Abstract
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.
Highlights
Among the different phenomena that paved the way to the emergence of the quantum world, the Compton effect certainly played a key role in the acceptance of the photon as the quantum counterpart of an electromagnetic wave (Stuewer, 1975)
In 1927, by the use of non-relativistic quantum mechanics, Wentzel (Wentzel, 1927a) (Wentzel, 1927b) succeeded in obtaining a generalization of the Kramers-Heisenberg dispersion formula to low-energy X-rays and bound electrons, showing that the modified line for bound electron scattering is a small continuous spectral distribution ascribed to scattering electrons whose initial state is a discrete level and whose final state is one of the continuum levels
At the emergence of the quantum description of Nature, quite a relevant role—though not unique—was played by the Compton process for the direct detection of photons, i.e. the quanta of the electromagnetic field, as well as for the dynamical description of the effect, which called for a suitable theoretical prediction for the scattering cross section
Summary
Among the different phenomena that paved the way to the emergence of the quantum world, the Compton effect certainly played a key role in the acceptance of the photon as the quantum counterpart of an electromagnetic wave (Stuewer, 1975). In terms of the differential cross section, i.e. the ratio of the number of scattered photons into the unit solid angle Ω over the number of incident photons, the result was the following: This was a remarkable formula, since it was immediately realized that it turned out to be in agreement with the experimental data about the absorption of X-rays by matter (Compton & Allison, 1935). The main point was that, contrary to Klein and Nishina, the Compton scattering revealed to be a second-order effect, where electron intermediate states are present to bridge from the photon absorption process to that of re-emission of another photon by the electron This result followed from the application of the time-dependent perturbation scheme of Dirac (Dirac, 1927b), the intermediate states being required by the interaction term linear in the electromagnetic field that prevents a direct transition from the initial to the final state.
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