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A new method for solving the Z > 137 problem and determining hydrogen-like energy levels

All-Russian Scientific Research Institute of Experimental Physics — Federal Nuclear Centre, prosp. Mira 37, Sarov, Nizhny Novgorod region, 607190, Russian Federation

A new method for including finite nuclear size effects is suggested to overcome the “Z > 137 catastrophe” encountered in solving the Dirac equation for an electron in the field of a point charge Ze. In this method, the boundary condition for the numerical solution of the equations for the Dirac radial wave functions is taken to be that the components of the electron current density are zero at the boundary of the nucleus. As a result, for all of the nuclei of the periodic table the calculated energy levels practically coincide with those obtained in a standard way from the Dirac equation for a Coulomb point charge potential. For Z > 105, the calculated energy level functions E(Z) prove to be smooth and monotonic. The ground energy level reaches E = −mc2 (i.e., the electron drops into the nucleus) at Zc = 178. The proposed method for accounting for the finite size of nuclei can be useful in numerically calculating the energy levels of many-electron atoms.

Fulltext is available at IOP
PACS: 03.65.Ge, 03.65.Pm, 12.20.−m (all)
DOI: 10.3367/UFNe.0184.201402i.0200
Citation: Neznamov V P, Safronov I I "A new method for solving the Z > 137 problem and determining hydrogen-like energy levels" Phys. Usp. 57 189–193 (2014)
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Received: 1st, November 2013, 5th, June 2013

Оригинал: Незнамов В П, Сафронов И И «Новый метод решения проблемы ‘Z > 137’ и определения уровней энергии водородоподобных атомов» УФН 184 200–205 (2014); DOI: 10.3367/UFNr.0184.201402i.0200

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