Second-order shifts

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

The g- and 14N hyperfine matrices are approximately axial for this radical, but the g axis lies close to the perpendicular plane of the hyperfine matrix. If the g axis was exactly in the A plane, the three negative-going gN, A features, corresponding to resonant field maxima, would be evenly spaced. In fact, the spacings are very uneven - far more so than can be explained by second-order shifts. The effect can be understood, and the spectrum simulated virtually exactly, if the gN axis is about 15° out of the A plane. 

In these cases, the g-matrix is nearly isotropic, but the principal axes of the two 59Co hyperfine matrices are non-coincident. The largest hyperfine matrix component (ay = 66.0 G in the case of the Co-Co-Fe-S cluster) results in 15 features, evenly spaced (apart from small second-order shifts). Another series of features, less widely spaced, shows some variation in spacing and, in a few cases, resolution into components. This behavior can be understood as follows Suppose that the hyperfine matrix y-axes are coincident and consider molecular orientations with the magnetic field in the vz-plane. To first order, the resonant field then is ... 

First order ENDOR frequencies of nonequivalent nuclei or of pairs of magnetically equivalent nuclei are given by Eq. (3.3) which is derived from the direct product spin base. To obtain correct second order shifts and splittings, however, adequate base functions have to be used. We start the discussion of second order contributions with the most simple case of a single nucleus and will then proceed to more complex nuclear spin systems. 

The quadrupole perturbed NMR spectriun of Rb (I = 312) consists of the central hne and two satelUte Unes. In D-RADP-x the nuclear quadrupole couphng constant decreases linearly with x from 9.6 MHz (x = 0) to 6.6 MHz (x = 0.95) [10], whereas the Larmor frequency amounts to 98.163 MHz in a Bo field of 7 T. Due to the substitutional disorder, and the corresponding loss of the translational invariance, the satelHte transitions are inhomogeneously broadened that much, that only the central transition can be directly observed. The position of the corresponding resonance line can be calculated using standard NMR-NQR theory [16], where the second order shift is given by ... 

This may be derived from experimental data by analyzing the first order inductive shift in non alternants and second order shift in alternants. Both substituent constants are therefore intimately related to substituent effects on the it - n dipole excited states. Figure 44 shows the correlation with the shake up intensities. The trends displayed are quite striking and leave little doubt that the satellites arise from it->it excitations. 

The second-order shift is not scaled by (3 cos2 0 — 1). It increases with Vq and is inversely proportional to the magnetic field (through vL). Bearing in mind that the dispersion of the chemical shift, which is normally the... 

We have thus far treated the K (n + 2)s states as having no Stark shifts. While they have no first order Stark shift, they do have a second order shift due to their dipole interaction with the p states, which are removed from the s states by energies large compared to the microwave frequency. The microwave field does not produce appreciable sidebands of the s state since it has no first order Stark shift. However, it does induce a Stark shift to lower energy. Not surprisingly the Stark shift produced by a low frequency microwave field of amplitude E is the same as the second order Stark shift produced by a static field Es/j2, they have the same value of (E2). Careful inspection of Fig. 10.9 reveals that the resonances observed with high microwave powers shift with power. 

Figure 23 Second order shifts in spin-1/2 spectra arising from coupling to a quadrupolar nucleus with S = 5/2, as a function of the parameter %D lu,...

A perturbation theory approach that simplifies the expression for second order shifts of resonance frequencies has been developed [59,60] for the analysis of the splitting pattern. This approach is valid only when the term K, defined... 

Rb NMR has been used to measure the temperature dependence of the second-order shifts of the central transition of RbSCN, a compound which undergoes an... 

The term AWi (= Ajr ) is the first-order energy shift, and AW, (= Cjr ) is the second-order shift. Equation 16 has the form of (2), and in the region of validity of the rotational Stark effect the simple Langevin theory applies. However, the effective angular momentum is determined by the first two terms of (16), and the effective polarizability is determined by the last two terms. 

The magnitude of the second-order shifts to each line may be worked out as follows. If X = A j4 H, the coefficient of x for various nuclear spins is given by... 

Anisotropy of g tensor and of hyperfine coupling tensor(s) A for 1 or 2 nuclei with second order shift of resonance lines. The principal axes of these tensors are... 

X-band (v = 9.5000 GHz) ESR spectrum of the hydrogen atom. The displacement of the 1st order simulated spectrum (dashed) compared to the exact simulated and experimental spectra is due to a second order shift in the tine positions... 

K. Only the /r/quartet due to 7 = 3/2 (Aj lines) is visible at 6 K. The doublet due to 7 = 1/2 E lines) marked as a star ( ) increases in intensity with increasing temperature. Because of the high resolution spectra the A1 and E lines are resolved by a small second-order shift (0.024 mT) of the H hf splitting. The figure is adapted from [75] with permission from the American Chemical Society... 

Bastow performed the first Ti and Ba SSNMR experiments on tetragonal BaTiOa in 1989 by using single-crystal techniques [118]. Bastow also studied the temperature dependence of the second-order shifts... 

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