A very effective Hamiltonian in nuclear physics

Atomic nuclei are complex quantum many-body systems. Effective theories have helped provide a better understanding of them. The best-known are the shell model, the (Aage) Bohr-Mottelson theory of non-spherical nuclei, and the liquid drop model. Here I introduce the Interacting Boson Model (IBM), which provides somewhat of a microscopic basis for the Bohr-Mottelson theory. Other effective theories in nuclear physics are chiral perturbation theory, Weinberg's theory for nucleon-pion interactions, and Wigner's random matrix theory.

The shell model has similarities to microscopic models in atomic physics. A major achievement is it explains the origins of magic numbers, i.e., nuclei with atomic numbers 2, 8, 20, 28, 50, 82, and 126 are particularly stable because they have closed shells. Other nuclei can then be described theoretically as an inert closed shell plus valence nucleons that interact with a mean-field potential due to the core nuclei and then with one another via effective interactions.

For medium to heavy nuclei the Bohr-Mottelson model describes collective excitations including transitions in the shape of nuclei.

An example of the trends in the low-lying excitation spectrum  to explain is shown in the figure below. The left spectrum is for nucleus with close to a magic number of nuclei and the right one for an almost half-filled shell. R_4/2 is the ratio of the energies of the J=4+ state to that of the 2+ state, relative to the ground state. B(E2) is the strength of the quadrupole transition between the 2+ state and the ground state.

The Interacting Boson Model (IBM) is surprisingly simple and successful. It illustrates the importance of quasi-particles, builds on the stability of closed shells, and neglects many degrees of freedom. It describes even-even nuclei, i.e., nuclei with an even number of protons and an even number of neutrons. The basic entities in the theory are pairs of nucleons. These are taken to be either an s-wave state or a d-wave state. There are five d-wave states (corresponding to the 2J+1 possible states of total angular momentum with J=2). Each state is represented by a boson creation operator and so the Hilbert space is six-dimensional. If the states are degenerate [which they are not] the model has U(6) symmetry.

The IBM Hamiltonian is written in terms of the most general possible combinations of the boson operators. This has a surprisingly simple form.

Note that it involves only four parameters. For a given nucleus these parameters can be fixed from experiment, and in principle calculated from the shell model. The Hamiltonian can be written in a form that gives physical insight, connects to the Bohr-Mottelson model and is amenable to a group theoretical analysis that makes calculation and understanding of the energy spectrum relatively simple.

Central to the group theoretical analysis is considering subalgebra chains as shown below

 

An example of an energy spectrum is shown below.

The fuzzy figures are taken from a helpful Physics Today article by Casten and Feng from 1984 (Aside: the article discusses an extension of the IBM involving supersymmetry, but I don't think that has been particularly fruitful).

The figure below connects the different parameter regimes of the model to the different subalgebra chains.

The nucleotide chart below has entries that have colour shading corresponding to their parameter values for the IBM model according to the symmetry triangle above.

The different vertices of the triangle correspond to different nuclear geometries and allow a connection to Aage Bohr's model for the surface excitations. 

This is discussed in a nice review article, which includes the figure above.

Quantum phase transitions in shape of nuclei

Pavel Cejnar, Jan Jolie, and Richard F. Casten

Aside: one thing that is not clear to me from the article concerns questions that arise because the nucleus has a finite number of degrees of freedom. Are the symmetries actually broken or is there tunneling between degenerate ground states?