Solving the production–diffusion equation for finite diffusion domains of various shapes: Part I. Implications for low-temperature (U–Th)/He thermochronology

We propose an accurate, fast and easy-to-use method to derive numerical solutions for production–diffusion equations for finite diffusion domains of various shapes and arbitrary cooling histories. Previous studies provide solutions for spheres, infinite cylinders and infinite sheets. We extend this range and provide solutions for finite bodies, i.e. finite cylinders and rectangular blocks of any aspect ratio. This approach is important as recently, it has become clear that, for example, the physical grain is the diffusion domain for He diffusion in apatite and titanite [J. Geophys. Res. 105 (2000) 2903; Geochim. Cosmochim. Acta 63 (1999) 3845]. We discuss the use of the new approach for forward modelling (U–Th)/He production–diffusion in apatite. Taking results with finite cylinders as a good approximation for apatite crystals, it is found that approximating instead with spheres or infinite cylinders having the same radius yields differences in calculated ages that can easily be as large as 20–35%. The relative differences are most pronounced in thermal histories that spend significant time at or near the closure temperature. On the other hand, reasonable agreement is found with spheres having the same surface to volume ratio.