Resonance condition overlap

Here Psa is the transfer probability. Esa represents the resonance condition (in practise the spectral overlap of the emission of S and the relevant absorption of A) and occurs in both formulas. The quantity gsA comprises the optical strengths of the relevant transitions and a distance-dependence of the type rsA n=6,8, etc.). The quantity /sa, however, is proportional to the wave function overlap of S and A and comprises an exponential distance-dependence. 

As shown in Fig. 6b, for a dump laser pulse overlapping with the pump laser pulse no net population transfer occurs. It is particularly interesting that the intermediate level 2 is not significantly populated at any time although level 3 is weakly populated during the interaction. This surprising population dynamics can be exploited to check whether the dump laser frequency is in resonance with the 2 - 3 transition and thus the double-resonance condition is fulfilled As in SEP experiments it is possible to monitor the population of level 2 either by fluorescence from level 2 or by ionization after absorption of an additional photon (see Fig. 4). In a simple model the ionization process from level 2 can be introduced by a time-dependent decay rate of level 2 [6, 60] that is proportional to the intensity of the laser pulses, whereas the fluorescence is only proportional to the population in level 2 after interaction with both laser pulses [54]. 

The integral is the spectral overlap integral of the donor emission f (v) with the acceptor absorption profile f (v) for resonance condition. F(R) summarizes the essential mechanisms, like the Dexter (1953) exchange or the Forster (1951) multipole mechanism with their specific R distance dependences. 

Here Me is the pure electronic transition moment for the resonant excited state e, of which v is a vibrational level of band width Tv, vvi is the transition frequency from the ground state vibrational level (0 to the excited vibrational level (v). The /1-term becomes larger as the denominator becomes smaller (resonance condition) and as Me becomes larger (stronger electronic absorption). The numerator contains the product of two overlap integrals of vibrational wavefunctions (Franck-Condon overlap) involving the /, j and v states. Because of the orthogonality of vibrational wavefunctions (Section 1.2),... 

There is another way to separate the overlapping signals, i.e., to work at higher magnetic fields. This follows from the resonance condition hu = gpeBo and implies, at the same time, the use of a higher microwave fre-... 

Dexter, following the classic work by Forster, considered energy transfer between a donor (or a sensitizer) S and an acceptor (or activator) A in a solid. This process occurs if the energy difference between the ground and excited states of S and A are equal (resonance condition) and if a suitable interaction between both systems exists. The interaction may be either an exchange interaction (if we have wave function overlap) or an electric or magnetic multipolar interaction. In practice the resonance condition can be tested by considering the spectral overlap of the S emission and the A absorption spectra. The Dexter result looks as follows ... 

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... 

The motion along the one-dimensional resonance line called Arnold diffusion is prominent at lower-order resonances when nonlinearity is weak. In fact, the motion with the residence time distribution of the power 3/2 is observed for low-order resonance with weak nonlinearity. On the other hand, overlapped resonances allow the motion across resonances which leads to Brownian motion at a two-dimensional region. Indeed, the distribution with the power 2 is observed at higher-order resonances, and it is more frequently observed with stronger nonlinearity. Hence, one can distinguish clearly the Arnold diffusion from the motion induced by resonance overlaps by the power of the residence time distribution at each resonance condition. 

Figure 2.12 Resonance condition for energy transfer from D to A. The thick horizontal bars in the left hand diagram represent electronic and the thin bars vibrational states. The dark area on the right represents the spectral overlap integral J...

In order to understand which is the principal parameter involved in the spatial overlap, let us consider the Nd " free-ion states (fig. 31). Cross-relaxation can take place only if the energy released by ion S(Nd ) is about equal to the energy which can be accepted by ion A(Nd), For two near but free ions the energy difference (e2 - ei) being negative, such transfer could not take place. For ions in a crystal, the Stark effect can provide the resonance condition . E2M - im = 0 which, for phonon-assisted transfers, can be extended to 0. 

It is possible to determine experimentally whether the levels involved are in resonance with one another by comparing the emission band of S with that of the relevant absorption band of A. The more these bands overlap, the better the resonance condition is fulfilled. 

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