Empirical determination of transmission attenuation curves in mass-thickness contrast TEM imaging

The thickness dependence of electron transmission, observed as mass-thickness contrast in transmission electron microscopy (TEM) images, was precisely measured for polystyrene and amorphous carbon. In the early stages of transmission attenuation, a slight increase in the attenuation coefficient was observed, although a constant value is generally expected according to Beer's law. In contrast, as generally known as nonlinear behavior due to multiple scattering, the coefficient decreased during the intermediate stages. In the later stages, an asymptotic behavior in which the transmission approached a constant value was observed. Based on these results, a function containing three parameters was proposed to express the nonlinear transmission attenuation with increasing thickness. Results obtained using this new model and other previously proposed models were compared with experimental data measured over a wide range of conditions: acceleration voltage of 200-3000 kV, objective aperture radius of 0.85-30 nm^-1, and thickness of 0.25-10 μm. It was confirmed that our model can well reproduce all of the measurements with a high degree of precision, while the other models were valid only under limited imaging conditions and/or for limited thickness ranges. Thus, a quantitative description of transmission attenuation under normal TEM observation conditions, that is, over a thickness range without physical absorption and a scattering angular range of a few degrees, is finally obtained after more than a century since early studies on β-rays. Based on a simplified model of multiple scattering, the characteristic behavior of the attenuation curves is intuitively explained.