Strong field approximation

Foyt et al. [137] interpreted the quadrupole-splitting parameters of low-spin ruthenium(II) complexes in terms of a crystal field model in the strong-field approximation with the configuration treated as an equivalent one-electron problem. They have shown that, starting from pure octahedral symmetry with zero quadrupole splitting, A q increases as the ratio of the axial distortion to the spin-orbit coupling increases. 

The strong-field approximation, on which the present formalism relies, is not gauge invariant see. e.g. [32]. This problem has been present not only for... 

In the ENDOR methods, the spin system is irradiated by a microwave field to partially saturate the EPR transition while simultaneously sweeping the sample with radiofrequency radiation through nuclear resonance transitions (Makinen, 1998 Makinen et al. 1998). ENDOR provides a means of precisely measuring the hyperfine interaction (Ahf) between electron and nuclear spins. Within the strong-field approximation, the observed A f is given by Eq. 1.16 ... 

When 0 = 0, the orbital is parallel to the nuclear magnetic moment which itself will be parallel to the external magnetic field if the latter field is much stronger than the field at the nucleus due to the electron (see Fig. 6). This is referred to as the strong-field approximation and it follows that for a p orbital, AanUo ( z ) = f-f -iid when 0 = 90°, Aaniso A x,m) = -jP- The tensor thus has the form - P,- P,- P, the principal value being positive. 

In the above description, it was assumed that the nucleus lines up with the applied field in all orientations of the orbital, i.e., the applied field Hq is much stronger than the field due to the electron at the nucleus. In practice, however, this strong field approximation is only a good one when (a) the applied field Po is large (e.g., at Q-band frequencies) (b) the anisotropic hyperfine coupling is small (and so the field at the nucleus is still relatively small) and (c) the isotropic hyperfine coupling is large (since the field due to the electron at the nucleus reinforces the applied field). Often when electrons are confined to p or d orbitals, the... 

Fig. 6. The strong field approximation. The resultant field lies close to the applied field.

The paramagnetic contributions to the chemical shielding of octahedral low spin d Co(III) complexes have commonly been interpreted through the strong-field approximation of the Ramsey shielding model ... 

For central ions with an octahedral ligand system the five d states are split into t2 and e states. According to crystal-field theory, the t2 state is stabilized by —4 Dq, whereas the e state is destabilized by +6 Dq. Therefore, in the strong-field approximation for an octahedral complex ion with electron system, the CFSE... 

Conditions (2) and (3) are equivalent to the Fraunhofer or far-field approximations in ordinary optics. The coherently illuminated region with usual laboratory X-ray sotuces is a few micrometres across. We therefore expect this theory to be useful in the cases of weak scattering but to be seriously awry for strong scattering. 

We have used here that X is either CO or a vacant site and (CO) + ( ) = . The main problem with this mean-field approximation is that at temperatures below the order-disorder transition there is a strong correlation between the occupations of sites. As will be shown in Section 4.3 (pco,co for neighboring sites is about 24 kJ/mol, which corresponds to a thermal energy at 2880 K. This means that neighboring sites will not be occupied simultaneously at any realistic temperatures, and a more sophisticated approach is needed that describes this well. For weak interactions eqn. (11) may be acceptable, but one should be aware that there is some correlation in the occupation of sites even if there is no long-range order. 

The Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

Eq. (15.29) is only valid in the strong field regime. It is clear that iiJm(KEmJco) is small, using Eq. (15.26) we compute a small value of bm(Emv/). When this value approaches the size of the atom, Eq. (15.29) is no longer valid, for the integral of Eq. (15.27) is dominated by contributions for b bm(Emvi), where neither the assumption of the energies being independent of r nor the dipole approximation... 

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