Hamiltonian magnetic-field term

The aim of [47] in their Zeeman measurements of Li in CdTe was not only to determine the -factors of this acceptor, but also values of the VB parameters of CdTe by the method proposed and used in the case of GaAs [128]. This method, which is self-consistent, is indirect and is based on the adjustment of the VB parameters of the acceptor Hamiltonian including magnetic field terms... 

The remainder of this section is devoted to a more detailed derivation of (5) from (3). We first transform (3) into a Luttinger Hamiltonian, following Luther and Peschel,6 to first order in the coefficient Jz of PjPj (This is, in fact, the limit of accuracy of the Luther-Peschel replacement of the HI Hamiltonian by a Luttinger model. The equivalence breaks down completely as Jz approaches unity where the HI model is singular.) We drop the constant magnetic field term, because when the Hamiltonian is cast in the Luttinger form it is evident that this term merely alters the fermi level and cannot affect the exponents. 

Contributions to the average V )o could arise from the magnetic field terms coming from the nuclear spins. The aim here is to obtain a spin hamiltonian acting on both nuclear and electronic spin states so one would in general be interested in the terms obtained before summing over electronic spin coordinates to obtain a spin hamiltonian... 

The magnetic field term gives rise to the Zeeman effect. These terms all arise from the nonrelativistic spin Hamiltonian, (4.22). 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

When we include the Zeeman interaction term, gpBB-S, in the spin Hamiltonian a complication arises. We have been accustomed to evaluating the dot product by simply taking the direction of the magnetic field to define the z-axis (the axis of quantization). When we have a strong dipolar interaction, the... 

The significance of the terms in the g tensor can best be illustrated by means of a simple example. Let us redefine our coordinates such that the spin is quantized along the z axis. The magnetic field vector is now at some different position in space as shown in Fig. 11. Expanding the Hamiltonian in Eq. (1C) gives,... 

Hamiltonian with the energy from appropriate terms in the true Hamiltonian. The latter terms include the interaction between the external field and the magnetic moment produced by the orbiting electron, the interaction between the external field and the magnetic moment due to electron spin, and the interaction between the orbital magnetic moment and the spin magnetic moment. These interactions may be expressed as a perturbation to the total Hamiltonian for the system where... 

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... 

In order to discuss the origin of these terms we need to allow the spins to have anisotropic shielding tensors. Molecular tumbling in solution makes the chemical shielding in the direction of the external magnetic field a stochastic function of time and acts therefore as a relaxation mechanism, called the chemical shielding anisotropy (CSA) mechanism. The Hamiltonian for each of the two spins, analogous to Eq. (5), contains therefore two... 

In yet another recent investigation, Kruk and Kowalewski considered the case when the static ZFS was smaller than the transient ZFS and the latter term should be considered as the unperturbed Hamiltonian at low magnetic fields (125). The validity conditions for the theory derived in that case were rather difficult to realize in experimentally relevant situations. The aqueous solution of Ni(II), a difficult case treated previously by the slow-motion theory (92,93), was however found to be possible to describe in a reasonable way using the new approach (125). 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

Consideration of the spin-orbit interaction and the effect of an external magnetic field on the electronic ground state of an ion in a CF allows evaluation of the various terms in the spin-Hamiltonian of Equation (31). In addition, the interaction of the nucleus of the paramagnetic ion and ligand nuclei with the d-electron cloud must be considered. In this way the experimentally determined terms of the spin-Hamiltonian may be related to such parameters as the energy differences between levels of the ion in the CF and amount of charge transfer between d-electrons and ligands. 

One other 1-electron density is of great imprortance - the current density J(r), which is again a pseudovector density. This is everywhere zero until we apply an infinitely weak magnetic field, so as to define a quantization axis and to introduce an imaginary term in the wavefunction i. The usual kinetic energy operator in the Hamiltonian is then replaced by... 

In cases where the classical energy, and hence the quantum Hamiltonian, do not contain terms that are explicitly time dependent (e.g., interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms discussed later in this text), the separations of variables techniques can be used to reduce the Schrodinger equation to a time-independent equation. 

The presence of a magnetic field along direction adds the following term to the Hamiltonian describing the system of interest ... 

Though the ESR Hamiltonian is typically expressed in terms of effective electronic and nuclear spins, it can, of course, also be derived from the more fundamental Breit-Pauli Hamiltonian, when the magnetic fields produced by the moving nuclei are explicitly taken into account. In order to see this, we shall recall that in classical electrodynamics the magnetic dipole equation can be derived in a multipole expansion of the current density. For the lowest order term the expansion yields (59)... 

The nuclear Zeeman term describes the interaction of the nuclear spins with the external magnetic field. Just as the hyperfine splitting, this term is not incorporated in the original purely electronic Breit-Pauli Hamiltonian as presented in Eqs. (59) and (60) but becomes relevant for ESR spectroscopy. 

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