The sphere with accumulation of a radiogenic isotope

This problem, in which a radiogenic element is allowed to leak out of its host mineral as it forms, has found important applications in geochronology, particularly for the K-Ar method (Wasserburg, 1954) and the U-Pb method (Tilton, 1960 Wasserburg, 1963) with the so-called continuous loss model. The equation for radial diffusion of a radiogenic element in a sphere with radius a and uniform parent isotope concentration P = P0 at t = 0 can be written 

The same derivation as that used for the accumulation of radiogenic isotope in a slab would lead to the solution but we will take advantage of this case to fully develop an application of Duhamel s principle (Appendix 8C). The assumption of zero initial and surface concentration of the radiogenic isotope is equivalent to 

Introducing the concentration deficit due to loss of radiogenic element times r as the new variable v(r, t), not to be confused with a velocity, we obtain 

Since the second derivative of v(r,t)+u(r,t) with respect to r is zero, we can rewrite the diffusion equation as 

In order to apply the Duhamel principle, we must retrieve the solution v(r, t) for constant v(a,t)=vs(t) = vs from equation (8.6.10). Upon multiplication by r/a, the solution reads 

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