Abstract Analytical expressions for the differential cross-sections for K, L, M, N, O and P shell atomic ionization in a solid are derived using a single-electron transition model, and are presented for use in electron energy loss spectroscopy. The semi-empirical model treats the initial target state as a bound, hydrogen-like atomic orbital, with an effective nuclear charge. The final ionization state in the solid is taken as an electron gas, where the wavefunction is a plane wave, with an effective mass for the ejected electron of the target. An empirical phase shift is introduced into the plane wave, by replacing the effective electron mass in the exponential part of the plane wave with the electron mass. The calculated cross-sections compare favorably with those determined experimentally, for a wide variety of representative edges. The limitations of the model are discussed, and a comparison with the hydrogenic model is given. Finally, the implementation of the model for quantitative elemental analysis, using standards for calibration, is discussed. The analytical expressions are given in an appendix.