Zeeman interaction

The interaction between the electron spin (with magnetic quantum number Wj = 5) and an external magnetic field (fio) is termed the Zeeman interaction and is expressed as 

In EPR experiments, the g-factor for a particular radical represents its chemical shift which greatly helps with the identification of different paramagnetic species. However, the parameter can often be difficult to calculate accurately as it depends on a number of different factors such as structure of radical and strength of spin-orbit coupling. Some typical g-factors for radicals can be found in the literature [10]. 

In a single radical the magnetic moment of the electron processes around Bo and the frequency of this precession is commonly termed the Larmor frequency ( uo)-The difference in the Larmor frequency for a radical pair can be found using the expression 

The oscillation of the 5 — To can be shown more formally by starting with the time-dependent Schrodinger equation 

Differentiating the expression for dCs /dt again and substituting into the expression for dCr/dt gives the second-order differential equation of the form 

Matrix elements of the isotropic exchange for a general triad 

The tensor ranks entering the decoupling formula for the evaluation of the reduced matrix elements are assigned as follows 

The reduced matrix elements of the constituent spins are expressed as follows R = (51S2Sl2535 f1(S1) s152512535) 

The 97-symbols containing one zero or three zero collapse into 67-symbols (Table 11.2), hence 

The triangular restrictions for the 67-symbols imply that 5 = 5,5 1 and 5,2 = S,2, S,2 1. Therefore the reduced matrix elements fill the places according to the recipe 

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The large static applied magnetic field (Bq) produces the Zeeman interaction (= where is the z-... 

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. 

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 combined effect of strong nuclear magnetic (Zeeman) interaction and weak electric quadmpole interaction for the excited state of Fe is demonstrated in... 

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 ... 

The spin state of a paramagnetic system with total spin S wiU lift its (25 + l)-fold degeneracy under the influence of ligand fields (zero-field interaction) and applied fields (Zeeman interaction). The magnetic hyperfine field sensed by the iron nuclei is different for the 25 + 1 spin states in magnitude and direction. Therefore, the absorption pattern of a particular iron nucleus for the incoming synchrotron radiation and consequently, the coherently scattered forward radiation depends on how the electronic states are occupied at a certain temperature. 

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)... 

An exception to this rule arises in the ESR spectra of radicals with small hyperfine parameters in solids. In that case the interplay between the Zeeman and anisotropic hyperfine interaction may give rise to satellite peaks for some radical orientations (S. M. Blinder, J. Chem. Phys., 1960, 33, 748 H. Sternlicht,./. Chem. Phys., 1960, 33, 1128). Such effects have been observed in organic free radicals (H. M. McConnell, C. Heller, T. Cole and R. W. Fessenden, J. Am. Chem. Soc., 1959, 82, 766) but are assumed to be negligible for the analysis of powder spectra (see Chapter 4) where A is often large or the resolution is insufficient to reveal subtle spectral features. The nuclear Zeeman interaction does, however, play a central role in electron-nuclear double resonance experiments and related methods [Appendix 2 and Section 2.6 (Chapter 2)]. 

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 ... 

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 ... 

This corresponds to an EPR-silent sample that gives no detectable ESR spectrum at X-band frequencies because it possesses a zero-field splitting larger than the Zeeman interaction (see Chapter 6), and the energy spacing between the two lowest levels is too large to be spanned by a microwave quantum at X-band. Nevertheless, higher frequencies are able to induce transitions. Since... 

For routine studies with the ESR spectrometer, it is most convenient to work at X-band frequencies ( 9.5 MHz or 3 cm). The sample is usually contained in a 4 or 5 mm diameter quartz tube having a sensitive region about 2 cm in length. An alternative frequency is at Q-band ( 35,000 MHz or 1 cm). Here, the cavity dimensions are much smaller and the diameter of the sample tube is less than 2 mm. This creates some problems in handling and degassing powder samples. By varying the frequency it is possible to determine which features in a spectrum are due to Zeeman interactions... 

The resonance field Hr as a function of orientation is determined from the appropriate spin Hamiltonian. For example, assume that the radical of interest is experiencing only Zeeman interactions and that it has one unique symmetry axis. For this case gxx = gvy = gx and gzz = g. At any orientation the g value is described by... 

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. 

Let us rewrite the resonance condition of an S = 1/2 system subject to the Zeeman interaction only as... 

When the hyperfine interaction is much smaller than the Zeeman interaction ( much means approximately two orders of magnitude or more), as is usually the case in X-band, then the resonance condition is... 

In any metalloprotein, be it tumbling in water or fixed in a frozen solution, not only the Zeeman interaction but also the hyperfine interaction will be anisotropic, so the resonance held in Equation 5.10 becomes a function of molecular orientation in the external held (or alternatively of the orientation of B in the molecular axes system) ... 

Formally, this procedure is correct only for spectra that are linear in the frequency, that is, spectra whose line positions are caused by the Zeeman interaction only, and whose linewidths are caused by a distribution in the Zeeman interaction (in g-values) only. Such spectra do exist low-spin heme spectra (e.g., cytochrome c cf. Figure 5.4F) fall in this category. But there are many more spectra that also carry contributions from field-independent interactions such as hyperfine splittings. Our frequency-renormalization procedure will still be applicable, as long as two spectra do not differ too much in frequency. In practice, this means that they should at least be taken at frequencies in the same band. For a counter-example, in Figure 5.6 we plotted the X-band and Q-band spectra of cobalamin (dominated by hyperfine interactions) normalized to a single frequency. To construct difference spectra from these two arrays obviously will generate nonsensical results. 

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... 

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