Vector potential magnetic moment

Here Iais the magnetic moment of nucleus A and Ra is the position (the nucleus is the natural Gauge origin). Adding this to the external vector potential in eq. (10.62) and expanding as in (10.63) gives... 

Table 1.1 Conjugate pairs of variables in work terms for the fundamental equation for the internal energy U. Here/is force of elongation, Z is length in the direction of the force, <7 is surface tension, As is surface area, , is the electric potential of the phase containing species i, qi is the contribution of species i to the electric charge of a phase, E is electric field strength, p is the electric dipole moment of the system, B is magnetic field strength (magnetic flux density), and m is the magnetic moment of the system. The dots indicate scalar products of vectors.

The electron coupled interaction of nuclear magnetic moments with themselves and also with an external magnetic field is responsible for NMR spectroscopy. Since the focus of this study is calculation of NMR spectra within the non-relativistic framework, we will take a closer look at the Hamiltonian derived from equation (76) to describe NMR processes. In this regard, we retain all the terms, which depend on nuclear magnetic moments of nuclei in the molecule and the external magnetic field through its vector potential in addition to the usual non-relativistic Hamiltonian. The result is... 

The vector potential created by spin of a nueleus with a magnetic moment p... 

Assuming an axially symmetric potential, the anisotropy energy of n) will be an even function of the longitudinal component of the magnetic moment s n. The averages we need to calculate are aU products of the form = (n =i (cn ))a> where the c are arbitrary constant vectors. Introducing the polar and azimuthal angles of the spin d, tp), we can write as... 

The binding corrections to h q)erfine splitting as well as the main Fermi contribution are contained in the matrix element of the interaction Hamiltonian of the electron with the external vector potential created by the muon magnetic moment (A = V X /Lx/(47rr)). This matrix element should be calculated between the Dirac-Coulomb wave functions with the proper reduced mass dependence (these wave functions are discussed at the end of Sect. 1.3). Thus we see that the proper approach to calculation of these corrections is to start with the EDE (see discussion in Sect. 1.3), solve it with the convenient... 

Larmor Precession. When a nucleus with magnetic moment fiN is placed in an external magnetic field B0, and the angle between the two vectors is 8, the orientational potential energy is... 

The electronic magnetic multipoles (25)-(27) are unperturbed, or permanent, moment operators. In the presence of a vector potential A(r, t) (we simplify the notation, omitting the index), the canonical momentum is replaced by the mechanical momentum... 

Within the Bloch gauge for the vector potential, the perturbed magnetic moments become, according to Eqn. (29),... 

In principle, a chemical shift calculation represents a perturbation theory, because of the presence of an external field Bz and magnetic moments due to the dipole character of nuclei. Therefore, perturbations to the Hamiltonian and the wave function have to be considered. The next important point is that the origin of the vector potential Az is not fixed due to the relation Bz = rot Az- Any change of the gauge origin Rq should not change any measurable observable. Therefore, a gauge transformation of the wave function 1%) and Hamilton operator h is essential... 

Another important quantity related to the current density distribution is the nuclear magnetic moment density distribution (or magnetization density distribution) m(r) = r x j(r), which integrates to the magnetic moment = f d r m(r) briefly mentioned above. Finally, the magnetic induction field, generated by the nuclear current density distribution, can be obtained from the vector potential or from the current density distribution as... 

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

Thus, at distances considerably greater than the loop radius, the vector potential and the corresponding components of the magnetic field are the same as those for a magnetic dipole located at the center of the loop with its moment directed perpendicularly to the loop. 

Let us choose the cylindrical system of coordinates (r, 0, z) and the vertical magnetic dipole is placed at the origin of this system (Fig. 4.1). The moment of the magnetic dipole is oriented along the z-axis. We will look for a solution using only the z-component of the vector-potential, A. As follows from Maxwell s equations the vector potential must satisfy several conditions ... 

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