In 1953, the first experimental examples were found of rotational bands in nuclei, with their energy levels following the same J(J+1) pattern of energies as in rotating molecules.
Quantum mechanically, it is impossible to have a collective rotation of a sphere, so this implied that the shape of these nuclei was nonspherical.
In principle, these rotational states could have been described as coherent superpositions of particle-hole excitations in the basis consisting of single-particle states of the spherical potential.
But in reality, the description of these states in this manner is intractable, due to the large number of valence particles—and this intractability was even greater in the 1950s, when computing power was extremely rudimentary.
For these reasons, Aage Bohr, Ben Mottelson, and Sven Gösta Nilsson constructed models in which the potential was deformed into an ellipsoidal shape.
It is essentially a nuclear shell model using a harmonic oscillator potential, but with anisotropy added, so that the oscillator frequencies along the three Cartesian axes are not all the same.
Typically the shape is a prolate ellipsoid, with the axis of symmetry taken to be z.
Here m is the mass of the nucleon, N is the total number of harmonic oscillator quanta in the spherical basis,
This is conventionally expressed in terms of a deformation parameter δ so that the harmonic oscillator part of the potential can be written as the sum of a spherically symmetric harmonic oscillator and a term proportional to δ.
Positive values of δ indicate prolate deformations, like an American football.
Considering the success of the nuclear liquid drop model, in which the nucleus is taken to be an incompressible fluid, the harmonic oscillator frequencies are constrained so that
remains constant with deformation, preserving the volume of equipotential surfaces.
Reproducing the observed density of nuclear matter requires
The remaining two terms in the Hamiltonian do not relate to deformation and are present in the spherical shell model as well.
term is to mock up the flat profile of the nuclear potential as a function of radius.
For nuclear wavefunctions (unlike atomic wavefunctions) states with high angular momentum have their probability density concentrated at greater radii.
Typical values of κ and μ for heavy nuclei are 0.06 and 0.5.
The difference between the spherical and deformed Hamiltonian is proportional to
, and this has matrix elements that are easy to calculate in this basis.
Eigenstates of the deformed Hamiltonian have good parity (corresponding to even or odd N) and Ω, the projection of the total angular momentum along the symmetry axis.
In the absence of a cranking term (see below), time-reversal symmetry causes states with opposite signs of Ω to be degenerate, so that in the calculations only positive values of Ω need to be considered.
In an odd, well-deformed nucleus, the single-particle levels are filled up to the Fermi level, and the odd particle's Ω and parity give the spin and parity of the ground state.
Usually the angular frequency vector ω is taken to be perpendicular to the symmetry axis, although tilted-axis cranking can also be considered.
has the desired value set by the Lagrange multiplier.
Often one wants to calculate a total energy as a function of deformation.
Minima of this function are predicted equilibrium shapes.
Adding the single-particle energies does not work for this purpose, partly because kinetic and potential terms are out of proportion by a factor of two, and partly because small errors in the energies accumulate in the sum.
For this reason, such sums are usually renormalized using a procedure introduced by Strutinsky.
Single-particle levels can be shown in a "spaghetti plot," as functions of the deformation.
Any such gap, at a zero or nonzero deformation, indicates that when the Fermi level is at that height, the nucleus will be stable relative to the liquid drop model.