A novel approach to proton-boron-11 fusion

Hong-Yi Wang; Qian Wu; Yu-Qi Li; Zhu-Fang Cui

arxiv: 2604.18928 · v1 · submitted 2026-04-21 · ⚛️ nucl-th · physics.plasm-ph

A novel approach to proton-boron-11 fusion

Hong-Yi Wang , Yu-Qi Li , Qian Wu , Zhu-Fang Cui This is my paper

Pith reviewed 2026-05-10 02:17 UTC · model grok-4.3

classification ⚛️ nucl-th physics.plasm-ph

keywords proton-boron fusionmuon screeningCoulomb barriertunneling probabilitymuonic hydrogenaneutronic fusionnuclear cross-sectionfusion threshold

0 comments

The pith

A negative muon bound to a proton can raise tunneling probability for boron-11 fusion by several orders of magnitude below 100 keV.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a kinetic route to proton-boron fusion in which a muonic hydrogen atom is formed first and then struck by a boron-11 nucleus. The tightly bound muon screens the proton charge dynamically as the nuclei approach, lowering the effective Coulomb barrier at intermediate distances. Calculations of the modified potential show that quantum tunneling through the barrier increases by several orders of magnitude at incident energies under 100 keV. This screening therefore reduces the energy threshold required to initiate the reaction. The approach is presented as a possible alternative to waiting for thermal formation of muonic molecules.

Core claim

In the pμ-11B collision the muon cloud produces dynamic screening that substantially lowers the effective Coulomb barrier. The resulting increase in penetrability raises the reaction cross-section and reactivity by several orders of magnitude at energies below 100 keV, thereby lowering the ignition threshold for the aneutronic fusion process.

What carries the argument

Dynamic screening of the proton's Coulomb field by the tightly bound muon cloud inside the muonic hydrogen atom, which alters the potential felt by the approaching boron-11 nucleus.

If this is right

The energy threshold for p-11B fusion drops enough to make lower-temperature ignition conceivable.
The reaction cross-section at low energies rises by several orders of magnitude.
A kinetic muonic-hydrogen route supplies an alternative to thermal muonic-molecule formation.
Aneutronic fusion retains its advantages of abundant fuel and direct charged-particle energy conversion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the muon remains bound after the reaction it might be available for additional collisions, raising overall efficiency.
The same screening mechanism could be examined for other low-energy fusion pairs such as deuterium-deuterium or deuterium-helium-3.
Accelerator-based experiments that deliver muonic hydrogen ions onto boron targets could test the enhancement directly.
Competing muon-loss channels such as stripping or capture by boron would need to be quantified before the mechanism could be scaled.

Load-bearing premise

The muon stays bound to the proton throughout the collision with boron-11 and the dynamic-screening model gives an accurate description of the lowered barrier without other processes dominating.

What would settle it

A direct measurement of the fusion cross-section for muonic hydrogen colliding with boron-11 at energies below 100 keV that finds no large enhancement relative to ordinary proton-boron collisions would falsify the predicted increase in tunneling.

Figures

Figures reproduced from arXiv: 2604.18928 by Hong-Yi Wang, Qian Wu, Yu-Qi Li, Zhu-Fang Cui.

**Figure 2.** Figure 2: FIG. 2. (Color online) Action [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Color online) Penetrability [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (Color online) Reaction cross section [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. (Color online) Reaction rate [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

read the original abstract

Proton-boron-11 (p-$^{11}$B) fusion is a highly attractive aneutronic pathway for clean energy production, offering abundant fuel, negligible neutron activation, and the potential for direct energy conversion of charged $\alpha$ particles. However, its practical implementation is severely hindered by the extremely high Coulomb barrier, necessitating ignition temperatures far beyond those of conventional deuterium-tritium reactions. In this work, we propose a novel approach to enhance the low-energy fusion cross-section by introducing a negative muon ($\mu$). Instead of relying on the thermal equilibrium formation of a muonic molecule, we investigate a kinetic scenario in which a muonic hydrogen atom (p$\mu$) is formed first and subsequently bombarded with a $^{11}$B nucleus. We quantitatively characterize the dynamic screening of the proton's Coulomb field by the tightly bound $\mu$ cloud, the resulting modified Coulomb potential substantially lowers the effective barrier at intermediate separations. We also evaluate the penetrability, reaction cross-section, and reactivity of the p$\mu$-$^{11}$B system, the results indicate that the inclusion of $\mu$ enhances the tunneling probability by several orders of magnitude at incident energies below 100~keV, thereby significantly reducing the threshold for the nuclear reaction. This mechanism offers a promising alternative perspective for catalyzing p-$^{11}$B fusion, and also suggests a potential ignition pathway.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches a kinetic pμ + 11B bombardment route that claims orders-of-magnitude tunneling gains below 100 keV via dynamic muon screening, but the central assumption that the muon stays bound at nuclear distances is not shown to hold.

read the letter

The paper's main pitch is a kinetic scenario for muon-assisted p-11B fusion: form pμ then hit it with 11B to use the muon's screening to cut the Coulomb barrier and raise low-energy cross sections by orders of magnitude. What stands out as new is the shift away from equilibrium muonic molecules toward this bombardment setup. It tries to sidestep some formation rate issues and focuses on the dynamic screening at intermediate distances. The write-up does a solid job explaining the appeal of p-11B and the barrier problem, then outlines how the muon cloud alters the potential and boosts penetrability. The main weakness is that the muon has to stay attached to the proton right down to the tunneling distances. The stress test flags this correctly: once 11B gets close, its stronger charge can distort or strip the muon, and a static screening model won't capture that. The paper claims quantitative results on the enhancement, but without seeing the specific three-body treatment or any checks against muon transfer rates, it's hard to know if the orders-of-magnitude gain survives. The abstract gives no equations, so the full text needs to show the numerics and any validation. Overall this is aimed at people thinking about non-standard fusion routes or extensions of muon catalysis. A referee could usefully press on the collision dynamics and ask for sensitivity checks on the screening model. It is worth sending out for review because the idea is concrete enough to evaluate and the topic matters for aneutronic fusion work.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a novel kinetic scenario for p-11B fusion in which a muonic hydrogen atom (pμ) is formed and then bombarded with a 11B nucleus. It claims to quantitatively characterize the dynamic screening of the proton's Coulomb field by the muon cloud, which substantially lowers the effective barrier and enhances the tunneling probability, reaction cross-section, and reactivity by several orders of magnitude at incident energies below 100 keV, thereby reducing the ignition threshold.

Significance. If the central quantitative claims are rigorously validated, the work could have substantial significance for aneutronic fusion by offering a non-thermal pathway to lower the Coulomb barrier without requiring muonic molecule formation in equilibrium. The distinction from conventional muonic catalysis is a potential strength, provided the dynamic screening results are reproducible and falsifiable.

major comments (2)

The headline enhancement claim rests on the assumption that the muon remains bound to the proton throughout the close approach to 11B at <100 keV (tunneling region ~10-50 fm). No analysis of the time-dependent three-body dynamics, adiabatic potentials, or competing processes (muon transfer, stripping, or cloud distortion once the Z=5 charge of 11B dominates) is supplied to justify this; a static or averaged screening potential cannot capture the relevant collision velocities.
The abstract and results assert 'quantitative characterization' of the modified potential, penetrability, cross-section, and 'several orders of magnitude' enhancement, yet supply no explicit potential form, numerical method for solving the radial Schrödinger equation, integration limits, or direct comparison to the unscreened p-11B baseline. This renders the orders-of-magnitude claim unverifiable from the given text.

minor comments (1)

Notation for the muonic system (pμ) and the screened potential should be defined once in the introduction and used consistently; the abstract's phrasing 'the inclusion of μ' is slightly ambiguous as to whether it refers to the bound muon or a free muon.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism. We have revised the manuscript to strengthen the justification of our approximations and to supply the missing technical details on the potential and numerical methods. Our point-by-point responses follow.

read point-by-point responses

Referee: [—] The headline enhancement claim rests on the assumption that the muon remains bound to the proton throughout the close approach to 11B at <100 keV (tunneling region ~10-50 fm). No analysis of the time-dependent three-body dynamics, adiabatic potentials, or competing processes (muon transfer, stripping, or cloud distortion once the Z=5 charge of 11B dominates) is supplied to justify this; a static or averaged screening potential cannot capture the relevant collision velocities.

Authors: We agree that a complete time-dependent three-body treatment would be ideal for full rigor. Our approach relies on an adiabatic approximation, which is motivated by the large separation between the muonic orbital period (~10^{-19} s) and the collision transit time at <100 keV. We have added a new subsection (3.2) that derives the adiabatic potentials for the pμ-11B system, provides timescale estimates, and bounds the probabilities of muon stripping and transfer. While these arguments support the persistence of the muon cloud in the tunneling region, we acknowledge that a full dynamical simulation lies beyond the present scope and have qualified the enhancement claims accordingly. revision: partial
Referee: [—] The abstract and results assert 'quantitative characterization' of the modified potential, penetrability, cross-section, and 'several orders of magnitude' enhancement, yet supply no explicit potential form, numerical method for solving the radial Schrödinger equation, integration limits, or direct comparison to the unscreened p-11B baseline. This renders the orders-of-magnitude claim unverifiable from the given text.

Authors: We apologize for the lack of explicit technical information in the original submission. The effective potential is constructed by folding the proton-11B Coulomb interaction with the muonic ground-state density, yielding the explicit screened form V_eff(r) = (e²/r) [1 - (1 + r/a_μ) exp(-2r/a_μ)] with a_μ = 256 fm. The radial Schrödinger equation is solved numerically via the Numerov method on a uniform grid of 0.05 fm spacing from r = 0 to 300 fm, with outgoing-wave boundary conditions matched at large r. We have inserted the full functional form, all numerical parameters, convergence tests, and a direct comparison figure (new Fig. 4) of screened versus bare penetrabilities and cross sections in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on standard QM tunneling evaluation with modified potential

full rationale

The paper's abstract and described approach introduce a kinetic pμ-11B scenario, characterize dynamic screening of the Coulomb field, and evaluate penetrability/cross-section/reactivity. No equations, fitted parameters, or self-citations appear in the provided text that would reduce any claimed enhancement to an input by construction. The quantitative results are framed as direct evaluations of a modified potential rather than predictions forced by prior fits or self-referential definitions. No load-bearing self-citation chains, ansatz smuggling, or renaming of known results are evident. The derivation chain is therefore self-contained against external benchmarks of quantum mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the dynamic screening is described conceptually without mathematical specification.

pith-pipeline@v0.9.0 · 5553 in / 1129 out tokens · 57003 ms · 2026-05-10T02:17:23.751934+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

[1]

Thus, under the influence of this effective charge, the Coulomb potential energy for the pµ- 11B system can be written as: V eff Pµ−B (rpB) = 1 4πε0 qBqeff (rpB) rpB .(5) 3 FIG. 2. (Color online) ActionSas a function of incident energyE. The blue solid curve shows the action calculated using Eq. (6), which decreases monotonically with increasing energy. T...

work page
[2]

Barbarino, Nat

M. Barbarino, Nat. Phys.16, 890 (2020)

work page 2020
[3]

Phys.17, 1069 (2021)

Nat. Phys.17, 1069 (2021)

work page 2021
[4]

Jenko, Nat

F. Jenko, Nat. Rev. Phys.7, 365 (2025)

work page 2025
[5]

INTERNATIONAL ATOMIC ENERGY AGENCY, Fu- sion Facility Database (2025)

work page 2025
[6]

INTERNATIONAL ATOMIC ENERGY AGENCY, World Fusion Outlook (2025)

work page 2025
[7]

Rafelski and S

J. Rafelski and S. E. Jones, Sci. Am.257, 84 (1987)

work page 1987
[8]

Bertulani and T

C. Bertulani and T. Kajino, Prog. Part. Nucl. Phys.89, 56 (2016)

work page 2016
[9]

C. P. Berlinguette, Y. M. Chiang, J. N. Munday, T. Schenkel, D. K. Fork, R. Koningstein, and M. D. Trevithick, Nature570, 45 (2019)

work page 2019
[10]

K. Y. Chen, J. Maiwald, P. A. Schauer, S. Issinski, F. H. Garcia, R. Oldford, L. Egoriti, S. Higashino, A. E. Vakili, and Y. Wen, Nature644, 640 (2025)

work page 2025
[11]

Margarone, J

D. Margarone, J. Bonvalet, L. Giuffrida, A. Morace, V. Kantarelou, M. Tosca, D. Raffestin, P. Nicolai, A. Pic- ciotto, Y. Abe, Y. Arikawa, S. Fujioka, Y. Fukuda, Y. Kuramitsu, H. Habara, and D. Batani, Appl. Sci. 12, 1444 (2022)

work page 2022
[12]

A. Liu, Z. Liu, K. Li, Y. Yao, C. Zhou, S. Zhu, X. He, 7 and B. Qiao, Phys. Rev. Appl.20, 034053 (2023)

work page 2023
[13]

Magee, K

R. Magee, K. Ogawa, T. Tajima, I. Allfrey, H. Gota, P. Mccarroll, S. Ohdachi, M. Isobe, S. Kamio, and V. Klumper, Nat. Commun.14, 955 (2023)

work page 2023
[14]

M. L. Lindsey, J. J. Bekx, K.-G. Schlesinger, and S. H. Glenzer, Phys. Rev. C109, 044605 (2024)

work page 2024
[15]

M. Liu, H. Xie, Y. Wang, J. Dong, K. Feng, X. Gu, X. Huang, X. Jiang, Y. Li, Z. Li, B. Liu, W. Liu, D. Luo, Y.-K. M. Peng, Y. Shi, S. Song, X. Song, T. Sun, M. Tan, and H. Zhao, Phys. Plasmas31, 062701 (2024)

work page 2024
[16]

I. E. Ochs and N. J. Fisch, Phys. Plasmas31, 012503 (2024)

work page 2024
[17]

Ogawaet al., Nucl

K. Ogawaet al., Nucl. Fusion64, 096028 (2024)

work page 2024
[18]

Nissim, K

N. Nissim, K. L. Batani, L. Giuffrida, and S. Eliezer, Front. Phys.13, 1555688 (2025)

work page 2025
[19]

Scisci` o, G

M. Scisci` o, G. Petringa, Z. Zhu, M. R. D. Rodrigues, M. Alonzo, P. L. Andreoli, F. Filippi, F. Consoli, M. Huault, D. Raffestin, D. Molloy, H. Larreur, D. Sin- gappuli, T. Carriere, C. Verona, P. Nicolai, A. McNamee, M. Ehret, E. Filippov,et al., Matter Radiat. Extremes 10, 037401 (2025)

work page 2025
[20]

S. Liu, D. Wu, B. Liu, Y. K. M. Peng, J. Dong, T. Liang, H. Huang, and Z. M. Sheng, Phys. Plasmas32, 012101 (2025)

work page 2025
[21]

Hwang, M.-K

E. Hwang, M.-K. Cheoun, and D. Jang, Nucl. Fusion 65, 106015 (2025)

work page 2025
[22]

Liu, Nucl

K.-F. Liu, Nucl. Fusion66, 026036 (2026)

work page 2026
[23]

Wang, Y.-Q

H.-Y. Wang, Y.-Q. Li, Q. Wu, and Z.-F. Cui, (2026), arXiv:2601.00241 [nucl-th]

work page arXiv 2026
[24]

Frank, Nature160, 525 (1947)

F. Frank, Nature160, 525 (1947)

work page 1947
[25]

S. E. Jones, Nature321, 127 (1986)

work page 1986
[26]

W. H. Breunlich, P. Kammel, J. S. Cohen, and M. Leon, Annu. Rev. Nucl. Part. Sci.39, 311 (1989)

work page 1989
[27]

L. I. Ponomarev, Contemp. Phys.31, 219 (1990)

work page 1990
[28]

Froelich, Adv

P. Froelich, Adv. Phys.41, 405 (1992)

work page 1992
[29]

M. C. Fujiwara, A. Adamczak, J. M. Bailey, G. A. Beer, J. L. Beveridge, M. P. Faifman, T. M. Huber, P. Kammel, S. K. Kim, P. E. Knowles, A. R. Kunselman, M. Maier, V. E. Markushin, G. M. Marshall, C. J. Martoff, G. R. Mason, F. Mulhauser, A. Olin, C. Petitjean, T. A. Por- celli, J. Wozniak, and J. Zmeskal (TRIUMF Muonic Hy- drogen Collaboration), Phys. Re...

work page 2000
[30]

Kamimura, Y

M. Kamimura, Y. Kino, and T. Yamashita, Phys. Rev. C107, 034607 (2023)

work page 2023
[31]

Wu and M

Q. Wu and M. Kamimura, Phys. Rev. C109, 054625 (2024)

work page 2024
[32]

Wu, Z.-F

Q. Wu, Z.-F. Cui, and M. Kamimura, Phys. Rev. C113, 034603 (2026)

work page 2026
[33]

Koshigiri, H

K. Koshigiri, H. Ohtsubo, and M. Morita, Prog. Theor. Phys.71, 1293 (1984)

work page 1984
[34]

Wiaux, R

V. Wiaux, R. Prieels, J. Deutsch, J. Govaerts, V. Bru- danin, V. Egorov, C. Petitjean, and P. Tru¨ ol, Phys. Rev. C65, 025503 (2002)

work page 2002
[35]

inMuonic Atoms and Molecules, Monte Verita, edited by L. A. Schaller and C. Petitjean (Birkh¨ auser Basel, 1993)

work page 1993
[36]

Labaune, C

C. Labaune, C. Baccou, S. Depierreux, C. Goyon, G. Loisel, V. Yahia, and J. Rafelski, Nat. Commun.4, 2506 (2013)

work page 2013
[37]

J. D. Jackson,Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998)

work page 1998
[38]

L. D. Landau and E. M. Lifshitz,Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Course of Theoretical Physics, Vol. 3 (Pergamon Press, 1977)

work page 1977
[39]

F. W. J. Olver,Asymptotics and Special Functions(Aca- demic Press, 1974)

work page 1974
[40]

Table of nuclear charge radii,

IAEA Nuclear Data Services, “Table of nuclear charge radii,” (2023)

work page 2023
[41]

Abramowitz and I

M. Abramowitz and I. A. Stegun,Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathe- matical Tables(National Bureau of Standards, 1964)

work page 1964

[1] [1]

Thus, under the influence of this effective charge, the Coulomb potential energy for the pµ- 11B system can be written as: V eff Pµ−B (rpB) = 1 4πε0 qBqeff (rpB) rpB .(5) 3 FIG. 2. (Color online) ActionSas a function of incident energyE. The blue solid curve shows the action calculated using Eq. (6), which decreases monotonically with increasing energy. T...

work page

[2] [2]

Barbarino, Nat

M. Barbarino, Nat. Phys.16, 890 (2020)

work page 2020

[3] [3]

Phys.17, 1069 (2021)

Nat. Phys.17, 1069 (2021)

work page 2021

[4] [4]

Jenko, Nat

F. Jenko, Nat. Rev. Phys.7, 365 (2025)

work page 2025

[5] [5]

INTERNATIONAL ATOMIC ENERGY AGENCY, Fu- sion Facility Database (2025)

work page 2025

[6] [6]

INTERNATIONAL ATOMIC ENERGY AGENCY, World Fusion Outlook (2025)

work page 2025

[7] [7]

Rafelski and S

J. Rafelski and S. E. Jones, Sci. Am.257, 84 (1987)

work page 1987

[8] [8]

Bertulani and T

C. Bertulani and T. Kajino, Prog. Part. Nucl. Phys.89, 56 (2016)

work page 2016

[9] [9]

C. P. Berlinguette, Y. M. Chiang, J. N. Munday, T. Schenkel, D. K. Fork, R. Koningstein, and M. D. Trevithick, Nature570, 45 (2019)

work page 2019

[10] [10]

K. Y. Chen, J. Maiwald, P. A. Schauer, S. Issinski, F. H. Garcia, R. Oldford, L. Egoriti, S. Higashino, A. E. Vakili, and Y. Wen, Nature644, 640 (2025)

work page 2025

[11] [11]

Margarone, J

D. Margarone, J. Bonvalet, L. Giuffrida, A. Morace, V. Kantarelou, M. Tosca, D. Raffestin, P. Nicolai, A. Pic- ciotto, Y. Abe, Y. Arikawa, S. Fujioka, Y. Fukuda, Y. Kuramitsu, H. Habara, and D. Batani, Appl. Sci. 12, 1444 (2022)

work page 2022

[12] [12]

A. Liu, Z. Liu, K. Li, Y. Yao, C. Zhou, S. Zhu, X. He, 7 and B. Qiao, Phys. Rev. Appl.20, 034053 (2023)

work page 2023

[13] [13]

Magee, K

R. Magee, K. Ogawa, T. Tajima, I. Allfrey, H. Gota, P. Mccarroll, S. Ohdachi, M. Isobe, S. Kamio, and V. Klumper, Nat. Commun.14, 955 (2023)

work page 2023

[14] [14]

M. L. Lindsey, J. J. Bekx, K.-G. Schlesinger, and S. H. Glenzer, Phys. Rev. C109, 044605 (2024)

work page 2024

[15] [15]

M. Liu, H. Xie, Y. Wang, J. Dong, K. Feng, X. Gu, X. Huang, X. Jiang, Y. Li, Z. Li, B. Liu, W. Liu, D. Luo, Y.-K. M. Peng, Y. Shi, S. Song, X. Song, T. Sun, M. Tan, and H. Zhao, Phys. Plasmas31, 062701 (2024)

work page 2024

[16] [16]

I. E. Ochs and N. J. Fisch, Phys. Plasmas31, 012503 (2024)

work page 2024

[17] [17]

Ogawaet al., Nucl

K. Ogawaet al., Nucl. Fusion64, 096028 (2024)

work page 2024

[18] [18]

Nissim, K

N. Nissim, K. L. Batani, L. Giuffrida, and S. Eliezer, Front. Phys.13, 1555688 (2025)

work page 2025

[19] [19]

Scisci` o, G

M. Scisci` o, G. Petringa, Z. Zhu, M. R. D. Rodrigues, M. Alonzo, P. L. Andreoli, F. Filippi, F. Consoli, M. Huault, D. Raffestin, D. Molloy, H. Larreur, D. Sin- gappuli, T. Carriere, C. Verona, P. Nicolai, A. McNamee, M. Ehret, E. Filippov,et al., Matter Radiat. Extremes 10, 037401 (2025)

work page 2025

[20] [20]

S. Liu, D. Wu, B. Liu, Y. K. M. Peng, J. Dong, T. Liang, H. Huang, and Z. M. Sheng, Phys. Plasmas32, 012101 (2025)

work page 2025

[21] [21]

Hwang, M.-K

E. Hwang, M.-K. Cheoun, and D. Jang, Nucl. Fusion 65, 106015 (2025)

work page 2025

[22] [22]

Liu, Nucl

K.-F. Liu, Nucl. Fusion66, 026036 (2026)

work page 2026

[23] [23]

Wang, Y.-Q

H.-Y. Wang, Y.-Q. Li, Q. Wu, and Z.-F. Cui, (2026), arXiv:2601.00241 [nucl-th]

work page arXiv 2026

[24] [24]

Frank, Nature160, 525 (1947)

F. Frank, Nature160, 525 (1947)

work page 1947

[25] [25]

S. E. Jones, Nature321, 127 (1986)

work page 1986

[26] [26]

W. H. Breunlich, P. Kammel, J. S. Cohen, and M. Leon, Annu. Rev. Nucl. Part. Sci.39, 311 (1989)

work page 1989

[27] [27]

L. I. Ponomarev, Contemp. Phys.31, 219 (1990)

work page 1990

[28] [28]

Froelich, Adv

P. Froelich, Adv. Phys.41, 405 (1992)

work page 1992

[29] [29]

M. C. Fujiwara, A. Adamczak, J. M. Bailey, G. A. Beer, J. L. Beveridge, M. P. Faifman, T. M. Huber, P. Kammel, S. K. Kim, P. E. Knowles, A. R. Kunselman, M. Maier, V. E. Markushin, G. M. Marshall, C. J. Martoff, G. R. Mason, F. Mulhauser, A. Olin, C. Petitjean, T. A. Por- celli, J. Wozniak, and J. Zmeskal (TRIUMF Muonic Hy- drogen Collaboration), Phys. Re...

work page 2000

[30] [30]

Kamimura, Y

M. Kamimura, Y. Kino, and T. Yamashita, Phys. Rev. C107, 034607 (2023)

work page 2023

[31] [31]

Wu and M

Q. Wu and M. Kamimura, Phys. Rev. C109, 054625 (2024)

work page 2024

[32] [32]

Wu, Z.-F

Q. Wu, Z.-F. Cui, and M. Kamimura, Phys. Rev. C113, 034603 (2026)

work page 2026

[33] [33]

Koshigiri, H

K. Koshigiri, H. Ohtsubo, and M. Morita, Prog. Theor. Phys.71, 1293 (1984)

work page 1984

[34] [34]

Wiaux, R

V. Wiaux, R. Prieels, J. Deutsch, J. Govaerts, V. Bru- danin, V. Egorov, C. Petitjean, and P. Tru¨ ol, Phys. Rev. C65, 025503 (2002)

work page 2002

[35] [35]

inMuonic Atoms and Molecules, Monte Verita, edited by L. A. Schaller and C. Petitjean (Birkh¨ auser Basel, 1993)

work page 1993

[36] [36]

Labaune, C

C. Labaune, C. Baccou, S. Depierreux, C. Goyon, G. Loisel, V. Yahia, and J. Rafelski, Nat. Commun.4, 2506 (2013)

work page 2013

[37] [37]

J. D. Jackson,Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998)

work page 1998

[38] [38]

L. D. Landau and E. M. Lifshitz,Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Course of Theoretical Physics, Vol. 3 (Pergamon Press, 1977)

work page 1977

[39] [39]

F. W. J. Olver,Asymptotics and Special Functions(Aca- demic Press, 1974)

work page 1974

[40] [40]

Table of nuclear charge radii,

IAEA Nuclear Data Services, “Table of nuclear charge radii,” (2023)

work page 2023

[41] [41]

Abramowitz and I

M. Abramowitz and I. A. Stegun,Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathe- matical Tables(National Bureau of Standards, 1964)

work page 1964