scholarly journals THE RELATIVISTIC MEAN-FIELD EQUATIONS OF THE ATOMIC NUCLEUS

2012 ◽  
Vol 24 (04) ◽  
pp. 1250008 ◽  
Author(s):  
SIMONA ROTA NODARI
Keyword(s):  
Nuclear Physics ◽  
Existence Of Solutions ◽  
Mean Field Theory ◽  
Mean Field ◽  
Static Case ◽  
Concentration Compactness ◽  
Field Equations ◽  
Relativistic Mean Field ◽  
Dirac Equations ◽  
Mean Field Equations

In nuclear physics, the relativistic mean-field theory describes the nucleus as a system of Dirac nucleons which interact via meson fields. In a static case and without nonlinear self-coupling of the σ meson, the relativistic mean-field equations become a system of Dirac equations where the potential is given by the meson and photon fields. The aim of this work is to prove the existence of solutions of these equations. We consider a minimization problem with constraints that involve negative spectral projectors and we apply the concentration-compactness lemma to find a minimizer of this problem. We show that this minimizer is a solution of the relativistic mean-field equations considered.

Properties of Relativistic Mean-Field Theory

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Mean Field Theory ◽  
Mean Field ◽  
Field Method ◽  
Basis Sets ◽  
Electron Interaction ◽  
Relativistic Mean Field ◽  
Dirac Equations ◽  
Electrons And Positrons ◽  
The One ◽  
Atomic Structure Calculations

We have previously seen how the Dirac equation for one particle requires some rather special consideration and interpretation in order to arrive at a form that is able to treat electrons and positrons on an equal footing. These problems persist also when we go to systems with more than one electron. One might think that the extension to several electrons should not introduce dramatic changes. After all, we noted that even the one-electron problem must be viewed as a many-electron (and -positron) system in order to arrive at a consistent description. The problem with introducing more electrons is that electron–electron interactions that were previously small—for the one-electron case typically arising from vacuum polarization and self-interaction—now occur to the same order as the kinetic energy and the interaction with the potential. So while a perturbative approach such as QED can use the solutions of the one-electron Dirac equations as a very good starting approximation to a more accurate description of the full system, the same would not work for a system with more electrons because it would mean neglecting interactions of the same magnitude as the zeroth-order energy. For applications to quantum chemistry, the treatment of the entire electron–electron interaction as a perturbation would be hopelessly impractical, as it is even in manyelectron relativistic atomic structure calculations. The technique for dealing with this problem is well known from nonrelativistic calculations on many-electron systems. One-particle basis sets are developed by considering the behavior of the single electron in the mean field of all the other electrons, and while this neglects a smaller part of the interaction energy, the electron correlation, it provides a suitable starting point for further variational or perturbational treatments to recover more of the electron–electron interaction. It is only natural to pursue the same approach for the relativistic case. Thus one may proceed to construct a mean-field method that can be used as a basis for the perturbation theory of QED.

TWO-OSCILLATOR BASIS EXPANSION FOR THE SOLUTION OF RELATIVISTIC MEAN FIELD EQUATIONS

2000 ◽  
Vol 09 (06) ◽  
pp. 507-520
Author(s):  
S. V. S. SASTRY ◽  
ARUN K. JAIN ◽  
Y. K. GAMBHIR
Keyword(s):  
Ground State ◽  
Mean Field ◽  
Surface Region ◽  
Binding Energies ◽  
Basis Functions ◽  
Field Equations ◽  
Basis Expansion ◽  
Relativistic Mean Field ◽  
Mean Field Equations ◽  
Spherical Nuclei

In the relativistic mean field (RMF) calculations usually the basis expansion method is employed. For this one uses single harmonic oscillator (HO) basis functions. A proper description of the ground state nuclear properties of spherical nuclei requires a large (around 20) number of major oscillator shells in the expansion. In halo nuclei where the nucleons have extended spatial distributions, the use of single HO basis for the expansion is inadequate for the correct description of the nuclear properties, especially that of the surface region. In order to rectify these inadequacies, in the present work an orthonormal basis composed of two HO basis functions having different sizes is proposed. It has been shown that for a typical case of (A=11) the ground state constructed using two-HO wave functions extends much beyond the second state or even third excited state of the single HO wave function. To demonstrate its usefulness explicit numerical RMF calculations have been carried out using this procedure for a set of representative spherical nuclei ranging from 16 O to 208 Pb . The binding energies, charge radii and density distributions have been correctly reproduced in the present scheme using a much smaller number of major shells (around 10) in the expansion.

scholarly journals Existence of Solutions to Mean Field Equations on Graphs

2020 ◽  
Vol 377 (1) ◽  
pp. 613-621
Author(s):  
An Huang ◽  
Yong Lin ◽  
Shing-Tung Yau
Keyword(s):  
Existence Of Solutions ◽  
Mean Field ◽  
Field Equations ◽  
Mean Field Equations

Calculation of the propagation amplitude in the case of a Coulomb interaction via time-independent mean-field theory

2007 ◽  
Vol 85 (7) ◽  
pp. 787-796
Author(s):  
R Yekken ◽  
F Mekideche
Keyword(s):  
Coulomb Interaction ◽  
Mean Field Theory ◽  
Mean Field ◽  
Numerical Comparison ◽  
Many Body ◽  
Field Equations ◽  
Range Interaction ◽  
Approximate Methods ◽  
Mean Field Equations ◽  
Many Body Systems

The exact study of many-body microscopic systems is impossible when the number of particles is large (N ≥ 3). Approximate methods are then used. The time-independent mean-field (TIMF) approximation has been proposed for the description of collisions in many-body systems. Collision amplitudes are derived by the use of a variational principle and the choice of trial functions as products of single-particle orbitals. Resulting mean-field equations with a nonvanishing right-hand side turn out to be a generalization of the traditional Hartree or Hatree–Fock type equations. These TIMF equations are successfully solved numerically for the case of short-range forces. In this paper, we test the validity of this theory for the Coulomb interaction between two particles, that is, a long-range interaction. A numerical comparison between the exact and the mean-field solutions is conducted PACS Nos.: 31.15.Ne, 31.15.Pf, 21.45.+v,25.10.-i

scholarly journals Mean field equations on a closed Riemannian surface with the action of an isometric group

2020 ◽  
Vol 31 (09) ◽  
pp. 2050072
Author(s):  
Yunyan Yang ◽  
Xiaobao Zhu
Keyword(s):  
Field Equation ◽  
Positive Integer ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Riemannian Surface ◽  
Sufficient Condition ◽  
Field Equations ◽  
Mean Field Equations ◽  
The Mean ◽  
Mean Field Equation

Let [Formula: see text] be a closed Riemannian surface, [Formula: see text] be an isometric group acting on it. Denote a positive integer [Formula: see text], where [Formula: see text] is the number of all distinct points of the set [Formula: see text]. A sufficient condition for existence of solutions to the mean field equation [Formula: see text] is given. This recovers results of Ding–Jost–Li–Wang, Asian J. Math. (1997) 230–248 when [Formula: see text] or equivalently [Formula: see text], where Id is the identity map.

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