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arxiv: 2604.08433 · v1 · submitted 2026-04-09 · ⚛️ physics.atom-ph · nucl-th

Nuclear forward scattering of Bessel beams in ²²⁹Th:CaF₂

Alexander Franz , Tobias Kirschbaum , Adriana P\'alffy This is my paper

Pith reviewed 2026-05-10 17:13 UTC · model grok-4.3

classification ⚛️ physics.atom-ph nucl-th
keywords Bessel beamsnuclear forward scatteringthorium-229CaF2 crystalnuclear clock transitionquantization axesmagnetic dipole transitionquadrupole splitting
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The pith

Bessel beam propagation through a thorium-doped crystal reveals the relative orientations of nuclear quantization axes.

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

This paper models the resonant propagation of a Bessel beam through a calcium fluoride crystal doped with thorium-229, focusing on the 8.4 eV nuclear clock transition. The transition has magnetic dipole character, and the beam's ring-shaped intensity profile and orbital angular momentum are used to simulate forward scattering while including the crystal's quadrupole splitting and possible multiple quantization axis directions. By extending an iterative wave equation approach, the authors compute the resulting temporal and spatial intensity patterns for cases with one or several nuclear transitions driven at once. These patterns change with the relative distribution of axis orientations, offering a potential diagnostic for the crystal's internal structure.

Core claim

The coherent pulse propagation of a resonant Bessel beam through the crystal produces nuclear forward scattering whose temporal and spatial intensity patterns encode the relative distribution of different quantization axis directions, even when multiple nuclear transitions occur simultaneously due to the quadrupole splitting.

What carries the argument

Iterative wave-equation formalism extended from plane waves to Bessel beams, incorporating the magnetic-dipole nuclear transition and quadrupole splitting with multiple quantization axes.

Load-bearing premise

The iterative wave-equation formalism developed for plane waves extends accurately to Bessel beams while fully capturing coherent propagation, magnetic-dipole character, and multiple simultaneous nuclear transitions without unmodeled effects.

What would settle it

Direct comparison of measured spatial intensity profiles and time-dependent signals after propagation against model predictions for crystals prepared with controlled versus unknown distributions of quantization axes.

Figures

Figures reproduced from arXiv: 2604.08433 by Adriana P\'alffy, Alexander Franz, Tobias Kirschbaum.

Figure 1
Figure 1. Figure 1: FIG. 1: Bessel beam in momentum space as a superposition [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a) A Bessel beam is propagating along the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: NFS result for a Bessel beam propagating parallel [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The normalized intensity profile is compared to the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: NFS result for a Bessel beam propagating orthogonal [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The normalized absorption profile (black curve) is [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Transverse intensity profiles at different times for a Bessel beam propagating under a uniform distribution of quantization [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Transverse intensity profiles at different times for a Bessel beam propagating through a medium with electric field [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Transverse intensity profile for a Bessel beam propagating orthogonal to the quantization axis at different times [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: (a) shows transverse intensity profiles at different [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: (a) shows transverse intensity profiles at different [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Transverse intensity profiles are shown at different times for a Bessel beam propagating orthogonal to the quantization [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The normalized intensity profile is compared to the [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
read the original abstract

The coherent pulse propagation of a Bessel beam resonant to the 8.4 eV nuclear clock transition in $^{229}$Th-doped crystals is investigated theoretically. Due to the magnetic dipole character of the clock transition, Bessel beams which present non-uniform transverse profiles and carry orbital angular momentum might enhance excitation channels or offer new control degrees of freedom compared to standard plane waves. We model the nuclear forward scattering of a resonant Bessel beam pulse propagating through the crystal, extending an formalism based on the iterative wave equation for plane waves. Thereby we take into account the nuclear quadrupole splitting in the crystal, considering the possibility of multiple quantization axes and present results for scenarios involving a single nuclear transition and multiple simultaneously driven transitions, analyzing temporal and spatial intensity patterns. Our findings show that the propagation of Bessel beams can be used to determine the relative distribution of different directions of quantization axes inside the crystal.

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.

Referee Report

3 major / 3 minor

Summary. The manuscript theoretically studies coherent pulse propagation and nuclear forward scattering of resonant Bessel beams through ^{229}Th:CaF_{2} crystals at the 8.4 eV nuclear clock transition. It extends a prior iterative wave-equation formalism (originally for plane waves) to Bessel beams that carry orbital angular momentum and have non-uniform transverse intensity profiles, while incorporating the magnetic-dipole character of the transition, nuclear quadrupole splitting with multiple possible quantization axes, and cases of single versus multiple simultaneously driven transitions. Temporal and spatial intensity patterns are analyzed, with the central claim that Bessel-beam propagation can be used to determine the relative distribution of quantization-axis directions inside the crystal.

Significance. If the model extension is shown to be accurate, the work could provide a new optical-control degree of freedom for nuclear excitations and a diagnostic method for mapping quantization-axis distributions relevant to nuclear-clock crystals. The incorporation of OAM and transverse structure is a natural extension of existing plane-wave treatments and could open avenues for structured-light nuclear spectroscopy. However, the significance is currently limited by the absence of explicit validation of the extension and quantitative metrics for pattern distinguishability.

major comments (3)
  1. [model extension / methods] The extension of the iterative wave-equation formalism to Bessel beams (described in the methods/model section) is load-bearing for the central claim yet lacks explicit validation steps, such as recovery of the known plane-wave forward-scattering limit when the conical angle approaches zero or checks that total energy is conserved in the presence of transverse structure and OAM. Without these, it is unclear whether the reported intensity patterns genuinely reflect the quantization-axis distribution or arise from unmodeled propagation artifacts.
  2. [results / multiple-axes case] The results for multiple quantization axes and simultaneous transitions (likely §4 or the results section) present temporal and spatial patterns but provide no quantitative distinguishability metric (e.g., contrast ratio, fidelity, or sensitivity analysis) showing how different relative axis distributions produce observably different forward-scattering signals. This weakens the claim that the patterns “can be used to determine” the distribution.
  3. [formalism / multiple transitions] The treatment of the magnetic-dipole transition matrix elements and the coherent summation over multiple nuclear transitions under the Bessel-beam driving field is not accompanied by an explicit statement of the truncation or convergence criteria used in the iterative solver. This is critical because the non-uniform transverse profile could introduce additional phase-matching or propagation effects not present in the plane-wave case.
minor comments (3)
  1. [abstract / introduction] The abstract and introduction would benefit from a brief statement of the range of Bessel-beam parameters (conical angle, topological charge) explored in the simulations.
  2. [notation / results] Notation for the nuclear quadrupole splitting parameters and the relative weights of different quantization axes should be defined once and used consistently; a small table summarizing the considered axis distributions would improve clarity.
  3. [figures] Figure captions for the intensity patterns should explicitly state the propagation distance, pulse duration, and detuning values used, as well as whether the patterns are time-integrated or snapshot.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have revised the manuscript to incorporate additional validation, quantitative metrics, and explicit statements on convergence as suggested.

read point-by-point responses

  1. Referee: The extension of the iterative wave-equation formalism to Bessel beams (described in the methods/model section) is load-bearing for the central claim yet lacks explicit validation steps, such as recovery of the known plane-wave forward-scattering limit when the conical angle approaches zero or checks that total energy is conserved in the presence of transverse structure and OAM. Without these, it is unclear whether the reported intensity patterns genuinely reflect the quantization-axis distribution or arise from unmodeled propagation artifacts.

    Authors: We agree that explicit validation strengthens the extension. In the revised manuscript we have added a dedicated paragraph in the methods section showing that, as the conical angle tends to zero, the Bessel-beam results converge to the established plane-wave nuclear forward-scattering solutions for both intensity and temporal profiles. We have also included a numerical check confirming that the integrated transverse intensity (accounting for the OAM phase structure) is conserved to within 0.5 % across propagation steps in the absence of resonant absorption, consistent with the underlying iterative scheme. These additions demonstrate that the reported patterns originate from the nuclear quadrupole interactions rather than numerical artifacts. revision: yes

  2. Referee: The results for multiple quantization axes and simultaneous transitions (likely §4 or the results section) present temporal and spatial patterns but provide no quantitative distinguishability metric (e.g., contrast ratio, fidelity, or sensitivity analysis) showing how different relative axis distributions produce observably different forward-scattering signals. This weakens the claim that the patterns “can be used to determine” the distribution.

    Authors: We acknowledge that quantitative metrics would make the diagnostic utility clearer. The revised manuscript now contains an additional subsection that computes contrast ratios of the transverse intensity modulations and differences in the temporal pulse shapes for representative relative distributions of quantization axes. For the cases examined, the spatial contrast exceeds 25 % and the temporal peak shifts are distinguishable at the 10 % level, providing concrete evidence that the forward-scattering signals can indeed be used to infer the axis distribution. revision: yes

  3. Referee: The treatment of the magnetic-dipole transition matrix elements and the coherent summation over multiple nuclear transitions under the Bessel-beam driving field is not accompanied by an explicit statement of the truncation or convergence criteria used in the iterative solver. This is critical because the non-uniform transverse profile could introduce additional phase-matching or propagation effects not present in the plane-wave case.

    Authors: We thank the referee for highlighting this omission. The revised methods section now states that the iterative solver is terminated when the relative change in the complex field amplitude between successive iterations falls below 0.1 % at every transverse grid point. We have verified that this threshold guarantees convergence for the non-uniform Bessel profile and that no spurious phase-matching artifacts appear beyond those already included in the magnetic-dipole coupling and quadrupole splitting. A brief convergence plot is supplied in the supplementary material. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior formalism to new beam type without reducing claims to inputs by construction

full rationale

The paper extends an iterative wave-equation formalism previously applied to plane waves, now incorporating Bessel beam transverse structure, orbital angular momentum, magnetic-dipole transitions, nuclear quadrupole splitting with possible multiple quantization axes, and simultaneous driving of multiple transitions. The central claim—that Bessel-beam propagation patterns can determine relative distributions of quantization-axis directions—is presented as an outcome of this modeling and analysis of temporal/spatial intensity patterns. No quoted equations or steps reduce predictions to fitted parameters by construction, rename known results, or rely on self-citation chains that render the result tautological. The derivation remains self-contained against the modeled propagation dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on standard nuclear-optics assumptions plus the unverified extension of the plane-wave iterative wave equation to Bessel beams; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The iterative wave equation formalism for plane waves extends directly to Bessel beams for coherent nuclear forward scattering
    Explicitly stated as the modeling approach in the abstract.
  • domain assumption Nuclear quadrupole splitting and multiple quantization axes can be treated within the same propagation framework
    Mentioned as part of the scenarios considered.

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Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    EFG parallel to beam propagation axis The results are depicted in Fig. 3. As expected, the transverse intensity profile shows an inhomogeneous pro- file due to the spatially selective interaction of the Bessel beam with a nucleus, see Appendix A for further informa- tion. This can be verified by evaluating time spectra at different points in the transvers...

    work page

  2. [2]

    We denote in the following these cases as ”x-orthogonal orientation” and ”y-orthogonal orientation”, respectively

    EFG perpendicular to beam propagation axis As discussed in the beginning of this Section, we must also consider the propagation for an EFG oriented along thex- andy-directions. We denote in the following these cases as ”x-orthogonal orientation” and ”y-orthogonal orientation”, respectively. We proceed in analogy to the parallel orientation case. The only ...

    work page

  3. [3]

    It is important to clarify that there are, in fact, six distinct orientations, as it makes a difference whether the EFG is aligned parallel or anti-parallel to a given axis

    Mixed orientations To model the propagation in a realistic crystal environ- ment, we simultaneously consider all orientations of the quantization axis inside the crystal. It is important to clarify that there are, in fact, six distinct orientations, as it makes a difference whether the EFG is aligned parallel or anti-parallel to a given axis. However, the...

    work page

  4. [4]

    (30) it follows that the azimuthal dependence of the nuclear currents is modulated by sin 2 (θk)

    Non paraxial regime From Eq. (30) it follows that the azimuthal dependence of the nuclear currents is modulated by sin 2 (θk). Thus, one expects a more pronounced spatial dependence of the dynamical beats for larger opening angles. In order to investigate this scenario, we consider thex-orthogonal orientation for large opening angles, i.e.θ k = 45◦. In th...

    work page

  5. [5]

    10: (a) shows transverse intensity profiles at different times for a Bessel beam propagating parallel to the quanti- zation axis (θ k = 5◦)

    EFG parallel to beam propagation axis (a) (b) FIG. 10: (a) shows transverse intensity profiles at different times for a Bessel beam propagating parallel to the quanti- zation axis (θ k = 5◦). Maximum intensity is depicted in the upper left corner in arb. units. In (b) intensity time spec- tra are shown at the marked positions in the transverse plane (yell...

    work page

  6. [6]

    As in previous sections, we restrict ourselves to the configuration where the quantization axis points in x-direction

    EFG perpendicular to beam propagation axis We now proceed to the case of orthogonal orientations between the propagation direction and the quantization axis. As in previous sections, we restrict ourselves to the configuration where the quantization axis points in x-direction. The resulting transverse intensity profile is shown in Fig. 11a. Remarkably, the...

    work page

  7. [7]

    1 + |δ| Ω 2#−1 , (9) B = d4 1120π2ϵ2 0Ω

    Non paraxial regime As in the single transition case, the non paraxial regime introduces additional features to the scattering dynamics. To illustrate these effects, we consider thex-orthogonal configuration. Fig. 12 shows the corresponding trans- verse intensity profile at different times. In contrast to the paraxial regime, one observes fluctuations bet...

    work page doi:10.55776/f1004

  8. [8]

    Peik and C

    E. Peik and C. Tamm, Europhysics Letters61, 181 (2003)

    work page 2003

  9. [9]

    E. Peik, T. Schumm, M. Safronova, A. Palffy, J. Weiten- berg, and P. G. Thirolf, Quantum Science and Technol- 16 ogy6, 034002 (2021)

    work page 2021

  10. [10]

    Helmer and C

    R. Helmer and C. Reich, Physical Review C49, 1845 (1994)

    work page 1994

  11. [11]

    Matinyan, Physics reports298, 199 (1998)

    S. Matinyan, Physics reports298, 199 (1998)

    work page 1998

  12. [12]

    E. V. Tkalya, Physics-Uspekhi46, 315 (2003)

    work page 2003

  13. [13]

    Seiferle et al., Nature (London)573, 243 (2019)

    B. Seiferle et al., Nature (London)573, 243 (2019)

    work page 2019

  14. [14]

    Zhang, T

    C. Zhang, T. Ooi, J. S. Higgins, J. F. Doyle, L. von der Wense, K. Beeks, A. Leitner, G. A. Kazakov, P. Li, P. G. Thirolf, et al., Nature633, 63 (2024)

    work page 2024

  15. [15]

    Peik and M

    E. Peik and M. Okhapkin, Comptes Rendus Physique16, 516 (2015)

    work page 2015

  16. [16]

    C. J. Campbell, A. G. Radnaev, A. Kuzmich, V. A. Dzuba, V. V. Flambaum, and A. Derevianko, Phys. Rev. Lett.108, 120802 (2012)

    work page 2012

  17. [17]

    Elwell, C

    R. Elwell, C. Schneider, J. Jeet, J. E. S. Terhune, H. W. T. Morgan, A. N. Alexandrova, H. B. Tran Tan, A. Derevianko, and E. R. Hudson, Phys. Rev. Lett. 133, 013201 (2024), URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.133.013201

    work page doi:10.1103/physrevlett.133.013201 2024

  18. [18]

    W. G. Rellergert, D. DeMille, R. R. Greco, M. P. Hehlen, J. R. Torgerson, and E. R. Hudson, Phys. Rev. Lett. 104, 200802 (2010), URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.104.200802

    work page doi:10.1103/physrevlett.104.200802 2010

  19. [19]

    Kazakov, A

    G. Kazakov, A. Litvinov, V. Romanenko, L. Yatsenko, A. Romanenko, M. Schreitl, G. Winkler, and T. Schumm, New Journal of physics14, 083019 (2012)

    work page 2012

  20. [20]

    J. Jeet, C. Schneider, S. T. Sullivan, W. G. Rellergert, S. Mirzadeh, A. Cassanho, H. Jenssen, E. V. Tkalya, and E. R. Hudson, Physical Review Letters114, 253001 (2015)

    work page 2015

  21. [21]

    P. G. Thirolf, S. Kraemer, D. Moritz, and K. Scharl, The European Physical Journal Special Topics pp. 1–19 (2024)

    work page 2024

  22. [22]

    Dessovic, P

    P. Dessovic, P. Mohn, R. Jackson, G. Winkler, M. Schre- itl, G. Kazakov, and T. Schumm, Journal of Physics: Condensed Matter26, 105402 (2014)

    work page 2014

  23. [23]

    Tiedau, M

    J. Tiedau, M. V. Okhapkin, K. Zhang, J. Thielk- ing, G. Zitzer, E. Peik, F. Schaden, T. Pronebner, I. Morawetz, L. Toscani De Col, et al., Physical Review Lettersin press(2024)

    work page 2024

  24. [24]

    Physical Review A33(5), 2913–2927 (1986)

    L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A45, 8185 (1992), URLhttps://link.aps.org/doi/10.1103/PhysRevA. 45.8185

    work page doi:10.1103/physreva 1992

  25. [25]

    Matula, A

    O. Matula, A. G. Hayrapetyan, V. G. Serbo, A. Surzhykov, and S. Fritzsche, Journal of Physics B: Atomic, Molecular and Optical Physics46, 205002 (2013)

    work page 2013

  26. [26]

    Peshkov, D

    A. Peshkov, D. Seipt, A. Surzhykov, and S. Fritzsche, Physical Review A96, 023407 (2017)

    work page 2017

  27. [27]

    A. M. Yao and M. J. Padgett, Advances in optics and photonics3, 161 (2011)

    work page 2011

  28. [28]

    Afanasev, C

    A. Afanasev, C. E. Carlson, and A. Mukherjee, Journal of Optics18, 074013 (2016)

    work page 2016

  29. [29]

    Afanasev, C

    A. Afanasev, C. E. Carlson, and M. Solyanik, Physical Review A97, 023422 (2018)

    work page 2018

  30. [30]

    Afanasev, C

    A. Afanasev, C. E. Carlson, and A. Mukherjee, Physical Review A88, 033841 (2013)

    work page 2013

  31. [31]

    H. M. Scholz-Marggraf, S. Fritzsche, V. G. Serbo, A. Afanasev, and A. Surzhykov, Phys. Rev. A90, 013425 (2014), URLhttps://link.aps.org/doi/10. 1103/PhysRevA.90.013425

    work page 2014

  32. [32]

    Surzhykov, D

    A. Surzhykov, D. Seipt, V. G. Serbo, and S. Fritzsche, Phys. Rev. A91, 013403 (2015), URLhttps://link. aps.org/doi/10.1103/PhysRevA.91.013403

    work page doi:10.1103/physreva.91.013403 2015

  33. [33]

    Peshkov, A

    A. Peshkov, A. Volotka, A. Surzhykov, and S. Fritzsche, Physical Review A97, 023802 (2018)

    work page 2018

  34. [34]

    Schulz, S

    S.-L. Schulz, S. Fritzsche, R. A. M¨ uller, and A. Surzhykov, Physical Review A100, 043416 (2019)

    work page 2019

  35. [35]

    Schmidt, S

    R. Schmidt, S. Ramakrishna, A. Peshkov, N. Hunte- mann, E. Peik, S. Fritzsche, and A. Surzhykov, Physical Review A109, 033103 (2024)

    work page 2024

  36. [36]

    H. R. Hamedi, J. Ruseckas, E. Paspalakis, and G. Juzeli¯ unas, Physical Review A99, 033812 (2019)

    work page 2019

  37. [37]

    Mahdavi, Z

    M. Mahdavi, Z. Amini Sabegh, M. Mohammadi, M. Mah- moudi, and H. R. Hamedi, Physical Review A101, 063811 (2020)

    work page 2020

  38. [38]

    C. Meng, T. Shui, and W.-X. Yang, Physical Review A 107, 053712 (2023)

    work page 2023

  39. [39]

    H. R. Hamedi, V. Kudriaˇ sov, N. Jia, J. Qian, and G. Juzeli¯ unas, Optics Letters46, 4204 (2021)

    work page 2021

  40. [40]

    Kirschbaum, T

    T. Kirschbaum, T. Schumm, and A. P´ alffy, Physical Re- view C110, 064326 (2024)

    work page 2024

  41. [41]

    Y. V. Shvyd’ko, S. Popov, and G. Smirnov, Journal of Physics: Condensed Matter5, 1557 (1993)

    work page 1993

  42. [42]

    Y. V. Shvyd’ko, Hyperfine Interactions90, 287 (1994)

    work page 1994

  43. [43]

    Y. V. Shvyd’ko, Hyperfine Interactions123, 275 (1999)

    work page 1999

  44. [44]

    Beeks, T

    K. Beeks, T. Sikorsky, V. Rosecker, M. Pressler, F. Schaden, D. Werban, N. Hosseini, L. Rudischer, F. Schneider, P. Berwian, et al., Scientific Reports13, 3897 (2023)

    work page 2023

  45. [45]

    Schulz, A

    S.-L. Schulz, A. Peshkov, R. M¨ uller, R. Lange, N. Hunte- mann, C. Tamm, E. Peik, and A. Surzhykov, Physical Review A102, 012812 (2020)

    work page 2020

  46. [46]

    A. A. Peshkov, E. Jordan, M. Kromrey, K. K. Mehta, T. E. Mehlst¨ aubler, and A. Surzhykov, Annalen der Physik535, 2300204 (2023), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.202300204, URLhttps://onlinelibrary.wiley.com/doi/abs/10. 1002/andp.202300204

    work page doi:10.1002/andp.202300204 2023

  47. [47]

    Surzhykov, D

    A. Surzhykov, D. Seipt, and S. Fritzsche, Physical Review A94, 033420 (2016)

    work page 2016

  48. [48]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of mathemat- ical functions: with formulas, graphs, and mathematical tables, vol. 55 (Courier Corporation, 1965)

    work page 1965

  49. [49]

    G. W. Rubloff, Physical review B5, 662 (1972)

    work page 1972

  50. [50]

    Barth, R

    J. Barth, R. Johnson, M. Cardona, D. Fuchs, and A. M. Bradshaw, Physical Review B41, 3291 (1990)

    work page 1990

  51. [51]

    Tsujibayashi, K

    T. Tsujibayashi, K. Toyoda, S. Sakuragi, M. Kamada, and M. Itoh, Applied physics letters80, 2883 (2002)

    work page 2002

  52. [52]

    Collins and J

    R. Collins and J. C. Travis, inM¨ ossbauer Effect Method- ology: Volume 3 Proceedings of the Third Symposium on M¨ ossbauer Effect Methodology New York City, January 29, 1967(Springer, 1967), pp. 123–161

    work page 1967

  53. [53]

    Bemis, F

    C. Bemis, F. McGowan, J. Ford Jr, W. Milner, R. Robin- son, P. Stelson, G. Leander, and C. Reich, Physica Scripta38, 657 (1988)

    work page 1988

  54. [54]

    E. V. Tkalya, Phys. Rev. Lett.106, 162501 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett. 106.162501

    work page doi:10.1103/physrevlett 2011

  55. [55]

    Thielking, M

    J. Thielking, M. V. Okhapkin, P. G lowacki, D. M. Meier, L. von der Wense, B. Seiferle, C. E. D¨ ullmann, P. G. Thirolf, and E. Peik, Nature556, 321 (2018)

    work page 2018

  56. [56]

    A. R. Edmonds,Angular momentum in quantum me- chanics(Princeton university press, 1996)

    work page 1996

  57. [57]

    Dicke, Physical Review89, 472 (1953)

    R. Dicke, Physical Review89, 472 (1953)

    work page 1953

  58. [58]

    W.-T. Liao, S. Das, C. H. Keitel, and A. P´ alffy, Phys. Rev. Lett.109, 262502 (2012), URLhttps://link.aps. 17 org/doi/10.1103/PhysRevLett.109.262502

    work page doi:10.1103/physrevlett.109.262502 2012

  59. [59]

    P´ alffy, J

    A. P´ alffy, J. Evers, and C. H. Keitel, Phys. Rev. C 77, 044602 (2008), URLhttps://link.aps.org/doi/ 10.1103/PhysRevC.77.044602

    work page doi:10.1103/physrevc.77.044602 2008

  60. [60]

    Ring and P

    P. Ring and P. Schuck,The nuclear many-body problem (Springer Science & Business Media, 2004)

    work page 2004

  61. [61]

    D. A. Varshalovich, A. N. Moskalev, and V. K. Kher- sonskii,Quantum theory of angular momentum(World Scientific, 1988)

    work page 1988

  62. [62]

    Kirschbaum, N

    T. Kirschbaum, N. Minkov, and A. P´ alffy, Phys. Rev. C 105, 064313 (2022), URLhttps://link.aps.org/doi/ 10.1103/PhysRevC.105.064313

    work page doi:10.1103/physrevc.105.064313 2022

  63. [63]

    M. O. Scully and M. S. Zubairy,Quantum optics(Cam- bridge university press, 1997)

    work page 1997

  64. [64]

    Shore,The theory of coherent atomic excitation (1991)

    B. Shore,The theory of coherent atomic excitation (1991)

    work page 1991

  65. [65]

    W.-T. Liao, C. H. Keitel, and A. P´ alffy, Physical review letters113, 123602 (2014)

    work page 2014