Numerical evaluation of eigenvalues of the sheet diffusion problem in the surface/diffusion mixed regime

In relaxation experiments where both bulk diffusion and surface processes are rate controlling, the theoretical transient depends on the roots βn of the transcendental equation βn tan βn=L, in which L contains the surface rate constant, the diffusion constant and sample dimensions. A method is presented to numerically evaluate the roots βn to high precision, which is necessary in the evaluation of the surface and bulk transport parameters through curve fitting. The method is simple, fast and accurate. It is demonstrated that large errors in these parameters can occur if the roots are not evaluated to high precision.