Subject Zeeman interactions

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

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... 

Note that the Zeeman interaction for a cubic system results in an isotropic g-value, but the combination with strain lowers the symmetry at least to axial (at least one of the 7 -, 0), and generally to rhombic. In other words, application of a general strain to a cubic system produces a symmetry identical to the one underlying a Zeeman interaction with three different g-values. In yet other words, a simple S = 1/2 system subject to a rhombic electronic Zeeman interaction only, can formally be described as a cubic system deformed by strain. 

Let us calculate the frequencies of transitions between Zeeman eigenstates s) and r), assuming that the nuclei are only subjected to an isotropic chemical shift and the first- and second-order quadrupolar interaction. As seen in Sect. 2.1, the Hamiltonian that governs the spin system in the frame of the Zeeman interaction (the rotating frame) is... 

If an electron (or nuclear) spin S is subjected to a magnetic field Hq in the direction z then the Zeeman interaction... 

Figure 2 Spectra for a polycrystalline sample for nuclei with different site symmetries and subject only to the Zeeman interaction. (A) Ogg ( bb cci ( ) aa bb cc> ( ) aa cci (D) Oggi Obb OcO-...

Fig. 3. Energy level diagram for a spin f nucleus showing the effect of the first-order quadrupolar interaction on the Zeeman energy levels. Frequency of the central transition (shown in bold lines) is independent of the quadrupolar interaction to first order, but is subject to second-order quadrupolar effects (see text).

In order to limit the size of the book, we have omitted from discussion such advanced topics as transformation theory and general quantum mechanics (aside from brief mention in the last chapter), the Dirac theory of the electron, quantization of the electromagnetic field, etc. We have also omitted several subjects which are ordinarily considered as part of elementary quantum mechanics, but which are of minor importance to the chemist, such as the Zeeman effect and magnetic interactions in general, the dispersion of light and allied phenomena, and most of the theory of aperiodic processes. 

EMR spectra correspond to transitions among the electronic Zeeman levels, subject to selection rule Amg = 1, Amj = 0 (for all rtij of the system), and it follows that these comparatively large transition energies are difficult to correlate with weak molecular interactions or subtle changes in the nuclear hyperfine energies due to substituent effects. 

---