Nilsson model

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 1950's, 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. The first successful model of this type is the one now known as the Nilsson model. 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.

Because the potential is not spherically symmetric, the single-particle states are not states of good angular momentum J. However, a Lagrange multiplier $-\omega \cdot J$, known as a "cranking" term, can be added to the Hamiltonian. Usually the angular frequency vector ω is taken to be perpendicular to the symmetry axis, although tilted-axis cranking can also be considered. Filling the single-particle states up to the Fermi level then produces states whose expected angular momentum along the cranking axis $\langle J_{x}\rangle$ has the desired value set by the Lagrange multiplier.

Hamiltonian

For an axially symmetric shape with the axis of symmetry being the z axis, the Hamiltonian is

$H={\frac {1}{2}}m\omega _{z}^{2}z^{2}+{\frac {1}{2}}m\omega _{\perp }^{2}\rho ^{2}-c_{1}\ell \cdot s-c_{2}(\ell ^{2}-\langle \ell ^{2}\rangle _{N}).$

Here m is the mass of the nucleon, N is the total number of harmonic oscillator quanta in the spherical basis, $\rho$ is the distance from the z axis, $\ell$ is the orbital angular momentum operator, $\ell ^{2}$ is its square (with eigenvalues $\ell (\ell +1)$), $\langle \ell ^{2}\rangle _{N}=(1/2)N(N+3)$ is the average value of $\ell ^{2}$ over the N shell, and s is the intrinsic spin.

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 $\omega _{z}\omega _{\perp }^{2}=\omega _{0}^{3}$ remains constant with deformation, preserving the volume of equipotential surfaces. The deformation parameter δ is introduced, with $(\omega _{\perp }/\omega _{z})^{2}=(1+{\frac {2}{3}}\delta )/(1-{\frac {4}{3}}\delta )$. Positive values of δ indicate prolate deformations. Reproducing the observed density of nuclear matter requires $\hbar \omega _{0}\approx (42\ {\text{MeV}})A^{-1/3}$.

The purpose of the $\ell ^{2}$ term is to flatten 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. The term $\langle \ell ^{2}\rangle _{N}$ prevents this from shifting the spherical energy levels up or down.

The two adjustable constants are conventionally parametrized as $c_{1}=2\kappa \hbar \omega _{0}$ and $c_{2}=\mu \kappa \hbar \omega _{0}$. Typical values of κ and μ for heavy nuclei are 0.06 and 0.5.

Choice of basis and quantum numbers

For ease of computation using the computational resources of the 1950's, Nilsson used a basis consisting of eigenstates of the spherical hamiltonian. The difference between the spherical and deformed Hamiltonian is proportional to $r^{2}Y_{20}$, and this has matrix elements that are easy to calculate in this basis. They couple the different N shells. 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, 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. The Nilsson quantum numbers are $\{N,\ell ,m_{\ell },m_{s}\}$.

References

Olivius, P., "Extending the nuclear cranking model to tilted axis rotation and alternative mean field potentials," doctoral thesis, Lund University, 2004, http://www.matfys.lth.se/staff/Peter.Olivius/thesis.pdf

Sven Gösta Nilsson, "Binding states of individual nucleons in strongly deformed nuclei," doctoral thesis, 1955