Radical pair survival probability

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 pair survival probability was called IIa(f). Assuming that the distances of separation of the reactant pairs is independent of all other distances (this is not strictly true) and that only one pair reacts at any one time, Clifford et al. developed the probability that a spur 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,... 

To connect the experimental results of Chuang et al. [266] and Khudyakov et al. [267] and others with the diffusion equation (122), it is necessary to define the probability that the radical pair is extant at a time t, given that it was formed at t0 with separation r0 i.e. the survival probability, p(t r0, f0) is given by... 

It has already been noted (Chap. 6, Sect. 2.2 and Sect. 2.1 of this chapter) that the survival probability of a pair of reactive molecules, radicals or ions is closely related to the density distribution of a vast excess of one of these reactants about the other (the homogeneous case which... 

With diffusive propagators, the survival probability of a pair of radicals was found to be... 

More rigorous treatments of the geminate combination also take into consideration the probability that the radicals of a pair escape from each other, reencounter in a later event, and finally recombine (Scheme 13.2). This model leads to time-dependent radical pair combination rates and, accordingly, they predict that P t) does not follow a simple exponential decay. For instance, even for the simple case of a contact-start recombination process (ro = o), the survival probabihty is a complex function as shown in Equation 13.2... 

The survival probability relates tbe concentration of tbe radical pairs, [RE], at a given elapsed time t, RP i i, to tbe initial concentration of geminate pairs, [RE] q according to... 

When the radicals encounter at r=d, the singlet pairs react with probability of A. The surviving radicals diffuse apart to r=ro and start a new reencounter cycle. By defining g as the average density matrix at the w-th reencounter and as the average density matrix of surviving radical pairs at the n-th reencounter, we obtain from Eq. (11-48)... 

A recent study of iodine atom recombination in solution by Luther et al. [294] used a dye laser (wavelength 590nm, pulse duration 1.5ps) to photodissociate iodine molecules in n-heptane, -octane or methyl cyclohexane at pressures from 0.1 to 300 MPa. Over this pressure range, the viscosity increases four-fold. The rate of free-radical recombination was monitored and the second-order rate coefficient was found to be linearly dependent on inverse viscosity. This provides good reason to believe that the recombination of free iodine atoms is diffusion-limited, especially as the rate coefficient is typically 10 °dm mol s . The recombination of primary and secondary pairs is too rapid to be monitored by such equipment as was used by Luther etal. [294] (see below). Instead, the depletion of molecular iodine absorption just after the laser pulse was used to estimate the yield of (free) photodissociated iodine atoms in solution. They found that the photodissociation quantum yields (survival probability) were about 2.3 times smaller than had been measured by Noyes and co-workers [291, 292] and also by Strong and Willard [295]. This observation raises doubts as to the accuracy of the iodine atom scavenging method used by Noyes et al. or perhaps points to the inherent difficulties of doing steady-state measurements. In addition, Luther et al. 

Northrup and Hynes [103] have remarked that the effects of the potential of mean force as well as hydrodynamic repulsion are very much more apparent in their effect on the survival (and escape) probability of a reactant pair of radicals than their effect on the rate coefficient. For instance, considering the escape probability of Fig. 20, suppose that an escape probability of 0.75 had been determined experimentally. Initial distances of separation Tq = 4i or 312 would have been deduced from the diffusion equation analysis alone or from the diffusion equation with the potential of mean force and hydrodynamic repulsion included. Again, the effect of a moderately slow rate of reaction of encounter pairs further reduces the recombination probability. Consequently, as the inherent uncertainty in the magnitudes of U r), D(r) and feact may be as much as a factor of 2, the estimation of an initial separation distance, Tq, of a radical pair from experimental measurements of escape probabilities may be in doubt by a factor of 30% or more. Careful and detailed analysis of the recombination of radical pairs has been made by Northrup and Hynes... 

Fig. 5.12 a Intensity of the observed polarisation at various initial hydroxyl radical separations ro. For all encounters involving the hydroxyl radical, the probability of reaction was obtained by projecting out the singlet component of the wavefimction. Polarisation on the surviving radical pairs b OH + OH c OH + R and d R + R using a spin relaxation time of 20 ps and an initial separation of 5 A. Standard errors on the polarisation are quoted to one standard deviation... 

Fig. 8.5 Singlet probability of the surviving radical pairs as a function of the number of pairs (e + Solv ) using an external field of 0.33 T. A mutual dififusion coefficient (D ) of 0.5 ps was used for S / c-RH+. Cations were arranged on aline 20 A apart, with anions Gaussian distributed from their respective cations with a standard deviation of 80 A. Hyperfine coupling constant was averaged using Eq. (8.23)...

In order to solve for the survival and recombination probabilities, p and q in eqn. (126), it is necessary to solve eqn. (122) for p(r, f]r0, f0) and use eqn. (123) to find p or eqn. (125) for q. Again, the boundary and initial conditions are required. Before the pair is formed (f < and ttf is slightly less than f0), the density p is zero, of necessity. The boundary conditions are closely related to the Smoluchowski conditions [eqns. (5), (22), (46) and (47)]. As the radicals approach each other they have a probability of reacting, which can be related to an effective second-order rate coefficient, fcact> f°r the activation-limited process of recombination by... 

Cr[1 - q] An A radical, having escaped from its geminate pair, may encounter several different B radicals and thus form several F-pairs before finally reacting. We have therefore to sum the F-polarization over all these pairs to find the total contribution from the F-pair process. The polarization in A arising from the (n l)the F-palr formed by an A radical is simply Up multiplied by the probability that A has survived the previous n pairs, i.e. [1 - 4 p]. The total polarization per mole of A is therefore given by the summation ... 

---