We present a mathematical formulation and numerical simulations that allow one to evaluate the convolution effect in the measurement of compositional or diffusion profiles due to spatial averaging in microprobe analysis. The analytical relation between the true and measured (convolved) compositions is reduced to a simple form that can be applied to calculate the convolution effect in microprobe step-scanning with the aid of an ordinary scientific calculator.
It is assumed that the excitation intensity of the sample volume has a Gaussian distribution with radial symmetry about the beam axis. The error standard deviation ∈ of the Gaussian, which is required to evaluate the convolution effect, can be determined by analyzing the smoothing of compositional discontinuity in a standard sample. The convolution effect on the measurement of a concentration profile vanishes when the latter has a constant slope and increases with the increase in the curvature of the profile. For a given value of ∈, the calculated convolution of a diffusion profile with constant diffusion coefficient almost exactly agrees with that predicted from probability theory.
The convolution effect on a diffusion coefficient retrieved directly from an experimentally measured diffusion profile decreases with the increasing length and decreasing value of ∈ and is insignificant for a profile about 15 μm long in a modem electron microprobe. Our formulation should also prove useful in retrieving true composition near a grain boundary from a microprobe spot analysis that suffers from spatial averaging of compositions on both sides of the boundary.