closed shell). The boson creation and annihilation operators satisfy:

The six operators $\{s^\dagger, d_m^\dagger\}$ span a 6-dimensional space. The bilinear products of creation and annihilation operators generate the Lie algebra of U(6), which is the starting group of the IBM.

U(6) Group Structure

The 36 generators of U(6) are the bilinear operators $\{s^\dagger s, s^\dagger \tilde{d}_m, d_m^\dagger s, d_m^\dagger \tilde{d}_{m'}\}$. The most general IBM-1 Hamiltonian conserving boson number and angular momentum can be written in terms of Casimir operators of various subgroups. The group chain starts with U(6) and must end with the rotational group:

There are exactly three subgroup chains that yield analytical solutions (dynamical symmetries):

The general IBM Hamiltonian interpolates between these limits, forming a "symmetry triangle" (Casten triangle) in parameter space. Most real nuclei lie between the symmetry limits.

Casimir Operators & Energy Formulas

Each dynamical symmetry has an energy formula expressed in terms of Casimir operators of the subgroups in the chain. The Casimir operators are invariants that commute with all generators of their respective group.

U(5) Vibrational Limit

Quantum numbers: $n_d$ (d-boson number), $v$ (O(5) seniority),$L$ (angular momentum). This gives equally-spaced multiplets characteristic of a vibrational nucleus with $R_{4/2} = E(4^+_1)/E(2^+_1) \approx 2.0$.

SU(3) Rotational Limit

The quantum numbers $(\lambda,\mu)$ label the SU(3) irreducible representation. The ground-state band has $(\lambda,\mu) = (2N,0)$, giving $R_{4/2} = 10/3 \approx 3.33$, the rigid rotor value.

O(6) Gamma-unstable Limit

This limit describes gamma-unstable nuclei (free to oscillate in the $\gamma$ degree of freedom). The characteristic ratio is $R_{4/2} = 5/2 = 2.50$. The quantum number $\sigma$ labels O(6) representations, with $\sigma = N, N-2, \ldots$

E2 Transitions in IBM

The E2 transition operator in the IBM is:

where $e_B$ is the boson effective charge and $\chi$ is a structure parameter. The value of $\chi$ determines which dynamical symmetry limit is realized:

• - $\chi = 0$: O(6) limit
• - $\chi = -\sqrt{7}/2$: SU(3) limit (prolate)
• - $\chi = +\sqrt{7}/2$: $\overline{\text{SU}(3)}$ limit (oblate)

The B(E2) values provide powerful tests of the model. Key predictions for the $2^+_1 \to 0^+_1$ transition:

These formulas show that B(E2) values scale linearly with N in the vibrational limit but quadratically in the rotational limit, reflecting enhanced collectivity in deformed nuclei.

Comparison with Experimental Data

The IBM has been remarkably successful in describing nuclear spectra. Key experimental signatures for identifying the dynamical symmetry of a nucleus:

Classic examples of each symmetry include:

• - U(5): $^{110}$Cd ($R_{4/2} = 2.14$), near-vibrational
• - SU(3): $^{156}$Gd ($R_{4/2} = 3.24$), good rotor
• - O(6): $^{196}$Pt ($R_{4/2} = 2.47$), gamma-unstable

IBM-2: Proton-Neutron Bosons

The IBM-2 distinguishes between proton ($\pi$) and neutron ($\nu$) bosons:

The IBM-2 Hamiltonian includes a crucial proton-neutron interaction term (the Majorana interaction) that controls the F-spin structure:

where $\hat{Q}_\rho$ is the quadrupole operator for proton/neutron bosons and$\hat{M}_{\pi\nu}$ is the Majorana operator. IBM-2 introduces the concept of F-spin (analogous to isospin for bosons). States with maximum F-spin ($F = F_{\max} = N/2$) are fully symmetric under proton-neutron boson exchange and correspond to the low-lying collective states. Mixed-symmetry states with $F < F_{\max}$ include the scissors mode ($1^+$ state), first observed in $^{156}$Gd at ~3 MeV.

Python Simulation: IBM Energy Spectra

Energy level diagrams for the three dynamical symmetry limits of IBM-1, showing the characteristic spectral patterns and quantum numbers.

Fortran Implementation: IBM Calculator

Computes energy levels and quantum numbers for all three IBM-1 dynamical symmetry limits, including energy ratios and comparison across boson numbers.