Resonant case

This constitutes the essential difference from the nonresonance case in which one solution p 0) goes into the other (p = 0) without any possible multiplicity of choices. Here, in the resonance case, in view of the multiplicity (family) of periodic solutions for p = 0, one has to narrow down this choice by the conditions stated in Eqs. (6-70). 

We may observe that the two sub-bands of the damped spectral density (110), as well as the two peaks involved by the undamped case, must be of the same intensity in the resonant case (A = 0), which may be verified by looking at the... 

Figure 10. Pure Fermi coupling within or beyond the exchange approximation. Left column spectra were obtained from expression (110) of the spectral density /sf (m, V. = 0) ex (a) Resonant case A — 0. A — 60 cm-1 (b) nonresonant case A — 120cm 1 with A — 60 cm-1 (dotted line),...

As the lattice interacts with light only through electrons, both DECP and ISRS should rely on the electron-phonon coupling in the material. Distinction between the two models lies solely in the nature of the electronic transition. In this context, Merlin and coworkers proposed DECP to be a resonant case of ISRS with the excited state having an infinitely long lifetime [26,28]. This original resonant ISRS model failed to explain different initial phases for different coherent phonon modes in the same crystal [21,25]. Recently, the model was modified to include finite electronic lifetime [29] to have more flexibility to reproduce the experimental observations. 

Being mainly concerned with the derivation and treatment of the dynamical and symmetry properties of the relativistic standard map, some papers (Nomura et.al., 1992) do not concern with the kinetical aspects of this map. However, the kinetical properties are interesting for particle transport and acceleration processes. Here we calculate the time-dependence of the energy for various values of (3 including the resonance case. 

In the resonance case the diffusive growth of the energy can be observed while it is highly suppressed for the value of (3 = 0.1 which is less than (3 = 1/2-7T. However, the diffusion is not unlimited even for the resonance case. 

Figure 1. The average energy for various parameter of 3 a) classical case 3 = 0 1 P = l/2 r (resonance case) b) quantum case / = 0.1, / = 0.5 at the fixed K = 5.

One thus identifies the adiabatic states of 9.33, with the 0 ) bound states of a Q space within the PT projection formalism given in Section 9.2 these state become resonances, since they are superpositions of the I q) and IA2) states, which interact with continuum states c,E 1) in the P space. Following Section 9.2, it follows that the E - QhQ matrix in such a case of two overlapping resonances case is given by... 

A simple case where the general a constants in Table 8.5 do not succeed in correlating acidity constants is when the acid or base function is in direct resonance with the substituent. This may occur in cases such as substituted phenols, anilines, and pyridines. For example, owing to resonance (see Fig. 8.4), a para nitro group decreases the pKa of phenol much more than would be predicted from the o para constant obtained from the dissociation of p-nitrobenzoic acid. In such resonance cases (another example would be the anilines), a special set of o values (denoted as oJpara) has been derived (Table 8.5) to try to account for both inductive and resonance... 

If a sample contains equivalent nuclei A (13C) subject to spin-spin coupling with nuclei X ( H), the transverse magnetization arises from two or more Larmor frequencies, depending on the multiplicity. The corresponding magnetization vectors periodically rephase and dephase with the field vector B, as in the off-resonance case with one Larmor frequency (Section 2.4.1). The FID signal is thus modulated by the frequency of the coupling constant JAX [7,13] as illustrated in Fig. 2.5 (a) for hexadeuteriodimethyl sulfoxide. 

Fig. 2.8. The components v(t) and (/) of the transverse magnetization in the rotating frame of reference for the off-resonance case...

When carrying out the integration in (4.48), let us keep in mind that in the resonance case a(w) is nonzero only in a very narrow region around the resonance frequency to0n, and so... 

There are two different temperature regimes of diffusive behavior they are analogous to those described by Holstein [1959] for polaron motion. At the lowest temperatures, coherent motion takes place in which the lattice oscillations are not excited transitions in which the phonon occupation numbers are not changed are dominant. The Frank-Condon factor is described by (2.51), and for the resonant case one has in the Debye model ... 

The sensitivity to the control parameters is evident by changing 3

near-resonance cases as the control variables a,- - are altered. Numerous examples are provided in Ref. [214],... 

Consider a resonance case > = >. For the global system, there are two states 0) nB = 1) and 11)1 = 0) with equal energy note the qualitative difference for 0) nB = 1) the energy is in the field while l) nB = 0), energy localizes at the material system the photon field has not available energy (it is "empty"). 

The second type of predissociation observed for diatomic molecules is known as electronic predissociation the principles are illustrated in figure 6.28. A vibrational level v of a bound state E lies below the dissociation asymptote of that state, but above the dissociation asymptote of a second state E2. This second state, E2, is a repulsive state which crosses the bound state E as shown. The two states are mixed, and the level v can predissociate via the unbound state. It is not, in fact, necessary for the potential curves of the two states to actually cross. It is, however, necessary that they be mixed and there are a number of different interaction terms which can be responsible for the mixing. We do not go into the details here because electronic predissociation, though an important phenomenon in electronic spectroscopy, seldom plays a role in rotational spectroscopy. Since it involves excited electronic states it could certainly be involved in some double resonance cases. 

In the following numerical experiment we show that it is possible to demonstrate the difference between locahzed and resonant rotor dynamics with Csl molecules. At time t = 0 the molecules are prepared in their rotational ground state ( 4 (t = 0)) = J = 0, M = 0)). For t > 0 they are exposed to a string of microwave pulses. The control parameters r and k determine the repetition frequency and the strength of the pulses. In order to be able to compare with the results for the planar kicked rotor discussed in Section 5.3, we choose k = 5 and r = 1 (for the nonresonant case) and r = 7t/3 (for the resonant case). This choice of control parameters translates into a driving frequency oi u = 1/T w 9 GHz and a field strength of q w IkV/cm. For the pulse shape we choose... 

E.E.Nikitin, Charge exchange in the accidental resonance case, Izvesiya AN SSSR, 27, 996 (1963)... 

The one-photon resonance case induces an equal sharing of the dynamics along the two eigenstate branches, which allows to recover the 7i-pulse formula... 

In order to simplify the treatment, we look at the resonant case Aw = 0. The equations are then given by... 

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