Resonant term, elimination

This resonant term cannot be eliminated by the KAM transformation. Instead we can treat it with a different type of transformation. We define a unitary transformation... 

As before, if we want to eliminate the perturbation by a KAM iteration for values of >2oo, we first have to deal with the resonant term (119). This can be... 

We will apply specific rotating wave transformations R that will allow us to identify resonant terms and to eliminate the nonresonant ones. We obtain an effective one-mode Floquet Hamiltonian of the form... 

They are such that R and ai are constant (except for the short period terms eliminated by the averaging and for contributions of higher orders). Constant R s mean semi-major axes and eccentricities constant in these solutions o and 02 constant mean that Aw = ai — <72 is constant - that is, the periapses are moving with same velocities so that their mutual separation do not vary. This frozen relative state in resonant systems is known as Apsidal corotation. 

Here 2tt Aft = trm is imposed to eliminate the effect by the zero-order interaction of the RF field, otherwise a term containing Ix will remain and it will distort the longitudinal magnetization, transverse magnetization, as well as the BSPS of the 13C . For the on-resonance condition, 5 = 0, all the higher-order average Hamiltonians vanish since [7f(/ ), = 0 for arbitrary l and... 

One can therefore transform away a resonance condition by going over to a new set of n action variables, n— 1 of which are conserved. Having eliminated a primary resonance [i.e., one having a large coefficient Vm in Eq. (3.29)], one can eliminate the next one in turn, etc. Of course, with each additional term in Eq. (3.29) that is included in the Hamiltonian, a larger range of actions become accessible to the dynamics. The books cited in Note 2 will all provide more details on this point. 

Resonance-ionization mass-spectrometry is still in the development stage in terms of its application to cosmochemistry. The Charisma instrument, which is operated by Argonne National Laboratories, uses multiple lasers to resonantly ionize only the elements of interest, which are then analyzed with a time-of-flight mass spectrometer. The Charisma instrument has made isotopic measurements of molybdenum, zirconium, strontium, ruthenium, barium and other elements in presolar grains. These measurements are made possible by the high ionization efficiency of the RIMS technique and its ability to completely eliminate isobaric interferences. Work is now underway to build a RIMS instrument that can be operated by an individual investigator in a university laboratory. If this succeeds, RIMS will play an increasing role in analysis of extraterrestrial materials. 

More convenient approaches for the elimination of undesired coherences are possible in the case of frequency-selective irradiation schemes. If the spins that are involved in zero-quantum coherences resonate in well-separated spectral regions, the spins can be manipulated separately by selective (or semiselective) pulses (Vincent et al., 1992,1993). For example, a selective 90°(7) pulse transforms the antiphase combination (lyS — I Sy), which corresponds to zero-quantum coherence in the tilted frame, into (I S -t- lySy), whereas (—I S - lySy) is obtained if a 90° (7) pulse is used instead. Hence, a two-step phase cycle eliminates the antiphase terms... 

An interesting problem of interpretation is associated with the presence of 1,4-elimination in the spectral data generated from 5,10-dihydrometalloanthracene derivatives. In early work on 5,10-dihydro-phenazastannins and phenoxastannins, the driving force for loss of an exocyclic M—R group was interpreted in terms of formation of the resonance-stabilized cation (XXVI) 211, 212) ... 

M. Hohwy and N. C. Nielsen, Elimination of high order terms in multiple pulse nuclear magnetic resonance specti oscopy Application to homonuclear decoupling in solids. J. Chem. Phys., 1997, 106. 7571-7586. 

To explain their results, Hart-Davies and Graham proposed to depict manganese complexes 172 in terms of resonance hybrids between representations 178 and 179 (261) (Figure 18). However, the possibility that the close Si-H distance is due to steric constraints has been raised by Bennett and Simpson (264). Steric effects favor a cis orientation of silyl and hydrido ligands and thus facilitate reactions involving reductive elimination pathways. 

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