Nuclear structure
Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics.
Models
The cluster model
The cluster model describes the nucleus as a molecule-like collection of proton-neutron groups with one or more valence neutrons occupying molecular orbitals.The liquid drop model
The liquid drop model is one of the first models of nuclear structure, proposed by Carl Friedrich von Weizsäcker in 1935. It describes the nucleus as a semiclassical fluid made up of neutrons and protons, with an internal repulsive electrostatic force proportional to the number of protons. The quantum mechanical nature of these particles appears via the Pauli exclusion principle, which states that no two nucleons of the same kind can be at the same state. Thus the fluid is actually what is known as a Fermi liquid.In this model, the binding energy of a nucleus with protons and neutrons is given by
where is the total number of nucleons. The terms proportional to and represent the volume and surface energy of the liquid drop, the term proportional to represents the electrostatic energy, the term proportional to represents the Pauli exclusion principle and the last term is the pairing term, which lowers the energy for even numbers of protons or neutrons.
The coefficients and the strength of the pairing term may be estimated theoretically, or fit to data.
This simple model reproduces the main features of the binding energy of nuclei.
The assumption of nucleus as a drop of Fermi liquid is still widely used in the form of Finite Range Droplet Model, due to the possible good reproduction of nuclear binding energy on the whole chart, with the necessary accuracy for predictions of unknown nuclei.
The shell model
The expression "shell model" is ambiguous in that it refers to two different items. It was previously used to describe the existence of nucleon shells according to an approach closer to what is now called [|mean field theory].Nowadays, it refers to a formalism analogous to the configuration interaction formalism used in quantum chemistry.
Introduction to the shell concept
Systematic measurements of the binding energy of atomic nuclei show systematic deviations with respect to those estimated from the liquid drop model. 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. These nuclei are called singly/doubly magic. This observation led scientists to assume the existence of a shell structure of nucleons within the nucleus, like that of electrons within atoms.Indeed, nucleons are quantum objects. Strictly speaking, one should not speak of energies of individual nucleons, because they are all correlated with each other. However, as an approximation one may envision an average nucleus, within which nucleons propagate individually. Owing to their quantum character, they may only occupy discrete energy levels. These levels are by no means uniformly distributed; some intervals of energy are crowded, and some are empty, generating a gap in possible energies. A shell is such a set of levels separated from the other ones by a wide empty gap.
The energy levels are found by solving the Schrödinger equation for a single nucleon moving in the average potential generated by all other nucleons. Each level may be occupied by a nucleon, or empty. Some levels accommodate several different quantum states with the same energy; they are said to be degenerate. This occurs in particular if the average nucleus exhibits a certain symmetry, like a spherical shape.
The concept of shells allows one to understand why some nuclei are bound more tightly than others. This is because two nucleons of the same kind cannot be in the same state. Werner Heisenberg extended the principle of Pauli exclusion to nucleons, via the introduction of the iso-spin concept. 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. So the lowest-energy state of the nucleus is one where nucleons fill all energy levels from the bottom up to some level. Nuclei that exhibit an odd number of either protons or neutrons are less bound than nuclei with even number. A nucleus with full shells is exceptionally stable, as will be explained.
As with electrons in the electron shell model, protons in the outermost shell are relatively loosely bound to the nucleus if there are only few protons in that shell, because they are farthest from the center of the nucleus. Therefore, nuclei which have a full outer proton shell will be more tightly bound and have a higher binding energy than other nuclei with a similar total number of protons. This is also true for neutrons.
Furthermore, the energy needed to excite the nucleus 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. Otherwise, if the highest occupied energy level lies in a partly filled shell, much less energy is required to raise a nucleon to a higher state in the same shell.
Some evolution of the shell structure observed in stable nuclei is expected away from the valley of stability. 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. 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.
Basic hypotheses
Some basic hypotheses are made in order to give a precise conceptual framework to the shell model:- The atomic nucleus is a quantum n-body system.
- The internal motion of nucleons within the nucleus is non-relativistic, and their behavior is governed by the Schrödinger equation.
- Nucleons are considered to be pointlike, without any internal structure.
Brief description of the formalism
The next step consists in defining a basis of single-particle states, i.e. a set of wavefunctions describing all possible nucleon states. Most of the time, this basis is obtained via a Hartree–Fock computation. With this set of one-particle states, Slater determinants are built, that is, wavefunctions for Z proton variables or N neutron variables, which are antisymmetrized products of single-particle wavefunctions.
In principle, the number of quantum states available for a single nucleon at a finite energy is finite, say n. The number of nucleons in the nucleus must be smaller than the number of available states, otherwise the nucleus cannot hold all of its nucleons. There are thus several ways to choose Z states among the n possible. In combinatorial mathematics, the number of choices of Z objects among n is the binomial coefficient C. If n is much larger than Z, this increases roughly like nZ. Practically, this number becomes so large that every computation is impossible for A=''N+Z'' larger than 8.
To obviate this difficulty, the space of possible single-particle states is divided into core and valence, by analogy with chemistry. 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. They do not appear in the Slater determinants, contrary to the states in the valence space, which is the space of all single-particle states not in the core, but possibly to be considered in the choice of the build of the N-body wavefunction. The set of all possible Slater determinants in the valence space defines a basis for N-body states.
The last step consists in computing the matrix of the Hamiltonian within this basis, and to diagonalize it. In spite of the reduction of the dimension of the basis owing to the fixation of the core, the matrices to be diagonalized reach easily dimensions of the order of 109, and demand specific diagonalization techniques.
The shell model calculations give in general an excellent fit with experimental data. They depend however strongly on two main factors:
- The way to divide the single-particle space into core and valence.
- The effective nucleon–nucleon interaction.
Mean field theories
The independent-particle model (IPM)
The interaction between nucleons, which is a consequence of strong interactions and binds the nucleons within the nucleus, exhibits the peculiar behaviour of having a finite range: it vanishes when the distance between two nucleons becomes too large; it is attractive at medium range, and repulsive at very small range. This last property correlates with the Pauli exclusion principle according to which two fermions cannot be in the same quantum state. This results in a very large mean free path predicted for a nucleon within the nucleus.The main idea of the Independent Particle approach is that a nucleon moves inside a certain potential well independently from the other nucleons. This amounts to replacing an N-body problem by N single-body problems. This essential simplification of the problem is the cornerstone of mean field theories. These are also widely used in atomic physics, where electrons move in a mean field due to the central nucleus and the electron cloud itself.
The independent particle model and mean field theories 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.