Solving the production–diffusion equation for finite diffusion domains of various shapes: Part II. Application to cases with α-ejection and nonhomogeneous distribution of the source

In this paper, numerical methods derived in Part I (Chem. Geol. (2002)) are expanded to accommodate the effects of α-emission and zonation of U and Th as relevant for (U–Th)/He thermochronology. Expressions for the required coefficients are derived for spheres, finite and infinite cylinders and rectangular blocks, for parent nuclide distributions, which are either uniform or have stepwise zoning. We show that the effects of α-emission and zonation on the He retention in a nonspherical crystal of modest aspect ratio can be approximated by a sphere of identical surface-to-volume ratio (S/V) if the zonation is transposed on to the sphere. rrently usual correction method is to perform first a diffusive calculation without considering α-emission and zonation and to multiply afterward with a correction factor that has been derived originally for nondiffusive calculations. Our results demonstrate that more accurate calculations yield different results and that the differences are often too large to be ignored in practice. Especially for thermal histories that include a protracted residence in the partial retention zone, the difference between the usual correction and our correction can be as high as ∼20% for homogeneous sources. For zoned crystals, the difference can be even larger.