Dirac vector model

Serber[15] has contributed to the analysis of symmetric group methods as an aid in dealing with the twin problems of antisymmetrization and spin state. In addition, Van Vleck espoused the use of the Dirac vector model[16] to deal with permutations. [17] Unfortunately, this becomes more difficult rapidly if permutations past binary interchanges are incorporated into the theory. Somewhat later the Japanese school involving Yamanouchi[18] and Kotani et al.[19] also published analyses of this problem using symmetric group methods. 

Models for the Signs of Reduced Coupling Constants 3.4.1. The Dirac Vector Model... 

Fig. 10. The Dirac Vector Model showing the orientation of nuclear and electron spins.

A physical picture of spin-spin coupling, the way in which the spin of one nucleus influences that of another, is not easy to develop. Several theoretical models are available. The best theories we have are based on the Dirac vector model. This model has limitations, but its predictions are substantially correct. According to the Dirac model, the electrons in the intervening bonds between the two nuclei transfer spin information from one nucleus to another by means of interaction between the nuclear and electronic spins. An electron near the nucleus is assumed to have the lowest energy of interaction with the nucleus when the spin of the electron (small arrow) has its spin direction opposed to (or paired with) that of the nucleus (heavy arrow). 

There are three main mechanisms for indirect or J coupling, all mediated by the valence electrons. The most important involves the Fermi contact interaction of an s electron with the nucleus (with which p, d,... electrons have no contact). This can be envisaged in terms of the Dirac vector model. For simplicity we can consider the HF molecule, since both nuclei have spin 1/2 and positive magnetogyric ratios y. The bonding pair of electrons must have antiparallel spins (a]8) by the Pauli principle, and their motions are correlated such that if the one with a spin is near one nucleus, the one with )S spin is likely to be near the other. Within each nucleus the magnetic moments of the nucleus and the electron are more stable when parallel, so that the spins are antiparallel (since the electron has a negative y). The nuclear spins are... 

A simple model which accounts for the signs of couplings over two, three, or more bonds relative to one-bond coupling is the Dirac vector model, used very early by McConnelland nicely explained by Lynden-Bell and Harris. With this model, it is easy to understand why the AT(HCH) is opposite in sign to AT(CH) and /f(HH) in ethane. 

Besides the methods discussed in the previous sections there are others yielding useful results, some of which will be briefly outlined in the following sections. Several methods have been proposed which are beyond the scope of this book, notably the Dirac -Van Vleck2 vector model, which yields results similar to those given by the method of Slater of Section 30. 

Correction vectors[40], as introduced by Dirac[41], provide a model-exact approach to dynamical NLO coefficients of Hubbard or PPP models with a large but... 

The origin of these interactions, called exchange, was first realized by Heisenberg and Dirac in 1926. The interpretation of the exchange effect as formally equivalent to the coupling between spins permits the use of a vector-coupling scheme to model the quantitative behavior of coupled spins with the... 

Presently it is widely accepted that the relativistic mean-field (RMF) model [40] gives a good description of nuclear matter and finite nuclei [41]. Within this approach the nucleons are supposed to obey the Dirac equation coupled to mean meson fields. Large scalar and vector potentials, of the order of 300 MeV, are necessary to explain the strong spin-orbit splitting in nuclei. The most debated... 

In 1982-1983 a relativistic scattering model was introduced [Me 83b, Sh 83, Cl 83b] which used the MRW five-term, Lorentz invariant NN operator, the scalar and vector densities of the Walecka model for finite nuclei [Ho 81d], and the Dirac equation for the scattered proton. The model could be used to calculate proton-nucleus scattering observables throughout the medium energy range. The first predictions [Sh 83, Cl 83b] at 500 and 800 MeV were startlingly successful. This model, which is now called the relativistic impulse approximation model, or the RIA model, has been used extensively. 

Fig. 28. Real optical potential strengths in infinite nuclear matter of the scalar and vector potentials for medium energy protons based on the RIA model with pseudoscalar invariant (solid curves) and the one-meson exchange model using the pseudovector invariant with explicit direct and exchange terms (dashed curves). Results of Dirac phenomenology [Ha 90] for p + °Ca elastic scattering (potentials evaluated at r = 0) are shown by the dotted curves.

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