RSS feeds
ÐóññêèéEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2015

 / 

August

←  →

Physics of our days


Coulomb problem for a Z>Zcr

 a,  a,  a,  a,  b,  c
a National Research Nuclear University ‘MEPhI’, Kashirskoe shosse 31, Moscow, 115409, Russian Federation
b Institute of Spectroscopy, Russian Academy of Sciences, ul. Fizicheskaya 5, Troitsk, Moscow, 108840, Russian Federation
c Russian Federation State Scientific Center ‘A.I. Alikhanov Institute of Theoretical and Experimental Physics’, ul. Bolshaya Cheremushkinskaya 25, Moscow, 117259, Russian Federation

A closed form equation is derived for the critical nucleus charge, i.e. for the value Z=Zcr at which the discrete level with the Dirac quantum number touches the lower continuum of Dirac equation solutions. For a Coulomb potential cut off rectangularly at a small distance r0 = Rħ/(mc), R ⪡ 1, the critical nucleus charge values are obtained for several values of κ and R. It is shown that the partial scattering matrix of the elastic positron-nucleus scattering, Sκ = exp (2iδκ(εp)), is also unitary for Z>Zcr. For this range, the scattering phase δκ(εp) is calculated as a function of the positron energy Ep = εpmc2, as are the positions and widths of quasidiscrete levels corresponding to the scattering matrix poles. The implication is that the single-particle approximation for the Dirac equation is valid not only for ZZcr and that there is no spontaneous creation of e+e− pairs from vacuum.

Fulltext pdf (529 KB)
Fulltext is also available at DOI: 10.3367/UFNe.0185.201508d.0845
Keywords: the Coulomb problem, pointlike nucleus, boundary conditions, critical charge, scattering phase, Breit—Wigner resonance
PACS: 03.65.Pm, 12.20.−m, 73.22.Pr (all)
DOI: 10.3367/UFNe.0185.201508d.0845
URL: https://ufn.ru/en/articles/2015/8/d/
000364717300004
2-s2.0-84947746838
2015PhyU...58..785K
Citation: Kuleshov V M, Mur V D, Narozhnyi N B, Fedotov A M, Lozovik Yu E, Popov V S "Coulomb problem for a Z>Zcr" Phys. Usp. 58 785–791 (2015)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 16th, June 2015, 23rd, June 2015

Îðèãèíàë: Êóëåøîâ Â Ì, Ìóð Â Ä, Íàðîæíûé Í Á, Ôåäîòîâ À Ì, Ëîçîâèê Þ Å, Ïîïîâ Â Ñ «Êóëîíîâñêàÿ çàäà÷à ñ çàðÿäîì ÿäðà Z>Zcr» ÓÔÍ 185 845–852 (2015); DOI: 10.3367/UFNr.0185.201508d.0845

References (46) ↓ Cited by (32) Similar articles (5)

  1. Sommerfeld A Ann. Physik 51 1 (1916)
  2. Dirac P A M Proc. R. Soc. Lond. A 117 610 (1928)
  3. Dirac P A M Proc. R. Soc. Lond. A 118 351 (1928)
  4. Darwin C G Proc. R. Soc. Lond. A 118 654 (1928)
  5. Gordon W Z. Phys. 48 11 (1928)
  6. Bethe H Quantenmechanik Der Ein Und Zwei Elektronenprobleme (Handbuch Der Physik, Bd. 24, Tl. 1) (Berlin: Springer, 1933)
  7. Berestetskii V B, Lifshits E M, Pitaevskii L P Kvantovaya Elektrodinamika (M.: Nauka, 1980); Berestetskii V B, Lifshitz E M, Pitaevskii L P Quantum Electrodynamics (Oxford: Butterworth-Heinemann, 1999)
  8. Dirac P A M The Principles Of Quantum Mechanics (Oxford: Clarendon Press, 1958); Dirak P Printsipy Kvantovoi Mekhaniki (M.: Nauka, 1979)
  9. Voronov B L, Gitman D M, Tyutin I V Teor. Mat. Fiz. 150 41 (2007); Voronov B L, Gitman D M, Tyutin I V Theor. Math. Phys. 150 34 (2007)
  10. Pomeranchuk I, Smorodinsky Ya J. Phys. USSR 9 97 (1945)
  11. Popov V S Pis’ma ZhETF 11 254 (1970); Popov V S JETP Lett. 11 162 (1970)
  12. Popov V S Yad. Fiz. 12 429 (1970); Popov V S Sov. J. Nucl. Phys. 12 235 (1971)
  13. Schiff L I, Snyder H, Weinberg J Phys. Rev. 57 315 (1940)
  14. Gershtein S S, Zel’dovich Ya B Zh. Eksp. Teor. Fiz. 57 654 (1969); Gershtein S S, Zel’dovich Ya B Sov. Phys. JETP 30 358 (1970)
  15. Gershtein S S, Zel’dovich Ya B Nuovo Cimento Lett. 1 835 (1969)
  16. Pieper W, Greiner W Z. Phys. 218 327 (1969)
  17. Popov V S Zh. Eksp. Teor. Fiz. 59 965 (1970); Popov V S Sov. Phys. JETP 32 526 (1971)
  18. Popov V S Zh. Eksp. Teor. Fiz. 60 1228 (1971); Popov V S Sov. Phys. JETP 33 665 (1971)
  19. Zel’dovich Ya B, Pitaevskii L P, Popov V S, Starobinskii A A Usp. Fiz. Nauk 105 780 (1971); Zel’dovich Ya B, Pitaevskii L P, Popov V S, Starobinskii A A Sov. Phys. Usp. 14 811 (1972)
  20. Zel’dovich Ya B, Popov V S Usp. Fiz. Nauk 105 403 (1971); Zel’dovich Ya B, Popov V S Sov. Phys. Usp. 14 673 (1972)
  21. Popov V S, Karnakov B M Usp. Fiz. Nauk 184 27 (2014); Popov V S, Karnakov B M Phys. Usp. 57 257 (2014)
  22. Gershtein S S, Popov V S Lett. Nuovo Cimento 6 593 (1973)
  23. Okun L B Comments Nucl. Part. Phys. 6 25 (1974)
  24. Greiner W, Müller B, Rafelski J Quantum Electrodynamics Of Strong Fields (Berlin: Springer-Verlag, 1985)
  25. Grib A A, Mamaev S G, Mostepanenko V M Vakuumnye Kvantovye Effekty v Sil’nykh Polyakh (M.: Energoatomizdat, 1988)
  26. Novoselov K S et al. Nature 438 197 (2005)
  27. Pereira V M, Kotov V N, Castro Neto A H Phys. Rev. B 78 085101 (2008)
  28. Castro Neto A H et al. Rev. Mod. Phys. 81 109 (2009)
  29. Kuleshov V M i dr. Pis’ma ZhETF 101 282 (2015); Kuleshov V M et al. JETP Lett. 101 264 (2015)
  30. Richtmyer R D Principles Of Advanced Mathematical Physics (New York: Springer-Verlag, 1978); Rikhtmaier R Printsipy Sovremennoi Matematicheskoi Fiziki (M.: Mir, 1982)
  31. Von Neumann J V Mathematische Grundlagen Der Quantenmechanik (Berlin: J. Springer, 1932); Von Neumann J V Mathematical Foundations Of Quantum Mechanics (Princeton, N.J.: Princeton Univ. Press, 1955); fon Neiman Dzh Matematicheskie Osnovy Kvantovoi Mekhaniki (M.: Mir, 1964)
  32. Case K M Phys. Rev. 80 797 (1950)
  33. Bateman H Higher Transcendental Functions Vol. 1, 2 (Director A Erdélyi) (New York: McGraw-Hill, 1953, 1954); Beitmen G, Erdeii A Vysshie Transtsendentnye Funktsii Vol. 1, 2 (M.: Nauka, 1973, 1974)
  34. Taylor J R Scattering Theory. The Quantum Theory On Nonrelativistic Collisions (New York: Wiley, 1972); Teilor Dzh Teoriya Rasseyaniya. Kvantovaya Teoriya Nerelyativistskikh Stolknovenii (M.: Mir, 1975)
  35. Mur V D, Popov V S Teor. Mat. Fiz. 27 204 (1976); Mur V D, Popov V S Theor. Math. Phys. 27 429 (1976)
  36. Popov V S, Eletskii V L, Mur V D Zh. Eksp. Teor. Fiz. 71 856 (1976); Popov V S, Eletskii V L, Mur V D Sov. Phys. JETP 44 451 (1976)
  37. Furry W H Phys. Rev. 81 115 (1951)
  38. Klein O Z. Phys. 53 157 (1929)
  39. Nikishov A I Zh. Eksp. Teor. Fiz. 57 1210 (1970); Nikishov A I Sov. Phys. JETP 30 660 (1970)
  40. Nikishov A I Nucl. Phys. B 21 346 (1970)
  41. Narozhnyi N B, Nikishov A I Yad. Fiz. 11 1071 (1970)
  42. Hansen A, Ravndal F Phys. Scripta 23 1036 (1981)
  43. Migdal A B Kachestvennye Metody v Kvantovoi Teorii (M.: Nauka, 1975); Migdal A B Qualitative Methods In Quantum Theory (Cambridge, Mass.: Advanced Book Program, Perseus Pub., 2000)
  44. Popov V S Yad. Fiz. 64 421 (2001); Popov V S Phys. Atom. Nucl. 64 367 (2001)
  45. Khriplovich I B Pis’ma ZhETF 100 552 (2014); Khriplovich I B JETP Lett. 100 494 (2014)
  46. Zhou S Y et al. Nature Mater. 6 770 (2007)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions