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.

Figure 1

Rhenium EDS spectrum for a 20 keV electron beam. Dots: experimental; red solid and green dashed lines: overall fits choosing different values for $E_{M_5}$ edge; gray lines: diagram transitions; red thin dotted line: background choosing $E_{M_5}=1883$ eV; green dotted line: background choosing $E_{M_5}=1907\mathrm{eV}$.

Figure 2

Rhenium WDS spectrum. Dots: experimental; pink lines: fit (solid) and background (dotted) choosing $E_{M_5}=1883$ eV; green lines: fit (dashed) and background (solid) choosing $E_{M_5}=1907$ eV; gray solid lines: diagram transitions; gray dashed lines: satellite transitions.

Figure 3

Theoretical $M_4$ (lines) and $M_5$ (symbols) ionization cross sections $\sigma$ based on the distorted wave Born approximation approach [27] for several elements, normalized to their corresponding maxima ${\sigma }^{\mathrm{m}}$, as a function of the overvoltage $U$.

Figure 4

Values obtained for the $Q$ ratios as a function of the atomic number, using the $E_{M_5}$ values from Refs. [7, 17] (spheres), and modified values with $E_{M_5}$ above the energy of the $M\beta$ line (open circles).

Figure 5

Rhenium photoionization cross section calculated with the autostructure code. The positions of the $M_5$ and $M_4$ edges obtained are also displayed.