Hyperfine second order

Cahbration spectra must be measured at defined temperamres (ambient temperature for a-iron) because of the influence of second-order Doppler shift (see Sect. 4.2.1) for the standard absorber. After folding, the experimental spectrum should be simulated with Lorentzian lines to obtain the exact line positions in units of channel numbers which for calibration can be related to the hteramre values of the hyperfine splitting. As shown in Fig. 3.4, the velocity increment per channel, Ostep, is then obtained from the equation Ustep = D,(mm s )/D,(channel numbers). Different... 

Nuclear hyperfine coupling results in a multi-line ESR spectrum, analogous to the spin-spin coupling multiplets of NMR spectra. ESR spectra are simpler to understand than NMR spectra in that second-order effects normally do not alter the intensities of components on the other hand, ESR multiplets can be much more complex when the electron interacts with several high-spin nuclei, and, as we will see in Chapter 3, there can also be considerable variation in line width within a spectrum. 

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... 

Once a hyperfine pattern has been recognized, the line position information can be summarized by the spin Hamiltonian parameters, g and at. These parameters can be extracted from spectra by a linear least-squares fit of experimental line positions to eqn (2.3). However, for high-spin nuclei and/or large couplings, one soon finds that the lines are not evenly spaced as predicted by eqn (2.3) and second-order corrections must be made. Solving the spin Hamiltonian, eqn (2.1), to second order in perturbation theory, eqn (2.3) becomes 4... 

Second-order effects on hyperfine structure in organometallic compounds are discussed in Chapter 3. 

Figure 3.3 Stick spectrum showing hyperfine pattern for coupling to three equivalent 59Co nuclei (1=1/2) computed to (a) first-order and (b) second-order in perturbation theory. (Adapted from ref. 7.) (c) Isotropic ESR spectrum of [PhCCo3(CO)9r in THF solution at 40°C.

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 ... 

The formal treatment is quite similar to the derivation of the principal g values as developed in Eqs. (7C) through (18C). The second-order energy term is set equal to the hyperfine term from the spin Hamiltonian, and for the z direction... 

FIGURE 5.1 Isotropic hyperfine pattern for 51VIV in S-band. The spectrum is from V0S04 in aqueous solution. Use of the low frequency enhances the second-order effect of unequal splitting between the eight hyperfine lines. 

Some transition ions have central hyperfine splittings somewhat greater than this value, for example, for copper one typically finds Az values in the range 30-200 gauss, and so in these systems the perturbation is not so small, and one has to develop so-called second-order corrections to the analytical expression in Equation 5.12 or 5.13 that is valid only for very small perturbations. The second-order perturbation result (Hagen 1982a) for central hyperfine splitting is ... 

Note that, just like for the first-order expression in Equation 5.12 also the second-order expression in Equation 5.18 applies to field-swept spectra, and a different expression found in EPR textbooks (Pake and Estle 1973) applies to frequency-swept spectra. The effect of including a second-order contribution to the central hyperfine splitting is illustrated in Figure 5.7 on the spectrum of a not uncommon contaminant of metalloprotein preparations Cu(II) ion coordinated by nitrogens of tris-hydroxy-ethyl aminomethane or Tris buffer. 

FIGURE 5.7 Second-order hyperfine shift in the X-band EPR of the Cu(II)-Tris complex. The thin solid line is the experimental spectrum of 1.5 mM CuS04 in 200 mM Tris-HCl buffer, pH 8.0 taken at v = 9420 MHz and T = 61 K. Tris is tris-(hydroxymethyl)aminomethane or 2-amino-2-hydroxymethyl-l,3-propanediol. The broken lines are simulations using the parameters g = 2.047, gN = 2.228, Atl = 185 gauss. In the lower trace the second-order correction has been omitted. 

The superhyperfine splittings are sufficiently small to ignore second-order effects at X-band, and for adducts of the nitrone compounds splitting from the nitrone-N and the beta-H are the only resolved hyperfine interactions, thus affording the extremely simple resonance condition (cf. Equation 5.10)... 

Relative signs of the hyperfine splittings of two nuclei. In single crystal ENDOR studies the relative signs between hfs constants of different nuclei may sometimes be determined from the cross-term (3.18). If this second order term between nuclei I and K is... 

In this paper, the main features of the two-step method are presented and PNC calculations are discussed, both those without accounting for correlation effects (PbF and HgF) and those in which electron correlations are taken into account by a combined method of the second-order perturbation theory (PT2) and configuration interaction (Cl), or PT2/CI [100] (for BaF and YbF), by the relativistic coupled cluster (RCC) method [101, 102] (for TIF, PbO, and HI+), and by the spin-orbit direct-CI method [103, 104, 105] (for PbO). In the ab initio calculations discussed here, the best accuracy of any current method has been attained for the hyperfine constants and P,T-odd parameters regarding the molecules containing heavy atoms. 

In addition to g tensor anisotropy, EPR spectra are often strongly affected by hyperfine interactions between the nuclear spin I and the electron spin S. These interactions take the form T A S, where A is the hyperfine coupling tensor. Like the g tensor, the A tensor is a second-order third-rank tensor that expresses orientation dependence, in this case, of the hyperfine coupling. The A and g tensors need not be colinear in other words, A is not necessarily diagonal in the coordinate systems which diagonalize g. 

Spectrum simulation treated the hyperfine interactions by second order perturbation theory and there were distributions in D and E/D, because strain in these parameters dominated the spectra. Spectral features grow in up to 1 equivalent of added Mn(II) at geff = 15.4, 5.3, 3.0 and 2.0 (Bi 1B) and a broad signal with a... 

This term is important only in that it gives a second-order contribution to the hyperfine interaction by allowing the nuclear spin and electron spin to couple indirectly through the orbital momentum. 

To first order, the (27+ 1) lines are separated by K. The second-order terms lead to uneven spacings of the hyperfine lines and to displacement of the center of the pattern from gpeH. 

Frozen-solution ESR spectra of Tc2G in mixed aqueous hydrochloric acid and ethanol provided data consistent with equal coupling of the unpaired electron to both technetium nuclei (101). IsotopicaUy pure "Tc (/ = 9/2) in 99Tc2Cl leads to a large number of lines in the X-band spectrum owing to second-order effects, in addition to the hyperfine lines presence for this dimeric axially symmetric system. The Q-band spectrum obtained at 77°K with a microwave frequency of 35.56 GHz exhibited fewer lines, and computer-simulated spectra were generated to correspond to the experimental spectrum withgit = 1.912, gi = 2.096, An = 166 x 10 4 cm"1, IAL = 67.2 x 10 4 cm 1, and gav = 2.035. 

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