Diffusion-controlled reactions. Black sphere model

5 Diffusion-controlled reactions. Black sphere model 

The integro-differential equations (5.1.2) to (5.1.4) could be considerably simplified (which is of great importance for actual computer calculations) in the black sphere approximation (3.2.16). Let us consider the particular case of instant recombination cr(r) = ao6 ro r), cro 00 (9 x) is the Heaviside 

For great values of (Tq equation (5.1.23) is nothing but a differential equation with a small parameter multiplying the derivative, whose main term of the asymptotic expansion is the solution of the degenerate equation (ctq = 0) 

When deriving equation (5.1.24), we took into account that the functional (5.1.7) remains finite as ctq — 0. 

Due to the instant recombination all the dissimilar particles with relative distances r tq disappear, which results in the Smoluchowski boundary condition 

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