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Diffusion of muonic hydrogen in hydrogen gas and the measurement of the 1$s$ hyperfine splitting of muonic hydrogen

by J. Nuber, A. Adamczak, M. Abdou Ahmed, L. Affolter, F. D. Amaro, P. Amaro, P. Carvalho, Y. -H. Chang, T. -L. Chen, W. -L. Chen, L. M. P. Fernandes, M. Ferro, D. Goeldi, T. Graf, M. Guerra, T. W. Hänsch, C. A. O. Henriques, M. Hildebrandt, P. Indelicato, O. Kara, K. Kirch, A. Knecht, F. Kottmann, Y. -W. Liu, J. Machado, M. Marszalek, R. D. P. Mano, C. M. B. Monteiro, F. Nez, A. Ouf, N. Paul, R. Pohl, E. Rapisarda, J. M. F. dos Santos, J. P. Santos, P. A. O. C. Silva, L. Sinkunaite, J. -T. Shy, K. Schuhmann, S. Rajamohanan, A. Soter, L. Sustelo, D. Taqqu, L. -B. Wang, F. Wauters, P. Yzombard, M. Zeyen, J. Zhang, A. Antognini

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Pedro Amaro · Thomas Graf · Jonas Nuber
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-07-11
Date submitted: 2023-05-31 10:26
Submitted by: Nuber, Jonas
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Experiment
  • High-Energy Physics - Experiment
Approaches: Experimental, Computational


The CREMA collaboration is pursuing a measurement of the ground-state hyperfine splitting (HFS) in muonic hydrogen ($\mu$p) with 1 ppm accuracy by means of pulsed laser spectroscopy. In the proposed experiment, the $\mu$p atom is excited by a laser pulse from the singlet to the triplet hyperfine sub-levels, and is quenched back to the singlet state by an inelastic collision with a H$_2$ molecule. The resulting increase of kinetic energy after this cycle modifies the $\mu$p atom diffusion in the hydrogen gas and the arrival time of the $\mu$p atoms at the target walls. This laser-induced modification of the arrival times is used to expose the atomic transition. In this paper we present the simulation of the $\mu$p diffusion in the H$_2$ gas which is at the core of the experimental scheme. These simulations have been implemented with the Geant4 framework by introducing various low-energy processes including the motion of the H$_2$ molecules, i.e. the effects related with the hydrogen target temperature. The simulations have been used to optimize the hydrogen target parameters (pressure, temperatures and thickness) and to estimate signal and background rates. These rates allow to estimate the maximum time needed to find the resonance and the statistical accuracy of the spectroscopy experiment.

Author comments upon resubmission

Dear Editor,

thank you for your message. With this, we are resubmitting the article after careful revision following the suggestions of the reviewer. As we have already mentioned in our answer to the reviewer, we fully agree with all their suggestions. Hence, we have modified the manuscript accordingly.
Below you find a list of changes we made to the manuscript. We stick to the numbering scheme of the referee. Note, that the page numbering changed slightly in the new version so that the locations given by the referee are not fully consistent with the new version any more.

Best regards,
Jonas Nuber on behalf of the authors

List of changes

(1) We changed the wording from “Entrance counter” to “Entrance detector” in Figure 1.
(2) We changed “X-rays" to "x rays” in figure caption 1.
(3) A reference to Fig. 3 was added in the last sentence of section 6.2 to redirect the reader to the figure.
(4) In figure caption 5, “incoupled” was changed to “in-coupled”.
(5) The axis label of the graph in Fig. 5c was corrected to “x [mm]” since it shows the profile in x direction (as noticed by the referee).
(6) We have introduced a second axis on the right of Figure 6, labeled as “Probability distribution [arbitrary scale]”. The new label and second axis were given in gray to better differentiate them from the left axis that shows the mean-free path. Furthermore, we added small arrows to indicate which axis should be considered for which plot.
(7) As suggested by the reviewer, we have added a paragraph in the end of Section 6.3 which briefly describes the x-ray detection scheme and also introduces the efficiencies used later-on in Section 7. Additionally, we have improved Figure 1 to sketch the detection system more precisely. The caption of Figure 1 was adjusted accordingly.
(8) We changed from the notation “1/s” to the notation “s-1“ as suggested by the referee. In Table 1, the notation “[1/s]” was kept in agreement with the reviewer’s suggestion.
(9) The reference for Figure 10a on page 19 of the new version was changed to “Fig. 10a”.
(10) The reference for Figure 10b on page 20 of the new version was changed to “Fig. 10b”.
(11) References [15], [23] and [41] were updated. The missing doi was added to reference [27].

Published as SciPost Phys. Core 6, 057 (2023)

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