Relativistic and nonrelativistic finite-basis-set calculations of low-lying levels of hydrogenic atoms in intense magnetic fields

Abstract

A finite-basis-set method is used to calculate relativistic and nonrelativistic binding energies of an electron in a static Coulomb field and in magnetic fields of arbitrary strength (0 ( B & 10 G).The basis set is composed of products of Slaterand Landau-type functions, and it contains the exact solutions at both the Coulomb limit (B = 0) and the Landau limit (Z = 0). Relativistic variational collapse is avoided and highly accurate results are obtained with the basis set. The relativistic corrections obtained for intense magnetic fields (B + 10 G) differ from the previous relativistic calculations based on the adiabatic approximations.It is found that the sign of the relativistic correction changes from negative to positive near B 10 ' 0 for the ground state and near B 10 G for the 2psyz(ti = -3/2) excited state of hydrogen. The method is checked to be very accurate by means of the virial theorem, sum rules, and the relativistic low-B limit where comparison can be made with perturbation results. In the nonrelativistic limit of the Dirac equation, our results agree with other accurate nonrelativistic calculations available and with our own calculations based on the Schrodinger equation, which converge to more significant digits than previous calculations for the whole range of magnetic fields.PACS number(s): 32.60.+i, 31.30.Jv, 31.10.+z

Ground states of H, He,…, Ne, and their singly positive ions in strong magnetic fields: The high-field regime

“…Relativistic calculations for the hydrogen atom and hydrogen-like ions were performed by Lindgren and Virtamo [22] and Chen and Goldman [23]. Our considerations are based on the work by Chen and Goldman [23] which contains results for the 1s and 2p −1 states for a broad range of magnetic field strengths.…”

“…Our considerations are based on the work by Chen and Goldman [23] which contains results for the 1s and 2p −1 states for a broad range of magnetic field strengths. Interpolating their results for the 1s state and using well known scaling transformations we can conclude that in the least favorable case of Z = 10 relativistic corrections δE = (E relativistic − E non−relativistic )/ |E non−relativistic | have to be of the order 4 · 10 −4 for γ = 200 and 2 · 10 −4 for γ = 10 4 .…”

Dual-kinetic-balance approach to the Dirac equation for axially symmetric systems: Application to static and time-dependent fields

“…The basis set of 78 radial B-splines of order k = 9 and 17 Legendre polynomials (of orders from 0 to 16) is enough to obtain all the results presented in Tables I − VI. [37,38] (they both coincide to all the presented digits) is given. The complete perturbation-theory result ∆EPT and the individual contributions are listed as well.…”

Hydrogen atom in strong magnetic field: A high accurate calculation in spheroidal coordinates

“…For the calculation of the low-lying states of a hydrogen atom in an ultra-high magnetic field (10 5 −10 9 T), which is the case for atoms in the atmosphere of white dwarfs and neutron stars, existent effective methods includes basis set method using different basis functions [3][4][5][6][7][8], eigenvalue analysis method [9][10][11], finite element method [12], adiabatic approximate method [13][14][15], and series method [16][17][18]. Among these methods, the series method introduced by Kravchenko and Liberman give the most accurate results for the low-lying spectra of hydrogen atom in a magnetic field [16][17][18].…”