20 FREILING, CROCKER, AND ADAMS the identity becomes apparent, with the result that solutions for heat flow can be easily adapted to the analogous cases for diffusion. Carslaw and Jaeger’? treat the case of a “radiating” solid sphere of radius R with initial temperature distribution f(r). Their solution is readily adaptable to the case of n°’ atoms of a fission product evenly distributed on the surface of a spherical particle and diffusing inward. We first write Carslaw and Jaeger’s Eq. 9.4(9) in diffusion notation in the form 2X n= » exp (-D at) C,I, sin a,r n=l where a,R is the nth root of the transcendental equation tan @nR = a,R For the case of no radiation (no mass transfer), C, reduces to R2a4+ 1 Cn = Rad which, by the use of the transcendental restriction, can be written — 1 n™ (sin @,R)? aR a,R—sin a,R cos aR These various forms of C, have different limit properties, and for evaluation of C, by letting a, approach its value of zero, the last form must be used. Thus C, equals °,R’qj. The integrals I,, have the general form R I= f, r’ f(r’) sin a,r’ dr’ For the case where n° atoms are deposited in a surface layer of thickness g, n° I,= GRal [sin a,(R~g) — a,(R—g) cos a,(R-g)] The equilibrium concentration at t= will be n;= 3n°/4cR°. Writing I, in terms of n; and letting g approach zero,