# Structure of the Balmer jump. The isolated hydrogen atom

## TL;DR

This paper analytically investigates the hydrogen atom's spectral behavior near the Balmer limit, revealing a smooth transition without a true discontinuity, and emphasizes the influence of broadening mechanisms on the Balmer jump.

## Contribution

It provides a semi-analytical model of the Balmer limit, demonstrating the continuity of the cross-section and defining the jump's location considering broadening effects.

## Key findings

- No discontinuity at the Balmer limit when considering all processes.
- The shape and position of the Balmer jump depend on broadening mechanisms.
- Analytical expression for the cross-section at the Balmer limit.

## Abstract

Context. The spectrum of the hydrogen atom was explained by Bohr more than one century ago. We revisit here some of the aspects of the underlying quantum structure, with a modern formalism, focusing on the limit of the Balmer series. Aims. We investigate the behaviour of the absorption coefficient of the isolated hydrogen atom in the neighbourhood of the Balmer limit. Methods. We analytically computed the total cross-section arising from bound-bound and bound-free transitions in the isolated hydrogen atom at the Balmer limit, and established a simplified semi-analytical model for the surroundings of that limit. We worked within the framework of the formalism of Landi Degl'Innocenti & Landolfi (2004, Astrophys. Space Sci. Lib., 307), which permits an almost straight-forward generalization of our results to other atoms and molecules, and which is perfectly suitable for including polarization phenomena in the problem. Results. We analytically show that there is no discontinuity at the Balmer limit, even though the concept of a "Balmer jump" is still meaningful. Furthermore, we give a possible definition of the location of the Balmer jump, and we check that this location is dependent on the broadening mechanisms. At the Balmer limit, we compute the cross-section in a fully analytical way. Conclusions. The Balmer jump is produced by a rapid drop of the total Balmer cross-section, yet this variation is smooth and continuous when both bound-bound and bound-free processes are taken into account, and its shape and location is dependent on the broadening mechanisms.