Rydberg states of hydrogenlike ions in a braneworld model

F Dahia,A S Lemos,E Maciel

doi:10.1140/epjc/s10052-018-6007-6

F Dahia, A S Lemos + Show 1 more

Open Access

https://doi.org/10.1140/epjc/s10052-018-6007-6

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Abstract

Precise measurements of optical transition frequencies between Rydberg states of hydrogen-like ions could be used to obtain an improved value of the Rydberg constant, by avoiding the uncertainties about the proton radius. Motivated by this perspective, we investigate the influence of the gravitational interaction on the energy levels of Hydrogen-like ions in Rydberg states in a braneworld model. As it is known, in this scenario, the gravitational interaction is amplified in short distances. We show that, for Rydberg states, the main contribution for the gravitational potential energy does not come from the rest energy concentrated on the nucleus but from the energy of the electromagnetic field created by its electric charge. The reason is connected to the fact that, when the ion is in a Rydberg state with high angular momentum, the gravitational potential is not computable in zero-width brane approximation due to the gravitational influence of the electrovacuum in which the lepton is moving. Considering a thick brane scenario, we calculate the gravitational potential energy associated to the nucleus charge in terms of the confinement parameter of the electric field in the brane. We show that the gravitational effects on the energy levels of a Rydberg state can be amplified by hidden dimensions even when the compactification scale is shorter than the Bohr radius.

Highlights

Direct laboratory tests of the inverse square law impose that the compactification radius (R) should be less than 44 μm [5,6], for instance
An interesting point to mention here is that when the space has more than two extra dimensions, the mean gravitational potential energy of a hydrogen-like atom in a S-state, U S, diverges if the brane structure is not taken into account
Considering a Hydrogen-like ion in a Rydberg state, we show that the extra dimensions can amplify the gravitational potential energy of the ion even when the compactification scale R is smaller than the Bohr radius, due to the behavior of χs

Summary

Introduction

Direct laboratory tests of the inverse square law impose that the compactification radius (R) should be less than 44 μm [5,6], for instance. An interesting point to mention here is that when the space has more than two extra dimensions, the mean gravitational potential energy of a hydrogen-like atom in a S-state, U S, diverges if the brane structure is not taken into account This is connected to fact that the gravitational potential, φ, produced by the proton mass is not computable in the interior of the nucleus in the approximation of zero-width brane [9,22,31]. In order to compute χ , the distribution of the electric energy inside the brane should be considered Addressing this problem in the thick brane scenario, we consider the Green function for the gravitational potential associated to length scales smaller than R, where the effects of extra dimensions are stronger. We estimate the effects of higher-dimensional gravity on particular optical transitions of Hydrogen-like ions

The gravitational field produced by the nucleus

The electrostatic potential on the brane

Dirac equation on the brane

Results and discussion

Final remarks