Nuclear structure
The cluster model describes the nucleus as a molecule-like collection of proton-neutron groups (e.g., alpha particles) with one or more valence neutrons occupying molecular orbitals.
The assumption of nucleus as a drop of Fermi liquid is still widely used in the form of Finite Range Droplet Model (FRDM), due to the possible good reproduction of nuclear binding energy on the whole chart, with the necessary accuracy for predictions of unknown nuclei.
It was previously used to describe the existence of nucleon shells according to an approach closer to what is now called mean field theory.
In particular, some nuclei having certain values for the number of protons and/or neutrons are bound more tightly together than predicted by the liquid drop model.
This observation led scientists to assume the existence of a shell structure of nucleons (protons and neutrons) within the nucleus, like that of electrons within atoms.
Werner Heisenberg extended the principle of Pauli exclusion to nucleons, via the introduction of the iso-spin concept.
[7] Nucleons are thought to be composed of two kind of particles, the neutron and the proton that differ through their intrinsic property, associated with their iso-spin quantum number.
This concept enables the explanation of the bound state of Deuterium, in which the proton and neutron can couple their spin and iso-spin in two different manners.
Furthermore, the energy needed to excite the nucleus (i.e. moving a nucleon to a higher, previously unoccupied level) is exceptionally high in such nuclei.
Whenever this unoccupied level is the next after a full shell, the only way to excite the nucleus is to raise one nucleon across the gap, thus spending a large amount of energy.
For example, observations of unstable isotopes have shown shifting and even a reordering of the single particle levels of which the shell structure is composed.
[8] This is sometimes observed as the creation of an island of inversion or in the reduction of excitation energy gaps above the traditional magic numbers.
The core is a set of single-particles which are assumed to be inactive, in the sense that they are the well bound lowest-energy states, and that there is no need to reexamine their situation.
The set of all possible Slater determinants in the valence space defines a basis for (Z-) N-body states.
The independent particle model and mean field theories (we shall see that there exist several variants) have a great success in describing the properties of the nucleus starting from an effective interaction or an effective potential, thus are a basic part of atomic nucleus theory.
One should also notice that they are modular enough, in that it is quite easy to extend the model to introduce effects such as nuclear pairing, or collective motions of the nucleon like rotation, or vibration, adding the corresponding energy terms in the formalism.
This implies that in many representations, the mean field is only a starting point for a more complete description which introduces correlations reproducing properties like collective excitations and nucleon transfer.
[12] In a seminal paper[13] by Dominique Vautherin and David M. Brink it was demonstrated that a Skyrme force that is density dependent can reproduce basic properties of atomic nuclei.
The second step consists in assuming that the wavefunction of the system can be written as a Slater determinant of one-particle spin-orbitals.
To be more precise, there should be mentioned that the energy is a functional of the density, defined as the sum of the individual squared wavefunctions.
Practically, the algorithm is started with a set of individual grossly reasonable wavefunctions (in general the eigenfunctions of a harmonic oscillator).
The calculation stops – convergence is reached – when the difference among wavefunctions, or energy levels, for two successive iterations is less than a fixed value.
Born first in the 1970s with the works of John Dirk Walecka on quantum hadrodynamics, the relativistic models of the nucleus were sharpened up towards the end of the 1980s by P. Ring and coworkers.
The main simplification consists in replacing in the equations all field terms (which are operators in the mathematical sense) by their mean value (which are functions).
Qualitatively, these spontaneous symmetry breakings can be explained in the following way: in the mean field theory, the nucleus is described as a set of independent particles.
This implies that each nucleon binds with another one to form a pair, consequently the system cannot be described as independent particles subjected to a common mean field.
Conversely, in the case of odd number of protons or neutrons, there exists an unpaired nucleon, which needs less energy to be excited.
The Hartree–Fock–Bogolyubov (HFB) method is a more sophisticated approach,[16] enabling one to consider the pairing and mean field interactions consistently on equal footing.
Peculiarity of mean field methods is the calculation of nuclear property by explicit symmetry breaking.
These correlations can be introduced taking into account the coupling of independent particle degrees of freedom, low-energy collective excitation of systems with even number of protons and neutrons.
