This page is an adapted translation of the corresponding fr:Structure nucléaire  As will be noted, there remain void paragraphs, as on the original. Competent Wikipedians are welcome to enrich it, and their contributions will be translated back to fr:
Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. This article is written from a nuclear physics perspective; as such, it is suggested that a casual reader first read the main nuclear physics article.
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This is one of the first models of nuclear structure, proposed by Carl Friedrich von Weizsäcker in 1935. It describes the nucleus as a classical fluid made up of neutrons and protons , with an internal repulsive electric 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 fluid. This simple model reproduces the main features of the binding energy of nuclei.
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 (protons and neutrons) 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. To be able to speak of a shell structure, one envisions first an average nucleus, within which nucleons propagate individually. Owing to their quantum character, they can have then only discrete values of energy levels. These levels are by no means uniformly distributed : some intervals of energy are crowded, but they are separated by almost empty gaps. A shell is such a set of levels separated from the other ones by a wide empty gap.
The determination of the energy levels is done via quantum mechanics, more precisely by diagonalization of the singlenucleon Hamiltonian. 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 has some symmetry.
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 (Pauli exclusion principle). So the lowestenergy state of the nucleus is one where nucleons fill all energy levels from the bottom up to some level. 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. All this is also true for neutrons.
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. 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.
The expression "shell model" is ambiguous in that it refers to two different eras in the state of the art. It was previously used to describe the existence of nucleon shells in the nucleus according to an approach closer to what is now called mean field theory. Nowadays, it refers to a set of techniques which help solving some variants of the nuclear nbody problem. We shall introduce these here.
Several basic hypotheses are made in order to give a precise conceptual framework to the shell model :
The general process used in the shell model calculations is the following. First a Hamiltonian for the nucleus is defined. As mentioned before, only 1 and 2body terms are taken into account in this definition. The interaction is an effective theory : it contains free parameters which have to be fitted with experimental data.
The next step consists in defining a basis of singleparticle 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 1particle states, Slater determinants are built, that is, wavefunctions for Z proton variables or N neutron variables, which are antisymmetrized products of singleparticle wavefunctions (antisymmetrized meaning that under exchange of variables for any pair of nucleons, the wavefunction only changes sign).
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 (or N) states among the n possible. In combinatorial mathematics, the number of choices of Z objects among n is the binomial coefficient C_{n}^{Z} . If n is much larger than Z (or N), this increases roughly like n^{Z}. 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 singleparticle states is divided into a core and a valence shell, by analogy with chemistry. The core is a set of singleparticles which are assumed to be inactive, in the sense that they are the well bound lowestenergy 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 singleparticle states not in the core, but possibly to be considered in the choice of the build of the (Z) Nbody wavefunction. The set of all possible Slater determinants in the valence space defines a basis for (Z) Nbody 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 10^{9}, 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 interaction between nucleons, which is a consequence of strong interactions, and binds the nucleons within the nucleus exhibits the peculiar behaviour to have a finite range : it vanishes when the distance among 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 (nucleons are fermions) cannot be in the same quantum state. This results, in theory, in a very large mean free path predicted for a nucleon within the nucleus. However, this prediction of the shell model is not confirmed by particle scattering experiments (see Cook, 2006, Models of the Atomic Nucleus). Experimental results on nucleonnucleon scattering indicate frequent elastic collisions implying a free mean path very much shorter than the nucleus radius. The suggestion of Weisskopf to invoke Pauli blocking has now been shown experimentally to do little to raise the free mean path anywhere near the length required for nucleons to orbit in energy shells before collision. This paradox of the shell model has lead Cook (2006) to conclude that "the independent orbiting of nucleons within the dense nuclear interior is a fiction".
The main idea of the Independent Particle approach is that a nucleon moves inside a certain potential well (which keeps it bound to the nucleus) independently from the other nucleons. In theory, this amounts to replacing a Nbody problem (N particles interacting) by N singlebody 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. However, as discussed by Cook (2006) one cannot apply the quantum results from atomic (electron) interactions to those within the nucleus to imply that nucleons move independently within shells.
Although the shell model hypothesis looks grossly simplifying, it led to big successes, and mean field theories (we shall see that there exist several variants) are now a basic part of atomic nucleus theory. One should also notice that they are modular enough (in terms of programming theory), in that it is quite easy to introduce certain effects such as nucleon pairing, or collective motions of the nucleon like rotation, or vibration, adding "by hand" the corresponding energy terms in the formalism. However, one cannot on the one hand have nucleons closely bound within clusters, such as shown experimentally using the alphacluster model formalism, and on the other hand require a large free mean path. Thus the extensive experimental data for nucleons showing short free mean path length and nuclear clustering effects indicate that the shell model is at best an incomplete explanation of nuclear structure.
A large part of the practical difficulties met in mean field theories is the definition (or calculation) of the potential of the mean field itself. One can very roughly distinguish between two approaches :
In the case of the Hartree–Fock approaches, the trouble is not to find the mathematical function which describes best the nuclear potential, but that which describes best the nucleonnucleon interaction. Indeed, in contrast with atomic physics where the interaction is known (it is the Coulomb interaction), the nucleonnucleon interaction within the nucleus is not known analytically.
There are two main reasons for this fact. First, the strong interaction acts essentially among the quarks forming the nucleons. The nucleonnucleon interaction in vacuum is a mere consequence of the quarkquark interaction. While the latter is well understood in the framework of the standard model at high energies, it is much more complicated in low energies due to color confinement and asymptotic freedom. Thus there is no fundamental theory allowing one to deduce the nucleonnucleon interaction from the quarkquark interaction. Further, even if this problem were solved, there would remain a large difference between the ideal (and conceptually simpler) case of two nucleons interacting in vacuo, and that of these nucleons interacting in the nuclear matter. To go further, it was necessary to invent the concept of effective interaction. The latter is basically a mathematical function with several arbitrary parameters, which are adjusted to agree with experimental data.
In the Hartree–Fock approach of the nbody problem, the starting point is a Hamiltonian containing n kinetic energy terms, and potential terms. As mentioned before, one of the mean field theory hypotheses is that only the 2body interaction is to be taken into account. The potential term of the Hamiltonian represents all possible 2body interactions in the set of n fermions. It is the first hypothesis.
The second step consists in assuming that the wavefunction of the system can be written as a Slater determinant. This statement is the mathematical translation of the independentparticle model. This is the second hypothesis.
There remains now to determine the components of this Slater determinant, that is, the individual wavefunctions of the nucleons. To this end, it is assumed that the total wavefunction (the Slater determinant) is such that the energy is minimum. This is the third hypothesis.
Technically, it means that one must compute the mean value of the (known) 2body Hamiltonian on the (unknown) Slater determinant, and impose that its mathematical variation vanishes. This leads to a set of equations where the unknowns are the individual wavefunctions: the Hartree–Fock equations. Solving these equations gives the wavefunctions and individual energy levels of nucleons, and so the total energy of the nucleus and its wavefunction.
This short account of the Hartree–Fock method explains why it is called also the variational approach. At the beginning of the calculation, the total energy is a "function of the individual wavefunctions" (a socalled functional), and everything is then made in order to optimize the choice of these wavefunctions so that the functional has a minimum – hopefully absolute, and not only local. 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. Let us note also that the Hartree–Fock method is also used in atomic physics and condensed matter physics as Density Functional Theory, DFT.
The process of solving the Hartree–Fock equations can only be iterative, since these are in fact a Schrödinger equation in which the potential depends on the density, that is, precisely on the wavefunctions to be determined. Practically, the algorithm is started with a set of individual grossly reasonable wavefunctions (in general the eigenfunctions of a harmonic oscillator). These allow to compute the density, and therefrom the Hartree–Fock potential. Once this done, the Schrödinger equation is solved anew, and so on. The calculation stops – convergence is reached – when the difference among wavefunctions, or energy levels, for two successive iterations is less than a fixed value. Then the mean field potential is completely determined, and the Hartree–Fock equations become standard Schrödinger equations. The corresponding Hamiltonian is then called the Hartree–Fock Hamiltonian.
Born first in the 1970’s with the works of D. Walecka on quantum hadrodynamics, the relativistic models of the nucleus were sharpened up towards the end of the 1980’s by P. Ring and coworkers. The starting point of these approaches is the relativistic quantum field theory. In this context, the nucleon interactions occur via the exchange of virtual particles called mesons. The idea is, in a first step, to build a lagrangian containing these interaction terms. Second, by an application of the least action principle, one gets a set of equations of motion. The real particles (here the nucleons) obey the Dirac equation, whilst the virtual ones (here the mesons) obey the Klein–Gordon equations.
In view of the nonperturbative nature of strong interaction, and also in view of the fact that the exact potential form of this interaction between groups of nucleons is relatively badly known, the use of such an approach in the case of atomic nuclei requires drastic approximations. 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). In this way, one gets a system of coupled integrodifferential equations, which can be solved numerically, if not analytically.
One of the focal points of all physics is symmetry. The nucleonnucleon interaction and all effective interactions used in practice have certain symmetries. They are invariant by translation (changing the frame of reference so that directions are not altered), by rotation (turning the frame of reference around some axis), or parity (changing the sense of axes) in the sense that the interaction does not change under any of these operations. Nevertheless, in the Hartree–Fock approach, solutions which are not invariant under such a symmetry can appear. One speaks then of spontaneous symmetry breaking.
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. Most additional correlations among nucleons which do not enter the mean field are neglected. They can appear however by a breaking of the symmetry of the mean field Hamiltonian, which is only approximate. If the density used to start the iterations of the Hartree–Fock process breaks certain symmetries, the final Hartree–Fock Hamiltonian may break these symmetries, if it is advantageous to keep these broken from the point of view of the total energy.
It may also converge towards a symmetric solution. In any case, if the final solution breaks the symmetry, for example, the rotational symmetry, so that the nucleus appears not to be spherical, but elliptic, all configurations deduced from this deformed nucleus by a rotation are just as good solutions for the Hartree–Fock problem. The ground state of the nucleus is then degenerate.
A similar phenomenon happens with the nuclear pairing, which violates the conservation of the number of baryons (see below).
Historically, the observation that the nuclei with an even number of nucleons are systematically more bound than those with an odd one led to propose the nuclear pairing hypothesis. The very simple idea is that each nucleon binds with another one to form a pair. When the nucleus has an even number of nucleons, each one of them finds a partner. To excite such a system, one must at least use such an energy as to break a pair. Conversely, in the case of odd number of nucleons, there exists a bachelor nucleon, which needs less energy to be excited.
This phenomenon is closely analogous to that of superconductivity in solid state physics (at least the lowtemperature superconductivity). The first theoretical description of nuclear pairing was proposed at the end of the 1950’s by Aage Bohr and Ben Mottelson (which led to their Nobel Prize in Physics in 1975). It was close to the BCS theory of Bardeen, Cooper and Schrieffer, which accounts for metal superconductivity. Theoretically, the pairing phenomenon as described by the BCS theory combines with the mean field theory : nucleons are both subject to the mean field potential and to the pairing interaction, but these are independent.
It is tempting to interpret the pairing interaction as a residual interaction. To build the mean field interaction, only some terms of the nucleonnucleon interaction are taken into account. Everything else is qualified as residual interaction. The value of the mean field theory rests on the fact that the residual interaction is numerically much less than what is taken into account for the mean field. There should be however a link between them, since they proceed from the same nucleonnucleon interaction. This is not taken into account in the BCS theory. To circumvent this point, the Hartree–Fock–Bogolyubov (HFB) approach has been developed, to include into a unified formalism the mean field, the pairing and their mutual links.
Let us note finally that one big difference between superconductivity and nuclear pairing resides in the number of particles. In a metal, the number of free electrons is very large, compared to the number of nucleons in a nucleus. The BCS (and HFB) approach describes the system wavefunction as a superposition of components with different numbers of particles. In the case of a metal, this violation of the conservation of the number of particles is of no consequence, in view of the huge statistics. But in nuclear physics, this leads to a real problem. Specific techniques for restoring the number of particles have been developed, in the framework of the restoration of broken symmetries.
