Nuclear structure

Encyclopedia

*This page is an adapted translation of the corresponding :fr:Structure nucléaire - As will be noted, there remain void paragraphs, as on the original (which provides no sources). Competent Wikipedians are welcome to enrich it, and their contributions will be translated back to French*

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

Nuclear physics

Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...

article.

## The liquid drop model

of yeahh :DThis is one of the first models of

**nuclear structure**, proposed by Carl Friedrich von Weizsäcker

Carl Friedrich von Weizsäcker

Carl Friedrich Freiherr von Weizsäcker was a German physicist and philosopher. He was the longest-living member of the research team which performed nuclear research in Germany during the Second World War, under Werner Heisenberg's leadership...

in 1935. It describes the nucleus as a classical fluid

Fluid mechanics

Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

made up of neutron

Neutron

The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

s and proton

Proton

The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s, with an internal repulsive electric force proportional to the number of protons. The quantum mechanical

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

nature of these particles appears via the Pauli exclusion principle

Pauli exclusion principle

The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, 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

Fermi energy

The Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature....

. This simple model reproduces the main features of the binding energy

Binding energy

Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together—often this means that energy is released upon the creation of a bound state...

of nuclei.

### Introduction to the shell concept

Systematic measurements of the binding energyBinding energy

Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together—often this means that energy is released upon the creation of a bound state...

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

Magic number (physics)

In nuclear physics, a magic number is a number of nucleons such that they are arranged into complete shells within the atomic nucleus...

. This observation led scientists to assume the existence of a shell structure of nucleon

Nucleon

In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...

s (protons and neutrons) within the nucleus, like that of electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s within atoms.

Indeed nucleons are quantum objects

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. 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 level

Energy level

A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

s. 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

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, more precisely by diagonalization of the single-nucleon Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

. 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

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

.

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

Pauli exclusion principle

The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

). So the lowest-energy 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

Electron shell

An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....

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

Island of inversion

An island of inversion is a region of the chart of nuclides that contains isotopes with a non-standard ordering of single particle levels in the nuclear shell model. Such an area was first described in 1975 by French physicists carrying out spectroscopic mass measurements of exotic isotopes of...

or in the reduction of excitation energy gaps above the traditional magic numbers.

### Basic hypotheses

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

*n*-body problem. We shall introduce these here.

Several basic hypotheses are made in order to give a precise conceptual framework to the shell model :

- The atomic nucleus is a quantumQuantum mechanics

*n*-body system.

- The nucleus is not a relativisticSpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

object. The equation of motion giving the system wavefunctionWavefunctionNot to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

(which contains the whole of the information about it in quantum mechanicsQuantum mechanics

), is the Schrödinger equationSchrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

(which is non-relativistic).

- The nucleons interact only via a two-body interaction. This limitation is in effect a practical consequence of the Pauli exclusion principlePauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

: the mean free pathMean free pathIn physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

of a nucleon being large with respect to the nucleus size, the probability that three nucleonsThree-body forceA three-body force is a force that does not exist in a system of two objects but appears in a three-body system. In general, if the behaviour of a system of more than two objects cannot be described by the two-body interactions between all possible pairs, as a first approximation, the deviation is...

interact simultaneously is considered as small enough to be negligible.

- Nucleons are considered to be pointlike, without any structure, in this model, for the sake of simplicity.

### Brief description of the formalism

The general process used in the shell model calculations is the following. First a HamiltonianHamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

for the nucleus is defined. As mentioned before, only one- and two-body terms are taken into account in this definition. The interaction is an effdive theory

Effective field theory

In physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...

: it contains free parameters which have to be fitted with experimental data.

The next step consists in defining a basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

of single-particle states, i.e. a set of wavefunction

Wavefunction

Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

s 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 determinant

Slater determinant

In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...

s are built, that is, wavefunctions for

*Z*proton variables or

*N*neutron variables, which are antisymmetrized products of single-particle 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

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

C. 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 single-particle states is divided into a core and a

valence shell, 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 (

*Z*-)

*N*-body wavefunction. The set of all possible Slater determinants in the valence space defines a basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

for

(

*Z*-)

*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 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 way to divide the single-particle space into core and valence.

- The effective nucleon-nucleon interaction.

### The independent-particle model

The interaction between nucleons, which is a consequence of strong interactionStrong interaction

In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

s and binds the nucleons within the nucleus, exhibits the peculiar behaviour of having 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

Pauli exclusion principle

according to which two fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s (nucleons are fermions) cannot be in the same quantum state. This results, in theory, in a very large mean free path

Mean free path

In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

predicted for a nucleon within the nucleus. However, this prediction of the shell model is not confirmed by particle scattering experiments (see Cook, 2010, Models of the Atomic Nucleus). Experimental results on nucleon-nucleon 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 led Cook 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

*N*-body problem (

*N*particles interacting) 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

Atomic physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

, where electrons move in a mean field due to the central nucleus and the electron cloud itself. However, as discussed by Cook (2010), 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

Interacting boson model

The interacting boson model is a model in nuclear physics in whichnucleons pair up, essentiallyacting as a single particle with boson properties, withintegral spin of 0, 2 or 4....

, or collective motions of the nucleon like rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

, or vibration

Vibration

Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...

, 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 alpha-cluster 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.

### Nuclear potential and effective interaction

A large part of the practical difficulties met in mean field theories is the definition (or calculation) of the potentialSchrödinger equation

The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

of the mean field itself. One can very roughly distinguish between two approaches :

- The
**phenomenological**approach is a parameterization of the nuclear potential by an appropriate mathematical function. Historically, this procedure was applied with the greatest success by Sven Gösta Nilsson, who used as a potential a (deformed) harmonic oscillatorQuantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

potential. The most recent parameterizations are based on more realistic functions, which account more accurately for scattering experiments, for example. In particular the form known as the Woods Saxon potentialWoods Saxon potentialThe Woods–Saxon potential is a mean field potential for the nucleons inside the atomic nucleus, which is used to approximately describe the forces applied on each nucleon, in the shell model for the structure of the nucleus....

can be mentioned.

- The
**self-consistent**or Hartree–Fock approach aims to deduce mathematically the nuclear potential from the nucleon-nucleon interaction. This technique implies a resolution of the Schrödinger equationSchrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

in an iterative fashion, since the potential depends there upon the wavefunctions to be determined. The latter are written as Slater determinantSlater determinantIn quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...

s.

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 nucleon-nucleon interaction. Indeed, in contrast with atomic physics

Atomic physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

where the interaction is known (it is the Coulomb interaction), the nucleon-nucleon interaction within the nucleus is not known analytically.

There are two main reasons for this fact. First, the strong interaction acts essentially among the quark

Quark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s forming the nucleons. The nucleon-nucleon interaction

Strong interaction

In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

in vacuum is a mere

*consequence*of the quark-quark interaction. While the latter is well understood in the framework of the standard model

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

at high energies, it is much more complicated in low energies due to color confinement and asymptotic freedom

Asymptotic freedom

In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...

. Thus there is no fundamental theory allowing one to deduce the nucleon-nucleon interaction

Strong interaction

In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

from the quark-quark 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.

### The self-consistent approaches of the Hartree–Fock type

In the Hartree–Fock approach of the*n*-body problem

Many-body problem

The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system...

, the starting point is a Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

containing

*n*kinetic energy

Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

terms, and potential terms. As mentioned before, one of the mean field theory hypotheses is that only the two-body interaction is to be taken into account. The potential term of the Hamiltonian represents all possible two-body interactions in the set of

*n*fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s. It is the first hypothesis.

The second step consists in assuming that the wavefunction

Wavefunction

Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

of the system can be written as a Slater determinant

Slater determinant

In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...

of one-particle spin-orbitals. This statement is the mathematical translation of the independent-particle model. This is the second hypothesis.

There remains now to determine the components of this Slater determinant, that is, the individual wavefunction

Wavefunction

s 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) two-body Hamiltonian

Hamiltonian (quantum mechanics)

on the (unknown) Slater determinant, and impose that its mathematical variation

Calculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

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

Calculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

approach. At the beginning of the calculation, the total energy is a "function of the individual wavefunctions" (a so-called 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

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

, defined as the sum of the individual squared wavefunctions. Let us note also that the Hartree–Fock method is also used in atomic physics

Atomic physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

and condensed matter physics

Condensed matter physics

Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

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

Schrödinger equation

in which the potential depends on the density

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

, that is, precisely on the wavefunction

Wavefunction

s to be determined. Practically, the algorithm is started with a set of individual grossly reasonable wavefunctions (in general the eigenfunctions of a harmonic oscillator

Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

). 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.

### The relativistic mean field approaches

Born first in the 1970s with the works of D. Walecka on quantum hadrodynamicsStrong interaction

, the relativistic

Special relativity

Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

models of the nucleus were sharpened up towards the end of the 1980s by P. Ring and coworkers. The starting point of these approaches is the relativistic quantum field theory

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

. In this context, the nucleon interactions occur via the exchange of virtual particle

Virtual particle

In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...

s called meson

Meson

In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...

s. The idea is, in a first step, to build a Lagrangian

Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

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

Dirac equation

The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...

, whilst the virtual ones (here the mesons) obey the Klein–Gordon equations.

In view of the non-perturbative

Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

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

Operator (physics)

In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

in the mathematical sense) by their mean value (which are functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

). In this way, one gets a system of coupled integro-differential equations, which can be solved numerically, if not analytically.

### Spontaneous breaking of symmetry in nuclear physics

One of the focal points of all physics is symmetrySymmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

. The nucleon-nucleon interaction

Strong interaction

and all effective interactions used in practice have certain symmetries. They are invariant by translation

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

(changing the frame of reference so that directions are not altered), by rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

(turning the frame of reference around some axis), or parity

Parity (physics)

In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

(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

Spontaneous symmetry breaking

Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....

.

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).

### Nuclear pairing phenomenon

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

Superconductivity

Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

in solid state physics (at least the low-temperature 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

BCS theory

BCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...

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 nucleon-nucleon 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 nucleon-nucleon interaction

Strong 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.

### Introductory texts

- Luc Valentin ;
*Le monde subatomique - Des quarks aux centrales nucléaires*, Hermann (1986). - Luc Valentin ;
*Noyaux et particules - Modèles et symétries*, Hermann (1997). - David Halliday ;
*Introduction à la physique nucléaire*, Dunod (1957).

### Fundamental texts

- Irving Kaplan;
*Nuclear physics*, the Addison-Wesley Series in Nuclear Science & Engineering, Addison-Wesley (1956). 2nd edition (1962). - A. Bohr & B. Mottelson ;
*Nuclear Structure*, 2 vol., Benjamin (1969-1975). Volume 1 :*Single Particle Motion*; Volume 2 :*Nuclear Deformations*. Réédité par World Scientific Publishing Company (1998), ISBN 981-02-3197-0. - P. Ring & P. Schuck;
*The nuclear many-body problem*, Springer Verlag (1980), ISBN 3-540-21206-X - A. de Shalit & H. Feshbach;
*Theoretical Nuclear Physics*, 2 vol., John Wiley & Sons (1974). Volume 1:*Nuclear Structure*; Volume 2:*Nuclear Reactions*, ISBN 0-471-20385-8

## External links

English Gesellschaft für Schwerionenforschung (GSI), Allemagne Joint Institute for Nuclear Research (JINR), Russie Argonne National Laboratory (ANL), États-Unis Riken, Japon National Superconducting Cyclotron Laboratory, Michigan State University, USA Facility for Rare Isotope Beams, Michigan State University, USAFrench Centre de Spectrométrie Nucléaire et de Spectrométrie de Masse (CSNSM), France Service de Physique Nucléaire CEA/DAM, France Institut National de Physique Nucléaire et de Physique des Particules (In2p3), France Grand Accélérateur National d'Ions Lourds (GANIL), France Commissariat à l'Energie Atomique (CEA), France Centre Européen de Recherches Nucléaires, Suisse