Diffusion of a radioactive component

The diffusion of a radioactive component is a relatively easy problem. It is discussed here to illustrate how coupled diffusion and homogeneous reaction can be treated, and to prepare for the more difficult problem of the diffusion of a radiogenic component. The diffusion of a radiogenic component, which is dealt with in Section 3.5.2, is an important geological problem because of its application in geochronology and thermochronology. 

A radioactive component is consumed by its decay (homogeneous reaction). For example, one-dimensional diffusion of in zircon along the crystallographic axis c or along any direction in the a-b plane can be described by 

Because the above equation is identical to the diffusion equation of a stable component, it can be solved the same way. After solving for w, then C can be found as we . For diffusion of two isotopes, one stable and one radioactive, because they have the same diffusivity, the concentration profile for the radioactive nuclide is simply the concentration profile of the stable isotope multiplied by either (i) Toe , where Fq is the initial isotopic ratio, or (ii) F, where F is the isotopic ratio at the time of measurement of the profiles. 

Consider the diffusion of Ar in hornblende. Hornblende is anisotropic but the anisotropy is ignored here. Along a principal axis x, the one-dimensional diffusion can be described by 

Substituting the solution for K into Equation 3-107a allows the Ar profile to be solved, most likely numerically. Assume uniform initial K concentration and ignore its diffusion. Then Xe( K) = Xe( Ko)e , and Equation 3-107a becomes 

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