Boltzmann distribution energy transfer

If the conditions for Forster transfer are not applicable, then the theory must be extended. There is recently experimental evidence that coherent energy transfer participates in photosynthesis [74, 75], In this case, the participating molecules are very close together. The excited state of the donor does not completely relax to the Boltzmann distribution before the energy can be shared with the acceptor, and the transfer can no longer be described by a Forster mechanism. We will not discuss this case. There has been active discussion of coherent transfer and very strong interactions in the literature for a longer time [69], and references can be found in some more recent papers [70-72, 76, 77],... 

Just as above, we can derive expressions for any fluorescence lifetime for any number of pathways. In this chapter we limit our discussion to cases where the excited molecules have relaxed to their lowest excited-state vibrational level by internal conversion (ic) before pursuing any other de-excitation pathway (see the Perrin-Jablonski diagram in Fig. 1.4). This means we do not consider coherent effects whereby the molecule decays, or transfers energy, from a higher excited state, or from a non-Boltzmann distribution of vibrational levels, before coming to steady-state equilibrium in its ground electronic state (see Section 1.2.2). Internal conversion only takes a few picoseconds, or less [82-84, 106]. In the case of incoherent decay, the method of excitation does not play a role in the decay by any of the pathways from the excited state the excitation scheme is only peculiar to the method we choose to measure the fluorescence (Sections 1.7-1.11). 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

Chemical processes, in contrast, are processes that are not limited by rates of energy transfer. In thermal processes, chemical reactions occur under conditions in which the statistical distribution of molecular energies obey the Maxwell-Boltzmann form, i.e., the fraction of species that have an energy E or larger is proportional to e p(—E/RT). In other words, the rates of intermolecular collisions are rapid enough that all the species become thermalized with respect to the bulk gas mixture (Golden and Larson, 1984 Benson, 1976). 

Thus, at high I, the pair population is a considerably smaller fraction of the total OH population than the initial fraction given by a Boltzmann distribution at the flame temperature. For example, for the nominal values of 14 and 0.4 A for Oq and Oy, the infinite-intensity fraction is < 1% of the total while the zero-intensity value is 4%. This result is generally valid for the entire range of parameters inserted into the model, which represent physically realistic energy transfer rates. However, the precise numerical values depend sensitively on the actual parameters inserted. These facts form the central conclusions of this study (4). A steady state model with no dummy level and a different set of rate constants and level structure (5) shows some similar features. 

The rationale for INEPT can best be understood by looking first at a somewhat simpler continuous wave experiment for transferring polarization, selective population transfer (SPT). SPT can be understood simply in terms of the populations of energy levels, whereas INEPT requires consideration of coherent precessing magnetization. Figure 9.9 shows the energy levels and populations of an AX spin system, which we take to be H and 13C, respectively, in this example. At equilibrium the populations conform to a Boltzmann distribution. Because the H... 

Assuming that thermal equilibrium has been obtained, a simple two-level diagram enables us to calculate q(V). Let us assume that the undoped and doped states can be represented by two levels, with energy 0 and Eox, and populations n0 and nox, respectively. If the doped state has a charge of m electrons (m electrons per site are transferred for the doping process), when the potential V is applied, its level will be shifted by -meV, as represented in Fig. 4. The Boltzmann distribution for the populations can be written as... 

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