588 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



51 



The continuity equation, in spherical coordinates takes the form 



r^ dr dt 



Substitution of (10.16) into (10.22) gives, finally, 



L I h 1^ + 7^ A = 1 % (10.23) 



r^dr\ dr '^ j Do dt 



Equations (10.23), (10.21), (10.20) and (10.19) form a set defining a 

 boundary value problem, the solution of which is p(7\ t), from which, in 

 turn, J*(r, t) can be computed. It then remains to compute (dn/dt) in 

 (10.2) from J*. The former is not simply AttR^J* (as in (10.8)) because 

 now J* is not defined unambiguously, being a function of r. J*{R, t) 

 might be employed but then the method is no less arbitrary than the 

 simple one described above. 



Fortunately, nature eliminates the dilemma. It is a peculiarity of 

 spherical diffusion, when the sink radius is much smaller than the radius 

 of the diffusion field, that after a brief transient period, 47rr'J*(r), except 

 near the boundaries of the field, becomes practically independent of r, 

 and depends only on t. This feature is elaborated in Appendix C. Since 

 in our case the radius of the field is of order, L, and the effective radius of 

 the sink is of order, R, and L » R, it may be expected that this phe- 

 nomenon will be observed. In fact its existence has been assumed previ- 

 ously in the derivation of (10.4). 



Under such conditions it does not matter how the radius of the sink 

 is defined so long as 4:irR^ is multiplied by J*{R) and not the value of 

 J* at some other location. 



The boundary value problem, (10.23), (10.21), (10.20), (10.19) is 

 solved in Appendix C, and it is shown there that the value of 47rr"J*(?') 

 obtained after the transient has passed is closely approximated by 



4xrV*(r) = -^C^° e-"' (10.24) 



with 



where 



M 



^ .kTiN-M) ,^) 



47r5W2/)o 



= l/47r [ r exp [q/KkTr] dr (10.26) 



