Observation of relativistic corrections to Moseley's law at high atomic number

Abstract

Transitions between low-lying electron states in atoms of heavy elements lead to electromagnetic radiation with discrete energies between about 0.1~keV and 100~keV (x rays) that are characteristic of the element. Moseley's law --- an empirical relation first described by Moseley in 1913 which supported predictions of the then-new Bohr model of atomic energy levels while simultaneously identifying the integer atomic number $Z$ as the measure of nuclear charge --- predicts that the energy of these characteristic x rays scales as $Z^2$. The foundational nature of Moseley's experiment has led to the popularity of Moseley's law measurements in undergraduate advanced laboratory physics courses. We report here observations of deviations from Moseley's law in the characteristic $K_{\alpha}$ x-ray emission of 13 elements ranging from $Z=29$ to $Z=92$. %: Cu, Rb, Mo, Ag, In, Ba, Tb, Ta, W, Pb, Au, Pt, and U. While following the square-law predictions of the Bohr model fairly well at low $Z$, the deviations become larger with increasing $Z$ (negligible probability of the Bohr model fitting data by a $\chi^2$ test). We find that relativistic models of atomic structure are necessary to fit the full range of atomic numbers observed (probability value of $0.20$ for the relativistic Bohr-Sommerfeld model). As has been argued by previous authors, measurements of the relativistic deviations from Moseley's law are both pedagogically valuable at the advanced laboratory level and accessible with modern but modest apparatus. Here, we show that this pedagogical value can be be extended even further --- to higher $Z$ elements, where the effects are more dramatically observable --- using apparatus which is enhanced relative to more modest versions, but nevertheless still accessible for many teaching laboratories.