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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 ofTheoretical 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 (2iδκp)), is also unitary for Z>Zcr. For this range, the scattering phase δκp) 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.

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

:   ,   ,   ,   ,   ,    « Z>Zcr» 185 845–852 (2015); DOI: 10.3367/UFNr.0185.201508d.0845

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