Diffusion processes in lead-uranium systems

Abstract

The ratio r_λ of radiogenic daughter to residual parent (with decay constant λ) was investigated for a continuous diffusion model for a diffusion coefficient that is an arbitrary function of time. A solution for simultaneous diffusion of both parent and daughter is given in closed form. A theory is also developed that describes the trajectories rλ, rλ′, for paired decay schemes (with decay constants λ and λ′) in the neighborhood of concordia. A wide class of systems is shown to be linear in the r_λ, r_(λ′), diagram when ∫_0^τ D(η) dη/a^2≪1. A radiation damage model is proposed foi the diffusion coefficient that relates D(τ) to the integrated irradiation damage and to the uranium (and thorium) contents. The functional relationship between r_λ and the uranium content is found for three important cases. The model D(τ) = D_(1τ) yields a r_λ, r_(λ′), curve that is very similar to that obtained by Tilton for D = D_0. The slope is somewhat greater and the linear region more extended. The theoretical curves are found to be in reasonable agreement with the existing experimental data, and they fit the existing data at least as well as the constant D model.