Full-range solution for the measurement of thin-film surface densities with proton-excited x rays

A Fredholm-integral expression for characteristic x-ray production by proton bombardment is developed relating multicomponent elemental thin-film concentration profiles to proton-stopping power, x-ray-production cross section, and x-ray attenuation. The thickness of Al2O3 and Ta2O5 films on their respective metallic substrates were measured with 20- to 80-keV protons with an accuracy of 5% for thick films (1200 Å) and 25% for thinner films (30 Å). Nonlinear regression analysis of the data indicated that all parameters of interest, including the zero-voltage anodization potential, can be determined from the raw data. With exception to the oxygen K-shell x-ray attenuation coefficient, the parameters were determined with good to reasonable accuracy. A general concentration profile could not be determined by numerical methods as a result of insufficient variation in the kernel of the integral equation.