Close nucleon encounters
The nucleon—nucleon interaction in vacuum is a mere consequence of the quark—quark 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 yet no fundamental theory allowing one to deduce the nucleon—nucleon interaction from the quark—quark interaction.
Furthermore, even if this problem were solved, there would remain a large difference between the ideal and conceptually simpler case of two nucleons interacting in vacuum, 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.
Most modern interaction are zero-range so they act only when the two nucleons are in contact, as introduced by Tony Skyrme. In the Hartree—Fock approach of the n -body 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 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 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 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 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 two-body 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 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 , defined as the sum of the individual squared wavefunctions.
Practically, the algorithm is started with a set of individual grossly reasonable wavefunctions in general the eigenfunctions of a harmonic oscillator. These allow to compute the density, and therefrom the Hartree—Fock potential. The calculation stops — convergence is reached — when the difference among wavefunctions, or energy levels, for two successive iterations is less than a fixed value.
The corresponding Hamiltonian is then called the Hartree—Fock Hamiltonian. Born first in the s with the works of John Dirk Walecka on quantum hadrodynamics , the relativistic models of the nucleus were sharpened up towards the end of the s by P. Ring and coworkers.
Nuclear structure - Wikipedia
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 non- perturbative 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 integro-differential equations , which can be solved numerically, if not analytically.
The interacting boson model IBM is a model in nuclear physics in which nucleons are represented as pairs, each of them acting as a boson particle, with integral spin of 0, 2 or 4. This makes calculations feasible for larger nuclei. There are several branches of this model - in one of them IBM-1 one can group all types of nucleons in pairs, in others for instance - IBM-2 one considers protons and neutrons in pairs separately.
One of the focal points of all physics is symmetry. The nucleon—nucleon 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.
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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.
The most common extension to mean field theory is the nuclear pairing.
Nuclei with an even number of nucleons are systematically more bound than those with an odd one. This implies that each nucleon binds with another one to form a pair, consequently the system cannot be described as independent particles subjected to a common mean field. When the nucleus has an even number of protons and neutrons, 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 protons or neutrons, there exists an unpaired nucleon, which needs less energy to be excited. This phenomenon is closely analogous to that of Type 1 superconductivity in solid state physics. The first theoretical description of nuclear pairing was proposed at the end of the s by Aage Bohr , Ben Mottelson , and David Pines which contributed to the reception of the Nobel Prize in Physics in by Bohr and Mottelson. 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.
The Hartree—Fock—Bogolyubov HFB method is a more sophisticated approach,  enabling one to consider the pairing and mean field interactions consistently on equal footing. HFB is now the de facto standard in the mean field treatment of nuclear systems. Peculiarity of mean field methods is the calculation of nuclear property by explicit symmetry breaking.
The calculation of the mean field with self-consistent methods e. Hartree-Fock , breaks rotational symmetry, and the calculation of pairing property breaks particle-number. Several techniques for symmetry restoration by projecting on good quantum numbers have been developed. Mean field methods eventually considering symmetry restoration are a good approximation for the ground state of the system, even postulating a system of independent particles. Higher-order corrections consider the fact that the particles interact together by the means of correlation.
These correlations can be introduced taking into account the coupling of independent particle degrees of freedom, low-energy collective excitation of systems with even number of protons and neutrons. In this way, excited states can be reproduced by the means of random phase approximation RPA , also eventually consistently calculating corrections to the ground state e.
From Wikipedia, the free encyclopedia. Nuclides ' classification. Nuclear stability. Radioactive decay. Nuclear fission. Capturing processes.
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Unsourced material may be challenged and removed. Main article: Semi-empirical mass formula. Main article: Nuclear shell model. Nuclear technology portal Physics portal. This is pretty straightforward at high energies. First, one needs to select kinematics sufficiently far from the regions allowed for scattering off a free nucleon, i. Here Q 2 is the square of the four momenta transferred to the nucleus, and q 0 is the energy transferred to the nucleus.
In addition, one needs to restrict Q 2 to values of less than a few giga-electron-volts squared; in this case, nucleons can be treated as partons with structure, since the nucleon remains intact in the final state due to final phase-volume restrictions. First, the photon is absorbed by a nucleon in the SRC with momentum opposite to that of the photon; this nucleon is turned around and two nucleons then fly out of the nucleus in the forward direction figure 1.
Fermi function beta decay
The inclusive nature of the process ensures that the final-state interaction does not modify the ratios of the cross-sections. However, they were in somewhat different kinematic regions for the lightest and heavier nuclei. Only in did the sustained efforts of Donal Day and collaborators to interpolate these data to the same kinematics lead to the first evidence for scaling, but the accuracy was not very high. An experiment with the CLAS detector at JLab was the first to take data on 3He and several heavier nuclei, up to iron, with identical kinematics, and the collaboration reported their first findings in Egiyan et al.
Using the 4. The next step was to look for the even more elusive SRC of three nucleons. It is practically impossible to observe such correlations in intermediate energy processes. However, at high Q2, it is straightforward to suppress scattering off both slow nucleons and two-nucleon SRCs. Again, a scaling of the ratios was expected. This is because there is a high probability for a nucleon to have two nearby nucleons in a heavier, denser nucleus. Hence, one expected to find two steps. This is exactly what the CLAS experiment observed in data recently reported for these kinematics and shown in figure 3 Egiyan et al.
More data for exploring SRCs have already been taken at JLab, and several more efforts are already planned to study this interesting region of nuclear physics, which has important implications for the dynamics of the cores of neutron stars. K S Egiyan et al. Type to search. Sign in Register. Enter e-mail address Show Enter password Remember me.
Related Nucleon Correlations in Nuclei
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