Resonance condition rotation number

A microwave cavity placed between SI and S2 can induce spin-flip transitions (F,Mp) = (1,1) —i (1,-1) if tuned to zvhf(H). In order to produce a positive signal, i.e. an increase in counting rate after S2 under resonance condition, S2 will be rotated by 180 degrees with respect to SI. Therefore, the (1, —1) state where Mp = —1 is defined with respect to the magnetic field direction in SI will be a (1,1) state in S2, while the (1,1) state of SI without spin flip would correspond to a (1, —1) state in S2. As a result, if the microwave frequency is off resonance, no H atoms will reach behind S2, while on resonance an increase in the number of atoms should be detected after S2. 

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

With the representations of these resonance layers in mind, we clarify quantitative differences between isolated resonances and overlapped resonances, by examining residence time distributions p(f) at each resonance layer. Since there are fluctuations in local rotation numbers due to the finite time average, we set some threshold W for each resonance condition, so that we compute the... 

Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

In laser-induced gas-phase reactions the first step is absorption of a vibrational quantum which, at moderate pressures, would only heat the gas since V-T energy transfer occurs. If this process were dominant the laser would be no more than a fancy (and expensive) Bunsen burner. At low pressures conditions are more favorable for multiple-photon absorption. The photon flux is large enough so that more than one quantum can be absorbed before significant V-T transfer occurs. The only trick is to maintain resonance between the laser frequency and the various transitions to be excited. If only pure vibrational transitions were involved anharmonicity would ensure that a laser tuned to the fundamental frequency ( 0 1) would be nonresonant for other transitions. However, since there are also changes in rotational quantum number resonance may be reestablished. The resonance conditions are illustrated in Fig. 6.9. 

Regarding quantitation in the CP/MAS experiment, for peak areas to accurately represent the number of nuclei resonating, one of the conditions that must be met is that the time constant for cross polarization must be significantly less than the time constant for proton spin lattice relaxation in the rotating fi ame, Tch or Tnh TipH. Other factors affecting quantitation in CP/MAS have been discussed in several reviews (28-33). Since no analyses of the spin dynamics were performed in this study, the solid state spectra presented in this manuscript will be interpreted only semiquantitatively. 

The differences in local fluorine environment are reflected in the i i [20] and T2 values reported in Table 19.3. In theory, the trifluoromethyl groups found in dexfenflura-mine, fluoxetine, fluvoxamine, and niflumic acid undergo less restrictive internal rotation and should also be less sensitive to their local environment than monofluorinated aromatic rings such as in tecastemizole or 5-FU. However, these compounds all have quite short in vivo T2 values except for niflumic acid [12], Because these T2 values are significantly shorter than those found in solution, it is fair to anticipate that all the compounds, except potentially niflumic acid, interact with their environment. However, as demonstrated with fluoxetine and FBAL (Table 19.3), reported T, values may vary considerably. This illustrates the difficulty of making a robust in vivo T, measurement under conditions of low SNR and broad resonance lines [8, 11, 12, 20, 24, 28, 31, 32, 35, 41 43, 50, 56, 68], In addition, the variable bioavailability, due to the small number of patients generally evaluated, compounds the measurement difficulty. 

However, the effect of a small perturbation in action-action-angle type flows is quite different. The two-parameter family of invariant cycles coalesce into invariant tori that are connected by resonant sheets defined by the u(h,l2) = 0 condition. The consequence of this is that contrary to action-angle-angle flows in this case a trajectory can cover the whole phase space and no transport barriers exist. Thus, in this type of flows global uniform mixing can be achieved for arbitrarily small perturbations. This type of resonance induced dispersion has been demonstrated numerically in a low-Reynolds number Couette flow between two rotating spheres by Cartwright et al. 

Garroway et al.(l ) have recently pointed out that the relative intensities of resonance lines of CP/MAS C NMR spectra are proportional to the number of carbons, when the time constant T y for H-13c cross polarization is much shorter than the spin-lattice relaxation times and tJp in the rotation frames of 1h and 13c nuclei, respectively. Since this condition has been found to be fulfilled under our experimental condition(15), such a quantitative analysis as shown in the text can be reasonably carried out for cellulose samples. 

---