Escape probabilities

Solving this diflfiision problem yields an analytical expression for the time-dependent escape probability q(t) ... 

Northrup S H and Hynes J T 1979 Short range caging effects for reactions in solution. II. Escape probability and time dependent reactivity J. Chem. Phys. 71 884... 

The small and weakly time-dependent CPG that persisLs at longer delays can be explained by the slower diffusion of excitons approaching the localization edge [15]. An alternative and intriguing explanation is, however, field-induced on-chain dissociation, a process that does not depend on the local environment but on the nature of the intrachain state. The one-dimensional Wannier exciton model describes the excited state [44]. Dissociation occurs because the electric field reduces the Coulomb barrier, thus enhancing the escape probability. This picture is interesting, but so far we do not have any clear proof of its validity. 

Williams (1964) derived the relation T = kBTrQV3De2, where T is the recombination time for a geminate e-ion pair at an initial separation of rg, is the dielectric constant of the medium, and the other symbols have their usual meanings. This r-cubed rule is based on the use of the Nernst-Einstein relation in a coulom-bic field with the assumption of instantaneous limiting velocity. Mozumder (1968) criticized the rule, as it connects initial distance and recombination time uniquely without allowance for diffusional broadening and without allowing for an escape probability. Nevertheless, the r-cubed rule was used extensively in earlier studies of geminate ion recombination kinetics. 

Pimblott (1993) has used MC and ME methods for the external field (E) dependence of the escape probability (Pesc) for multiple ion-pair spurs. At low fields, Pesc increases linearly with E with a slope-to-intercept ratio (S/I) very similar to the isolated ion-pair case as given by Onsager (1938). Therefore, from the agreement of the experimental S/I with the Onsager value, one cannot conclude that only isolated ion-pairs are involved. However, the near equality of S/I is contingent on small Pesc, which is not expected at high fields. 

With r = 28.45 nm, r = 3.0 nm, and r = 8.39 nm, Bartczak and Hummel (1986) compute the escape probability Pesc = 0.0336, 0.0261, and 0.0230 respectively for N = 1, 2, and 3. While the first is comparable to the Onsager value, the latter are new results. The kinetics of recombination for the isolated pair, found by Bartczak and Hummel (1987) using MC, is very similar to that obtained by Abell e al. (1972). For N > 1, these authors found the recombination kinetics to be faster than that for the isolated pair. For two pairs, the calculated escape probability increased with the external field, but not as strongly as for N = 1. 

In hydrocarbon liquids other than n-hexane, the procedure for obtaining the thermalization distance distribution could conceivably be the same. However, in practice, a detailed theoretical analysis is rarely done. Instead, the free-ion yield extrapolated to zero external field (see Chapter 9) is fitted to a one-parameter distribution function weighted with the Onsager escape probability, and the mean thermalization length (r ) is extracted therefrom (see Mozumder, 1974 ... 

Replacement of the upper limit of integration y by < causes little error since y is much greater than r. Similarly, replacing the lower limit r by zero causes correspondingly little error when used with the Onsager escape probability. The escape probability as a negative ion can now be expressed as... 

This expression can be generalized in the presence of an external field, and the ratio of the escape probability as a negative ion to that as an electron in the absence of a scavenger computed as a function of the external field. From such an analysis and taking L = 4 A, a typical intermolecular separation, Mozumder and Tachiya obtained electron attachment cross sections in NP as 4 x 10-16,5 X 10 17, and 1 X 1CF18 cm2, respectively, for SF6, CC14, and CS2 with -15% uncertainty. 

With this distribution, the escape probability is calculated using the Onsager (1938) formula (see Chapter 9),... 

Over the temperature interval 165 K to 300 K, the calculations of Silinsh and Jurgis (1985) indicate that the thermalization rate in pentacene decreases from 3 X 1012 to 0.8 X 1012 s 1. The trend is opposite to what would be expected in liquid hydrocarbons and may be attributed to the rapid increase of mcff with temperature. The calculated mean thermalization distance increases with incident photon energy fairly rapidly, from 3 nm at 2.3 eV to 10 nm at 2.9 eV, both at 204 K. With increasing temperature, (r) decreases somewhat. These thermalization distances have been found to be consistent with the experimental photogeneration quantum efficiency when Onsager s formula for the escape probability is used. 

Demanding continuity of n at r = rg, using the Kirchhoff relation I — l = 1, and identifying (for unit input current) as the escape probability 0, we get... 

Equation (9.2) shows the effect of the reaction radius rl on the escape probability, which, remarkably, is free of the diffusion coefficient. Normally rc r which reduces Eq. (9.2) to the celebrated Onsager formula 0 = exp(- r /rg) as given by Eq. (9.1). 

FIGURE 9.1 Simplified derivation of Onsager s escape probability formula. In the stationary state a unit electron current at ro partitions as I toward the reaction radius and as Is toward the sink at infinity the latter is the escape probability. Reproduced from Mozumder (1969a), with the permission of John Wiley Sons, Inc. ... 

For the few ion-pair spur, the escape probability is nearly the same if the inter-positive-ion separation remains -1 nm or less. 

FIGURE 9.2 Variation of the ion escape probability in a nonpolar liquid with incident electron energy according to the simulations of Bartczak and Hummel. The curve shown is for exponential intra-ionic separation with a b value of 5.12 nm. (See the original reference for other parametric values.) Agreement with various limited experiments in n-hexane is only approximate. Reproduced from Bartczak and Hummel (1997), with the permission of Am. Chem. Soc . 

Here, Vd = pE is the drift velocity. The recombination and escape probabilities are now given by PR = NR /n+° L0 and Pkc = 1 - Pr. Since Vd = i, but T /r1 these probabilities are independent of mobility. However, the initial separation r0 is expected to depend (increase) with electron mobility, thus making the escape probability indirectly dependent on the mobility. These effects are quite similar to those in the Onsager theory... 

Detailed comparison of calculated and experimental results for the variation of the escape probability with the external field in Lar, LKr, and LXe has been made by Mozumder (1995a, b, 1996) using the data on LET, W value, mobility, and so forth. Experiments are with MeV electrons or beta-emitters having minimum LET in these liquids. The external field generally does not have any preferred direction relative to the track axis. Mozumder (1995a) argues that in such... 

Values of rh in LAr, LKr, and LXe at their respective temperatures are 1568, 3600, and 4600 nm. Wide variation of thermalization length around the rms value is of course expected. However, the escape probability has been found to be insensitive to rg in these cases around their rms values. Therefore, averaging Pesc over the distribution of thermalization length has been deemed unnecessary when r() is taken equal to rth. 

FIGURE 9.4 Variation of the escape probability with the external field in LAr for a 1-MeV incident electron. Full curve, absolute calculation experimental points and calculated values normalized to 22 KV/cm are denoted by diamonds and circles, respectively. See text for explanation of parameter values used in the calculation. Reproduced from Mozumder, (1995a), with the permission of Elsevier . 

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