Zeeman interaction term

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... 

We now notice that we could write a Hamiltonian operator that would give the same matrix elements we have here, but as a first-order result. Including the electron Zeeman interaction term, we have the resulting spin Hamiltonian ... 

The extended magnetic term on the left-hand side, which is called the Zeeman interaction term, leads to the Zeeman effect, which is a splitting of spectral lines under the influence of a magnetic field. Meanwhile, it is easily proven for non-hybridized electron spins in the nonrelativistic case that the Pauli spin matrix, a, in Eq. (6.55) is just twice the spin operator, s, in Eq. (2.91). The Zeeman interaction term is, therefore, written as... 

Fortunately, the Zeeman interaction term in Eq. (6.102) can be assembled with the inner product of the vector potential and the momentum in Eq. (6.105) by using the magnetization density operator, m, corresponding to the magnetic dipole moment, m, whose expectation value is the reverse sign of the first energy derivative in terms of magnetic field, as... 

Vignale and Rasolt derived the Dirac-Kohn-Sham equation incorporating the vector potential from the nonrelativistic Dirac equation neglecting the magnetic effect, i.e., the Zeeman interaction term, in Eq. (6.101) as (Vignale and Rasolt 1987, 1988)... 

This produces cross-terms that give the enhanced nuclear Zeeman interaction term... 

OIDEP usually results from Tq-S mixing in radical pairs, although T i-S mixing has also been considered (Atkins et al., 1971, 1973). The time development of electron-spin state populations is a function of the electron Zeeman interaction, the electron-nuclear hyperfine interaction, the electron-electron exchange interaction, together with spin-rotational and orientation dependent terms (Pedersen and Freed, 1972). Electron spin lattice relaxation Ti = 10 to 10 sec) is normally slower than the polarizing process. 

Pulse techniques, coupled with the observation of the decay of enhancement (Atkins et al., 1970a, b Glarum and Marshall, 1970 Smaller etal., 1971) constitute the most sensitive procedure for detecting CIDEP. Both net and multiplet polarization have been described. As with CIDNP, the former is believed to arise essentially from the Zeeman interaction and the latter from the hyperfine term. Qualitative rules analogous to Kaptein s rules should be capable of development. 

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

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)... 

The spin Hamiltonian for a biradical consists of terms representing the electron Zeeman interaction, the exchange coupling of the two electron spins, and hyperfine interaction of each electron with the nuclear spins. We assume that there are two equivalent nuclei, each strongly coupled to one electron and essentially uncoupled to the other. The spin Hamiltonian is ... 

Thus far in this chapter we have considered single-spin systems only. The zero-field interaction that we worked out in considerable detail was understood to describe interaction between unpaired electrons localized all on a single paramagnetic site with spin S and with associated spin wavefunctions defined in terms of its m5-values, that is, (j) = I ms) or a linear combination of these. However, many systems of potential interest are defined by two or more different spins (cf. Figure 5.2). By means of two relatively simple examples we will now illustrate how to deal with these systems in situations where the strength of the interaction between two spins is comparable to the Zeeman interaction of at least one of them S Sh B Sa. 

Just like the Zeeman interaction (S B), the hyperfine interaction (.S /) is a bilinear term and its coupling to strain (T S I), which we will call A-strain (also, -strain ), should be formally similar to the g-strain (T S B) just discussed. In the early work of Tucker on the effective S = 1/2 system Co2+ in the cubic host MgO, a shift in central hyperfine splitting was found to be proportional to the strain-induced g-shift given by Equation 9.22 (Tucker 1966). 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

Recapitulating, the SBM theory is based on two fundamental assumptions. The first one is that the electron relaxation (which is a motion in the electron spin space) is uncorrelated with molecular reorientation (which is a spatial motion infiuencing the dipole coupling). The second assumption is that the electron spin system is dominated hy the electronic Zeeman interaction. Other interactions lead to relaxation, which can be described in terms of the longitudinal and transverse relaxation times Tie and T g. This point will be elaborated on later. In this sense, one can call the modified Solomon Bloembergen equations a Zeeman-limit theory. The validity of both the above assumptions is questionable in many cases of practical importance. 

Bertini and co-workers 119) and Kruk et al. 96) formulated a theory of electron spin relaxation in slowly-rotating systems valid for arbitrary relation between the static ZFS and the Zeeman interaction. The unperturbed, static Hamiltonian was allowed to contain both these interactions. Such an unperturbed Hamiltonian, Hq, depends on the relative orientation of the molecule-fixed P frame and the laboratory frame. For cylindrically symmetric ZFS, we need only one angle, p, to specify the orientation of the two frames. The eigenstates of Hq(P) were used to define the basis set in which the relaxation superoperator Rzpsi ) expressed. The superoperator M, the projection vectors and the electron-spin spectral densities cf. Eqs. (62-64)), all become dependent on the angle p. The expression in Eq. (61) needs to be modified in two ways first, we need to include the crossterms electron-spin spectral densities, and These terms can be... 

A more general theory for outer-sphere paramagnetic relaxation enhancement, valid for an arbitrary relation between the Zeeman coupling and the axial static ZFS, has been developed by Kruk and co-workers (96 in the same paper which dealt with the inner-sphere case. The static ZFS was included, along with the Zeeman interaction in the unperturbed Hamiltonian. The general expression for the nuclear spin-lattice relaxation rate of the outer-sphere nuclei was written in terms of electron spin spectral densities, as ... 

We now calculate the Zeeman interaction with these equations and get the same results as before except that g and g as given in Eq. (40) have the additional terms... 

The first term on the right-hand side arises from external electric fields. The second (B) term arises from external magnetic inductions interacting with electronic orbital motion. The SL term arises from electron spin-orbital motion interactions. The Z term arises from the Zeeman interaction between electron spin and the external electric field. H s arises from electron spin-electron spin interactions and includes all hyperfine terms arising from nuclear spins. 

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