V.S. Popovc aNational Research Nuclear University ‘MEPhI’, Kashirskoe shosse 31, Moscow, 115409, Russian Federation bInstitute of Spectroscopy, Russian Academy of Sciences, ul. Fizicheskaya 5, Troitsk, Moscow, 108840, Russian Federation cRussian 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.
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/ 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)