Practical formulas for inner-shell ionization cross sections by electron impact: Applications in quantitative Auger electron spectroscopy

We evaluated analytical formulas for the inner-shell ionization cross section from three sources in calculations of the backscattering factor (BF) for quantitative Auger electron spectroscopy: the Gryzinski expression from classical atomic physics, the Casnati [J. Phys. B 15, 155 (1982)] formula from fits to measured K-shell cross sections, and two expressions from Bote [At. Data Nucl. Data Tables (unpublished)] from fits to improved calculations of ionization cross sections by Bote and Salvat [Phys. Rev. A 77, 042701 (2008)]. BFs were calculated for five illustrative Auger transitions (Al KL23L23, Si KL23L23, Cu L3M45M45, Ag M4N45N45, and Au M5N67N67 in the respective elemental solids) for primary energies from the threshold for ionization to 30 keV. Differences between BFs calculated from the expressions of Bote and those from the expressions of Casnati were less than 2%, while the differences between BFs from the expressions of Bote and those from the Gryzinski expression were less than 5.4%. BFs from the expressions of Bote were also compared with BFs from a widely used formula of Shimizu with differences between 7% and 15% at 30 keV. The Shimizu [Jpn. J. Appl. Phys., Part 1 22, 1631 (1983)] formula was derived for primary energies from 3 to 10 keV but is widely employed in quantitative Auger applications for higher energies. The differences between BFs from the expressions of Bote and the Shimizu formula arise from the improved models and data now available.