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

Abstract

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]

Nuclear magnetization distribution and hyperfine splitting in Bi82+ ion

“…In order to compare our calculations with the data we need ∆E QED . It was calculated in several papers and the results are consistent [11,[16][17][18][19]. In [19] it was shown that ∆E QED is insensitive to details of the magnetization distribution.…”

“…The leading QED corrections originate from the one-loop self-energy and vacuum-polarization effects. The one-loop QED effects for hfs have been calculated by different groups and the results are consistent [11,[16][17][18][19].…”

“…There are two principal corrections to the Fermi-Breit formulae that is necessary to take into account: the magnetic moment distribution within the nucleus [9][10][11][12][13][14][15] and radiative corrections [11,[16][17][18][19]. Both of these corrections are of comparable magnitude.…”

“…High experimental precision attained in the laser spectroscopic measurement of the ground-state hfs in hydrogen-like 209 Bi 82+ [1] stimulates considerable theoretical activity in this field (see, e.g., [11][12][13][14][15][16][17][18][19] and references therein).…”

Theory of light hydrogenlike atoms

“…(394) we would obtain in the nonrecoil limit ∆E(2P − 2S) = 0.0264 meV (257) instead of the result in eq. (255). Comparing these numbers we see that the numbers discussed in this Section may be used for estimates of the proton size contribution of order (Zα) 5 , but when the results of the Lamb shift measurements become available, this correction will require some kind of a self-consistent consideration.…”