The knowledge of the structure of the atomic nuclei , or nuclear structure is one of the key chapters of the Nuclear physics. Taking into account his importance, one made a separated article of it, and one will consult with profit the article Nuclear physics to locate of them the context, the related chapters of physics and the applications of this large branch of physics.

Liquid drop model

One of the first models of the core, proposed by Weizsäcker in 1935, is that of the liquid drop (see the detail under Formula of Weizsäcker). The core is compared to a fluid (quantum) made up of Nucléon S (Proton S and Neutron S) which is confined in a finished volume of space by the strong interaction. The balance of this drop is the result of several contributions:

• a gravitational contribution: each nucleon interacts with its neighbors via the strong interaction. The sum of all these interactions makes it possible the core to exist. One can show that this term is proportional to the volume of the core: it is thus the term known as of volume;

• a repulsive contribution due for a purpose of surface: the nucleons located close to the surface of the core have less neighbors than those in the center of the core. They thus interact less and are thus less dependant. The total binding energy is decreased by it by as much: it is the term of surface;

• a second Coulomb repulsive contribution. The neutrons are neutral particles, but the protons have a electric charge (positive) and thus tend to be pushed back mutually. This effect also decreases the binding energy: it is the Coulombien term;

• a third repulsive contribution due to the coexistence within the core of two populations of particles (neutrons and protons), of which each one tends to disperse: it is the term of Fermi.

The liquid drop model was sophisticated in 1969 by Myers and Swiatecki. Terms corresponding to the nuclear pairing, the diffusivity of the surface of the core, etc were gradually added. This model was historically very important: it made it possible to reproduce the atomic masses with a rather good precision (90 to 95% of the binding energy corresponds to a liquid drop, the remainder coming from quantum effects purely ) and stimulated the first theoretical work on the Fission by Fermi, Bohr, etc, at the end of the years 1930. This simple model also makes it possible to explain why the cores can be deformed, in some limiting.

Nevertheless, the liquid drop model is entirely macroscopic and is unaware of consequently completely:

• quantum nature of the protons and the neutrons, as well as core itself,

• majority of the characteristics of the nucleon-nucleon interaction.

Consequently, though its simplicity and its physical contents are worth to him to be always abundantly used in physics of the nuclear structure, of the additional techniques must be employed to obtain a reliable quantum description.

Shell model

Introduction: the concept of layer

Systematic measurements of energy of connection of the atomic nuclei show a deviation compared to the liquid energy of drop. In particular, certain cores having of the particular numbers of protons and/or neutrons are much more dependant (stable) that does not predict it the liquid model of drop. These cores are known as magic. This observation resulted in postulating the existence of a structure in layers of the nucleons in the core following the example situation of the structure in layers of the electrons in the Atome S.

Concretely, the nucleons are quantum objects. In any rigor, one cannot speak about individual energies of the nucleons, because they all are correlated the ones with the others. To reveal the structure in layers, an average core is planned, within which independently the nucleons evolve/move. Energies which they can have there are discrete: they are energy levels. The distribution of these levels is not inevitably uniform: it may be that a certain beach in energy contains many levels whereas another energy beach is quasi-vacuum. Such a distribution is called structure in layers. A layer is a whole of levels close in separate energy to the other layers by a virgin space of any level (called " gap" in English).

The calculation of the energy levels calls upon the formalism of the quantum Mécanique. They are obtained by Diagonalisation Hamiltonien of the system. Each level can be occupied, and sometimes corresponds in several states of particles: it is said whereas it is degenerated. The degeneration of an energy level is generally related to the properties of Symétrie of the average core.

The concept of structure in layers makes it possible to include/understand why certain cores are more stable than others. It should be known that two nucleons of the same type cannot be in the same state (Principe of exclusion of Pauli). In the state of lower energy of the core, the nucleons fill all the levels, of low in energy up to a certain given level (called the level of Fermi). Let us suppose that this level of Fermi is the last occupied level of a layer. To excite the core, a nucleon should be forced to leave one of the occupied levels, and to promote it in a free level. As in our example the first free state is located in the following layer, this excitation is expensive énergétiquement (energy necessary being at least equal to the difference between the two layers considered). A contrario, if the last occupied level and the first free level belong to the same layer, energy necessary will be much weaker and the core will be thus less stable. All thus depends on the number of nucleons available, and the structure in layers of the core.

Fundamental assumptions

The term of shell model can lend to confusion, in the sense that it recovers two concepts which, although connected, are not less different. Historically, it was used to describe the existence of nucleonics layers in the core according to an approach which arose in fact more than one theory of average field. Nowadays, it indicates a whole of techniques which are used to solve a certain alternative of the problem with NR nuclear body. These are these last that we will introduce here.

Several fundamental assumptions are made to give a framework of precise work to the shell model:

• the atomic nucleus is a quantum system with NR body.

• the core is not a relativistic object. The equation of the movement giving the Fonction of wave system (quantity which in quantum Mécanique contains all information on this one) is the equation of Schrödinger.

• the Nucléon S interact only according to one interaction with two bodies. This limitation is in fact a practical consequence of the Principe of exclusion of Pauli: the Mean free path of a nucleon being large compared to the size of the core, the probability that three nucleons interact simultaneously is regarded as sufficiently weak being able to be neglected.

• the nucleons are specific objects and without structure.

Brief description of the formalism

The general procedure implemented in calculations of shell model is the following one. First of all, one defines some Hamiltonien which will describe the core. As it was mentioned higher, only the terms with two bodies are taken into account in the definition of the interaction. The latter is a effective Interaction: it contains free parameters which must be adjusted starting from experimental data.

The following stage consists in defining a bases individual quantum states, i.e. a whole of functions of wave describing all the possible states of a Nucléon. Most of the time, this base is obtained by a calculation Hartree-Fock. With this whole of individual states, one builds determining of Slater, i.e. functions of wave of Z or NR variables (according to whether one considers the protons or the neutrons) being written as a antisymetrized product of the individual functions (antisymetrized meaning that the exchange of the unspecified variables of two nucleons changes the sign of the function of wave).

In theory, the number of quantum states dependant (or discrete) available for only one nucleon is finished, call it N . The number of nucleons of the core is increasingly smaller than this number of possible states. There thus exist several possibilities of choosing a whole of Z (or NR) individual functions among possible N functions. In combinative Analysis, the number of choices of Z objects among N is called the number of Combinaison S. It is that if the number of possible states, N , is much larger than the number of functions to be chosen, Z or NR, the number of possible choices - i.e of determinants of Slater - quickly becomes very large. In practice, this number is so large that it makes any calculation almost impossible for cores beyond has equal to or higher than 8.

To mitigate this difficulty, one introduces a division of the space of the possible individual states in a heart and a Espace of valence named by analogy with the concept of Couche of valence in chemistry. The heart is a whole of individual levels which one will suppose inactive, with the direction where one avoids considering them to build our determinants of Slater. By opposition, the space of valence is the whole of all the individual states which can play a part in the construction of the functions of wave with Z (or NR) body. The whole of all the possible determinants of Slater in the space of valence defines a bases states with Z (or NR) body.

The last stage consists in calculating the matrix of the Hamiltonian with two bodies in this base with Z body, and the diagonaliser. In spite of the reduction of the size of the base due to the introduction of the space of valence, the matrices with diagonaliser reach dimensions about 10⁹ easily and require specific techniques of diagonalisation. Calculations of shell model in general reproduce the experimental data with an excellent precision. However, they strongly depend on two big factors:

• the subdivision of total space in heart and space of valence;
• the nucleon-nucleon effective interaction.

Associated problems

Theories of average field

The model with independent particles

The interaction between the nucleons, which derives from the strong interaction and which confines the nucleons inside the core with the characteristic to be of finished range: it is cancelled when the distance between two nucleons becomes too large; gravitational remotely average, it becomes repulsive when this distance tends towards 0. This last property illustrates the principle of Pauli which stipulates that two Fermion S (the nucleons are fermions) cannot be in same a quantum state. That has as a consequence that the Libre range of one nucleon inside the core very large is brought back to the size of this one. This result, confirmed by experiments of diffusions of particles, led to the development of the model with independent particles.

The central idea of this approach is that all occurs as if a nucleon moved in a certain well of potential (which would confine it in the core) independently of the presence of the other nucleons. On the theoretical level, this assumption amounts replacing the problem with NR body - NR particles in interaction - by NR problems to 1 body - a particle moving in a certain potential. This essential simplification of the problem is the angular stone of the theories of average field. Those are also abundantly used in physical atomic where, this time, in fact the electron S move in an average field created by the core and the electronic cloud itself.

Though this assumption can seem very coarse and far too simplifying, it led to great successes and the theories of average field (we will see that there are several alternatives) are now integral part of the theory of the atomic nucleus. It also should be noted that they are rather modular, to take again a term of data processing, in the sense that one can easily describe certain effects (nuclear pairing, collective movements of rotation or vibration) in " rajoutant" in the formalism terms necessary.

Nuclear potential and effective interaction

Most of the practical difficulties encountered in the theories of average field is the definition (or the calculation) of the potential of field average itself. One distinguishes, very summarily, two approaches:

• the phenomenologic approach consists with paramétriser the nuclear potential by a suitable mathematical function. Historically, this procedure was applied with the most success by Nilsson which used a potential of the type oscillating harmonic (deformed). More recent parameterizations rest on more realistic functions which more accurately describe for example the experiments of diffusion. Let us quote in particular the form of Woods-Saxon.

the purpose of

• the self-coherent approach, or Hartree-Fock, is mathematically to deduce the nuclear potential starting from the nucleon-nucleon interaction. This technique requires to solve the equation of Schrödinger by an iterative procedure, the potential appearing there depend on the functions of wave which one seeks to determine. Those are written in the form of a Déterminant of Slater.

In the case of the approaches Hartree-Fock, the difficulty is not the mathematical function which will describe best the nuclear potential, but that which will approach the most nucleon-nucleon interaction. Indeed, contrary to, for example, the atomic physical where the interaction is known (it is the interaction Coulombienne), the nucleon-nucleon interaction in the core is not known analytically.

One can advance two reasons with that. First of all, the strong interaction acts primarily between the Quark S which form the nucleons. The nucleon-nucleon interaction in the vacuum is only a consequence of the interaction quark-quark. However if the description of the latter is relatively well included/understood within the framework of the Standard Modèle, there does not exist theory allowing the passage of the interaction quark-quark towards the nucleon-nucleon interaction. In addition, even if this problem were solved, it would remain a great difference between the case ideal (and conceptually simpler) of two nucleons interacting in the vacuum, and that of these two same nucleons interacting in the middle of several tens of other nucleons. In other words, the nucleon-nucleon interaction in the vacuum is not the same one as within the core. To be able to advance, it was thus necessary to invent the concept of effective Interaction. The latter is primarily a mathematical function which contains several free parameters which are adjusted on experimental data.

Self-coherent approaches of Hartree-Fock type

In the approach Hartree-Fock of the Problem with NR body, the starting point is a Hamiltonien containing terms of kinetic energy (as much as particles of the system, let us say NR) and of the terms of potential. As it was mentioned higher, one of the assumptions of the theories of average field is that only the interaction with 2 bodies must be taken into account. The term of potential of the Hamiltonian thus counts all the interactions with 2 possible bodies in a whole of NR Fermion S. It is the first assumption.

The second phase consists in supposing that the Fonction of wave of the system can be written like a Déterminant of Slater. This postulate is the mathematical translation of the assumption of the model with independent particles. It is the second assumption. Remain to now determine the components of this Déterminant of Slater, i.e. the individual functions of wave of each Nucléon which are for the moment unknown factors. For that, one will suppose that the function of total wave of the core (the determinant of Slater) must be such as energy is minimal. It is the third assumption. Technically, that means that one will calculate the Median value Hamiltonien with 2 bodies (known) on the function of wave of the core (the Déterminant of Slater, unknown), and will impose that the variation, with the mathematical direction, of this quantity is null. This will lead us to a whole of equations which have as unknown factors the invividuelles functions of wave: equations of Hartree-Fock. The solution of these equations gives us the functions of wave and the individual energy levels, and extension the total energy of the core and its function of wave.

This small brief talk of the Hartree-Fock method makes it possible to include/understand why the latter is described as variational approach . At the beginning of calculation, total energy is a " function of the functions of wave individuelles" (one calls that a functional calculus), and all the technique consists in optimizing the choice of these functions of wave so that the functional calculus has a minimum (which one hopes for absolute and not relative). To be more precise, we should mention that energy is actually a functional calculus of the density, definite as the sum of the squares of the individual functions of wave. Let us note that the theory known under the name of Hartree-Fock approach in nuclear physics is also used in physical atomic and Physics the solid state under the name of Théorie of the functional calculus of the density (Density Functional Theory, DFT, in English).

The process of resolution of the Hartree-Fock equations can only be iterative, because those are actually a equation of Schrödinger in which the potential depends on the density, i.e. functions of wave that one seeks to determine. From a practical point of view, one starts the algorithm with a whole of individual functions of wave which approximately appear reasonable (in general clean functions of a oscillating harmonic). Those enable us to calculate the density, and thus the Hartree-Fock potential. Once this one determined, one can solve the equation of Schrödinger, which gives us a new whole of individual functions at stage 1, with which one determines a new density, etc the algorithm stops - it is said that convergence is reached - when the difference between the functions of wave (or energies individual) between two iterations is lower than a certain predetermined value. With convergence, the potential of average field is entirely given, and the Hartree-Fock equations are brought back to an equation of traditional Schrödinger. The Hamiltonian corresponding extremely logically is called Hartree-Fock Hamiltonian.

Approaches of relativistic average field

Initially appeared in the years 1970 with work of D. Walecka on the quantum chromodynamic , the relativistic models of the core were sophisticated towards the end of the year 1980 by P. Ring and his/her collaborators. The starting point of these approaches is the Quantum theory of the relativistic fields. In this context, the interaction of the nucleons is done via the exchange of virtual particles called Méson S. the formalism consists, in a first stage, to build a Lagrangien containing these terms of interaction. In the second time, the application of the principle of less action provides a whole of equations of the movement. The real particles (here nucleons) obey the equation of Dirac while the virtual particles (here mesons) obey the equations of Klein-Gordon.

Because of perturbative nature not of the strong interaction, as owing to the fact that the latter is relatively badly known, the use of such an approach in the case of the atomic nuclei requires strong approximations. Major simplification consisted in replacing in the equations all the terms of field (which are operators with the mathematical direction) by their Median value (which is function S). In this way, one brings back to a system equations coupled intégro-differentials which can be numerically solved failing to be it analytically.

Spontaneous crack of symmetry in nuclear physics

One of the central concepts of all physics is that of Symétrie. The nucleon-nucleon interaction and all the effective interactions used in practice have certain symmetries such as the invariance by translation (to move according to a straight line the reference frame does not change the interaction), by Rotation (to make a rotation of the reference frame does not change anything with the interaction) or of parity (inversion of the axes of the reference frame). Nevertheless, of the solutions of the Hartree-Fock approach can appear which break one or more these symmetries. One speaks then about spontaneous crack of symmetry.

Very qualitatively, these spontaneous cracks of symmetry can be explained in the following way: the goal of the approach of average field is to describe the core like a whole of independent particles. In the Hartree-Fock approach, that results in what the function of total wave of the core is a product antisymetrized of the individual functions of wave nucleonics, i.e. a Déterminant of Slater. The particular form of these functions of wave implies, inter alia, that the majority of the additional correlations between nucleons - all that does not return within the framework of the average field - will be automatically neglected. Nevertheless, the Hamiltonian with two bodies " contient" all these correlations. The only way so that these last appear in the final solution of the variational problem is that symmetries of the nucleon-nucleon interaction are broken on the level of the Hamiltonian of average field (the Hartree-Fock Hamiltonian). It is necessary to distinguish here between the Hamiltonian with two initial bodies, which preserve symmetries, and the Hartree-Fock Hamiltonian, obtained by variation of the total energy (functional of the density), which can possibly break these symmetries. Concretely, if the core presents correlations which are not included in the average field, and if the density used as starting point of the iterations of calculation breaks a certain number of symmetries, then the Hartree-Fock Hamiltonian will be able to also break these symmetries, but the iterative process can as well converge towards a symmetrical solution, if this one is preferable from the energy point of view.

It comes out from this short attempt at explanation that the concept of spontaneous crack of symmetry in nuclear physics is closely related to that of average field. It does not take place to be, for example, within the framework of the shell model. The most widespread example of such a phenomenon is the spontaneous crack of invariance by rotation, which results in the appearance of a deformation of the average field. Strictly speaking , a core is never deformed - it is an abuse language, but in the approximation of average field, all occurs as if the core were deformed. In fact, with regard to invariance by rotation, the average field obtained does not have symmetry by rotation, but all the versions deduced from this average field by rotation would give exactly equivalent solutions. And the physical statuses of the core correspond to any of these solutions.

Also let us quote the nuclear phenomenon of pairing, which corresponds to the not-conservation of the number of particles.

Extensions of the theories of average field

The phenomenon of nuclear pairing

Historically, the observation that the cores having an even number of nucleons are systematically more dependant than those by having an odd number resulted in proposing the assumption of nuclear pairing. The idea, extremely simple, is that each nucleon binds with another to form a pair. When the core has an even number of nucleons, each one of them finds a partner. To excite such a system, it is then necessary to provide an excitation at least equal to energy necessary to break a pair. On the contrary, in the case of the odd cores, there exists an unmarried nucleon, which requires less energy to be excited.

This phenomenon is very similar to that of the Supraconductivité in physics the solid state (all at least supraconductivity at low temperature). The first theoretical description of nuclear pairing, proposed at the end of the years 1950 by Aage Bohr and Ben Mottelson (and which contributed to be worth the to them Nobel Prize of physics in 1975), was the theory besides known as BCS, or Bardeen-Cooper-Schrieffer which describes supraconductivity in metals. On the theoretical level, the phenomenon of pairing as described by theory BCS is superimposed on the approximation of average field. In other words, the nucleons are at the same time subjected to the potential of average field and the interaction of pairing, but one is independent of the different one.

However it is trying to interpret the interaction of pairing like a residual Interaction. It was seen that in the approach Hartree-Fock, the potential of average field is given starting from the interaction with two nucleon-nucleon bodies. Without returning too much in the technical details, only a certain number of all the possible interactions are taken into account to build the average field in question indeed. All that " reste" is described as residual Interaction. In other words, the nucleon-nucleon interaction in the core breaks up into a potential of average field and a residual interaction. The validity of the approach of average field comes from what the latter is quantitatively much weaker than the average field and can thus be neglected at first approximation.

Nevertheless, if we interpret the interaction of pairing like a component of the residual interaction, then it should exist " liens" between pairing and average field, since both proceed of same the nucleon-nucleon interaction. These bonds are not taken into account in approach BCS. To mitigate this difficulty, it was developed an approach known as Hartree-Fock-Bogolyubov (HFB), which includes in the same formalism the average field, pairing, and the interaction between the two.

Let us note to finish that one of the great differences between supraconductivity in atomic physics and pairing in nuclear physics is due to the number of particles. The number of pairs of electrons in a metal is colossal compared to the number of nucleons in the core. However approach BCS (thus besides that approach HFB) described the function of wave of the system (metal or core) like a superposition of functions of wave corresponding to different numbers of particles. So in the case of a metal, this violation of the number of particles hardly has importance for statistical reasons, in nuclear physics, the problem is real. It was thus necessary to develop specific techniques of restoration of the number of particles, within the general framework of the restoration of broken symmetries mentioned above.

The description of the collective movements of the core

How to be " with-delà" average field

One of the major disadvantages of the approaches of field-means in Nuclear physics is the crack of symmetries of the interaction (crack of the invariance by rotation, translation, of the number of particles if the correlations of pairing are taken into account, etc). These symmetries can nevertheless be restored. A simple case is that of invariance by translation: the nucleon-nucleon interaction has this symmetry, but the core being an object of finished size, the breeze necessarily (if one moves to it reference frame according to a straight line, the system changes). To mitigate this difficulty, it is enough to make a change of reference frame and to pass in a reference frame related to the core (known as intrinsic). Thus, any translation of the reference frame also moving the core (since the reference frame is attached to this last), invariance by translation is preserved. Mathematically, of the terms of correction of the movement of the Center of mass of the core must be introduced into the Hamiltonian of the system.

The case of the violation of the number of particles in the approaches of the type HFB, and more still of invariance by rotation, is technically much more complex. One of the followed strategies is of '' to project '' the function of wave of the system which breaks the symmetry concerned on a subspace where this symmetry is preserved. Two possibilities arise then:

• is the projection is carried out at the end of self-coherent calculation (Hartree-Fock), case known as of projection after variation;
• is the projection is done before-even calculation. In this case, the Hartree-Fock equations are obtained in fact in a way different by a variation from the Hamiltonian projected . It is the case of projection before variation.

The first approach has the advantage of being numerically much lighter to implement. Nevertheless, in certain cases, it leads to not-physics results. The second approach is most rigorous, but also most expensive to numerically implement.

The second strategy consists in mixing functions of wave having the symmetry broken according to a variational Principe. It is the base of the method of the coordinate generating (Generator Coordinate Method in English, or GCM). The idea is to build a state having good symmetries like a Superposition of states of field-means which, them, break certain symmetries. The coefficients of this superposition are given in a self-coherent way by a variational calculation. This approach is relatively powerful and does not limit itself to the restoration of symmetries. It is indeed employed to describe the movement collectives of the cores.

Other approaches

No-core Shell Model

Models in clusters

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