Revisiting $M$-shell binding energies for elements with 72 $\le Z\le$ 83

Experimental $M_4$ and $M_5$ cross-section curves of some sixth-period elements were observed to exhibit an abnormal crossover above 20 keV, when the $M_5$ binding energy is taken from the most commonly used databases available in the literature for the assessment of absorption effects [Aguilar, Castellano, Segui, Trincavelli, and Carreras, J. Anal. At. Spectrom. 38, 751 (2023)]. This clearly suggests the need for a revision of the published binding-energy values [Bearden and Burr, Rev. Mod. Phys. 39, 125 (1967); Larkins, At. Data Nucl. Data Tables 20, 311 (1977)]. The self-absorption effects of the $M\beta$ line and of bremsstrahlung in an energy region close to the $M_5$ binding energy were analyzed by using energy and wavelength dispersive spectroscopy, respectively. An important inconsistency in the x-ray absorption was found when assessed accordingly with the literature, which led to shift the $M_5$-edge positions of Re, Os, Ir, and Pt. In the particular case of rhenium, theoretical calculations have been performed to assess $M$ binding energies. To this end, the many-electron Dirac equation was numerically solved to estimate the configuration energies associated with $M_5$ one-vacancy states. The binding energies obtained are consistent with the present experimental results, and with the most widely accepted characteristic-energy values.