Averaging of anisotropy

Much of the width arises from incomplete averaging of anisotropies in the g-and hyperfine matrices (Chapter 3). For radicals with axial symmetry the parameters of eqn (2.8) depend on Ag = - g , AA, = AiM - A and tr,... 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

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). 

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. 

If j Rf is exactly the magic angle and infinite spinning speed is assumed, the first-order anisotropic terms are zero for both single and DQ coherence (33). This does not hold true for finite spinning speed, but a complete averaging of the first-order effect occurs at the exact rotor cycles. Therefore, the x evolution time has to match exactly a multiple of the rotor period. The second-order anisotropy refocusing occurs for... 

To complete the calculation, we need to obtain averages of low-grade powers of s weighted by the noninteracting distribution (moments), which is the only place where one needs to specify the form of E. In the next section we will do that for systems with axially symmetric anisotropy. 

These expressions are formally identical to those for the average of a quantity involving the anisotropy axes n, when these are distributed at random vl3i/( i) =/ For instance, for arbitrary n-independent vectors... 

It is convenient to consider a model of an anisotropic recombination region the reflecting recombination sphere (white sphere) with black reaction spots on its surface [77, 78], The measure of the reaction anisotropy here is the geometrical steric factor Q which is a ratio of a black spot square to a total surface square. Such a model could be actual for reactions of complex biologically active molecules and tunnelling recombination when the donor electron has an asymmetric (e.g., p-like) wavefunction. Note the non-trivial result that at small Q, due to the partial averaging of the reaction anisotropy by rotational motion arising due to numerous repeated contacts of reactants before the reaction, the reaction rate is K() oc J 1/2 rather than the intuitive estimate Kq oc Q. 

---