Overlap integral Photon

FRET is a nonradiative process that is, the transfer takes place without the emission or absorption of a photon. And yet, the transition dipoles, which are central to the mechanism by which the ground and excited states are coupled, are conspicuously present in the expression for the rate of transfer. For instance, the fluorescence quantum yield and fluorescence spectrum of the donor and the absorption spectrum of the acceptor are part of the overlap integral in the Forster rate expression, Eq. (1.2). These spectroscopic transitions are usually associated with the emission and absorption of a photon. These dipole matrix elements in the quantum mechanical expression for the rate of FRET are the same matrix elements as found for the interaction of a propagating EM field with the chromophores. However, the origin of the EM perturbation driving the energy transfer and the spectroscopic transitions are quite different. The source of this interaction term... 

Most of the experiments to date have been carried out on doubly excited states converging to isolated low angular momentum states of the ion. In general the spectra are well characterized as two photon ICE spectra in which the observed spectrum is characterized by the product of an overlap integral and the spectral density of the doubly excited autoionizing states. As an example we show in Fig. 23.9 the Ba 6sl9d— 9dn d spectra observed by Camus et al.31 As shown in Fig. 

R — oo. As we will show below, the partial cross sections for absorbing the photon and producing the diatomic fragment in vibrational channel n are proportional to the square modulus of the overlap of these continuum wavefunctions with the nuclear wavefunction in the electronic ground state (indicated by the shaded areas). Since the bound wavefunction of the parent molecule is rather confined, only a very small portion of the continuum wavefunctions is sampled in the overlap integral. 

Now the overlap integral, Jl e dX, is a constant for a particular combination of photon source and absorbing substance, b is determined by the reaction vessel chosen, and cp is a characteristic of the reaction. By grouping the constant terms into an overall constant k, the expression is simplified to ... 

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. 

The excitation profile (EP) is obtained by measuring the Raman cross section for a vibrational mode as a function of the incident laser photon energy. By application of a theoretical model involving the Frank-Condon overlap integrals it is possible to obtain from the EP of a vibrational mode an estimate of the strength of the interaction with the electronic excited state, z ... 

Ta = life time of luminescent state of S q = quantum efficiency Zja = distance between centres S and A n = dielectric constant of the host lattice f,(E) = normalized emission band /a(E) = normalized absorption band Oa = integrated absorption of A E = photon energy Z = overlap integral r, = position coordinate of an electron a = lattice spacing ti, = average hopping time c.t. = charge transfer Ar = expansion of luminescent center... 

Spectral overlap In the context of radiative energy transfer, it is the integral, J = ff (a) 8a (o )da, which measures the overlap of the emission spectrum of the excited donor, D, and the absorption spectrum of the ground state acceptor, A. fr is the measured normalized emission of D, fo —fD decadic molar absorption coefticient of A at wavenumber a. 

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