Resonant limit

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

The positive charge in 28 is stabilized by /j-cr-C-C-hyperconj ligation with the C-C-ring bonds of the two cyclopropyl moieties. In the parlance of VB theory this is described by resonance of 28 with non-bonding resonance limiting structures, the homoallenyl cation type structure 28a, the homopropargyl cation type structure 28b and the Dewar-type limiting resonance structure 28c. 

Consider now the resonance limit, when only a small number of eigenstates is involved. In this limit a true quantum beat spectrum is obtained. For simplicity of presentation we consider a three-level system in which the closely spaced coherently excited levels fa and fa decay into the ground state. The total number of photons counted is just... 

Physical property Resonance limit Intermediate case Statistical limit... 

When pv 1, corresponding to a case intermediate between the statistical limit and the resonance limit, application of the general theory is more complicated but still formally possible. 

The tensors which enter theoretical expressions are transition tensors 7, for a transition between an initial state i and a final state /. The Placzek polarizability theory for vibrational Raman scattering [56], which we use here, is valid in the far from resonance limit, i and / are then vibrational states. If we assume that they differ for normal mode p, then the transition tensors can be written as... 

The only nonclassical aspect of the far off-resonance limit result, Eq, (12 is the dependence of the dynamic polarizability [Eq. (12.56)] on the flu coordinates X. This dependence is in accord with qualitative classical exp since we expect different molecular configurations to be more easily polarizabji others. However, it is only quantum mechanics, via Eq. (12.56), that advises compute this shape-dependent polarizability. 

In the interaction process the a+ and er photons experience phase shifts but are not absorbed. In the off-resonant limit the flux of er+ and er photons are individually conserved leading to Eq. (6b). By conservation of angular momentum along the z-dircction this leads to the constancy of Jz expressed by Eq. (6d). 

Returning now to Eqs.(28), it can be seen that, because of the antisymmetry relations, the resonant and non-resonant parts of each tensor element tend to cancel in the off-resonance limit Eq Ej > El > hco. In addition, the contributions from le> and s> states to each term in Eqs.(28) are of comparable magnitude off resonance. These contributions tend to cancel because of the difference in sign of the HT perturbation energy denominators, viz. (Ee - E )" = - (Es - Ee) . The net result is the disappearance of antisymmetric scattering in the off-resonance region. 

The solution gi fp00 is plotted in Fig. 4-13 for several values of (co/coo) and several values of ReIt will be noted that (4-243) does not reduce to (4-233) in the limit as Rem 0. This is because the roots, V and s2 have negative real parts for all Re / 0, so that the homogeneous terms in (4-140) vanish for sufficiently large times but are purely imaginary for Reml = 0 (s = i) and thus do not vanish even as t oo. Indeed, the second of the two terms in (4-233) is a homogeneous solution, which has no counterpart in (4-243) for Re, 0. In the resonant limit co [Pg.263]

Usually, the initial conditions (before the on-resonant absorption) are obtained from the far off-resonance limit of the steady-state results in Equations 37, where (aw)2 and ... 

The isocyanate group reactivity is based on the polarisation of these groups represented by the following resonance limit structures [1] (equation 1.1) ... 

The existence of several resonant limiting forms describing the structure of the formed nitromercaptobenzoic acid explains the color.)... 

All formulae are for the ferfrom resonance limit only. Q circularly polarized. II, 1 linearly polarized parallel and perpendicular to the scattering plane respectively. Other symbols explained in the text. 

The two principal resonance limits of the theory of ROA are the far-from-resonance (FFR) limit, the original form of the theory of ROA, and the single-electronic-state (SES) limit, for the case of strong resonance between a single excited electronic state and the incident laser radiation. In the case of FFR ROA, ab initio calculations have been carried out for direct comparison with experiment. The SES theory is so simple that the complete SES-ROA spectrum can be predicted from the parent resonance Raman spectrum and the electronic circular dichroism spectrum of the resonant electronic state. 

Nuclear magnetic resonance Limited sensitivity, applicable to a wide range of analytes, costly, still under development. 

This optical-optical double resonance has been already discussed in Sect.8.9 as a method of labelling molecular levels and identifying molecular transitions in complex spectra. There we had not, however, considered the line profiles of the double resonance limits, which will be the subject of this section. 

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