User:ANUnuclearhonoursclass/Nuclear Deformations

In nuclear physics, any nucleus with a non-spherical shape is said to be deformed.

Introduction
In many respects spherical nuclei are exceptional, as the vast majority of nuclei are deformed. Many nuclei are only slightly non-spherical, and so can still be well described by spherical nuclear models. However, for nuclei with a mass number $$A$$ in the range $$150 < A<190$$ and $$A>220$$ deformation can no longer be ignored.

To lowest order, most deformed nuclei can be approximated as quadrupole deformed (i.e. a prolate or oblate ellipsoid). Higher order deformations, such as octupole and hexadecapole are also possible, as are triaxial deformations and other more exotic shapes.

The single particle states of quadrupole deformed nuclei are described by the Nilsson model. Deformation alters the properties of single-particle wavefunctions from those of the spherical shell model. In particular, the breaking of spherical symmetry changes the degeneracies of energy levels, and allows for collective rotation of the nucleus. Additionally, as the charge distribution is no longer spherical, deformed nuclei can have large electric multipole moments. The presence of rotational excited states and the detection of large quadrupole moments are good indications that a nucleus is deformed.

Quadrupole deformation
A quadrupole deformed nucleus has a prolate or oblate ellipsoidal shape. This is the lowest order of deformation seen in nuclei. A prolate ellipsoid has two equal, short axes and a single long axis; an oblate ellipsoid has two equal long axes and one short axis. Both cases are illustrated below. For deformed nuclei, the axis of unequal length is often termed the deformation axis.

Deformation parameters
The degree to which a nucleus is deformed is described by a deformation parameter: $$\varepsilon$$ or $$\beta$$. $$\varepsilon$$ is the deformation parameter used in the Nilsson model for describing single particle properties of the nucleus, while $$\beta$$ is used when describing collective behaviour such as rotation. In each case, a positive deformation parameter indicates a prolate shape while a negative deformation parameter describes an oblate shape.

Nuclear shape
In terms of the deformation parameter $$\beta$$, the surface of a prolate or oblate nucleus can be described by
 * $$R(\theta,\phi) = R_{av}\left[1+\beta Y_{20}(\theta,\phi)\right]$$

where:  

Electric quadrupole moment
In the rest frame of the nucleus, the electric quadrupole moment, which gives the charge distribution of the nucleus, is
 * $$Q_{0} = \frac{3}{\sqrt{5\pi}}ZR_{av}^{2}\beta(1+0.16\beta) $$

The quadrupole moment in the laboratory frame is
 * $$Q=Q_{0}\left(\frac{3\Omega^{2}-j(j+1)}{(j+1)(2j+3)}\right)$$

where:  

Rotational energy levels
In quantum mechanics rotation about an axis of symmetry cannot occur. So, for instance, a spherical nuclei cannot rotate. The presence of deformation allows for collective rotation of a nucleus. In this case, a rotational band may be built on each particle- or vibrational-excitation of the nucleus. A rotational band refers to a sequence of states with increasing energy and rotational angular momentum.

Single particle properties
The single-particle states in quadrupole deformed nuclei are described by the Nilsson model. The Nilsson model accounts for deformation by including a deformed harmonic oscillator potential in the Hamiltonian, which is given by:
 * $$ \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)$$ deformation is incorporated into the second term, where different oscillator frequencies are used, and the deformation parameter $$\varepsilon$$ is defined by:
 * $$\omega_{z}=\omega_{0} \left( 1-\frac{2}{3} \varepsilon \right)$$
 * $$\omega_{\bot}=\omega_{0}\left( 1+\frac{1}{3} \varepsilon \right)$$

The inclusion of deformation in the Hamiltonian lifts the degeneracy of energy levels seen for spherical nuclei. Additionally, in a deformed nucleus the orbital and total angular momentum are no longer "good" quantum numbers. For deformed nuclei, only the projection of the total angular momenta onto the deformation axis and parity are good quantum numbers.

Prolate vs. Oblate deformation
A nucleus will take on a deformed shape rather than a spherical shape if in doing so it can lower its energy. The majority of quadrupole deformed nuclei are prolate rather than oblate deformed in their ground state. This can be understood by considering two details of the single-particle levels calculated using the Nilsson model. These two facts mean that prolate nuclei, where low $$\Omega$$ levels are lower in energy, have a large number of orbitals that are low in energy in the regions of interest, while oblate nuclei, where high $$\Omega$$ levels have lower energy, have only a few. This is illustrated for an i13/2 level. With more energetically favourable states possible for prolate nuclei, a nucleus is more likely to be prolate deformed.
 * 1) Firstly, the angular momentum of single particle states tends to be high just above shell gaps, where prolate deformation is common. This means these levels will split into many different levels when the nucleus becomes deformed, so the behaviour of these states is significant.
 * 2) Secondly, the energy spacing between levels with different angular momentum projections $$\Omega$$ in a deformed nucleus is proportional to $$\Omega$$, so small $$\Omega$$ levels are closely spaced, while large $$\Omega$$ levels are widely spaced.