Vector Potential, Nuclear Spin

In spin-lattice relaxation, the excited nuclei transfer their excitation energy to their environment. They do so via interaction of their magnetic vectors with fluctuating local fields of sufficient strengths and a fluctuation frequency of the order of the Larmor frequency of the nuclear spin type. Depending upon the atomic and electronic environment of a nucleus in a molecule and the motion of that molecule, there are five potential mechanisms contributing to spin-lattice relaxation of the nucleus. 

Derivation of nuclear spin magnetic interactions through the magnetic vector potential... 

Let us therefore re-examine the lowest-order terms in (3.140) containing the magnetic vector potential, including the effects of nuclear spin and an apphed magnetic field, but excluding A whose consequences were investigated hilly in chapter 3. The important terms are... 

It will be recalled that in section 3.10 we developed the nuclear Hamiltonian by calculating the magnetic vector potential Aea at nucleus a arising from the spin and orbital motion of the electrons. Clearly we should now also include the nuclear spin contribution to Aa, the complete magnetic vector potential being (3.249) plus the additional term from the other nucleus a, ... 

The vector potential arising from the nuclear spin 1 of nucleus K at position Rg- is normally given as... 

The internal vector potential due to the non-zero nuclear magnetic moment p,N (nuclear spin) is... 

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... 

This modified vector potential has to be inserted into the Hamiltonian of equation (3) and additional terms giving the energy of the interaction of the total magnetic induction field with the nuclear moments and also the nuclear spin-nuclear spin interaction included. The total magnetic field is derived from equation (8) and (9) as... 

As an example of the explicit expression we give the vector potential produced by the nuclear spin of a point nucleus... 

Doing so yields the so-called Schrddinger equivalent potential (SEP). The central, spin independent part of this potential is determined by the sum of the scalar and vector Dirac potentials plus smaller, but important, quadratic terms. The spin-orbit potential depends mainly on the vector-scalar difference. Thus the relatively small spin independent nuclear potential arises from the near cancellation between the strong scalar and vector potentials, while the relatively strong nuclear spin-orbit force comes about from the constructive addition of the two parts of the Dirac potential. 

Some of the terms included in the Breit-Pauli Hamiltonian also describe small interactions that can be probed experimentally by inducing suitable excitations in the electron or nuclear spin space, giving rise to important contributions to observable NMR and ESR parameters. In particular, for molecular properties for which there are interaction mechanisms involving the electron spin, also the spin-orbit interaction (O Eqs. 11.13 and O 11.14) becomes important The Breit-Pauli Hamiltonian in O Eqs. 11.5-11.22, however, only includes molecule-external field interactions through the presence of a scalar electrostatic potential 0 (and the associated electric field F) and the appearance of the magnetic vector potential in the mechanical momentum operator (O Eq. 11.23). In order to extract in more detail the interaction between the electronic structure of a molecule and an external electromagnetic field, we need to consider in more detail the form of the scalar and vector potentials. 

Both the Fermi contact and the spin-dipole operators can be derived considering the magnetic field induction created by the vector potential describing the nuclear magnetic moment... 

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