Finite-Length Internal Spherical Diffusion

Let us consider now diffusion inside a sphere neglecting the diffusion gradient outside the sphere. Such a case might be observed for hydrogen absorption or Li intercalation into spherical particles. Diffusion inside the sphere can go only to the sphere center and is called finite-length internal spherical diffusion. In the steady state in which the impedance measurements are carried out, dc concentration inside the sphere is uniform, and no dc current is flowing. Ac perturbation causes oscillations of concentration at the sphere surface, which diffuse inside the sphere. In such a case, two boundary conditions in Eq. (4.94) are changed  

Using the standard substitution, Eq. (4.92), and using the condition at r = 0 one obtains 

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