Dirac degeneracy

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy in the Dirac spectrum between levels with the same j and / = j 1/2. This means that the expression for the energy levels in (3.4) already predicts a nonvanishing contribution to the classical Lamb shift E 2Si) — E 2Pi). Due to the smallness of the electron-proton mass ratio this extra term is extremely small in hydrogen. The leading contribution to the Lamb shift, induced by the QED radiative correction, is much larger. 

Fermi and Dirac realized that most of the NA electrons do not contribute to Cyetal We must, instead, consider the electron gas as an FD system, with most probable occupation, Eq. (5.2.11) with slight changes in notation, and ignoring the degeneracy index g, ... 

It would be of interest to apply the method of March and Murray [12] to convert C, the electron density for non-degenerate electrons, into results applicable to intermediate degeneracy governed by Fermi-Dirac statistics. Unfortunately, without switching on the model potential F(r), this is already difficult to handle by purely analytically methods, as can be seen from the case of complete degeneracy for the harmonic oscillator alone. No doubt, numerical procedures will eventually enable present results to be transformed according to the route established in [12]. 

Nevertheless, it seems likely that the model treatment of atomic ions in hot, non-degenerate plasmas presented in this work, is well worth further study, the intermediate Fermi-Dirac degeneracy being of obvious importance. Under these conditions, an appropriate starting point to introduce the potential would be the elevated temperature Thomas-Fermi theory [46]. 

The Fermi-Dirac distribution law for the kinetic energy of the particles of a gas would be obtained by replacing p W) by the expression of Equation 49-5 for point particles (without spin) or molecules all of which are in the same non-degenerate state (aside from translation), or by this expression multiplied by the appropriate degeneracy factor, which is 2 for electrons or protons (with spin quantum number ), or in general 21 + 1 for spin quantum number I. This law can be used, for example, in discussing the behavior of a gas of electrons. The principal application which has been made of it is in the theory of metals,1 a metal being considered as a first approximation as a gas of electrons in a volume equal to the volume of the metal. 

This second point of view can be illustrated by an example from the late 1940 s that will play an important role in this chapter. At that time the Schrodinger equation was well established, and its relativistic generalization, the Dirac equation, appeared to describe the spectrum of hydrogen perfectly, though the question of how to apply the Dirac equation to many-electron systems was still open. However, when more precise experiments were carried out, most notably by Lamb and Retherford [1], a small disagreement with theory was found. The attempt to understand this new physics stimulated theoretical efforts that led to the modern form of the first quantum field theory. Quantum Electrodynamics (QED). This small shift, which removes the Dirac degeneracy between the 2si/2 and states, known as the Lamb shift, is an example of a radiative correction. 

An ordinary gas should, if it follows the Termi-Dirac statisti( , show degeneracy at low enough temperatures. But the effects cannot be detected, since they are masked by deviations from the gas laws due to van der Waals forces. 

The electronic conductivity increases proportional to the deviation from stoichiometry y, but for high values of y and <7e, there is a deviation caused by the beginning of electron degeneracy, that is, transition from Boltzmann- to Fermi-Dirac-statistics. The ion conductivity in Ag2S is much smaller than so it could be measured by using Agl/Ag-probes [13]. Because of the disorder and... 

Energies of the lowest levels of a 4f configuration on Eu and their degeneracies (d) in the O crystal field calculated within the Dirac-Hartree-Fock-Roothaan and complete open-shell configuration interaction scheme ... 

Each atomic spinor tp r) = tp r,RA) has its center at the position of the nucleus Ra of some atom A. In a first step, we include only those atomic spinors (r) which would be considered in an atomic Dirac-Hartree-Fock calculation on every atom of the molecule. Of course, if a given atom occurs more than once in the molecule, a set of atomic spinors of this atom is to be placed at every position where a nucleus of this t) e of atom occurs in the molecule. The number of basis spinors m is then smallest for such a minimal basis set. In this case, it can be calculated as the number of shells s per atom times the degeneracy d of these shells times the number of atoms M in the molecule, m = s A) x d s) x M. 

The first term, Hj, is the spin-orbit (one electron term) and spin-other-orbit (two electron term) couplings, which are the topic of the following subsection. The second term Hf contains the spin-spin coupling term and Fermi contact interaction. Both the Hj and f/ can lift degeneracy in multiplets. The parameter Hf is the Dirac correction term for electron spin and Ff is the classical relativistic correction to the interaction between electrons due to retardation of the electromagnetic field produced by an electron. The parameter H is the so-called mass-velocity effect, due to the variation of electron mass with velocity. Finally, H is the effect of external electric and magnetic fields. 

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