Zero field interactions

Just like the g-value and A-values also the zero-field interaction parameter can be anisotropic and have three values Dy, Dy and Dz. In contrast to g and A, however, the three D- s are not independent because D2 + Dy2 + D2 = 0, and so they can be reduced to two independent parameters by redefinition  

FIGURE 5.9 Effective g-values for half-integer spin systems in axial symmetry. The scheme gives the values for all transitions within the Kramer s doublets of S = nl2 systems assuming gmal = 2.00 and S S S B. 

For example the X-band spectra of high-spin ferric hemes (S = 5/2 g 2 E/D 0.1) show a single transition described by 

This expression rapidly loses its validity with E/D values decreasing from full rhombicity. Similarly, when starting from axial symmetry for the ms = 1/2 doublet at increased rhombicity values the Equations 5.31-5.33 become increasingly inaccurate, and for E/D 0.1 they are no longer valid. 

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over 

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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. 

We have seen in Chapter 2 that the electronic Zeeman term, the interaction between unpaired electrons in molecules and an external magnetic field, is the basis of EPR, but we have also discussed in Chapter 4 the fact that if a system has more than one unpaired electron, their spins can mutually interact even in the absence of an external field, and we have alluded to the fact that this zero-field interaction affords EPR spectra that are quite different from those caused by the Zeeman term alone. Let us now broaden our view to include many more possible interactions, but at the same time let us be systematic and realize that this plethora of possibilities is eventually reducible to five basic types only, two of which are usually so weak that they can be ignored. 

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. 

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... 

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. 

For higher integer spins the number of allowed zero-field interaction terms further increases, and so does the convolution of comparable effects, except once more for a unique term that directly splits the highest non-Kramer s doublet. For S > 3 we have the addition, valid in cubic (and, therefore, in tetragonal, rhombic, and triclinic) symmetry ... 

An—at least, theoretically—simple example is the S = 1 system in weak-field subject to a dominant zero-field interaction and a weakly perturbing electronic Zeeman interaction (similar to the S = 2 case treated above). The initial basis set is... 

This matrix is diagonal in the zero-field interaction, so the zeroth-order energy levels can be directly seen to be... 

The zero-field spin Hamiltonian parameters, D and E, are assumed to be distributed according to a normal distribution with standard deviations oD and aE, which we will express as a percentage of the average values (D) and (E). -Strain itself is not expected to be of significance, because the shape of high-spin spectra in the weak-field limit is dominated by the zero-field interaction. 

In their subsequent analysis Baker and Bleaney (ibidem) decided to ignore the last term on the assumption that gdl 3b hv. Although this is a reasonable approximation for lanthanide and actinide integer-spin ions doped in single crystals, it is not usually an acceptable assumption for the broad-line spectra from metalloproteins. Furthermore, the assumption of a A-distribution around zero (i.e., D 0 but all other zero-field interaction parameters are zero) is equally untenable for biomolecules. Therefore, we go for a later extension of the theory, based on a full Equation 12.9 and on (A) 0, for application to metalloproteins (Hagen 1982b). 

Hagen, W.R. 2007. Wide zero field interaction distributions in the high-spin EPR of metalloproteins. Molecular Physics 105 2031-2039. 

Special Hamiltonians. For some ions in which the zero-field interaction is very large, it is possible to see only a few of the allowed transitions. In these cases the absorption lines have sometimes been fitted to spin Hamiltonians of somewhat different character than that of Eq. (78). For example, in the 5= 1 system it may be that only the (— l ->+ 1) transition can be observed when D is very large. In this case it will be shown in Sec. IV that this absorption can be represented fairly well by solving the spin Hamiltonian... 

In the triplet model the spin polarization is with respect to the internal molecular states, TjJ>, Ty>, and T > of the triplet and evolves with time according to the time-dependent Schrodinger equation into a spin polarization with respect to the electron spin Zeeman levels Ti>, Tq>, and T i> in an external magnetic field Bq. Consider a simple case of axially symmetric zero-field splitting (i.e., D y 0 and E = 0 D and E are the usual zero-field parameters). Tx>, [Ty>, and TZ> are the eigenstates of the zero-field interaction Hzfs, where Z is the major principal axis. The initial polarization arising from the population differences among these states can be expressed as... 

For axiaUy symmetric molecules, the calculated shape of the AMs = 1 lines are given in Figure 1.8. The separation of the outer lines is 2D (where D = D/g(if) while that of the irmer lines is D (E is zero in this case). The theoretical Une shape for a randomly oriented triplet with E 0 is also shown in Figure 1.8. The separation of the outermost lines is again 2D whereas that of the intermediate and inner pairs is D + 31 /2 and D — 31 /2 respectively. As the zero-field interactions become comparable to and larger than the microwave energy, the Une shape exhibits severe distortions from the simulated case in Figure 1.8. 

The field separation and thus the resolution increases with the microwave frequency. Anisotropy of the g-factor occurring in solid materials is also better resolved gi and g2 could correspond to the axial (g ) and perpendicular (gj ) components in a cylindrically symmetric case, or two of the three g-factors of a general g-tensor (Chapter 4). Splittings due to hyperfine- or zero-field interactions are (to first order) independent of the microwave frequency. Measurements at different frequencies can therefore clarify if an observed spectrum is split by Zeeman interactions (different g-factors) or other reasons. 

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