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,