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Inelastic scattering of vortex electrons beyond the Born approximation

T0 review · 0 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that Coulomb distortion, not just the first Born plane-wave approximation, controls the inelastic scattering of vortex electrons by hydrogen at 20–100 eV, visibly changing wave-function phase and density as well as…

desk verdict First distorted-wave treatment of inelastic vortex-electron scattering; central claim holds, but the paper needs to clarify the distortion-potential channel choice and add convergence tests. read the letter →

arxiv 2412.08246 v1 pith:ZQ26R4YB submitted 2024-12-11 physics.atom-ph

S. Strnat , J. Sommerfeldt , A. K. Sahoo , L. Sharma , A. Surzhykov This is my paper · ORCID
classification physics.atom-ph PACS 34.80.Dp
keywords vortexelectronstwisteddistortedwaveapproximationfirstBornelectronimpactexcitationhydrogen1s-2porbitalangularmomentumCoulombdistortion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends the theory of inelastic scattering of vortex (twisted) electrons to include the Coulomb interaction between the projectile and the target atom, not just the plane-wave first Born approximation. It constructs vortex wave functions from distorted solutions of the Schrödinger equation and derives distorted-wave scattering amplitudes for the 1s→2p excitation of hydrogen at 20–100 eV. The central result is that the Coulomb interaction changes the squared transition amplitudes substantially, most strongly for the |2p m_f=0> sublevel and at large scattering angles, and also bends the vortex line and distorts the annular density and phase profile of the beam. A sympathetic reader would care because low-energy vortex-electron experiments and microscopy applications operate in exactly this regime, where the earlier Born-based description is no longer reliable.

What carries the argument

The load-bearing object is the displaced distorted vortex wave function of Eq. (14), $F^{{V,(±)}}$_{m_l κ p_z b}(r)=∫ $d^{3}$p/(2π) a_{m_l κ p_z}(p) $e^{{-i b·p}}$ $F^{{(±)}}$_p(r), built by superposing target-centered Coulomb-distorted waves $F^{{(±)}}$_p(r) with the Bessel-state amplitude a_{m_l κ p_z}(p) and a displacement phase $e^{{-i b·p}}$ that puts the vortex line at impact parameter b. Its multipole expansion, Eq. (15), contains the Bessel function J_{m_l-m}(κb), and inserting it into the distorted-wave amplitude yields Eq. (24), the central formula whose squared modulus is compared with the first Born result (19) throughout the paper. The radial functions $ϕ^{{(±)}}$_l(p,r) entering these expressions are obtained numerically by integrating the radial Schrödinger equation with the static electron-hydrogen distortion potential.

What would settle it

Solve the full Schrödinger equation for a 20–100 eV vortex electron interacting with a hydrogen atom (for instance with a numerical grid or close-coupling calculation) and compare the resulting wave function and 1s→2p amplitudes with Eq. (14) and Eq. (24); if the exact angular distributions differ from the distorted-wave curves near the predicted minima or at large angles, the displaced-ansatz construction is not the correct physical state. Experimentally, angle-resolved measurement of electrons scattered after exciting the |2p m_f=0> sublevel at b≈1 a0 would provide a direct test, since the two theories differ by orders of magnitude there.

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Extended reading notes

Core claim

For a hydrogen atom excited from the 1s ground state to the 2p sublevels by a vortex electron with kinetic energy between 20 and 100 eV, the Coulomb attraction between projectile and target significantly modifies the squared transition amplitudes compared with first Born predictions. The effect is strongest for the |2p m_f=0> sublevel and for large scattering angles, and it persists even at 100 eV; for b≠0 deviations appear already at small angles, including a minimum near θ_p'≈10° that the first Born approximation does not produce. The paper also finds that the Coulomb field bends the vortex line toward the atom and distorts the doughnut-shaped probability density and helical phase, effects that vanish at large |z| where free-space behavior is recovered.

Load-bearing premise

Everything in the distorted-wave calculation depends on the assumption that the physical vortex state in the presence of the Coulomb field is correctly represented by taking the target-centered distorted wave F_p(r) and simply inserting the phase $e^{{-ip·b}}$ into the superposition, without deriving this object as an actual solution of the Schrödinger equation.

Editorial extensions

If this is right

  • At b=0 and 20 eV, first Born and distorted wave agree for θ_p'≲30° but diverge strongly beyond about 50°, so any low-energy vortex-electron scattering analysis must include Coulomb distortion.
  • For an atom displaced from the vortex line (b≠0), the m_f=0 squared amplitude develops a minimum near θ_p'≈10° that the first Born approximation does not predict.
  • Strictly forward scattering obeys the selection rule m_i + m_l = m_f; for the 1s→2p transition it is allowed only when the incident OAM projection equals the final magnetic quantum number.
  • The impact-parameter dependence of the squared amplitude follows |J_{m_l-m_f}(κb)|^2 in the first Born approximation, but the distorted-wave oscillations become asynchronous for m_f=0, meaning beam-profile measurements can expose the distortion.
  • Even at 100 eV the Coulomb distortion affects small and large scattering angles, so high-energy Born descriptions of vortex-electron scattering are not automatically safe in this regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the displaced-ansatz construction of Eq. (14) is confirmed against full numerical solutions, the same recipe could be carried over to other inelastic channels (e.g., 1s→3p or ionization) and to heavier targets where the static distortion potential is stronger and the effects larger.
  • The predicted bending of the vortex line toward the Coulomb center implies that in low-energy electron microscopy the actual impact parameter of a vortex probe differs from the nominal beam displacement; measuring the doughnut-center position as a function of z near the sample would test this directly.
  • Because the Coulomb effect shows up most at large angles and in the m_f=0 channel, angle-resolved and magnetic-sublevel-resolved detection would be a sharper experimental probe than total cross sections; Kapitza-Dirac-produced beams in the tens-of-eV range may provide the natural testbed.
  • The same e^{-ip·b} ansatz could be stress-tested analytically by computing the first-order correction in the projectile-target interaction to the displaced wave function; if that correction renormalizes the impact parameter b, the asymptotic beam position assumed in Eq. (14) would need reinterpretation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. This paper presents a theoretical study of inelastic scattering of vortex (Bessel) electrons by a hydrogen atom at projectile energies of 20–100 eV. The authors construct two types of incident wave functions: a free-space Bessel beam (Eqs. 6–9) and a 'distorted vortex' wave function (Eq. 14) obtained by coherently superposing distorted waves F_p^{(+)}(r) with the phase e^{-ip·b}. For the 1s→2p excitation, they derive the first-Born amplitude (Eq. 19) and a distorted-wave amplitude (Eq. 24) using the multipole expansion of the electron–electron interaction and numerically computed radial partial waves. The numerical results are presented for the probability density and phase of the vortex beam in the presence of the target (Figs. 2–4) and for the squared transition amplitudes as functions of scattering angle, impact parameter, energy, and OAM projection (Figs. 5–8). The paper's central claim is that the Coulomb electron–target interaction strongly modifies the vortex wave function and the squared transition amplitudes relative to the first-Born prediction, most prominently for the |2p mf=0> sublevel and at large scattering angles.

Significance. The work addresses a timely question: whether the common first-Born treatment of vortex-electron inelastic scattering is adequate at low energies, where electron microscopes and Kapitza–Dirac sources operate. The paper's central comparison is meaningful because both amplitudes are derived from the same Schrödinger equation with stated potentials and contain no fitted parameters. The use of the standard distorted-wave approximation gives the study a solid theoretical basis, and the finding that Coulomb distortion changes the squared amplitudes by orders of magnitude at large angles is a useful, falsifiable prediction. The displaced vortex ansatz of Eq. (14), although presented opaquely, is justified by linearity and by the required incoming asymptotic boundary condition. The main limitations are that the numerical results are not benchmarked against other calculations or experiment and that the derivation of Eq. (14) is not written out clearly; neither limitation undermines the qualitative conclusion. If the clarity and reproducibility issues are fixed, the paper should be a useful reference for future low-energy vortex-electron collision studies.

minor comments (6)
  1. [Section 3.2, Eq. (14)] The derivation of Eq. (14) is difficult to follow. The step 'introduce an additional phase factor e^{-ip·b}' and the subsequent coordinate transformation obscure the argument; the result follows more directly from the linearity of the Schrödinger equation and the asymptotic form of F_p^{(+)}(r). Please rewrite this passage.
  2. [Section 5, Eq. (26)] The text says R(r1) denotes the radial part of the initial |1s> or final |2p> state, but it does not state which state is used in the initial and final channel distorted waves of Eq. (24). Please specify the assignment, and also state whether the 2p static potential is obtained by spherical averaging.
  3. [Section 5] A convergence test (variation of the number of partial waves, box size, and grid points) for at least one representative case would support the stated numerical parameters, which currently lack a convergence check.
  4. [Section 4.1, Eq. (19)] The sentence 'In the first line of this expression ψV is given by Eq. (6)' should refer to the shifted Bessel expression of Eq. (9), since the phase e^{-iκb cos φ_p} in the integrand is otherwise unexplained.
  5. [Section 4 heading] The word 'transistion' should be 'transition'; also, the reference note for Ref. [16] stating a factor-4π difference between the amplitudes should be either explained in the main text or moved to a footnote.
  6. [Section 2, Fig. 1 and throughout] The symbol b is used both as a vector and as its magnitude; please distinguish these uses (for example, bold b for the vector and b for the magnitude) to avoid ambiguity in Eqs. (14), (15), and (30).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the distorted-wave vortex amplitude is derived from the Schrödinger equation via an explicit superposition, with no fitted parameters and no prediction that reduces to an input.

full rationale

The paper's central comparison is between two independently constructed amplitudes: the first Born vortex amplitude (Eq. 19), which is a known result rederived from the free Bessel-state superposition, and the distorted-wave vortex amplitude (Eq. 24), which follows from inserting the distorted vortex wave function (Eq. 15) into the standard distorted-wave matrix element (Eq. 20). No parameter is fitted to data, no subset of results is used to predict a closely related quantity, and neither amplitude is defined in terms of the other. The construction of the displaced distorted vortex state in Eq. (14) is made explicitly in the text as a coherent superposition of distorted plane waves with a phase factor e^{-ip·b}; it is not imported as an unexamined premise from a self-citation. The citations to Refs. [13, 26, 27] occur after the derivation and merely note that similar expressions were used elsewhere, so they are not load-bearing. The comparison between Eq. (24) and Eq. (19) is a genuine calculation: the Coulomb-distorted radial functions are obtained numerically from the radial Schrödinger equation (Eq. 25) with a stated static potential (Eq. 26), and the Born amplitude is obtained from the Bethe integral. Whether Eq. (14) is the physically correct vortex state in the presence of the Coulomb field is a modeling assumption and a possible accuracy concern, but it does not constitute circularity because the conclusion does not reduce to that assumption by construction; the distorted-wave results could in principle disagree with the Born results, and the paper reports such disagreement. Self-citations in the elastic-scattering context are contextual and do not inject the inelastic result. The derivation chain is therefore self-contained with respect to its inputs, and no specific reduction of a prediction to an input can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or new entities. Central claim rests on the distorted-wave Born approximation framework, the static distorting potential, and the displaced-vortex ansatz in Eq. (14), which are prior or introduced modeling assumptions.

assumptions (4)
  • domain assumption The electron-atom interaction is described by a spherically symmetric static potential Ud(r0) (Eq. 26), neglecting exchange, target polarization, and channel coupling.
    Used to construct distorted waves in Sec. 5 and is central to the distorted-wave amplitudes in Eq. (24).
  • ad hoc to paper The displaced distorted vortex wave function of Eq. (14), obtained by inserting e^{-ip·b} into the superposition while leaving F_p(r) centered at the origin, represents the physical vortex beam with impact parameter b.
    This ansatz is introduced via a 'mathematical trick' in Sec. 3.2 and is not independently derived or compared with alternatives.
  • domain assumption The distorted-wave Born approximation, retaining only the first-order transition amplitude built from distorted waves, is accurate enough at 20 to 100 eV to isolate Coulomb effects.
    The paper compares first Born and distorted-wave approximations; higher-order and exchange contributions are omitted.
  • standard math Non-relativistic Schrodinger treatment with hydrogenic 1s and 2p wave functions is valid in the studied energy range.
    Used throughout for projectile and target states.

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Cite this review

Pith. "Pith review of Inelastic scattering of vortex electrons beyond the Born approximation." pith.science (2026). https://pith.science/paper/ZQ26R4YB

@misc{pith2026241208246,
  author       = {Pith},
  title        = {Pith review of: Inelastic scattering of vortex electrons beyond the Born approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQ26R4YB}},
  note         = {Machine review of arXiv:2412.08246}
}
abstract

We present a theoretical study of the inelastic scattering of vortex electrons by a hydrogen atom. In our study, special emphasis is placed on the effects of the Coulomb interaction between a projectile electron and a target atom. To understand these effects, we construct vortex electron wave functions both from free space and distorted solutions of the Schr\"odinger equation. These wave functions give rise to the first Born and distorted wave scattering amplitudes, respectively. The derived theory has been employed to investigate the $1s \rightarrow 2p$ transition of a hydrogen atom induced by electrons with the kinetic energies in the range from 20 to 100 eV. The results of the calculations have clearly indicated that the Coulomb interaction can significantly affect the phase pattern and probability density of a vortex electron beam as well as the squared transition amplitudes. For the latter, the most pronounced effect was found for the excitation to the $\ket{2p\, m_f=0}$ sublevel and large scattering angles.

Figures

Figures reproduced from arXiv: 2412.08246 by the authors.

Figure 1
Figure 1. The geometry of inelastic scattering of a vortex electron beam by a target hydrogen atom is considered. The incident electron, which is assumed to be in the Bessel state with well-defined longitudinal momentum pz and projection of orbital angular momentum mℓ on the propagation axis. This axis is shifted from the coordinate origin by the vector b. to the z axis and has a well-defined longitudinal momentum pz. The cen… view at source ↗
Figure 2
Figure 2. Upper panel: Probability density of a 20 eV vortex electron constructed from plane waves (6) in the xy plane. The columns show different impact parameters b. The opening angle is θp = 30◦ and the electron has an orbital angular momentum projection of +1. Lower panel: Phase of the electrons described above. from the middle and right columns of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Upper panel: Probability density of a 20 eV vortex electron constructed from distorted waves (15) in the xy plane. The columns show different impact parameters b. The opening angle is θp = 30◦ and the electron has an orbital angular momentum projection of +1. Lower panel: Phase of the electrons described above. distorted vortex electrons depends additionally on the distance to the vortex line. This effect can be att… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Schematic sketch of the behavior of the phase pattern in the transverse plane of during the propagation of the electron vortex beam along the z axis with (asymptotic) impact parameter b = 2 a0. The target atom is assumed to be located at the coordinate origin. 10 8 10 …
Figure 5
Figure 5. Figure 5: Squared amplitude as a function of the polar scattering angle θp′ for the 1s → 2p excitation of hydrogen. Predictions of the first Born and distorted wave approximations are displayed by red dashed and blue solid lines, respectively. The columns correspond to the impac…
Figure 6
Figure 6. Figure 6: The same as [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Squared amplitude as a function of the impact parameter b for the 1s → 2p excitation of hydrogen. Predictions of the first Born and distorted wave approximations are displayed by red dashed and blue solid lines, respectively. The columns correspond to the polar scatter…
Figure 8
Figure 8. Figure 8: Squared amplitude as a function of the polar scattering angle θp′ for the |1s⟩ → |2p mf = 0⟩ excitation of hydrogen. Predictions of the first Born and distorted wave approximations are displayed by red dashed and blue solid lines, respectively. Calculations have been p…

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Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Bliokh, Yury P

    Konstantin Yu. Bliokh, Yury P. Bliokh, Sergey Savel’ev, and Franco Nori. Semiclassical dynamics of electron wave packet states with phase vortices. Phys. Rev. Lett., 99:190404, Nov 2007

    work page 2007

  2. [2]

    Ivanov, G

    K.Y Bliokh, I.P. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. B´ ech´ e, R. Juchtmans, M.A. Alonso, P. Schattschneider, Nori, and J. Verbeeck. Theory and applications of free-electron vortex states. Physics Reports, 690:1–70, 2017

    work page 2017

  3. [3]

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

    work page 2017

  4. [4]

    Generation of electron beams carrying orbital angular momentum

    Masaya Uchida and Akira Tonomura. Generation of electron beams carrying orbital angular momentum. Nature, 464(7289):737–739, 2010

    work page 2010

  5. [5]

    Production and application of electron vortex beams

    Johan Verbeeck, He Tian, and Peter Schattschneider. Production and application of electron vortex beams. Nature, 467(7313):301–304, 2010

    work page 2010

  6. [6]

    Using electron vortex beams to determine chirality of crystals in transmission electron microscopy

    Roeland Juchtmans, Armand B´ ech´ e, Artem Abakumov, Maria Batuk, and Jo Verbeeck. Using electron vortex beams to determine chirality of crystals in transmission electron microscopy. Physical Review B, 91(9):094112, 2015

    work page 2015

  7. [7]

    Orbital angular momentum in electron diffraction and its use to determine chiral crystal symmetries

    Roeland Juchtmans and Jo Verbeeck. Orbital angular momentum in electron diffraction and its use to determine chiral crystal symmetries. Physical Review B, 92(13):134108, 2015

    work page 2015

  8. [8]

    Dichroism in the interaction between vortex electron beams, plasmons, and molecules

    A Asenjo-Garcia and FJ Garc ´ ıa de Abajo. Dichroism in the interaction between vortex electron beams, plasmons, and molecules. Physical review letters, 113(6):066102, 2014

    work page 2014

Show all 33 references
  1. [9]

    Elastic scattering of electron vortex beams in magnetic matter

    Alexander Edstr¨ om, Axel Lubk, and J´ an Rusz. Elastic scattering of electron vortex beams in magnetic matter. Physical Review Letters, 116(12):127203, 2016

    2016

  2. [10]

    Rutherford scattering of electron vortices

    Ruben Van Boxem, Bart Partoens, and Johan Verbeeck. Rutherford scattering of electron vortices. Physical Review A, 89(3):032715, 2014

    2014

  3. [11]

    Elastic scattering of twisted electrons by diatomic molecules

    A V Maiorova, S Fritzsche, RA M¨ uller, and A Surzhykov. Elastic scattering of twisted electrons by diatomic molecules. Physical Review A, 98(4):042701, 2018

    2018

  4. [12]

    Scattering of twisted relativistic electrons by atoms

    Valery Serbo, IP Ivanov, Stephan Fritzsche, Daniel Seipt, and Andrey Surzhykov. Scattering of twisted relativistic electrons by atoms. Physical Review A, 92(1):012705, 2015

    2015

  5. [13]

    Elastic scattering of twisted electrons by an atomic target: Going beyond the born approximation

    VP Kosheleva, V A Zaytsev, A Surzhykov, VM Shabaev, and Th St¨ ohlker. Elastic scattering of twisted electrons by an atomic target: Going beyond the born approximation. Physical Review A, 98(2):022706, 2018

    2018

  6. [14]

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

    2022

  7. [15]

    A distorted-wave approach to the elastic scattering of twisted electrons

    AL Harris and S Fritzsche. A distorted-wave approach to the elastic scattering of twisted electrons. arXiv preprint arXiv:2411.14558, 2024

    2024 arXiv

  8. [16]

    Inelastic electron-vortex-beam scattering

    Ruben Van Boxem, Bart Partoens, and Jo Verbeeck. Inelastic electron-vortex-beam scattering. Physical Review A, 91(3):032703, 2015. In this paper, the amplitude for the inelastic scattering of vortex electrons (24) differs from our scattering amplitude (19) by the factor of 4 π

    2015

  9. [17]

    Future directions in high-resolution electron microscopy: Novel optical components Inelastic scattering of vortex electrons 17 and techniques

    Peter Hawkes. Future directions in high-resolution electron microscopy: Novel optical components Inelastic scattering of vortex electrons 17 and techniques. Comptes Rendus Physique, 15(2):110–118, 2014. Seeing and measuring with electrons: Transmission Electron Microscopy toda...

    2014

  10. [18]

    Creating electron vortex beams with light

    Jonathan Handali, Pratistha Shakya, and Brett Barwick. Creating electron vortex beams with light. Opt. Express, 23(4):5236–5243, Feb 2015

    2015

  11. [19]

    Low energy electron microscopy

    E Bauer. Low energy electron microscopy. Reports on Progress in Physics, 57(9):895, 1994

    1994

  12. [20]

    Electron impact excitation of the 4 1P1 state of calcium

    Raj Kumar Chauhan, Rajesh Srivastava, and A D Stauffer. Electron impact excitation of the 4 1P1 state of calcium. Journal of Physics B: Atomic, Molecular and Optical Physics, 38(14):2385, 6 2005

    2005

  13. [21]

    Mathematical methods for physicists: a comprehensive guide

    George B Arfken, Hans J Weber, and Frank E Harris. Mathematical methods for physicists: a comprehensive guide. Academic press, 2011

    2011

  14. [22]

    Burke and C.J

    P.G. Burke and C.J. Joachain. Theory of electron—atom collisions: Part 1: Potential scattering. Springer Science & Business Media, 2013

    2013

  15. [23]

    ‘twisted’electrons

    Hugo Larocque, Ido Kaminer, Vincenzo Grillo, Gerd Leuchs, Miles J Padgett, Robert W Boyd, Mordechai Segev, and Ebrahim Karimi. ‘twisted’electrons. Contemporary Physics, 59(2):126– 144, 2018

    2018

  16. [24]

    A new way of producing electron vortex probes for stem

    Jo Verbeeck, He Tian, and Armand B´ ech´ e. A new way of producing electron vortex probes for stem. Ultramicroscopy, 113:83–87, 2012

    2012

  17. [25]

    Electron vortex beams with high quanta of orbital angular momentum

    Benjamin J McMorran, Amit Agrawal, Ian M Anderson, Andrew A Herzing, Henri J Lezec, Jabez J McClelland, and John Unguris. Electron vortex beams with high quanta of orbital angular momentum. science, 331(6014):192–195, 2011

    2011

  18. [26]

    V. A. Zaytsev, V. G. Serbo, and V. M. Shabaev. Radiative recombination of twisted electrons with bare nuclei: Going beyond the born approximation. Physical Review A, 95:012702, 2017

    2017

  19. [27]

    Radiative recombination of twisted electrons with hydrogenlike heavy ions: Linear polarization of emitted photons

    Anna V Maiorova, Anton A Peshkov, and Andrey Surzhykov. Radiative recombination of twisted electrons with hydrogenlike heavy ions: Linear polarization of emitted photons. Physical Review A, 104(2):022821, 2021

    2021

  20. [28]

    Lectures on ion-atom collisions: from nonrelativistic to relativistic velocities

    J¨ org Eichler. Lectures on ion-atom collisions: from nonrelativistic to relativistic velocities . Elsevier, 2005

    2005

  21. [29]

    Electron impact excitation of atoms

    Benjamin Lawrence Moiseiwitsch and Stephen Judson Smith. Electron impact excitation of atoms. Reviews of Modern Physics, 40(2):238, 1968

    1968

  22. [30]

    Application of a simplified second born approximation to the scattering of electrons and protons by hydrogen atoms

    AR Holt and BL Moiseiwitsch. Application of a simplified second born approximation to the scattering of electrons and protons by hydrogen atoms. Journal of Physics B: Atomic and Molecular Physics, 1(1):36, 1968

    1968

  23. [31]

    Elektrodynamik: Lehrbuch zur Theoretischen Physik II, volume 2

    Torsten Fließbach. Elektrodynamik: Lehrbuch zur Theoretischen Physik II, volume 2. Springer- Verlag, 2012

    2012

  24. [32]

    H.R.J. Walters. Perturbative methods in electron- and positron-atom scattering. Physics Reports, 116(1):1–102, 1984

    1984

  25. [33]

    Salvat, J.M

    F. Salvat, J.M. Fern´ andez-Varea, and W. Williamson. Accurate numerical solution of the radial schr¨ odinger and dirac wave equations.Computer Physics Communications, 90(1):151–168, 1995

    1995

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