Relevant contributions by Majorana regarding Compton scattering off free or bound electrons are
considered in detail, where a (full quantum) generalization of the Kramers-Heisenberg dispersion
formula is derived. The role of intermediate electronic states is appropriately pointed out in recovering
the standard Klein-Nishina formula (for free electron scattering) by making recourse to a
limpid physical scheme alternative to the (then unknown) Feynman diagram approach. For bound
electron scattering, a quantitative description of the broadening of the Compton line was obtained
for the first time by introducing a finite mean life for the excited state of the electron system. Finally,
a generalization aimed to describe Compton scattering assisted by a non-vanishing applied
magnetic field is as well considered, revealing its relevance for present day research.
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