Integral encounter theory recombination

VI. Pair Recombination—A Stochastic Approach A. Generalized Encounter Theory An Inelastic Collision Integral Atomic and Molecular Recombination Dynamics... 

Now we see that m(r, t) is exactly the same distribution of ions that was studied in the previous section in the absence of the pumping, Eq. (3.287) reduces to Eq. (3.216). However, the presence of a pumping term makes an essential extension of the theory, which now allows us to account not only for the recombination of ions but also for their accumulation. Using the initial condition m(r, 0) = 0, implies that there were no ion pairs at the beginning. All of them appear as ionization products in the encounters of excited donors with electron acceptors. Their accumulation and geminate recombination are described by the integral term in Eq. (3.277b), where the solution of Eq. (3.287) should be used. 

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