pith. sign in

arxiv: 2606.31818 · v1 · pith:YLRIMX2Znew · submitted 2026-06-30 · ⚛️ nucl-th

Nuclear excitation via inelastic scattering of low-energy vortex electrons

Jia-Lin Zhang , Zhi-Wei Lu , Mamutjan Ababekri , Yuanbin Wu , Jian-Xing Li This is my paper

Pith reviewed 2026-07-01 02:44 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords vortex electronsnuclear excitationinelastic scatteringorbital angular momentumangular distributionsCoulomb fieldtopological protectionDirac distorted-wave Born approximation
0
0 comments X

The pith

Vortex electrons produce angular distributions opposite to ordinary electrons when exciting nuclei.

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

This paper introduces a method to detect orbital angular momentum transfer from electrons to nuclei by using low-energy vortex electrons in inelastic scattering. The authors develop a theoretical framework based on the Dirac equation with distorted waves to handle the electron's orbital angular momentum and the strong Coulomb field from the nucleus without approximations. When applied to the nucleus of thorium-229, the calculations reveal that vortex electrons and regular electrons lead to opposite patterns in the angular distribution of the scattered particles. This difference arises from how the orbital angular momentum alters the allowed transitions and how the Coulomb force redistributes the contributing waves. The work also indicates that vortex electrons enjoy some form of protection against losing their topological character in the nuclear field.

Core claim

The central discovery is that inelastic scattering of low-energy vortex electrons off nuclei like 229Th produces angular distributions opposite to those from non-vortex electrons. This opposition stems from an orbital-angular-momentum-modified selection rule together with the Coulomb field's effect on partial-wave strengths. The framework also shows that vortex electrons maintain topological protection while traversing the nuclear Coulomb field.

What carries the argument

A Dirac distorted-wave Born approximation framework incorporating the incident electron's orbital angular momentum with nonperturbative Coulomb treatment.

If this is right

  • An angle-resolved measurement can serve as a signature for nuclear OAM transfer.
  • Vortex electrons offer a new probe for OAM effects at the nuclear scale.
  • The topological protection implies vortex states persist through strong fields.
  • This approach deepens understanding of vortex particle properties in nuclear interactions.

Where Pith is reading between the lines

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

  • Similar calculations could be performed for other nuclei to test whether the opposite distribution effect holds generally.
  • The method might be adapted to probe OAM transfer in scattering processes involving other charged particles.
  • Direct experiments with electron beams carrying controlled orbital angular momentum could verify the predicted angular patterns.

Load-bearing premise

The Dirac distorted-wave Born approximation accurately captures the effects of the electron's orbital angular momentum and the nonperturbative Coulomb interaction to predict the angular distributions.

What would settle it

Observation of identical angular distributions for vortex and non-vortex electrons in the excitation of 229Th would contradict the predicted opposition.

Figures

Figures reproduced from arXiv: 2606.31818 by Jia-Lin Zhang, Jian-Xing Li, Mamutjan Ababekri, Yuanbin Wu, Zhi-Wei Lu.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) A low-energy vortex electron, incident on the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The normalized di [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dependence of the angle-resolved signature on the beam and [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: σ(E2)/σ(M1) increasing from 0.36 to 15.6. By contrast, the M1 and E2 transitions are suppressed by comparable factors of ≃ 12 and ≃ 6.2 from the PW to the V-PW case so that E2 remains dominant and σ(E2)/σ(M1) merely grows from 3.40 to 6.36. In addition, the cross sections within the Dirac-DWBA framework are about four orders of magnitude larger than those within the Dirac-PWBA framework. The multipole inve… view at source ↗
read the original abstract

Vortex particles carrying orbital angular momenta (OAMs) have found important applications in broad fields. However, the experimental verification of OAM transfer at the nuclear scale remains a great challenge. Here, we put forward a novel method to probe such OAM transfer through nuclear excitation via inelastic scattering of low-energy vortex electrons. We develop a Dirac distorted-wave Born approximation framework that incorporates the incident-electron OAM and a nonperturbative treatment of the Coulomb field, and apply it to $^{229}\mathrm{Th}$. We find that the vortex and non-vortex electrons yield opposite angular distributions, attributed to the OAM-modified selection rule and the Coulomb-induced redistribution of partial-wave strengths, providing an angle-resolved signature. Moreover, the vortex electron exhibits topological protection in the nuclear Coulomb field. Our method offers a route to probing nuclear-scale OAM transfer and deepens our understanding of the topological properties of vortex particles.

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

2 major / 2 minor

Summary. The manuscript develops a Dirac distorted-wave Born approximation (DWBA) framework for inelastic scattering of low-energy vortex electrons that incorporates incident orbital angular momentum (OAM) and treats the nuclear Coulomb field nonperturbatively. Applied to 229Th, it predicts that vortex electrons produce angular distributions opposite to those from plane-wave electrons, attributed to an OAM-modified selection rule combined with Coulomb-induced redistribution of partial-wave strengths; it further claims topological protection of the vortex electron in the nuclear Coulomb field, offering an angle-resolved signature for nuclear-scale OAM transfer.

Significance. If the DWBA implementation of OAM and the resulting angular distributions hold, the work supplies a concrete, falsifiable route to experimental verification of OAM transfer at nuclear scales—an area where direct probes remain scarce. The nonperturbative Coulomb treatment and the reported sign reversal constitute potentially useful predictions, provided the framework is shown to be robust against higher-order corrections.

major comments (2)
  1. [framework description] The central claim of opposite angular distributions rests on the assertion that the Dirac DWBA accurately embeds the incident OAM into the distorted waves and that first-order matrix elements suffice. The manuscript provides no explicit derivation or validation (e.g., reduction to the plane-wave limit or comparison with known OAM-free cases) demonstrating that the OAM-modified selection rule is implemented without approximation in the partial-wave content.
  2. [results and discussion] At the low energies considered, Coulomb distortion is strong; the paper does not quantify the size of second-order Born or multiple-scattering contributions that could alter the reported redistribution of partial-wave strengths and thereby change the sign of the angular distribution difference.
minor comments (2)
  1. [methods] Notation for the vortex-electron wave function and the precise definition of the OAM quantum number should be stated explicitly in the methods section to allow independent reproduction.
  2. [introduction] The abstract and introduction would benefit from a brief statement of the nuclear transition considered (e.g., the specific multipolarity in 229Th) to contextualize the selection-rule modification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. We address each major comment below and will revise the manuscript accordingly to improve clarity and address concerns about validation and approximations.

read point-by-point responses

  1. Referee: [framework description] The central claim of opposite angular distributions rests on the assertion that the Dirac DWBA accurately embeds the incident OAM into the distorted waves and that first-order matrix elements suffice. The manuscript provides no explicit derivation or validation (e.g., reduction to the plane-wave limit or comparison with known OAM-free cases) demonstrating that the OAM-modified selection rule is implemented without approximation in the partial-wave content.

    Authors: We agree that an explicit derivation and validation of the OAM embedding would strengthen the manuscript. In the revised version, we will add a dedicated subsection deriving the partial-wave expansion of the vortex electron wave function in the Dirac DWBA, explicitly showing how the incident OAM is incorporated into the distorted waves and demonstrating the reduction to the standard plane-wave DWBA when the OAM quantum number vanishes. This will confirm that the OAM-modified selection rules are implemented without additional approximations. revision: yes

  2. Referee: [results and discussion] At the low energies considered, Coulomb distortion is strong; the paper does not quantify the size of second-order Born or multiple-scattering contributions that could alter the reported redistribution of partial-wave strengths and thereby change the sign of the angular distribution difference.

    Authors: We acknowledge the importance of assessing higher-order contributions at low energies where Coulomb distortion is significant. A complete numerical quantification of second-order Born terms lies beyond the scope of the present work. However, the partial-wave series in our calculations exhibits rapid convergence, and we will add a discussion in the revised manuscript comparing the magnitude of higher-order effects to literature results for similar low-energy electron scattering processes. We argue that the reported sign reversal originates primarily from the OAM selection rules and is unlikely to be overturned by higher-order corrections, but we will note the first-order DWBA as an approximation. revision: partial

Circularity Check

0 steps flagged

No circularity: framework developed and applied independently

full rationale

The paper develops a Dirac DWBA framework incorporating OAM and applies it to compute angular distributions for 229Th, yielding opposite patterns for vortex vs. plane-wave cases via OAM-modified selection rules and partial-wave redistribution. No equations or claims reduce by construction to fitted parameters, self-citations, or prior ansatze; the central results follow from the new framework's numerical implementation rather than tautological redefinition or load-bearing self-reference. This matches the default expectation of non-circularity for a computational application paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents detailed extraction; the framework relies on standard Dirac equation and DWBA assumptions not enumerated here.

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discussion (0)

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Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    K. Y . Bliokh, I. P. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. B ´ech´e, R. Juchtmans, M. A. Alonso, P. Schattschneider, F. Nori,et al., Theory and applications of free-electron vortex states, Phys. Rep.690, 1 (2017)

    work page 2017

  2. [2]

    S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, Electron vortices: Beams with orbital angular momentum, Rev. Mod. Phys.89, 035004 (2017)

    work page 2017

  3. [3]

    I. P. Ivanov, Promises and challenges of high-energy vortex states collisions, Prog. Part. Nucl. Phys.127, 103987 (2022)

    work page 2022

  4. [4]

    Allen, M

    L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Wo- erdman, Orbital angular momentum of light and the transforma- tion of Laguerre-Gaussian laser modes, Phys. Rev. A45, 8185 (1992)

    work page 1992

  5. [5]

    B. A. Knyazev and V . G. Serbo, Beams of photons with nonzero projections of orbital angular momenta: new results, Phys. Usp. 61, 449 (2018)

    work page 2018

  6. [6]

    A. L. Fetter and A. A. Svidzinsky, V ortices in a trapped dilute Bose-Einstein condensate, J. Phys. Condens. Matter13, R135 (2001)

    work page 2001

  7. [7]

    I. P. Ivanov, Creation of two vortex-entangled beams in a vortex- beam collision with a plane wave, Phys. Rev. A85, 033813 (2012)

    work page 2012

  8. [8]

    Edstr¨om, A

    A. Edstr¨om, A. Lubk, and J. Rusz, Elastic scattering of electron vortex beams in magnetic matter, Phys. Rev. Lett.116, 127203 (2016)

    work page 2016

  9. [9]

    Grillo, T

    V . Grillo, T. R. Harvey, F. Venturi, J. S. Pierce, R. Balboni, F. Bouchard, G. Carlo Gazzadi, S. Frabboni, A. H. Tavabi, Z.-A. Li,et al., Observation of nanoscale magnetic fields using twisted electron beams, Nat. Commun.8, 689 (2017)

    work page 2017

  10. [10]

    M. J. Padgett, Orbital angular momentum 25 years on, Opt. Express25, 11265 (2017)

    work page 2017

  11. [11]

    Hern ´andez-Garc´ıa, J

    C. Hern ´andez-Garc´ıa, J. Vieira, J. T. Mendon c ¸a, L. Rego, J. San Rom´an, L. Plaja, P. R. Ribic, D. Gauthier, and A. Pic´on, Generation and applications of extreme-ultraviolet vortices, Pho- tonics4, 28 (2017)

    work page 2017

  12. [12]

    Van Boxem, B

    R. Van Boxem, B. Partoens, and J. Verbeeck, Inelastic electron- vortex-beam scattering, Phys. Rev. A91, 032703 (2015)

    work page 2015

  13. [13]

    V . K. Ivanov, A. D. Chaikovskaia, and D. V . Karlovets, Studying highly relativistic vortex-electron beams by atomic scattering, Phys. Rev. A108, 062803 (2023)

    work page 2023

  14. [14]

    Han, J.-B

    M. Han, J.-B. Ji, T. Bal ˇci¯unas, K. Ueda, and H. J. W ¨orner, Attosecond circular-dichroism chronoscopy of electron vortices, Nat. Phys.19, 230 (2023)

    work page 2023

  15. [15]

    Lange, N

    R. Lange, N. Huntemann, A. A. Peshkov, A. Surzhykov, and E. Peik, Excitation of an electric octupole transition by twisted light, Phys. Rev. Lett.129, 253901 (2022)

    work page 2022

  16. [16]

    Babiker, D

    M. Babiker, D. L. Andrews, and V . E. Lembessis, Atoms in complex twisted light, J. Opt.21, 013001 (2018)

    work page 2018

  17. [17]

    Y . Wu, S. Gargiulo, F. Carbone, C. H. Keitel, and A. P´alffy, Dy- namical control of nuclear isomer depletion via electron vortex beams, Phys. Rev. Lett.128, 162501 (2022)

    work page 2022

  18. [18]

    Z.-W. Lu, L. Guo, Z.-Z. Li, M. Ababekri, F.-Q. Chen, C. Fu, C. Lv, R. Xu, X. Kong, Y .-F. Niu, and J.-X. Li, Manipulation of giant multipole resonances via vortex γ photons, Phys. Rev. Lett. 131, 202502 (2023)

    work page 2023

  19. [19]

    Z.-W. Lu, L. Guo, M. Ababekri, J.-L. Zhang, X.-F. Weng, Y . Wu, Y .-F. Niu, and J.-X. Li, Angular momentum resolved inelastic electron scattering for nuclear giant resonances, Phys. Rev. Lett. 134, 052501 (2025)

    work page 2025

  20. [20]

    Kirschbaum, T

    T. Kirschbaum, T. Schumm, and A. P´alffy, Photoexcitation of the 229Th nuclear clock transition using twisted light, Phys. Rev. C110, 064326 (2024)

    work page 2024

  21. [21]

    Z.-W. Lu, H. Zhang, T. Li, M. Ababekri, X. Wang, and J.-X. Li, Nuclear excitation and control induced by intense vortex lasers, Phys. Rev. C113, 044315 (2026)

    work page 2026

  22. [22]

    S. A.-L. Schulz, A. A. Peshkov, R. A. M ¨uller, R. Lange, N. Huntemann, C. Tamm, E. Peik, and A. Surzhykov, Gen- eralized excitation of atomic multipole transitions by twisted light modes, Phys. Rev. A102, 012812 (2020)

    work page 2020

  23. [23]

    Y . Duan, R. A. M¨uller, and A. Surzhykov, Selection rules for atomic excitation by twisted light, J. Phys. B: At. Mol. Opt. Phys. 52, 184002 (2019)

    work page 2019

  24. [24]

    H. M. Scholz-Marggraf, S. Fritzsche, V . G. Serbo, A. Afanasev, and A. Surzhykov, Absorption of twisted light by hydrogenlike atoms, Phys. Rev. A90, 013425 (2014)

    work page 2014

  25. [25]

    C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. 6 Poschinger, and F. Schmidt-Kaler, Transfer of optical orbital angular momentum to a bound electron, Nat. Commun.7, 12998 (2016)

    work page 2016

  26. [26]

    Uchida and A

    M. Uchida and A. Tonomura, Generation of electron beams carrying orbital angular momentum, Nature464, 737 (2010)

    work page 2010

  27. [27]

    Verbeeck, H

    J. Verbeeck, H. Tian, and P. Schattschneider, Production and application of electron vortex beams, Nature467, 301 (2010)

    work page 2010

  28. [28]

    B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, Electron vortex beams with high quanta of orbital angular momentum, Science 331, 192 (2011)

    work page 2011

  29. [29]

    C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, Controlling neutron orbital angular momentum, Nature525, 504 (2015)

    work page 2015

  30. [30]

    Maruyama, T

    T. Maruyama, T. Hayakawa, T. Kajino, and M.-K. Cheoun, Gen- eration of photon vortex by synchrotron radiation from electrons in Landau states under astrophysical magnetic fields, Phys. Lett. B826, 136779 (2022)

    work page 2022

  31. [31]

    Y . Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities, Light Sci. Appl.8, 90 (2019)

    work page 2019

  32. [32]

    A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, Observation of an x-ray vortex, Opt. Lett.27, 1752 (2002)

    work page 2002

  33. [33]

    Terhalle, A

    B. Terhalle, A. Langner, B. P ¨aiv¨anranta, V . A. Guzenko, C. David, and Y . Ekinci, Generation of extreme ultraviolet vortex beams using computer generated holograms, Opt. Lett.36, 4143 (2011)

    work page 2011

  34. [34]

    Gariepy, J

    G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, Creating high-harmonic beams with controlled orbital angular momentum, Phys. Rev. Lett.113, 153901 (2014)

    work page 2014

  35. [35]

    Hemsing, A

    E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A. Marinelli, C. Hast, and J. B. Rosenzweig, Coherent optical vortices from relativistic electron beams, Nat. Phys.9, 549 (2013)

    work page 2013

  36. [36]

    G. M. Vanacore, G. Berruto, I. Madan, E. Pomarico, P. Bia- gioni, R. J. Lamb, D. McGrouther, O. Reinhardt, I. Kaminer, B. Barwick,et al., Ultrafast generation and control of an electron vortex beam via chiral plasmonic near fields, Nat. Mater.18, 573 (2019)

    work page 2019

  37. [37]

    B´ech´e, R

    A. B´ech´e, R. Van Boxem, G. Van Tendeloo, and J. Verbeeck, Magnetic monopole field exposed by electrons, Nat. Phys.10, 26 (2014)

    work page 2014

  38. [38]

    Grillo, G

    V . Grillo, G. C. Gazzadi, E. Mafakheri, S. Frabboni, E. Karimi, and R. W. Boyd, Holographic generation of highly twisted elec- tron beams, Phys. Rev. Lett.114, 034801 (2015)

    work page 2015

  39. [39]

    ˇRih´aˇcek, M

    T. ˇRih´aˇcek, M. Hor ´ak, T. Schachinger, F. Mika, M. Mat ˇejka, S. Kr´atk´y, T. Foˇrt, T. Radliˇcka, C. W. Johnson, L. Nov´ak,et al., Beam shaping and probe characterization in the scanning elec- tron microscope, Ultramicroscopy225, 113268 (2021)

    work page 2021

  40. [40]

    M. Wei, S. Chen, Y . Wang, P.-L. He, X. Hu, M. Zhu, H. Xu, W. Zhou, J. Jia, X. Ge,et al., Experimental evidence of vortex γ photons in all-optical inverse Compton scattering, Phys. Rev. Lett.136, 025001 (2026)

    work page 2026

  41. [41]

    M. N. Harakeh and A. Woude,Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitation, V ol. 24 (Oxford University Press on Demand, Oxford, 2001)

    work page 2001

  42. [42]

    Col `o, Theoretical methods for giant resonances, inHand- book of Nuclear Physics(Springer Nature Singapore, Singapore,

    G. Col `o, Theoretical methods for giant resonances, inHand- book of Nuclear Physics(Springer Nature Singapore, Singapore,

    work page

  43. [43]

    Zilges, D

    A. Zilges, D. L. Balabanski, J. Isaak, and N. Pietralla, Photonu- clear reactions—from basic research to applications, Prog. Part. Nucl. Phys.122, 103903 (2022)

    work page 2022

  44. [44]

    S. S. Baturin, D. V . Grosman, G. K. Sizykh, and D. V . Karlovets, Evolution of an accelerated charged vortex particle in an inho- mogeneous magnetic lens, Phys. Rev. A106, 042211 (2022)

    work page 2022

  45. [45]

    G. K. Sizykh, A. D. Chaikovskaia, D. V . Grosman, I. I. Pavlov, and D. V . Karlovets, Transmission of vortex electrons through a solenoid, Phys. Rev. A109, L040201 (2024)

    work page 2024

  46. [46]

    Ababekri, J.-L

    M. Ababekri, J.-L. Zhou, R.-T. Guo, Y .-Z. Ren, Y .-H. Kou, Q. Zhao, Z.-P. Li, and J.-X. Li, Generation of ultrarelativistic vortex leptons with large orbital angular momenta, Phys. Rev. D 110, 076024 (2024)

    work page 2024

  47. [47]

    U. D. Jentschura and V . G. Serbo, Generation of high-energy photons with large Orbital Angular Momentum by Compton backscattering, Phys. Rev. Lett.106, 013001 (2011)

    work page 2011

  48. [48]

    P´alffy, W

    A. P´alffy, W. Scheid, and Z. Harman, Theory of nuclear excita- tion by electron capture for heavy ions, Phys. Rev. A73, 012715 (2006)

    work page 2006

  49. [49]

    Beeks, T

    K. Beeks, T. Sikorsky, T. Schumm, J. Thielking, M. V . Okhapkin, and E. Peik, The thorium-229 low-energy isomer and the nuclear clock, Nat. Rev. Phys.3, 238 (2021)

    work page 2021

  50. [50]

    C. J. Chiara, J. J. Carroll, M. P. Carpenter, J. P. Greene, D. J. Hartley, R. V . F. Janssens, G. J. Lane, J. C. Marsh, D. A. Matters, M. Polasik,et al., Isomer depletion as experimental evidence of nuclear excitation by electron capture, Nature554, 216 (2018)

    work page 2018

  51. [51]

    Dinget al., Isomer depletion of 93mMo triggered by inelas- tic nuclear scattering rather than nuclear excitation by electron capture, Phys

    B. Dinget al., Isomer depletion of 93mMo triggered by inelas- tic nuclear scattering rather than nuclear excitation by electron capture, Phys. Rev. Lett.136, 052502 (2026)

    work page 2026

  52. [52]

    Y . Wu, C. H. Keitel, and A. P´alffy,93mMo isomer depletion via Beam-Based nuclear excitation by electron capture, Phys. Rev. Lett.122, 212501 (2019)

    work page 2019

  53. [53]

    Guoet al., Probing 93mMo isomer depletion with an isomer beam, Phys

    S. Guoet al., Probing 93mMo isomer depletion with an isomer beam, Phys. Rev. Lett.128, 242502 (2022)

    work page 2022

  54. [54]

    Tiedau, M

    J. Tiedau, M. V . Okhapkin, K. Zhang, J. Thielking, G. Zitzer, E. Peik, F. Schaden, T. Pronebner, I. Morawetz, L. T. De Col, et al., Laser excitation of the Th-229 nucleus, Phys. Rev. Lett. 132, 182501 (2024)

    work page 2024

  55. [55]

    Elwell, C

    R. Elwell, C. Schneider, J. Jeet, J. E. S. Terhune, H. W. T. Mor- gan, A. N. Alexandrova, H. B. Tran Tan, A. Derevianko, and E. R. Hudson, Laser excitation of the 229Th nuclear isomeric transition in a solid-state host, Phys. Rev. Lett.133, 013201 (2024)

    work page 2024

  56. [56]

    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., Frequency ratio of the 229mTh nuclear isomeric transition and the 87Sr atomic clock, Nature633, 63 (2024)

    work page 2024

  57. [57]

    E. V . Tkalya, Excitation of229mTh at inelastic scattering of low energy electrons, Phys. Rev. Lett.124, 242501 (2020)

    work page 2020

  58. [58]

    Zhang, W

    H. Zhang, W. Wang, and X. Wang, Nuclear excitation cross section of 229Th via inelastic electron scattering, Phys. Rev. C 106, 044604 (2022)

    work page 2022

  59. [59]

    Xu, J.-H

    Y .-Y . Xu, J.-H. Cheng, Y .-T. Zou, Q. Xiao, and T.-P. Yu, Inelastic- electron-scattering-induced nuclear excitation rates and dynam- ics in 229Th, Phys. Rev. C110, 064621 (2024)

    work page 2024

  60. [60]

    Seiferle, L

    B. Seiferle, L. von der Wense, P. V . Bilous, I. Amersdorffer, C. Lemell, F. Libisch, S. Stellmer, T. Schumm, C. E. D¨ullmann, A. P´alffy, and P. G. Thirolf, Energy of the229Th nuclear clock transition, Nature573, 243 (2019)

    work page 2019

  61. [61]

    L. A. Kroger and C. W. Reich, Features of the low-energy level scheme of 229Th as observed in the α-decay of 233U, Nucl. Phys. A259, 29 (1976)

    work page 1976

  62. [62]

    P. G. Thirolf, S. Kraemer, D. Moritz, and K. Scharl, The thorium isomer 229mTh: review of status and perspectives after more than 50 years of research, Eur. Phys. J. Spec. Top.233, 1113 (2024)

    work page 2024

  63. [63]

    K. Y . Bliokh, M. R. Dennis, and F. Nori, Relativistic electron 7 vortex beams: Angular momentum and spin-orbit interaction, Phys. Rev. Lett.107, 174802 (2011)

    work page 2011

  64. [64]

    Salvat and J

    F. Salvat and J. M. Fern´andez-Varea, RADIAL: A Fortran sub- routine package for the solution of the radial Schr ¨odinger and Dirac wave equations, Comput. Phys. Commun.240, 165 (2019)

    work page 2019

  65. [65]

    Liu and X

    B. Liu and X. Wang, Isomeric excitation of 235U by inelastic scattering of low-energy electrons, Phys. Rev. C106, 064604 (2022)

    work page 2022

  66. [66]

    Serbo, I

    V . Serbo, I. P. Ivanov, S. Fritzsche, D. Seipt, and A. Surzhykov, Scattering of twisted relativistic electrons by atoms, Phys. Rev. A92, 012705 (2015)

    work page 2015

  67. [67]

    V . A. Zaytsev, V . G. Serbo, and V . M. Shabaev, Radiative recom- bination of twisted electrons with bare nuclei: Going beyond the Born approximation, Phys. Rev. A95, 012702 (2017)

    work page 2017

  68. [68]

    V . P. Kosheleva, V . A. Zaytsev, A. Surzhykov, V . M. Shabaev, and T. St ¨ohlker, Elastic scattering of twisted electrons by an atomic target: Going beyond the Born approximation, Phys. Rev. A98, 022706 (2018)

    work page 2018

  69. [69]

    M. E. Groshev, V . A. Zaytsev, V . A. Yerokhin, and V . M. Shabaev, Bremsstrahlung from twisted electrons in the field of heavy nuclei, Phys. Rev. A101, 012708 (2020)

    work page 2020

  70. [70]

    See Supplemental Material for a detailed derivation of the in- elastic electron–nucleus scattering differential cross sections and total cross sections for non-vortex and vortex electrons, and for additional numerical results for different opening angles and incident energies

    work page

  71. [71]

    B ´ech´e, R

    A. B ´ech´e, R. Juchtmans, and J. Verbeeck, Efficient creation of electron vortex beams for high resolution STEM imaging, Ultramicroscopy178, 12 (2017)

    work page 2017

  72. [72]

    Reimer and H

    L. Reimer and H. Kohl,Transmission Electron Microscopy (Springer Science+Business Media, New York, 2008)

    work page 2008

  73. [73]

    I. P. Ivanov, N. Korchagin, A. Pimikov, and P. Zhang, Kinematic surprises in twisted-particle collisions, Phys. Rev. D101, 016007 (2020)

    work page 2020