Zeeman-quadrupole interactions

Fig. 4.10. The left side of the scheme represents the starting situation of pure Zeeman splitting, as described by (4.48) and shown before in Fig. 4.9. In this example, the field B = (0,0,B), which defines the quantization axis, is chosen as the z-direction. The additional quadrupole interaction, as shown on the right side of Fig. 4.10, leads to a pair-wise shift of the Zeeman states with mj = 3/2 and mi = 1/2 up- and down-wards in opposite sense. In first order, all lines are shifted by the same energy as expected from the m/-dependence of the electric...

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)... 

In the first row of (3.1) the terms denote the electron Zeeman (2 EZ), the hf (2 hft), the nuclear Zeeman (XNZ) and the nuclear quadrupole interaction (CXQ) of the central (metal) ion. The second row represents the hf, the nuclear Zeeman and the nuclear quadrupole interactions for sets of magnetically equivalent ligand nuclei. Each particular set is denoted by the index k, the individual nuclei of set k by kx. 

HypB protein, 47 289 HypC protein, 47 289 Hyperfine coupling, 13 149-178 anisotropic, 13 150-161 Hyperfine coupling anisotropic dipolar, 13 150-154 nuclear Zeeman interaction, 13 155 quadrupole interaction, 13 154, 155 factors affecting magnitude of metal influence of charge on metal, 13 169-170 isotropic and anisotropic, 13 166-170 libration, 13 170... 

In the limit where the nuclear Zeeman term in the nuclear spin hamiltonian is much larger than the quadrupole interaction, it is only the secular part of Hq that contributes to the time-independent hamiltonian, H0. 

When the Zeeman interaction is much larger than the quadrupole interaction, nonsecular terms may be discarded to give a relatively simple expression for the quadrupole Hamiltonian ... 

Figure 5.1 The effect of quadrupole interactions on an / = 3/2 nucleus in a magnetic field. The Zeeman interaction splits the levels by an equal amount, cot, (the Larmor frequency in frequency units). The central +1 /2 to -1 /2 transition is unaffected by first order coupling co j however, second order coupling, co , affects all transitions.

In the high field limit, where the quadrupole interaction acts as a perturbation of the Zeeman states, the terms of this Hamiltonian which commute with L lead to the perturbation of first-order... 

The frequencies under each operator indicate the size of the interaction. The largest interaction next to the Zeeman interaction Hz is the quadrupole interaction Hq, followed by the coupling Hrf of the spins to the exciting rf field, the dipole-dipole coupling Hd, the chemical shift H , and the indirect coupling Hj. 

It is desirable to apply fields of strong enough amplitude so that dominates all other interaction Hamiltonians except for the Zeeman interaction. The rf pulses can then be treated as infinitely short delta pulses, and the analysis of the experimental spectra becomes comparatively simple. However, arcing in the probe limits useful amplitudes to the order of 200 kHz, so that in solid-state NMR the delta-pulse approximation must be treated with care for the dipole-dipole interaction among protons, and it breaks down for the quadrupole interaction. 

In strong magneticfields, the resonance frequencies are determined largely by the Zeeman interaction (A = Z in Tables 3.1.1 and 3.1.2). The other interactions can be treated as perturbations (cf. eqn (3.1.1)). The coupling to the rf field, the dipole-dipole interaction, the chemical shift, and the J coupling can be readily treated hy first-order perturbation theory. For the quadrupole interaction, this approximation holds true only for small quadrupole moments like those of Li and H. 

Under anisotropic conditions, NMR lineshapes for a quadrupolar nucleus are dominated by chemical shielding and (first and second order) quadrupolar interactions. Dipolar interaction is usually a minor contribution only. First-order quadrupole interaction lifts the degeneracy of the allowed 21 (i.e. seven in the case of V / = V2) Zeeman transitions as shown in Figure 3.7, giving rise to seven equidistant lines, viz. a central line (mj = + V2 -V2. unaffected by quadrupole interaction) and six satellite lines. The overall breadth of the spectrum is determined by the size of the nuclear quadrupole coupling constant Cq the deviations from axial symmetry and hence the shape of the spectral envelope are governed by the asymmetry parameter. Static solid-state NMR thus provides additional parameters, in particular the quadrupole coupling constant, which correlates with the electronic situation in a vanadium compound. [ 1 The central component reflects the anisotropy of the chemical shift. 

Bu4N)[Y(Pc)2l at 0.04 K and with apphed field (various sweep rates) along the easy axis, (b, lower) Expansion of low field area, (b, upper) Zeeman diagram for the M = 6 doublet of Tb including nuclear hyperfine and quadrupole interaction with the I = 3/2 b nucleus. The states are labelled by their Mj, Mi quantum numbers and correspondence between level anti-crossings and QTM steps are shown. Reprinted with permission from Ishikawa et al., 2005 [89]. Copyright (2005) Wiley-VCH... 

Nuclear magnetic resonance (NMR) is perhaps the simplest technique for obtaining deuterium quadrupole coupling constants in solids or in liquid crystalline solutions. In ordinary NMR experiments with a magnetic field Hq > 104 gauss, the nuclear quadrupole interaction [Eq. (6)1 for deuterium is much smaller than the Zeeman interaction and can be treated as a perturbation to the Hamiltonian... 

Of the NMR-active nuclei around three-quarters have / > 1 so that the quadrupole interaction can affect their spectra. The quadrupole inter action can be significant relative to the Zeeman splitting. The splitting of the energy levels by the quadrupole interaction alone gives rise to pure nuclear quadrupole resonance (NQR) spectroscopy. This chapter will only deal with the case when the quadrupole interaction can be regarded as simply a perturbation of the Zeeman levels. 

The nuclear interactions observed in ENDOR are three the nuclear Zeeman, electron-nuclear hyperfine, and (for I > i) the nuclear electric quadrupole interactions. The Hamiltonian H including these nuclear interactions in an external magnetic field B is given as... 

Summarizing, foiu different magnetic interactions may occiu, which influence the behavior of electrons in a magnetic field (a) the Zeeman interaction, Hu (b) the nuclear hyperfine interaction. Hup, (c) the dectrostatic quadrupole interaction, Hq and (d) the zero-field spHtting if S > V2> Hps- The sum of these interactions results in the total spin Hamiltonian, Hf. 

---