Pith. sign in

REVIEW 2 major objections 6 minor 45 references

Coulomb impurities in graphene driven by fast ions

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that a fast ultrarelativistic ion sweeping past a graphene layer ionizes a Coulomb-impurity artificial atom with a cross section two orders of magnitude larger than the equivalent three-dimensional process, and it…

desk verdict A clean, internally consistent application of Baltz's exact collision solution to a gapped graphene Coulomb impurity, but the load-bearing (1−σz) potential is imported from 3D ultrarelativistic physics and is not justified for graphene quasiparticles with vF ≈ c/300. read the letter →

arxiv 2411.13429 v1 pith:XNGPRM6B submitted 2024-11-20 cond-mat.mes-hall quant-ph

Saparboy Rakhmanov , Reinhold Egger , Doniyor Jumanazarov , Davron Matrasulov This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords grapheneCoulombimpurityartificialatomDiracequationionizationcrosssectionultrarelativisticionexactsolution2Dmaterials
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

The paper treats a charged impurity (adatom) in a graphene monolayer as a two-dimensional artificial atom and asks what happens when an ultrarelativistic ion passes parallel to the layer at impact parameter b. The authors solve the time-dependent 2D Dirac equation exactly for this setup, exploiting a light-cone coordinate trick that turns the ion's brief pulse into a simple phase factor. From that solution they obtain the survival probability of the ground state, excitation probabilities to bound states, and the total ionization probability via unitarity. Their central quantitative claim is that the ionization cross section is two orders of magnitude larger than the corresponding three-dimensional relativistic ion-atom collision. If correct, this makes collision-induced transitions in graphene Coulomb impurities observable with a table-top ion beam, offering a laboratory window into 2D relativistic atomic physics.

What carries the argument

The machinery is the reduction of the scattering problem to a light-cone instantaneous kick. The projectile potential is idealized as $V(x,z,t) = -\delta(z-ct)\,\alpha Z_P (1-\sigma_z)\ln(1+x^2/b^2)$, a delta-function pulse along the ion's worldline with a sublattice projection. Integrating the Dirac equation across the light-cone coordinate $z_-$ yields the boundary condition $(1-\sigma_z)\Psi = (1-\sigma_z) e^{-i\theta(ct-z)\alpha Z_P \ln(1+x^2/b^2)} \psi_i$, and substituting this into the interaction-picture equation gives the exact amplitude as a static matrix element. All probabilities follow from that one expression, and ionization is obtained without computing continuum wavefunctions by subtracting survival and excitation from unity.

What would settle it

Compare the predicted impact-parameter-resolved ionization probability $P(b)$ and the integrated cross sections (for example, about $2.7\times 10^9$ barn for $Z_P=10$) against a controlled experiment in which a fast ion beam of known charge passes over a graphene sample with isolated Coulomb impurities; a measured cross section differing by orders of magnitude, or a survival probability that does not saturate to unity at large $b$, would rule out the assumed potential.

Watch

Extended reading notes

Core claim

The central discovery is an exact closed-form transition amplitude for a Coulomb-impurity artificial atom in graphene struck by a fast ion. Working in light-cone coordinates, the authors reduce the time-dependent Dirac equation to a constraint that the spinor just before the pulse equals the initial spinor multiplied by a phase $e^{-i\alpha Z_P \ln(1+x^2/b^2)}$ on the sublattice selected by $(1-\sigma_z)$. The amplitude from initial state $\psi_i$ to final state $\psi_f$ is then a single matrix element, Eq. (13), whose modulus squared gives survival and excitation probabilities; ionization follows from unitarity as $1 - P_{\text{survival}} - P_{\text{excitation}}$. Numerically, for parameters $\alpha_{\rm gr} Z_T = 0.4$, $M\approx 0.1$ eV, the ionization cross sections range from $2.76\times 10^7$ barn for $Z_P=1$ to $2.74\times 10^9$ barn for $Z_P=10$, about two orders of magnitude above the 3D results from the corresponding exact Dirac calculation.

Load-bearing premise

The entire calculation rests on the assumed shape of the ion's potential, a delta-pulse with a logarithmic profile that acts only on one sublattice; if the real graphene coupling differs, the exact solution no longer applies to graphene.

Editorial extensions

If this is right

  • A fast ion beam passing over a graphene layer should ionize Coulomb-impurity artificial atoms with high efficiency: cross sections of order $10^7$ to $10^9$ barn for projectile charge $Z_P$ between 1 and 10.
  • Ionization dominates over excitation at large impact parameters, so the dominant observable signature is a free electron rather than a bound excited state.
  • The unitarity relation in Eq. (16) lets one compute total ionization without explicit continuum wavefunctions, a shortcut that extends to any transition sum.
  • The exact amplitude in Eq. (13) can be reused to compute particle-hole pair creation and angular distributions of differential cross sections in the same setup.

Reading between the lines

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

  • If the $(1-\sigma_z)$ factor is replaced by a sublattice-diagonal coupling, the exact solution is lost and the cross sections may change by orders of magnitude; a first-principles derivation of the graphene projectile potential would settle whether the 'two orders of magnitude' advantage survives.
  • The same light-cone technique should apply to other two-dimensional Dirac materials with different Fermi velocities, with cross sections expected to scale with the effective fine-structure constant and with $v_F/c$, offering a tunable family of table-top experiments.
  • The predicted dominance of ionization over bound excitation suggests that charge-sensing or current-noise measurements in a graphene flake could detect individual ionization events, providing a direct experimental test.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper proposes a theoretical model for electronic transitions in a Coulomb impurity in graphene (a '2D artificial atom') driven by a fast ultrarelativistic ion. The authors write down a time-dependent 2D Dirac equation with a projectile potential taken as a light-cone delta function with a specific sublattice operator (1-σ_z). They solve the equation exactly using light-cone coordinates and derive transition amplitudes, survival/excitation/ionization probabilities, and ionization cross sections. The central quantitative claim is that the ionization cross section is two orders of magnitude larger than the corresponding 3D result.

Significance. If the model were correct, this would provide a table-top probe of 2D relativistic atomic physics and strong-field collision phenomena in graphene, with a clean analytic exact solution. The paper is internally consistent mathematically, and the use of unitarity to obtain ionization probabilities without continuum wavefunctions is elegant. However, the physical input potential is not derived from the graphene Dirac Hamiltonian, and the predictive power of the results for graphene is therefore not established.

major comments (2)
  1. [Section 2, Eq. (4)] The projectile potential in Eq. (4) is taken from the 3D ultrarelativistic calculation of Ref. [23], where the operator (1-α_z) appears because the electron's Dirac velocity along the beam is c α_z. For the 2D Dirac-Weyl Hamiltonian in Eq. (1), the velocity operator along the beam direction is v_F σ_z with v_F ≈ c/300. Minimal coupling to the Liénard-Wiechert potentials of the ion, with A_z ≈ A_0 in the ultrarelativistic limit, gives V = e φ σ_0 - e v_F σ_z A_z ≈ e φ (σ_0 - (v_F/c) σ_z), which is essentially proportional to σ_0, not to (1-σ_z). The paper does not derive Eq. (4) from the graphene Hamiltonian; it merely states 'Following the arguments in Ref. [23]'. Since the exact solution in Eq. (11) and the transition amplitude in Eq. (13) rely on the projector property of (1-σ_z), the replacement of (1-σ_z) by the correct near-scalar coupling would invalidate the exact solution. Consequently, the ionization cross sections in Table I are not predictions for graphene unless Eq. (4) is independently justified.
  2. [Section 2, Eqs. (10)-(11)] The light-cone integration that produces Eq. (11) implicitly assumes that the quasiparticle propagates with velocity c in the z direction. For the graphene Hamiltonian (1), the characteristic velocity along z is v_F, so the jump condition across the moving delta function δ(z-ct) contains c - v_F σ_z rather than c(1-σ_z). Thus the derivation is internally consistent only for an artificially ultrarelativistic 2D Dirac fermion with v_F = c, not for graphene. This is a separate reason that the claimed exact solution does not apply to the stated physical system.
minor comments (6)
  1. [Section 2, Eq. (4)] The delta-function potential in Eq. (4) has implicit dimensions; please specify the natural units used and the origin of the dimensional factor in the coefficient so that the expression is dimensionally transparent.
  2. [Section 3, Eqs. (14)-(16)] In Fig. 4 and the sum in Eq. (16), specify how many bound states are included in the numerical sum and demonstrate convergence of the result with respect to the number of states.
  3. [Table I] Provide the 3D cross-section values from Ref. [16] alongside the 2D values so that the 'two orders of magnitude larger' claim can be verified quantitatively.
  4. [Conclusions] There are several typos, for example 'formed offers' and 'extenstion' in the Conclusions; please proofread the manuscript carefully.
  5. [References] Reference [28] is missing the article title; please complete the bibliographic entry.
  6. [Fig. 1 caption] The caption contains the typo 'artifical' instead of 'artificial'; this should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transition amplitudes and cross sections are derived analytically from a stated model Hamiltonian, with no fitting to data and no load-bearing self-citation.

full rationale

The paper's derivation chain is conditional on a stated model: the graphene Coulomb-impurity Hamiltonian H0 in Eq. (1), with bound-state wave functions and energies taken from Novikov's independent solution, and the projectile potential V(x,z,t) in Eq. (4), taken from Baltz's 3D ultrarelativistic treatment. The transition amplitude Eq. (13), the probabilities Eqs. (14)-(16), and the cross sections in Table I are obtained by exact algebraic manipulation of these inputs, not by fitting any parameter to the paper's own predictions. There is no self-definitional step: Eq. (4) defines the interaction, and Eq. (13) is its mathematical consequence rather than a restatement of Eq. (4). There is no fitted input called a prediction: no parameter is calibrated to a subset of data and then used to predict a closely related quantity. The only self-citation, Ref. [9], appears inside a general reference list [2-9] for Coulomb impurities in graphene as a testing ground for relativistic quantum mechanics; it is not invoked as the source of the potential, the wave functions, or the exact-solution method, and it is not load-bearing. The concern that the (1-sigma_z) factor in Eq. (4) is imported from 3D collisions where the electron moves at c, whereas graphene quasiparticles move at v_F ~ c/300, is a physical modeling objection about whether Eq. (4) is the correct Hamiltonian for graphene; it is not a circularity, because the paper explicitly derives its results from Eq. (4) and does not claim Eq. (4) follows from those results. The derivation is self-contained relative to its assumptions, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model introduces no new particles or fields. The visible free parameters are model inputs chosen for the numerics, not fitted to data. The main ad hoc element is the (1-σz) coupling in the projectile potential.

free parameters (3)
  • M (gap) = 0.1 eV
    Chosen by hand for the numerical plots and table; the cross sections depend on it.
  • alpha_gr * ZT = 0.4
    Chosen by hand to stay below the supercritical threshold 1/2; binds the ground state energies.
  • ZP (projectile charge) = 1, 5, 10
    Control parameter in the table; cross sections scale strongly with it.
assumptions (5)
  • domain assumption Low-energy Dirac equation (1) with a single K valley and one spin polarization describes the graphene artificial atom for smooth perturbations on scales larger than the lattice constant.
    Invoked after Eq. (1); neglects intervalley scattering and spin effects.
  • domain assumption The projectile moves at exactly the speed of light, so its potential is a delta function in (z-ct) with the form (4).
    The paper states v ≈ c and uses the strict light-cone limit following Ref. [23].
  • ad hoc to paper The ion potential couples to the Dirac quasiparticle through the operator (1-σz).
    This form is taken from the 3D relativistic case, where it follows from eφ(1-αz); for graphene the minimal coupling gives approximately eφ σ0 because vF << c. The paper does not derive or justify this replacement.
  • standard math The bound states of H0 form a complete basis together with the continuum, so unitarity can be used to extract the ionization probability.
    Used in Eq. (16); standard for a self-adjoint Hamiltonian with a regular potential.
  • domain assumption The bound-state spectrum (3) from Novikov is correct for α_gr ZT < 1/2.
    Taken from Ref. [27] as an external input.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Coulomb impurities in graphene driven by fast ions." pith.science (2026). https://pith.science/paper/XNGPRM6B

@misc{pith2026241113429,
  author       = {Pith},
  title        = {Pith review of: Coulomb impurities in graphene driven by fast ions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XNGPRM6B}},
  note         = {Machine review of arXiv:2411.13429}
}
read the original abstract

We provide a theoretical model for electronic transitions in a two-dimensional (2D) artificial atom in a graphene monolayer. The artificial atom is due to the presence of a charged adatom (Coulomb impurity) in the layer and interacts with a fast ultrarelativistic ion moving parallel to the layer. We compute the probability and cross sections for the corresponding electronic transitions by means of an exact solution of the time-dependent 2D Dirac equation describing the interaction of the planar atom with the electromagnetic field of the ultrarelativistic projectile.

Figures

Figures reproduced from arXiv: 2411.13429 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic sketch of the geometry studied in this pa [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Excitation probability ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Survival probability ( [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Sketch for a possible experimental realization of th [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 42 canonical work pages

  1. [23]

    A. J. Baltz, M. J. Rhoades-Brown, and J. Weneser, Bound-electron-positron pair production in relativistic heavy-ion collisions, Phys. Rev. A 47, 3444 (1993)

    work page 1993

  2. [1]

    A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009)

    2009

  3. [2]

    In Figure 2, the survival probability ( 14) of the target ground state is plotted as a function of the impact pa- rameterb for different values of the projectile charge ZP

    Note the logarithmic scale for the probabilities. In Figure 2, the survival probability ( 14) of the target ground state is plotted as a function of the impact pa- rameterb for different values of the projectile charge ZP . We observe that the survival probability grows rapidly within a small range of b, saturating at the maximal value Pi(b) = 1 at large b...

    work page

  4. [3]

    Y. Wang, D. Wong, A. V. Shytov, V. W. Brar, S. Choi, Q. Wu, H.-Z. Tsai, W. Regan, A. Zettl, R. K. Kawakami, S. G. Louie, L. S. Levitov, and M. F. Crommie, Observ- ing Atomic Collapse Resonances in Artificial Nuclei on Graphene, Science 340, 734 (2013)

    work page 2013

  5. [4]

    Luican-Mayer, M

    A. Luican-Mayer, M. Kharitonov, G. Li, C.-P. Lu, I. Skachko, A.-M. B. Gon¸ calves, K. Watanabe, T. Taniguchi, and E. Y. Andrei, Screening Charged Impuri- ties and Lifting the Orbital Degeneracy in Graphene by Populating Landau Levels, Phys. Rev. Lett. 112, 036804 (2014)

    work page 2014

  6. [5]

    V. M. Pereira, J. Nilsson, and A. H. Castro Neto, Coulomb Impurity Problem in Graphene, Phys. Rev. Lett. 99, 166802 (2007)

    2007

  7. [6]

    Y. Wang, V. W. Brar, A. V. Shytov, Q. Wu, W. Regan, H.-Z. Tsai, A. Zettl, L. S. Levitov, and M. F. Crommie, Mapping Dirac quasiparticles near a single Coulomb im- purity on graphene , Nature Physics 8, 653 (2012)

    work page 2012

  8. [7]

    (9) In order to obtain the exact solution of Eq

    thus comes with the initial condition af (t → −∞) = δf,i . (9) In order to obtain the exact solution of Eq. ( 7) with V (x,z,t ) in Eq. ( 4), it suffices to determine Ψ( r,t ) near z =ct [23]. We therefore transform to light-cone coordi- nates, z− = 1√ 2 (ct −z), z + = 1√ 2 (ct +z). (10) For t < z/c but very close to t = z/c , by integrating Eq. ( 5) over z...

    work page

Show all 45 references
  1. [8]

    O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin, Su- percritical Coulomb center and excitonic instability in graphene, Phys. Rev. B 80, 165429 (2009)

    2009

  2. [9]

    De Martino, D

    A. De Martino, D. Kl¨ opfer, D. Matrasulov, and R. Egger, Electric-Dipole-Induced Universality for Dirac Fermions in Graphene, Phys. Rev. Lett. 112, 186603 (2014)

    2014

  3. [10]

    Kl¨ opfer, A

    D. Kl¨ opfer, A. De Martino, and R. Egger, Bound States and Supercriticality in Graphene-Based Topological In- sulators, Crystals 3, 14 (2013)

    2013

  4. [11]

    Rakhmanov, R

    S. Rakhmanov, R. Egger, and D. Matrasulov, Coulomb impurities in graphene driven by ultrashort electromag- netic pulses: excitation, ionization, and pair creation, Phys. Scr. 99, 075953 (2024)

    2024

  5. [12]

    Renk and K

    T. Renk and K. J. Eskola, Hard dihadron correlations in heavy-ion collisions at energies available at the BNL Relativistic Heavy Ion Collider and CERN Large Hadron Collider, Phys. Rev. C 84, 054913 (2011)

    2011

  6. [13]

    M. I. Abdulhamid, and et al., Exclusive J/ψ , ψ (2s) and e+e− pair production in Au+Au ultraperipheral colli- sions at the BNL Relativistic Heavy Ion Collider, Phys. Rev. C 110, 014911 (2024)

    2024

  7. [14]

    Adam and el al., Beam energy dependence of net-Λ fluctuations measured by the STAR experiment at the BNL Relativistic Heavy Ion Collider, Phys

    J. Adam and el al., Beam energy dependence of net-Λ fluctuations measured by the STAR experiment at the BNL Relativistic Heavy Ion Collider, Phys. Rev. C 102, 024903 (2020)

    2020

  8. [15]

    St¨ ohlker, Y

    T. St¨ ohlker, Y. A. Litvinov, and for the SPARC Collab- oration, Atomic physics experiments at the high energy storage ring, Physica Scripta 2015, 014025 (2015)

    2015

  9. [16]

    P. M. Hillenbrand, S. Hagmann, T. St¨ ohlker, Y. Litvinov, C. Kozhuharov, U. Spillmann, V. Shabaev, K. Stiebing, M. Lestinsky, A. Surzhykov, A. Voitkiv, B. Franzke, D. Fischer, C. Brandau, S. Schippers, A. Mueller, D. Schnei- der, D. Jakubassa, A. Artiomov, E. DeFilippo, X. Ma...

    2013

  10. [17]

    Hagmann, T

    S. Hagmann, T. St¨ ohlker, Y. Litvinov, C. Kozhuharov, P.-M. Hillenbrand, U. Spillmann, V. Shabaev, K. Stiebing, M. Lestinsky, A. Surzhykov, A. Voitkiv, B. Franzke, D. Fischer, D. Schneider, D. Jakubassa, A. Ar- tiomov, E. DeFilippo, X. Ma, R. D¨ orner, and H. Rothard, Few-bod...

    2013

  11. [18]

    A. J. Baltz, Exact Dirac-equation calculation of ioniza- tion induced by ultrarelativistic heavy ions, Phys. Rev. A 61, 042701 (2000)

    2000

  12. [19]

    A. J. Baltz, Some exact analytical results and a semiem- pirical formula for single electron ionization induced by ultrarelativistic heavy ions, Phys. Rev. A 64, 022718 (2001)

    2001

  13. [20]

    A. J. Baltz, Coulomb corrections in the calculation of ultrarelativistic heavy ion production of continuum electron-positron pairs, Phys. Rev. C 68, 034906 (2003)

    2003

  14. [21]

    A. J. Baltz, Calculation of heavy ion pair production to all orders in Zα , Phys. Rev. C 71, 024901 (2005)

    2005

  15. [22]

    A. J. Baltz, M. J. Rhoades-Brown, and J. Weneser, En- ergy dependence of bound-electron-positron pair produc- tion at very-high-energy ion-ion transits, Phys. Rev. A 44, 5569 (1991)

    1991

  16. [24]

    A. J. Baltz, Coulomb potential from a particle in uniform ultrarelativistic motion, Phys. Rev. A 52, 4970 (1995)

    1995

  17. [25]

    A. J. Baltz, Exact Dirac Equation Calculation of Ion- ization and Pair Production Induced by Ultrarelativistic Heavy Ions, Phys. Rev. Lett. 78, 1231 (1997)

    1997

  18. [26]

    A. J. Baltz and L. McLerran, Two center light cone cal- culation of pair production induced by ultrarelativistic heavy ions, Phys. Rev. C 58, 1679 (1998)

    1998

  19. [27]

    A. J. Baltz, M. J. Rhoades-Brown, and J. Weneser, Cal- culation of the cross section for e+-e− (K orbit) pairs by very-high-energy fully stripped heavy ions at perturba- tional impact parameters, Phys. Rev. A 48, 2002 (1993)

    1993

  20. [28]

    V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, Electron-Electron Interactions in Graphene: Current Status and Perspectives, Rev. Mod. Phys. 84, 1067 (2012)

    2012

  21. [29]

    D. S. Novikov, Elastic scattering theory and transport in graphene, Phys. Rev. B 76, 245435 (2007)

    2007

  22. [30]

    V. R. Khalilov and C.-L. Ho, Mod. Phys. Lett. A 13, 615 (1998)

    1998

  23. [31]

    Allor, T

    D. Allor, T. D. Cohen, and D. A. McGady, Schwinger mechanism and graphene, Phys. Rev. D 78, 096009 (2008)

    2008

  24. [32]

    Lewkowicz and B

    M. Lewkowicz and B. Rosenstein, Dynamics of Particle- Hole Pair Creation in Graphene, Phys. Rev. Lett. 102, 106802 (2009)

    2009

  25. [33]

    Lewkowicz, H

    M. Lewkowicz, H. C. Kao, and B. Rosenstein, Signature of the Schwinger pair creation rate via radiation gener- ated in graphene by a strong electric current, Phys. Rev. B 84, 035414 (2011)

    2011

  26. [34]

    G. L. Klimchitskaya and V. M. Mostepanenko, Creation 6 of quasiparticles in graphene by a time-dependent electric field, Phys. Rev. D 87, 125011 (2013)

    2013

  27. [35]

    Fillion-Gourdeau and S

    F. Fillion-Gourdeau and S. MacLean, Time-dependent pair creation and the Schwinger mechanism in graphene, Phys. Rev. B 92, 035401 (2015)

    2015

  28. [36]

    I. Akal, R. Egger, C. M¨ uller, and S. Villalba-Ch´ avez, Sim- ulating dynamically assisted production of Dirac pairs in gapped graphene monolayers, Phys. Rev. D 99, 016025 (2019)

    2019

  29. [37]

    I. Akal, R. Egger, C. M¨ uller, and S. Villalba-Ch´ avez, Low-dimensional approach to pair production in an os- cillating electric field: Application to bandgap graphene layers, Phys. Rev. D 93, 116006 (2016)

    2016

  30. [38]

    Golub, R

    A. Golub, R. Egger, C. M¨ uller, and S. Villalba-Ch´ avez, Dimensionality-driven photoproduction of massive dirac pairs near threshold in gapped graphene monolayers, Phys. Rev. Lett. 124, 110403 (2020)

    2020

  31. [39]

    Villalba-Ch´ avez, O

    S. Villalba-Ch´ avez, O. Mathiak, R. Egger, and C. M¨ uller, Light-amplified Landau-Zener conductivity in gapped graphene monolayers: A simulacrum of photocatalyzed vacuum instability, Phys. Rev. D 108, 116007 (2023)

    2023

  32. [40]

    Eichler and T

    J. Eichler and T. St¨ ohlker, Radiative electron capture in relativistic ion-atom collisions and the photoelectric effect in hydrogen-like high- Z systems, Phys. Rep. 439, 1 (2001)

    2001

  33. [41]

    Eichler, Theory of relativistic ion-atom collisions, Phys

    J. Eichler, Theory of relativistic ion-atom collisions, Phys. Rep. 193, 165 (1990)

    1990

  34. [42]

    Dewangan and J

    D. Dewangan and J. Eichler, Charge exchange in ener- getic ion-atom collisions, Phys. Rep. 247, 59 (1994)

    1994

  35. [43]

    Eichler, Relativistic atomic collisions (San Diego: Aca - demic Press, 1995)

    J. Eichler, Relativistic atomic collisions (San Diego: Aca - demic Press, 1995)

    1995

  36. [44]

    Asorey and A

    M. Asorey and A. Santagata, The critical transition of coulomb impurities in gapped graphene, J. High Energ. Phys. 2020, 144 (2020)

    2020

  37. [45]

    V. M. Kuleshov, V. D. Mur, and N. B. Narozhny, Coulomb problem for graphene with supercritical impu- rity, J. Phys.: Conf. Ser. 788, 012044 (2017)

    2017

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.