Energy mismatch

E s are the unperturbed energies of the electronic and vibrational states, respectively, and Bm is a constant energy factor which depends on the M excited state. It appears from Eq. (6.5) that ungerade symmetry of inverse energy mismatch between the relevant levels. 

These, then, are the reasons why magnetic resonance methods, microwave or far-infrared laser, have had limited success with 2A diatomic radicals. Similar considerations apply to nonlinear polyatomic radicals in doublet states success in far-infrared laser magnetic resonance depends upon the magnitude of the spin-rotation coupling, and the size of the energy mismatch between the transition frequency and the laser frequency, since the mismatch has to be magnetically tuned. This becomes less of a limitation as more laser frequencies become available, except that one then needs to know in advance which laser frequency to choose. It becomes part of the search problem ... 

Molecules Temperature (K) Solvant Energy Mismatch (cm ) Nonlinearity 9 Reference... 

Fls will not combine with His because the energy of the fluorine orbital is so low. An energy mismatch also occurs between the F2S and His orbitals. The bond must form by combining the Hi6 and F2p orbitals. To produce... 

A(q) = e q) + q2/(2m ) — q.V being the energy mismatch between the the states in) and q). The second term in the square brackets in Eq. (7b) arises from the coupling-constant renormalization in Eq. (4) and compensates for the ultraviolet divergence of the first term. This compensation is completely analogous to that of the electron mass renormalization in calculations of the radiative shift of an atomic optical transition [Bethe 1947 Cohen-Tannoudji 1992],... 

Recall that the interaction form (13.52) was chosen to express the close encounter nature of a molecule-bath interaction needed to affect a transition in which the molecular energy change is much larger than Awd where cfD is the Debye cutoff frequency of the thermal environment. This energy mismatch implies that many bath phonons are generated in such transition, as will be indeed seen below. 

Fig. 2. Phonon-assisted energy transfer for ions with a transition energy mismatch of AE,...

Two different cases can be treated. The first is the case where the energy mismatch is small compared to the available phonon energies so the relevant phonon modes are those of small wave vector, k-Rga << 1. Thus, the phonon wavelength is large compared to the sensitizer-activator separation. A Debye distribution of phonon modes can be used to evaluate the sum in Eq. (30) and the coherence factor can be averaged over all angles. This leads to 2)... 

In the second case the energy mismatch is larger, so k R a > 1 and the wavelength of the phonons is shorter than the separation between the sensitizer and activator. Following the same procedure described above leads to )... 

The energy transfer rates given in Eqs. (32) and (34) predict the same temperature dependences which are contained in the phonon occupation number given in Eq, (33). The major difference in the two energy transfer rates is their dependence on energy mismatch. 

In certain cases involving very small energy mismatches two-phonon assisted transfer processes may be more important than the one-phonon processes. Numerous different types of two-phonon processes have been developed and found to predict a variety of different temperature dependences i2), The opposite limit of very large energy mismatch can be treated in a similar way to multiphonon relaxation processes 12,13). fhe matrix element in Eq. (2) is carried out to the Nth power where N is the number of phonons involved in the transfer process. The temperature dependence of the transfer rate for phonon emission processes then becomes... 

Assuming an average phonon energy to make up an energy mismatch hw = AE/N, the energy transfer rate is constant at low temperatures and rises steeply at high temperatures with the slope depending on the number of phonons as W [kg T/hcc]. It has been noted that each N-phonon rate differs from the previous (N—l)-phonon term by a characteristic constant factor e and thus is related to the 0-phonon transfer rate by... 

Ybp3 Tm " " has been investigated as well as the more dilute system Yy xYbxTmi yp3 with a wide range of concentrations ji e magnitudes of the transfer rates have been determined and the observed exponential dependence on energy mismatch indicates the occurrence of a multiphonon type of process (see Sect. 2.2). The mechanism for the energy transfer and the kinetics of the process have not been determined. 

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