The Electron Zeeman Interaction

The electron is a negahvely charged particle which possesses orbital angular momentum as it moves around the nucleus. The electron also possesses spin 

The most important physical consequence of the electron spin is the associated magnetic moment, Pe- This magnetic moment is directly proportional to the spin angular momentum and one may therefore write 

This magnetic moment interacts with the applied magnetic field. In classical terms the energy of the interaction between the magnetic moment (p) and the field (B) is described by 

For a quantum mechanical system one must replace p by the corresponding operator, giving the following simple spin Hamiltonian for a free electron in a magnetic field  

If the field is defined along the z direction, then the scalar product simplilies to the following Hamiltonian  

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The coupling constants of the hyperfme and the electron Zeeman interactions are scalar as long as radicals in isotropic solution are considered, leading to the Hamiltonian... 

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. 

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 ... 

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 ... 

By far the most important influence of a nuclear spin on the EPR spectrum is through the interaction between the electron spin S and the nuclear spin I. Usually, at X-band frequencies this interaction is weaker, by an order of magnitude or more, than the electronic Zeeman interaction, and so it introduces small changes in the EPR spectrum known as hyperfine structure. As a first orientation to these patterns, note that just like the electron spin S, also the nuclear spin / has a multiplicity ... 

FIGURE 5.2 A schematic model of multiple X Y interactions. Black dots are unpaired electrons the central, big black dot is the point of EPR observation. Straight lines are interactions a single straight line symbolizes the electronic Zeeman interaction S B double lines represent central and ligand hyperfine interactions S I triple lines are zero-field interactions S S between electrons (i) around a single metal (ii) at different centers within a molecule and (iii) at centers in different molecules. 

Now we can define the anisotropic resonance condition for an S = 1/2 system subject to the electronic Zeeman interaction only as... 

Systems with more than one unpaired electron are not only subject to the electronic Zeeman interaction but also to the magnetic-field independent interelectronic zero-field interaction, and the spin Hamiltonian then becomes... 

For biomolecular S = 1/2 systems subject to central hyperfine interaction the intermediate-field situation (B S S I) is not likely to occur unless the micro-wave frequency is lowered to L-band values. When v = 1 GHz, the resonance field for g = 2 is at B = 357 gauss. Some Cu(II) sites in proteins have Az 200 gauss, and this would certainly define L-band EPR as a situation in which the electronic Zeeman interaction is comparable in strength to that of the copper hyperfine interaction. No relevant literature appears to be available on the subject. An early measurement of the Cun(H20)6 reference system (cf. Figure 3.4) in L-band, and its simulation using the axial form of Equation 5.18 indicated that for this system... 

We can now extend the spin Hamiltonians by making combinations of T, with B, and/or S, and/or I, and since we are interested in the effect of strain on the g-value from the electronic Zeeman interaction (B S), the combination of interest here is T B S. 

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. 

The basic idea of the slow-motion theory is to treat the electron spin as a part of the lattice and limit the spin part of the problem to the nuclear spin rather than the IS system. The difficult part of the problem is to treat, in an appropriate way, the combined lattice, now containing the classical degrees of freedom (such as rotation in condensed matter) as well as quantized degrees of freedom (such as the electron Zeeman interaction). The Liouville superoperator formalism is very well suited for treating this type of problems. 

The simplest possible physical picture of the lattice contains the electron Zeeman interaction, the axially symmetric ZFS (whose principal axis coincides with the dipole-dipole axis) and the molecular rotation. The corresponding Liouvillian is given by ... 

Hamiltonian matrix for the cubic ligand held, spin-orbit coupling and the electronic Zeeman interaction in the real cubic bases of a 2D term. The g-factor for the free electron has been set to two for clarity (Table A.l). 

The first and second terms describe the electron and nuclear Zeeman interactions, where ys- and y are the gyromagnetic ratios of the electron and nucleus, respectively, and B0 is the externally applied magnetic field. This description of the electron Zeeman interaction is appropriate for a free electron or organic radical, but for metal ions or semiconductors it should be rewritten as gjuB(S B0) where g is the y-f actor of the unpaired electron and juB is the Bohr magnetron. The terms Hs and Ho in... 

Application of an external magnetic field alters the nature of some of the magnetic interactions in a RP and also leads to additional terms in the spin Hamiltonian through the electron Zeeman interaction. 

The forces which drive the intersystem crossing are the nuclear spin-dependent hyperfine interactions in the radicals and the electron Zeeman interactions. This becomes evident from the following after pair formation in the magnetic field of a NMR spectrometer, say, the two unpaired electron sfnns precess about the magnetic field axis starting from defined initial phase an es. These iititial phase angles are different for the four possible initial electronic states T Tq,... 

Fourthly, two unpaired electrons interact because of the overlap of their electronic orbitals. This gives rise to the so-called exchange energy, which again changes the resonance frequency of the individual electrons compared to that of the free electron. In Table 3 the four interactions are tabulated, together with their mathematical expressions. We have neglected the nuclear Zeeman interaction as this is more than six hundred times smaller than the electronic Zeeman interaction and, to first order, does not influence the EPR resonance. 

Consider the expression for the energy of a one-electron (S = -j, "is = one-proton (/ = mj = y) system, now including the nuclear Zeeman interaction (whose sign is opposite that of the electron Zeeman interaction) ... 

In Eq. (13) the first term is the electronic Zeeman interaction, which is of the form... 

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