Hyperfine first order

This expression for S, could include up to infinite order correction in by taking the limit a qo since is at least of first order in as well as via the iterative nature according of Eq. (3.7). However, will, in eral, give rise of very small effects, such as hyperfine sfditting, so it can be expected... 

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). 

Fig. 4.13 Combined magnetic hyperfine interaction for Fe with strong electric quadrupole interaction. Top left, electric quadrupole splitting of the ground (g) and excited state (e). Top right first-order perturbation by magnetic dipole interaction arising from a weak field along the main component > 0 of the EFG fq = 0). Bottom the resultant Mossbauer spectrum is shown for a single-crystal type measurement with B fixed perpendicular to the y-rays and B oriented along...

The eigenfunctions corresponding the E+ and E are mixtures of f —i) and -1 i). If the hyperfine coupling is sufficiently small, A < < gp B, the second term in brackets in eqns (1.8C) and (1.8D) are negligible, which corresponds to first-order in perturbation theory, and the energies become ... 

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.

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 ... 

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

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. 

An alternative model (Venable 1967) proposes that the main cause of inhomogeneous broadening is unresolved superhyperfine interactions and, therefore, that the linewidth expression should be equivalent to the Equation 5.12 for the angular dependence of first-order hyperfine splitting ... 

The ozonide ion has widely spaced energy levels, and to first order the g values are not influenced by the host lattice or the surface. Thus, the absolute values of the g values are useful in the identification of the ion. These g values, along with the hyperfine coupling constants, are given in Table I. The three sets of hyperfine constants indicate that the oxygen atoms are not equivalent, at least when the ozonide ion is formed according to reaction 2. The geometry of the ion on MgO is believed to be... 

Signs of the hyperfine principal values of a single nucleus. According to (3.3) and (3.4) the first order ENDOR frequencies of a single nucleus with spin I = 1/2 are given by... 

This shows that the FC operator arises as an artifact if one wants to describe the hyperfine interaction by first-order perturbation theory in terms of two-component spinors. In other words, when singular functions are involved, the boundary conditions cannot be ignored. These give rise to the FC operator. 

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. 

In the case of hyperfine interaction, the spectrum becomes more complex. From Eq. (96) we find that, to first order, the resonant magnetic field for one hyperfine line is given by... 

The similarity of the preceding first-order ESR treatment to the first-order NMR treatment of two coupled protons is evident. For an unpaired electron interacting with n equivalent nuclei of spin the hyperfine coupling term in the spin Hamiltonian is... 

Resonance gamma spectrometry or Mossbauer spectrometry can be used to study the hyperfine interactions between a nucleus and its chemical neighborhood [142], In order to examine these interactions with the help of a Mossbauer spectrometer, the first-order Doppler effect shift of the wave emitted by a moving source is applied. The arrangement used for a Mossbauer spectrometer consists of a radioactive source containing a Mossbauer isotope in an excited state (see Figure 4.54)... 

The only atomic wave-functions that do not have a node at the nucleus are s-functions. The isotropic coupling constant is thus a measure of the s-character of the wave-function of the unpaired electron at the nucleus in question. The coupling constant for an atomic s-electron can be either measured experimentally or calculated from Hartree-Fock atomic wave-functions so that, to a first approximation, the s-electron density may be calculated from the ratio of the experimental and atomic coupling constants. Should the first-order s-character of the wave-function of the unpaired electron be zero, as for example in the planar methyl radical, then a small isotropic coupling usually arises from second-order spin-polarization effects. The ESR spectra of solutions show only isotropic hyperfine coupling. 

So the first-order shift caused by the hyperfine operator Wp contributes to EM... 

The rate constant for the first-order term, a, is dependent on the radiation dose (therefore, on the radical concentraion), while the rate constant for the second-order term, m, is independent of radiation dose. The dose-independent term is attributed to the hyperfine interaction with the surrounding protons. 

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