Pair survival probability

The probability density w(r,t) contains the complete information about the behavior of the geminate pair. However, it is difficult to solve Eq. (2) in most cases of interest. Moreover, the probability density w(r,t) itself is usually experimentally unobservable. The quantities of greater practical importance are the pair survival probability defined as... 

By introducing an equation adjoint to Eq. (2), one can derive the following equation for the pair survival probability [3]... 

In the preceding part of this section, we have concentrated on the electron escape probability, which is an important quantity in the geminate phase of recombination, and can be experimentally observed. However, modern experimental techniques also give us a possibility to observe the time-resolved kinetics of geminate recombination in some systems. Theoretically, the decay of the geminate ion pairs can be described by the pair survival probability, W t), defined by Eq. (4). One method of calculating W t) is to solve the Smoluchowski equation [Eq. (2)] for w r,t) and, then, to integrate the solution over the space variable. Another method [4] is to directly solve Eq. (7) under relevant conditions. 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

It can be shown that for the totally absorbing boundary condition, and for R r, the long-time behavior of the pair survival probability W(l) is described by... 

This asymptotic form of W(t) is also plotted in Fig. 2 (as dashed lines). However, we see that this approximation is valid only when the pair survival probability slightly exceeds the escape probability, so its practical importance is rather limited. 

Bulk recombination is a many-body problem. It is not so obvious whether the approach based on Eqs. (29-34) correctly describes the many-body character of bulk recombination. Tachiya [29] formulated the rate of bulk recombination in terms of the pair survival probability of geminate recombination and showed that the approach described above is exact only when the minority reactants are fixed and the majority reactants are... 

In the independent pair model, the average or expectation value of N is the survival probability of the radical pairs when each is distributed as a Gaussian. It can be calculated by convolution and will not be unity even at time t = 0 because some pairs are formed with r0 initially containing N0 reactants [and hence M0 = (N0/2) (N0 — 1) pairs] and IV at time t [and hence M — (AT/2) (IV —1) pairs] does not react further for a time r and this was shown to be... 

Diffusion models of geminate pair combination connect the time-dependent pair survival probability, P t), with the macroscopic properties of the host solvent. Radicals are treated as spherical particles immersed in a uniformly viscous medium. The pair is assumed to undergo random Brownian movements that ultimately lead to either recombination or escape. The expression of P i) depends on the degree of sophistication of the theory chosen for analyzing the process. In the simplest theory,... 

Finally, the ensemble-averaged quantity < 1F(Z. .. z r, /)>yy gives the probability of donors surviving to time t. Monte Carlo calculations have shown that in many cases the pair survival probability is approximated well by a single exponential except at very short times [79a, 81]. The rate constant /cd is related to the geometric and dif-fusional characteristics of the system by the equation [79a]... 

Df = D is the mutual diffusion coefficient, and Uf r) is the interaction potential assumed to be zero for the process (9.117). Equation (9.121) should be solved with the uniform initial condition, p r, 0) = 1, and the reflecting boundary condition at encounter. An equivalent formulation defines P(t) in terms of the pair survival probability [335, 340], a function of time and initial separation, which also satisfies the reaction-diffusion equation but with the adjoint operator (.Sf = in the absence of interaction). 

In this section, the solution to the backward diffusion equation for S2(r, t) using two different types of boundary conditions for both neutral and charged species is presented. These solutions will then be used in the next section to demonstrate the link between the bulk reaction rate and the pair survival probability. Before presenting... 

Bulk reaction is the reaction between two particles, say A and B which are uniformly distributed in the chemical system. In this situation it becomes necessary to define the concentrations ca and Cb of both species. In this section the bulk reaction rate is derived in terms of the pair survival probability and it is demonstrated how Smoluchowski s time dependent rate constant can be obtained by making use of the independent pairs approximation. 

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