User:ANUnuclearhonoursclass/Nilsson Model

In nuclear physics and nuclear structure, the Nilsson model describes the single-particle states of deformed nuclei. It generalises the shell model, which describes only spherical nuclei. An important foundation for the theoretical framework of the Nilsson model is the definition and classification of nuclear deformation. Nilsson's original model dealt with only quadrupole deformations (prolate or oblate). It was cast in the form of a modified harmonic oscillator potential with a spin orbit coupling term. However, a single-particle treatment of deformation can also be achieved with a Woods Saxon potential. Later developments of the Nilsson model also allowed for generalisation to higher-order nuclear deformations.

In the Nilsson model, the single particle orbital angular momentum and total angular momentum are no longer good quantum numbers. Only the projection of the total angular momentum onto the symmetry axis and the parity are good quantum numbers. The energies of each wave function are determined by the principal quantum number, the projection of the total angular momentum onto the symmetry axis, the projection of the orbital angular momentum onto the symmetry axis and the parity.

History
Development of the Nilsson Model began in the late 1940s and early 1950s with the work of Leo James Rainwater, Aage Bohr and Ben Mottelson. It was first hypothesized by Rainwater that the limitations of contemporary models, such as the nuclear shell model and the liquid drop model, could be addressed if nuclei were not strictly viewed as spherical . Preliminary investigations by Bohr and Mottelson suggested that modelling some nuclei as ellipsoids would produce results that fitted better with experimental data. The task of calculating the effects deformation would have on known properties such as energy levels was given to Sven Gösta Nilsson after whom the final model was named.

In developing the model, Nilsson generalised the nuclear shell model, with its spherical potential, to include nuclei with prolate and oblate deformations. The resultant model, presented in 1955, was able to reproduce the magic numbers, produce correct energy level splitting, and match some of the other experimental data yet to be explained by previous models. The model is still used today to understand many deformed nuclei.

The Nilsson Model Hamiltonian
The Nilsson model employs a deformed harmonic oscillator potential, with corrections to account for angular momentum and  spin-orbit coupling. The Nilsson model Hamiltonian for a single particle in a deformed nucleus can be written as


 * $$ \hat{H}=-\frac{\mathbf{\hat{p}}^2}{2m}+

\frac{1}{2}m\left[\omega_\bot^2(\hat{x}^2+\hat{y}^2)+\omega_z^2\hat{z}^2\right]- \kappa\mathbf{\hat{l}}\cdot\mathbf{\hat{s}}- \mu'\left(\mathbf{\hat{l}}^2-\langle\mathbf{\hat{l}}^2\rangle_N\right)$$

where:

 

The first term describes the kinetic energy of the particle. The second term is the deformed harmonic oscillator potential. The deformation enters the equation through the oscillator frequencies, which are defined as:


 * $$\omega_{z}=\omega_0\left( 1-\frac{2}{3} \varepsilon \right)$$


 * $$\omega_{\bot}=\omega_0\left( 1+\frac{1}{3} \varepsilon

\right)$$

where:

 

The deformation parameter is defined such that:  

The third and forth terms are corrections to the harmonic oscillator potential that better reproduce experimental energy levels. The third term is a spin-orbit term. This lowers the energy of levels where the spin and orbital angular momentum are aligned. The fourth term lowers the energy appropriately at high angular momentum, as the harmonic oscillator potential is too shallow at larger radii. The parameters $$ \kappa$$ and $$ \mu '$$, which determine the strength of the third and fourth terms respectively, are normally determined experimentally. The best experimental fits are obtained by choosing different $$ \kappa$$ and $$ \mu '$$ values for different oscillator shells, $$ N$$. Ben Mottelson, a colleague of Nilsson, described the fourth term as the crucial difference between Nilsson and his competitors' work :

"[the fourth term] made it possible for Sven Gösta to incorporate the growing amount of experimental evidence that was becoming available concerning the sequence of single particle orbits in spherical nuclei.... only [Nilsson's] potential started out from the correct sequence in spherical nuclei."

Solving the Schrödinger equation with the Nilsson model Hamiltonian yields a series of energy levels with dependence on the deformation. The major difference between the deformed and undeformed levels is the presence of $\Omega$-splitting: the lifting of energy degeneracies as a result of deformation.

Nilsson wavefunctions and quantum numbers
The wavefunctions for the spherical shell model can be described by the quantum numbers [Nlj] where:
 * N is the principal quantum number
 * l is the orbital angular momentum
 * j = l + s is the total angular momentum

In the Nilsson model, l and j are no longer 'good' quantum numbers since, for deformed nuclei, orbitals with the same total angular momentum can have different energies. The quantum numbers used to describe the Nilsson model wavefunctions are $$\Omega^{\pi}[Nn_{z}\Lambda]$$ where It should be noted, however, that most of these quantum numbers are simply labels for the orbitals. The only good quantum numbers are $$\Omega$$ and $$\pi$$
 * z refers to the symmetry axis
 * $$\Omega$$ is the projection of the single particle total angular momentum onto the z axis. The notation K which is the quantum number describing the the projection of the total angular momentum onto the symmetry axis is also used.
 * $$\pi$$ is the parity ( +1 or -1)
 * N is the principal quantum number
 * $$n_{z}$$ is the number of nodes of the wavefunction in the z direction (the number of times the radial wavefunction crosses zero). Larger $$ n_{z}$$ values correspond to wavefunctions more extended in the z direction, which means lower energy orbitals.
 * $$\Lambda$$ is the projection of the orbital angular momentum, l, onto the z axis.

$$\Omega$$-splitting


In the shell model, energy levels with total angular momentum j have a 2j+1 degeneracy due to spherical symmetry. For deformed nuclei this degeneracy is lifted. To understand this, the valence nucleon can be thought of as a particle orbiting in a mean field potential generated by the bulk of the nucleus. For a deformed nucleus, the energy level depends on the spatial orientation of the orbit. If the total angular momentum is oriented nearly perpendicular to the symmetry axis, then the projection of the total angular momentum $$\Omega$$ onto the symmetry axis is small. For a prolate nucleus, the orbital with the smallest such projection has the lowest energy, and vice-versa for oblate nuclei. The two limiting cases are depicted for a prolate nucleus.

The energy levels from $$\Omega$$-splitting are two fold degenerate since the orbits of states with the same $$ |\Omega|$$ are equivalent (i.e. $$ \Omega$$ and $$ -\Omega$$ give the same energy). For example, in a prolate nucleus a state with j=5/2 (such as the d5/2 state as shown) will have three distinct energy levels with $$ |\Omega|$$= 1/2, 3/2, 5/2. The $$ \Omega$$=1/2 level has the lowest energy and the $$ \Omega$$=5/2 has the highest.

The degree of $$\Omega$$-splitting depends on the orientation of the nucleon orbit with respect to the symmetry axis
 * $$\theta=\sin^{-1}\frac{\Omega}{j}$$

This means that the energy splitting between adjacent $$\Omega$$ values is large for large values of $$\Omega$$ and smaller for small values of $$\Omega$$. For instance, for j=5/2 the spacing between the $$\Omega$$=1/2 and $$\Omega$$=3/2 levels is smaller than that between the $$\Omega$$=3/2 and $$\Omega$$=5/2.

$$\Omega$$-splitting in deformed nuclei is important because it means that a nucleon orbiting a deformed potential can have a lower energy than an equivalent nucleon orbiting in a spherical potential.

Configurational mixing and level-level repulsion


The quadrupole potential in the Nilsson model also alters the energy eigenfunctions compared to the shell model. The eigenfunctions of the Nilsson model are mixtures of states with the same angular momentum projection $$\Omega$$ and parity $$\pi$$, but different total $$j$$ and orbital $$l$$ angular momentum. A particular Nilsson wavefunction denoted by i is written


 * $$\Psi_{Nil}(i)=\sum_{N,j,l}C_j^i|Nlj\Omega\rangle$$

where:

 

The expansion coefficients are only significant for states with small energy separations, and are strongest for states with small differences in orbital angular momentum (generally $$\Delta l=2$$, since the states must have the same parity). The interaction also prefers states in which the orientation of the nucleon spin has not changed relative to the angular momentum. For example, s$$_{1/2}$$ (l=0, j=l+1/2) and d$$_{5/2}$$ (l=2, j=l+1/2) states will mix more strongly than d$$_{5/2}$$ (l=2, j=l+1/2) and d$$_{3/2}$$ (l=2, j=l-1/2) states.

This mixing alters the energy levels because two levels with the same quantum numbers $$\Omega$$ and $$\pi$$ cannot cross. As two levels approach each other in energy, they mix strongly, and the interaction shifts the energy of the higher state up, and the lower state down: the states repel. This is termed level-level repulsion; its effect is the apparent 'bending' of states which is seen in Nilsson diagrams.

Motivation for Deformed Models
Part of the motivation for considering deformed nuclei was the detection of larger than expected electric quadrupole moments. Deformation of the bulk of the nucleus would imply a different charge distribution than a spherical nucleus with only valence protons in deformed orbits, thus accounting for large observed quadrupole moments.

Testing the Nilsson Model States
Single nucleon transfer experiments provide a means with which the single nucleon states of a nucleus may be probed. In a single nucleon transfer experiment, a light beam is fired at a target of the nucleus to be studied. The energies and angular distributions of particles in the outgoing beam can be used to infer information about the structure of single particle states in the target nucleus.