First-order quadrupole shift

Fig. 4.11 Angular dependence (Scos d - 1 + 77sin 0cos2(/)) of the first-order quadrupole shift, Eq in high-field Mossbauer spectra. Left side > 0 and r/ = 0. Right side r] = 1 viewed in...

Fig. 4.12 Surface plots of the EFG tensor as determined from the angular dependence of the first-order quadrupole shift, Eq of high-field magnetic Mossbauer spectra. The plots visualize the value of the function (3cos 0 - 1 + rysin d cos20) for > 0 and ry = 0 (a), r] = 0.3 (b), and 77 = 1 (c)...

In contrast, soft magnetic solids and paramagnetic systems with weak anisotropy may be completely polarized by an applied field, that is, the effective field at the Mossbauer nucleus is along the direction of the applied field, whereas the EFG is powder-distributed as in the case of crystallites or molecules. In this case, first-order quadrupole shifts cannot be observed in the magnetic Mossbauer spectra because they are symmetrically smeared out around the unperturbed positions of hyperfine fines, as given by the powder average of EQ mj, d, in (4.51). The result is a symmetric broadening of all hyperfine fines (however, distinct asymmetries arise if the first-order condition is violated). 

In the case of an additional electric quadrupole interaction (when > 0), as is shown in (d), the substates of the excited state with m/ = 3/2 and 1/2 shown in (c) are pairwise shifted up and downwards in energy in opposite directions by the so-called first-order quadrupole shift This leads to an asymmetry of the resulting splitting pattern in the Mossbauer spectrum, as shown in the lower part of (d). 

The perturbation of the four substates of the excited 7 = 3/2 manifold by induces a typical asymmetry of the resulting magnetically split Mossbauer spectrum as pictured at the bottom of Fig. 4.10 for positive the inner four lines, 2-5, are shifted to lower velocities, whereas the outer two lines, 1 and 6, are shifted to higher velocities by equal amounts. In first order, the line intensities are not affected. For negative the line asymmetry is just inverted, as the quadmpole shift of the nuclear 1/2 and 3/2 states is opposite. Thus, the sign and the size of the EFG component along the field can be easily derived from a magnetic Mossbauer spectrum with first-order quadrupole perturbation. 

Various ID and 2D experiments related to first-order quadrupole broadening also benefit from the fast rotation frequency. For heavier nuclei, quadrupole interaction and chemical shift anisotropy averaging at higher speeds results in better focusing of signal intensity to the remaining rotation sidebands (Fig. 5). 

The sample is placed into a gas driven spinner, the rotation axis of which is inclined by an angle ij/ against the magnetic field Bo (Fig. 3.3.4). For MAS this angle is adjusted to the magic angle of 54.7°. Then the anisotropic parts of all interactions which are described by second-rank tensors can be averaged. These are the anisotropy of the chemical shift, the dipole-dipole interaction, and the first-order quadrupole interaction. 

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. 

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...

Distinct quadrupole shifts do occur as well in magnetically split spectra of single-crystals, poly crystalline powder or frozen solution samples. In all three cases, the line shifts obey the simple first-order expression at high-field condition. 

Fig. 14. The pulse sequence for recording the double-quantum 2H experiment.37 The entire experiment is conducted under magic-angle spinning. This two-dimensional experiment separates 2H spinning sideband patterns (or alternatively, static-like 2H quadrupole powder patterns) according to the 2H double-quantum chemical shift, so improving the resolution over a single-quantum experiment. In addition, the doublequantum transition frequency has no contribution from quadrupole coupling (to first order) so, the double-quantum spectrum is not complicated by spinning sidebands. Details of molecular motion are then extracted from the separated 2H spinning sideband patterns by simulation.37 All pulses in the sequence are 90° pulses with the phases shown (the first two pulses are phase cycled to select double-quantum coherence in q). The r delay is of the order 10 gs. The q period is usually rotor-synchronized.

This term is anisotropic and produces a powder pattern. It has been derived under the assumptions that first-order perturbation of the S-states is sufficient, that the J tensor is axially symmetric and that the unique axis of J is aligned with the intemuclear vector. Under MAS this term will be scaled but, as it is not proportional to P2(cos0), it cannot be completely removed. Hence the MAS spectrum will still have some residual width, but the most profound effect is to leave an isotropic term which can be calculated by averaging the powder lineshape. Hence for a J-coupled system with an axially symmetric quadrupole interaction, the spectrum is shifted from the isotropic chemical shift by ... 

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. 

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