Hopping length

All local concentrations C of particles entering the non-linear functions F in equation (2.1.40) are taken at the same space points, in other words, the chemical reaction is treated as a local one. Taking into account that for extended systems we shouldn t consider distances greater than the distinctive microscopic scale Ao, the choice of equation (2.1.40) means that inside infinitesimal volumes vo particles are well mixed and their reaction could be described by the phenomenological reaction rates earlier used for systems with complete reactant mixing. This means that Ao value must exceed such distinctive scales of the reaction as contact recombination radius, effective radius of a dynamical interaction and the particle hop length, which imposes quite natural limits on the choice of volumes v0 used for averaging. 

Here the diffusion operator DA is replaced by the operator L describing motion as stochastic hops in the continuous coordinate space. Let distribution function of hop lengths normalized as... 

The quasi-steady-state hopping recombination rate K(oo) = Kq is related to the coefficient i eff via equation (4.2.14) as in the diffusion-controlled case. As in equation (4.2.15), this Ifu is defined by the asymptotics of the solution, Y(r,oo) = y(r), as r —> oo. It is important, however, that R ff cannot generally be treated as the effective recombination radius. It holds provided that the hop length is much smaller than the distinctive scale ro of tunnelling recombination... 

If hop length is large enough, the predominant contribution into integral (4.3.5) comes from large distances r lying outside the effective recombination sphere, where Y(r, t) 1. Therefore we can estimate (4.3.5) as... 

To treat the problem of recombination kinetics for arbitrary hopping lengths 0 < A < oo, it is convenient to restrict ourselves to the exponential hopping length distribution known as the Torrey model [82]. In this particular case the integral kernel [Pg.211]

Fig. 4.10. The ratio of effective hopping reaction radius to the diffusion one vs. hop length [81]. 1 - exact calculation. Different approximations curve 2 - equation (4.3.26), 3 - (4.3.22), 4 - (4.3.23), 5 - (4.3.27). Diffusion mechanism holds between the axis x = 0 and the vertical line, hopping on the right hand side of it.

The analytical formalism just discussed has two shortcomings first, the usage of quite particular hop length distribution and, secondly, the restriction to the steady-state properties. The Torrey model becomes inadequate for point defects in crystals, where single hop lengths A between the nearest lattice sites takes place, p(r) = <5(r - A) in equation (4.3.4). This results in the... 

Qualitatively speaking, as the hop length tends to zero, the transient time can be estimated in the spirit of equation (4.2.21) as... 

The results obtained in these computer simulations of the hopping reactions were applied in [86] to the centre recombination in KC1-T10 stimulated by step-like temperature increase. As it is clearly seen from Fig. 4.15, taking directly into account finite hop lengths (A = ao/(2 /2) 2.2 A to be compared with ro < 1 A of the electron Tl° centre) permits us to obtain a much better agreement with the experimental data than the standard continuous diffusion approximation (curves 1 and 3, respectively). 

It is clear from eq. (15) that a modification of the density of phonon states in nanocrystals influences the efficiency of energy transfer. Because the energy transfer rate depends also on the distance between the donor and acceptor, the transfer in very small nanocrystals is restricted. This restriction may be understood based on the fact that the hopping length and the transfer probability are restricted for a donor to find a matching acceptor in the neighborhood of the nanoparticle. 

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