Basis-set approach to calculating the radiative self-energy in highly ionized atoms

Blundell, S. A.; Snyderman, N.J.

doi:10.1103/physreva.44.r1427

Cited by 112 publications

(47 citation statements)

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“…The nuclear-size self-energy (NSE) correction is defined as the difference between the matrix elements (6) evaluated with the point-Coulomb potential and the potential of the extended-charge nucleus. Numerical, all-order (in Zα) evaluation of the one-loop self-energy correction have been extensively discussed in the literature over past decades [8,[16][17][18][19][20], both for the case of the point-Coulomb and extended-nucleus potentials. In the present investigation, we employ the method developed in our previous work [21] for the case of the point nucleus.…”

Section: Ns Correction To Self Energymentioning

confidence: 99%

Nuclear-size correction to the Lamb shift of one-electron atoms

Yerokhin

2011

Phys. Rev. A

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The nuclear size effect on the one-loop self energy and vacuum polarization is evaluated for the 1s, 2s, 3s, 2p 1/2 , and 2p 3/2 states of hydrogen-like ions. The calculation is performed to all orders in the binding nuclear strength parameter Zα. Detailed comparison is made with previous all-order calculations and calculations based on the expansion in the parameter Zα. Extrapolation of the all-order numerical results obtained towards Z = 1 provides results for the radiative nuclear size effect on the hydrogen Lamb shift.PACS numbers: 31.30.jf, 12.20.Ds, 31.15.A- IntroductionThe distribution of charge of the nucleus influences the Dirac energies of atomic systems as well as the quantum electrodynamical corrections to the energy levels. This effect, termed as the nuclear size (NS) correction, is important for comparison of theoretical predictions with experimental data for the whole range of the nuclear charges Z, from hydrogen (Z = 1) up to uranium (Z = 92). The NS corrections to the one-loop self energy and vacuum polarization have been previously investigated both within the approach based on the expansion in the nuclear binding strength parameter Zα [1][2][3][4][5][6] (where α is the fine structure constant) and within the numerical approach that accounts the parameter Zα to all orders [7,8]. In the high-Z region, the numerical allorder approach provides accurate predictions for the NS effect on the radiative corrections. For lower Z, however, the NS effect diminishes and becomes increasingly more difficult to identify in a numerical calculation. On the contrary, the Zα-expansion approach provides accurate predictions for the low-Z ions and only the qualitative estimates in the high-Z region. A quantitative cross-check of the two complementary approaches is not simple and has not been accomplished up to now.The NS corrections became of particular interest recently, after the results of the muonic hydrogen Lambshift experiment were announced [9]. It turned out that the value for the proton charge radius r p deduced from the muonic hydrogen differs by five standard deviations from the spectroscopic value of r p derived from the hydrogen atom. This unexplained disagreement stimulates the scientific community to double-check all contributions originating from the nuclear charge distribution, both for the muonic and normal atoms.The aim of the present investigation is to perform an accurate numerical all-order calculation of the NS correction to the one-loop self energy and vacuum polarization and to make a detailed comparison with the Zα-expansion results available.The relativistic units (m = = c = 1) and the charge units α = e 2 /(4π) are used in this paper. I. NS CORRECTION TO DIRAC ENERGYThe leading-order NS correction to the energy levels of a hydrogen-like atom is defined as the difference of the corresponding eigenvalues of the Dirac equation with the point-Coulomb and the extended-nucleus potentials. The two most commonly used models of the nuclear charge distribution are the uniformly charged sphere ("sp...

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Section: Ns Correction To Self Energymentioning

confidence: 99%

Nuclear-size correction to the Lamb shift of one-electron atoms

Yerokhin

2011

Phys. Rev. A

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“…This renormalization is well-known and discussed in details in [23,24,25]. The ultraviolet divergence in the vertex and reducible contributions can be isolated by expanding the bound-electron propagator in terms of the interaction with the field of the nucleus.…”

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Radiative corrections to the magnetic-dipole transition amplitude in B-like ions

et al. 2006

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“…The infrared divergencies can explicitly be shown to cancel. The numerical evaluation of the finite parts is made along the same lines as in [12,18,19]. To compare our results with the Za expansion we introduce the functionF defined by…”

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Self-Energy Correction to the Hyperfine Structure Splitting of Hydrogenlike Atoms

Persson¹,

Schneider

Greiner

et al. 1996

Phys. Rev. Lett.

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A first testing ground for QED in the combined presence of a strong Coulomb field and a strong magnetic field is provided by the precise measurement of the hyperfine structure splitting of hydrogenlike 209 Bi. We present a complete calculation of the one-loop self-energy correction to the first-order hyperfine interaction for various nuclear charges. In the low-Z regime we almost perfectly agree with the Za expansion, but for medium and high Z there is a substantial deviation.PACS numbers: 31.30. Gs, 31.15.Ar, 31.30.Jv Very recently it was reported that for the first time the hyperfine structure splitting of a hydrogenlike high-Z atom was observed with a high relative accuracy of about 10 24 at the ESR at GSI, Darmstadt. The transition energy of the ground state hyperfine structure splitting of 209 Bi 821 was measured to be DE 5.0840͑8͒ eV [1]. This has challenged theory to perform calculations with comparable accuracy, including also one-loop QED corrections. The new experimental situation opens up a possibility to perform a novel test of QED in a combined strong magnetic field and a strong Coulomb field.The leading QED effects are of two types: vacuum polarization and self-energy corrections. The vacuum polarization correction is relatively straightforward to compute, using an Uehling-like approximation. This contribution was calculated to be DE VP 10.035 eV quite recently [2]. The remaining Wichmann-Kroll contribution is very small. The one-loop self-energy correction, on the other hand, is more difficult to elaborate, and earlier calculations using an Za expansion of the Coulomb field are correct only up to order a͑Za͒ 2 mc 2 [3-7]. For heavy elements such an expansion is not reliable. Therefore it is necessary to calculate the self-energy contribution to all orders in Za. In this Letter we present the first complete calculation of this type for different nuclear charges ranging from Z 1 to Z 92 using similar techniques as published earlier in Refs. [8][9][10][11][12].First we give a brief outline of the computation of the self-energy correction and later we discuss the numerical results. A more detailed analysis of the calculation of QED corrections to the hyperfine interaction will be presented in a forthcoming paper [13].In a previous paper it was shown that the unrenormalized self-energy correction for a bound electron state can be written in Feynman gauge as [9] E bou ͑a͒ 2 awhere a m a m ͑1 2 a ? a͒, C ͓l͔ q is the q component of the spherical angular tensor of order l, j l ͑kr͒ denotes the spherical Bessel function of order l, and ja͘ represents the reference state. There is also a corresponding mass counter term.To calculate the self-energy corrections to the hyperfine structure we treat the magnetic potential A͑r ͒ as a perturbation of the system. This perturbation will affect the binding energy of the bound electron, the wave function, and the bound propagator as already sketched out in Ref. [14]

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Basis-set approach to calculating the radiative self-energy in highly ionized atoms

Cited by 112 publications

References 17 publications

Nuclear-size correction to the Lamb shift of one-electron atoms

Nuclear-size correction to the Lamb shift of one-electron atoms

Radiative corrections to the magnetic-dipole transition amplitude in B-like ions

Self-Energy Correction to the Hyperfine Structure Splitting of Hydrogenlike Atoms

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