Physical Reference Data

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Theory and theoretical input data

The energy levels of hydrogen-like atoms are determined mainly by the Dirac eigenvalues, quantum electrodynamic (QED) effects such as self energy and vacuum polarization, and nuclear size and motion effects. The theory and theoretical data needed for the evaluation of the energy levels in hydrogen and deuterium are summarized on this page.

     The energy levels and transition frequencies are calculated using the results of the 2002 CODATA least-squares adjustment of the fundamental physical constants. The calculation is based on the formulation described by U.D. Jentschura, S. Kotochigova, E.O. Le Bigot, P.J. Mohr, and B.N. Taylor in Precise calculation of transition frequencies of hydrogen and deuterium based on a least-squares analysis, Phys. Rev. Lett. 95, 163003 (2005).     
     Matrices referred to in the above paper can be accessed from the fundamental constants Web site Although those complete matrices from the 2002 CODATA least-squares adjustment of the fundamental physical constants can provide the basis for the calculation of the energy levels, in this Web database, we use matrices from a least-squares adjustment that includes only data relevant to the hydrogen and deuterium energy levels. The results for the energy levels are essentially the same in either case.     
     The contributions to the theoretical values of the individual energy levels are summarized by P.J. Mohr and B.N. Taylor in CODATA recommended values of the fundamental physical constants: 2002, Rev. Mod. Phys. 77, 1 (2005), Appendix A, which is reprinted here.     
     The largest QED contribution is the one-photon self-energy correction. This correction includes the nonrelativistic Bethe logarithm which requires a separate calculation for each state. For states with principle quantum number n ≤ 20, we employ the values given by G.W.F. Drake and R.A. Swainson, Phys. Rev. A 41, 1243 (1990). For states with 21 ≤ n ≤ 200, we use the results given in the attached table of Bethe logarithms by U.D. Jentschura and P.J. Mohr, NIST Technical Note 1467. The calculation of the values in the latter table is described by U.D. Jentschura and P.J. Mohr in Calculation of hydrogenic Bethe Logarithms for Rydberg states, Phys. Rev. A 72, 012110 (2005). Higher-order contributions to the self energy are summarized by U.J. Jentschura, S. Kotochigova, E.O. Le Bigot, and P.J. Mohr in Precise theory of levels of hydrogen and deuterium: The one-photon self energy correction, NIST Technical Note 1469.     
     The second-largest QED contribution is the one-photon vacuum polarization. The calculated data needed to evaluate this contribution is given by S. Kotochigova and P. Mohr, Precise theory of levels of hydrogen and deuterium: The Coulomb vacuum polarization correction, NIST Technical Note 1468.     
     The two-photon corrections are somewhat smaller, but still very important at the level of accuracy considered here. A summary of the status of the theory and the data necessary for the present work is given by U.D. Jentschura and P.J. Mohr, Precise theory of levels of hydrogen and deuterium: The two-photon radiative corrections, NIST Internal Report 7217.     

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