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Influence of the nuclear magnetic field on electron-positron pair production in low-energy heavy-nuclei collisions

N. K. Dulaev1,*, D. A. Telnov1,†, V. M. Shabaev1,2,‡, Y. S. Kozhedub1, X. Ma3, I. A. Maltsev1, R. V. Popov2,1, and I. I. Tupitsyn1

  • *Contact author: st069071@student.spbu.ru
  • †Contact author: d.telnov@spbu.ru
  • ‡Contact author: v.shabaev@spbu.ru
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Phys. Rev. D 112, 116016 – Published 19 December, 2025

DOI: https://doi.org/10.1103/bnrx-lhf1

Abstract

Total and energy-angle differential probabilities of positrons created in slow collisions of two identical nuclei are calculated within a relativistic two-center approach. The time-dependent Dirac equation is solved in the rotating frame using the generalized pseudospectral method in modified prolate spheroidal coordinates. The magnetic interaction induced by the motion of the nuclei is included in the Hamiltonian. The rotational coupling term is also taken into account. Angle-integrated and angle-resolved energy spectra of the emitted positrons are calculated by projecting the propagated wave function onto positive-energy plane-wave states. Our results show that the magnetic interaction leads to a slight increase in the critical internuclear distance and enhances the total positron yield by up to several percentages. However, it does not qualitatively alter the energy or angular distributions of emitted positrons. The angular distributions remain nearly isotropic. Most important, the characteristic signatures of the supercritical regime identified in the previous calculations neglecting the magnetic interaction are preserved and can help to detect spontaneous vacuum decay.

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References (73)

  1. W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer-Verlag, Berlin, 1985).
  2. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Advances in QED with intense background fields, Phys. Rep. 1010, 1 (2023).
  3. I. Y. Kostyukov, Physics of extremely strong electromagnetic field: Status and prospects, Bulletin of the Lebedev Physics Institute 51, S653 (2024).
  4. I. Pomeranchuk and J. Smorodinsky, On the energy levels of systems with Z>137, J. Phys. USSR 9, 97 (1945).
  5. S. S. Gershtein and Y. B. Zeldovich, Positron production during the mutual approach of heavy nuclei and the polarization of the vacuum, Zh. Eksp. Teor. Fiz. 57, 654 (1969) [Sov. Phys. JETP 30, 358 (1970)].
  6. W. Pieper and W. Greiner, Interior electron shells in superheavy nuclei, Z. Phys. A 218, 327 (1969).
  7. V. S. Popov, Electron energy levels at Z>137, Pis’ma Zh. Eksp. Teor. Fiz. 11, 254 (1970) [JETP Lett. 11, 162 (1970)]; Collapse to center at Z greater than 137 and critical nuclear charge, Yad. Fiz. 12, 429 (1970) [Sov. J. Nucl. Phys. 12, 235 (1971)]; Position production in a coulomb field with Z>137, Zh. Eksp. Teor. Fiz. 59, 965 (1970) [Sov. Phys. JETP 32, 526 (1971)]; On the properties of the discrete spectrum for Z close to 137, Zh. Eksp. Teor. Fiz. 60, 1228 (1971) [Sov. Phys. JETP 33, 665 (1971)].
  8. Y. B. Zeldovich and V. S. Popov, Electronic structure of superheavy atoms, Usp. Fiz. Nauk 105, 403 (1971) [Sov. Phys. Usp. 14, 673 (1972)].
  9. B. Müller, H. Peitz, J. Rafelski, and W. Greiner, Solution of the Dirac equation for strong external fields, Phys. Rev. Lett. 28, 1235 (1972); B. Müller, J. Rafelski, and W. Greiner, Electron shells in over-critical external fields, Z. Phys. A 257, 62 (1972).
  10. V. D. Mur and V. S. Popov, Bound states near the limit of the lower continuum, Teor. Mat. Fiz. 27, 204 (1976) [Theor. Math. Phys. 27, 429 (1976)].
  11. V. S. Popov, V. L. Eletsky, and V. D. Mur, Properties of deep-lying levels in a strong electrostatic field, Zh. Eksp. Teor. Fiz. 71, 856 (1976) [Sov. Phys. JETP 44, 451 (1976)].
  12. B. Müller, Positron creation in superheavy quasi-molecules, Annu. Rev. Nucl. Sci. 26, 351 (1976).
  13. J. Reinhardt and W. Greiner, Quantum electrodynamics of strong fields, Rep. Prog. Phys. 40, 219 (1977).
  14. G. Soff, J. Reinhardt, B. Müller, and W. Greiner, Shakeoff of the vacuum polarization in quasimolecular collisions of very heavy ions, Phys. Rev. Lett. 38, 592 (1977).
  15. J. Rafelski, L. P. Fulcher, and A. Klein, Fermions and bosons interacting with arbitrarily strong external fields, Phys. Rep. 38, 227 (1978).
  16. V. S. Popov, D. N. Voskresensky, V. L. Eletskii, and V. D. Mur, WKB method at Z>137 and its applications to the theory of supercritical atoms, Zh. Eksp. Teor. Fiz. 76, 431 (1979) [Sov. Phys. JETP 49, 218 (1979)].
  17. K. Smith, H. Peitz, B. Müller, and W. Greiner, Induced decay of the neutral vaccum in overcritical fields occurring in heavy-ion collisions, Phys. Rev. Lett. 32, 554 (1974).
  18. J. Reinhardt, B. Müller, and W. Greiner, Theory of positron production in heavy-ion collisions, Phys. Rev. A 24, 103 (1981).
  19. U. Müller, T. de Reus, J. Reinhardt, B. Müller, W. Greiner, and G. Soff, Positron production in crossed beams of bare uranium nuclei, Phys. Rev. A 37, 1449 (1988).
  20. F. Bosch and B. Müller, Positron creation in heavy-ion collisions, Prog. Part. Nucl. Phys. 16, 195 (1986).
  21. U. Müller-Nehler and G. Soff, Electron excitations in superheavy quasimolecules, Phys. Rep. 246, 101 (1994).
  22. J. Reinhardt and W. Greiner, Supercritical fields and the decay of the vacuum, in Proceeding of the Memorial Symposium for Gerhard Sof, edited by W. Greiner and J. Reinhardt (EP Systema, Budapest, 2005), pp. 181–192.
  23. J. Rafelski, J. Kirsch, B. Müller, J. Reinhardt, and W. Greiner, Probing QED vacuum with heavy ions, in New Horizons in Fundamental Physics, edited by S. Schramm and M. Schäfer (Springer International Publishing, Cham, 2017), pp. 211–251.
  24. A. Gumberidze, T. Stöhlker, H. Beyer, F. Bosch, A. Bräuning-Demian, S. Hagmann, C. Kozhuharov, T. Kühl, R. Mann, P. Indelicato, W. Quint, R. Schuch, and A. Warczak, X-ray spectroscopy of highly-charged heavy ions at FAIR, Nucl. Instrum. Methods Phys. Res., Sect. B 267, 248 (2009).
  25. M. Lestinsky et al., Physics book: CRYRING@ESR, Eur. Phys. J. Special Topics 225, 797 (2016).
  26. M. Lestinsky, E. B. Menz, H. Danared, C. Krantz, E. Lindroth, Z. Andelkovic, C. Brandau, A. Bräuning-Demian, S. Fedotova, W. Geithner et al., First experiments with CRYRING@ESR, Atoms 10, 141 (2022).
  27. X. Ma et al., HIAF: New opportunities for atomic physics with highly charged heavy ions, Nucl. Instrum. Methods Phys. Res., Sect. B 408, 169 (2017).
  28. X. Zhou and J. Yang (The HIAF Project Team Collaboration), Status of the high-intensity heavy-ion accelerator facility in China, AAPPS Bull. 32, 35 (2022).
  29. G. M. Ter-Akopian, W. Greiner, I. N. Meshkov, Y. T. Oganessian, J. Reinhardt, and G. V. Trubnikov, Layout of new experiments on the observation of spontaneous electron–positron pair creation in supercritical Coulomb fields, Int. J. Mod. Phys. E 24, 1550016 (2015).
  30. G. Trubnikov, A. Butenko, V. Golovatyuk, A. Guskov, M. Kapishin, V. Kekelidze, R. Lednicky, I. Meshkov, and A. Sorin, NICA heavy-ion collider at JINR (Dubna). Status of accelerator complex and first physics at NICA., J. Phys. Conf. Ser. 2586, 012013 (2023).
  31. E. Ackad and M. Horbatsch, Numerical calculation of supercritical Dirac resonance parameters by analytic continuation methods, Phys. Rev. A 75, 022508 (2007).
  32. S. I. Godunov, B. Machet, and M. I. Vysotsky, Resonances in positron scattering on a supercritical nucleus and spontaneous production of e+e− pairs, Eur. Phys. J. C 77, 782 (2017).
  33. E. Ackad and M. Horbatsch, Supercritical Dirac resonance parameters from extrapolated analytic continuation methods, Phys. Rev. A 76, 022503 (2007).
  34. A. Marsman and M. Horbatsch, Calculation of supercritical Dirac resonance parameters for heavy-ion systems from a coupled-differential-equation approach, Phys. Rev. A 84, 032517 (2011).
  35. P. Grashin and K. Sveshnikov, Vacuum polarization energy decline and spontaneous positron emission in QED under coulomb supercriticality, Phys. Rev. D 106, 013003 (2022).
  36. A. Krasnov and K. Sveshnikov, Non-perturbative effects in the QED-vacuum energy exposed to the supercritical Coulomb field, Mod. Phys. Lett. A 37, 2250136 (2022).
  37. D. N. Voskresensky, Electron-positron vacuum instability in strong electric fields. relativistic semiclassical approach, Universe 7, 104 (2021).
  38. I. A. Maltsev, V. M. Shabaev, V. A. Zaytsev, R. V. Popov, Y. S. Kozhedub, and D. A. Tumakov, Calculation of the energy and width of supercritical resonance in a uranium quasimolecule, Opt. Spectrosc. 128, 1100 (2020).
  39. D. A. Telnov, N. K. Dulaev, Y. S. Kozhedub, I. A. Maltsev, R. V. Popov, I. I. Tupitsyn, and V. M. Shabaev, Positron supercritical resonances and spontaneous positron creation in slow collisions of heavy nuclei, Phys. At. Nucl. 87, 319 (2024).
  40. I. A. Maltsev, V. M. Shabaev, I. I. Tupitsyn, A. I. Bondarev, Y. S. Kozhedub, G. Plunien, and T. Stöhlker, Electron-positron pair creation in low-energy collisions of heavy bare nuclei, Phys. Rev. A 91, 032708 (2015).
  41. I. A. Maltsev, V. M. Shabaev, R. V. Popov, Y. S. Kozhedub, G. Plunien, X. Ma, T. Stöhlker, and D. A. Tumakov, How to observe the vacuum decay in low-energy heavy-ion collisions, Phys. Rev. Lett. 123, 113401 (2019).
  42. R. V. Popov, V. M. Shabaev, D. A. Telnov, I. I. Tupitsyn, I. A. Maltsev, Y. S. Kozhedub, A. I. Bondarev, N. V. Kozin, X. Ma, G. Plunien et al., How to access QED at a supercritical Coulomb field, Phys. Rev. D 102, 076005 (2020).
  43. I. Maltsev, V. Shabaev, I. Tupitsyn, Y. Kozhedub, G. Plunien, and T. Stöhlker, Pair production in low-energy collisions of uranium nuclei beyond the monopole approximation, Nucl. Instrum. Methods Phys. Res., Sect. B 408, 97 (2017).
  44. I. A. Maltsev, V. M. Shabaev, R. V. Popov, Y. S. Kozhedub, G. Plunien, X. Ma, and T. Stöhlker, Electron-positron pair production in slow collisions of heavy nuclei beyond the monopole approximation, Phys. Rev. A 98, 062709 (2018).
  45. R. V. Popov, A. I. Bondarev, Y. S. Kozhedub, I. A. Maltsev, V. M. Shabaev, I. I. Tupitsyn, X. Ma, G. Plunien, and T. Stöhlker, One-center calculations of the electron-positron pair creation in low-energy collisions of heavy bare nuclei, Eur. Phys. J. D 72, 115 (2018).
  46. R. V. Popov, V. M. Shabaev, I. A. Maltsev, D. A. Telnov, N. K. Dulaev, and D. A. Tumakov, Spontaneous vacuum decay in low-energy collisions of heavy nuclei beyond the monopole approximation, Phys. Rev. D 107, 116014 (2023).
  47. H. Backe, L. Handschug, F. Hessberger, E. Kankeleit, L. Richter, F. Weik, R. Willwater, H. Bokemeyer, P. Vincent, Y. Nakayama et al., Observation of positron creation in superheavy ion-atom collision systems, Phys. Rev. Lett. 40, 1443 (1978).
  48. C. Kozhuharov, P. Kienle, E. Berdermann, H. Bokemeyer, J. S. Greenberg, Y. Nakayama, P. Vincent, H. Backe, L. Handschug, and E. Kankeleit, Positrons from 1.4-GeV uranium-atom collisions, Phys. Rev. Lett. 42, 376 (1979).
  49. N. K. Dulaev, D. A. Telnov, V. M. Shabaev, Y. S. Kozhedub, I. A. Maltsev, R. V. Popov, and I. I. Tupitsyn, Angular and energy distributions of positrons created in subcritical and supercritical slow collisions of heavy nuclei, Phys. Rev. D 109, 036008 (2024).
  50. N. K. Dulaev, D. A. Telnov, V. M. Shabaev, Y. S. Kozhedub, X. Ma, I. A. Maltsev, R. V. Popov, and I. I. Tupitsyn, Three-dimensional calculations of positron creation in supercritical collisions of heavy nuclei, Phys. Rev. D 111, 016018 (2025).
  51. R. V. Popov, V. M. Shabaev, I. A. Maltsev, D. A. Telnov, N. K. Dulaev, A. D. Mironov, and A. M. Ryzhkov, Influence of the internuclear axis rotation on the probabilities of pair production in collisions of heavy nuclei, Phys. Part. Nucl. Lett. 22, 374 (2025).
  52. N. K. Dulaev, D. A. Telnov, R. V. Popov, V. M. Shabaev, Y. S. Kozhedub, X. Ma, I. A. Maltsev, A. D. Mironov, and I. I. Tupitsyn, Effect of electron–electron interaction on pair production in supercritical collisions of highly charged ions, Chin. Phys. C 49, 094105 (2025).
  53. J. Rafelski and B. Müller, Magnetic splitting of quasimolecular electronic states in strong fields, Phys. Rev. Lett. 36, 517 (1976).
  54. W. Greiner, B. Müller, and G. Soff, Delta electrons as a probe of strong magnetic fields in heavy ion collisions, Phys. Lett. A 69, 27 (1978).
  55. G. Soff, J. Reinhardt, and W. Greiner, Spin polarization of electrons induced by strong collisional magnetic fields, Phys. Rev. A 23, 701 (1981).
  56. G. Soff, J. Reinhardt, and W. Greiner, 1sσ ionization in high-energy collisions of very heavy ions, Phys. Lett. 83A, 158 (1981).
  57. G. Soff and J. Reinhardt, Positron creation in heavy ion collisions the influence of the magnetic field, Phys. Lett. B 211, 179 (1988).
  58. L. R. B. Elton, Nuclear Sizes (Oxford University Press, New York, 1961).
  59. I. Angeli and K. Marinova, Table of experimental nuclear ground state charge radii: An update, At. Data Nucl. Data Tables 99, 69 (2013).
  60. B. Müller, J. Rafelski, and W. Greiner, Auto-ionization of positrons in heavy ion collisions, Z. Phys. 257, 183 (1972).
  61. U. Heinz, W. Greiner, and B. Müller, Electron-translation effects in heavy-ion scattering, Phys. Rev. A 23, 562 (1981).
  62. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
  63. L. D. Landau and E. M. Lifshitz, Classical Theory of Fields, 4th ed. (Butterworth Heinemann, Amsterdam, 1975).
  64. B. Müller and W. Greiner, The two centre Dirac equation, Z. Naturforsch. A 31, 1 (1976).
  65. D. A. Telnov and S.-I. Chu, Ab initio study of the orientation effects in multiphoton ionization and high-order harmonic generation from the ground and excited electronic states of H2+, Phys. Rev. A 76, 043412 (2007).
  66. D. A. Telnov and S.-I. Chu, Effects of multiple electronic shells on strong-field multiphoton ionization and high-order harmonic generation of diatomic molecules with arbitrary orientation: An all-electron time-dependent density-functional approach, Phys. Rev. A 80, 043412 (2009).
  67. D. A. Telnov, D. A. Krapivin, J. Heslar, and S.-I. Chu, Multiphoton ionization of one-electron relativistic diatomic quasimolecules in strong laser fields, J. Phys. Chem. A 122, 8026 (2018).
  68. D. A. Telnov and S.-I. Chu, Relativistic ionization probabilities and photoelectron distributions of hydrogenlike ions in superstrong electromagnetic fields, Phys. Rev. A 104, 023111 (2021).
  69. J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Math. Proc. Cambridge Philos. Soc. 43, 50 (1947).
  70. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Pergamon Press, Oxford, 1982).
  71. V. Galitski, B. Karnakov, V. Kogan, and V. Galitski, Jr., Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers (Oxford University Press, Oxford, 2013).
  72. https://rscf.ru/project/22-62-00004/.
  73. A. Anikina, D. Belyakov, T. Bezhanyan, M. Kirakosyan, A. Kokorev, M. Lyubimova, M. Matveev, D. Podgainy, A. Rahmonova, S. Shadmehri et al., Structure and features of the software and information environment of the HybriLIT heterogeneous platform, in Distributed Computer and Communication Networks, edited by V. M. Vishnevsky, K. E. Samouylov, and D. V. Kozyrev (Springer Nature Switzerland, Cham, 2025), pp. 444–457.

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Phys. Rev. D 112, 116016– Published 19 December, 2025
  • Received 10 September 2025
  • Accepted 17 November 2025
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