Recoil correction to the ground-state energy of hydrogenlike atoms

Shabaev, V. M.; Artemyev, A. N.; Beier, Thomas; Plunien, G.; Yerokhin, V. A.; Soff, G.

doi:10.1103/physreva.57.4235

Cited by 72 publications

(85 citation statements)

References 23 publications

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“…[22][23][24]. The numerical accuracy of these studies, however, was not sufficient for making any conclusions about the higher-order fns recoil effect for hydrogen.…”

mentioning

confidence: 97%

“…We note that one should not use ∆E L in the denominator of Eq. (23) because of a cancellation of spurious terms between ∆E L and ∆E C [22]. It might be also mentioned that the full two-transverse-photon fns correction contains terms induced by virtual nuclear excitations [7,14].…”

mentioning

confidence: 99%

“…In the present Letter, we perform the first high-precision evaluation of this effect. The low-order part of the recoil correction for an extended nuclear charge is given by [25,26] (see also [24]) (8)(8) −0.000 000 01 −0.000 000 01 0.000 0 (2) a 2 1.6755 −0.000 628 (4)(4) −0.000 629 (4)(4) −0.000 000 06 (6)(0) −0.000 000 04 (4)(0) 3 2.4440 −0.001 28 (1)(1) −0.001 29 (1)(1) −0.000 0002 (4)(0) −0.000 000 1 (1)(0) 4 2.5190 −0.001 50 (2)(1) −0.001 50 (2)(1) −0.000 0003 (7)(0) −0.000 000 2 (2)(0) 5 2.4060 −0.001 56 (2)(1) −0.001 56 (2)(1) −0.000 0004 (10)(0) −0.000 0002 (2)(0) a Shabaev et al 1998 [22] where V ′ (r) = dV (r)/dr, W ′ (r) = dW (r)/dr,…”

mentioning

confidence: 99%

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Nuclear Recoil Effect in the Lamb Shift of Light Hydrogenlike Atoms

Yerokhin

Shabaev

2015

Phys. Rev. Lett.

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We report high-precision calculations of the nuclear recoil effect to the Lamb shift of hydrogen-like atoms to the first order in the electron-nucleus mass ratio and to all orders in the nuclear binding strength parameter Zα. The results are in excellent agreement with the known terms of the Zα expansion and allow an accurate identification of the nonperturbative higher-order remainder. For hydrogen, the higher-order remainder was found to be much larger than anticipated. This result resolves the long-standing disagreement between the numerical all-order and the analytical Zα-expansion approaches to the recoil effect and completely removes the second-largest theoretical uncertainty in the hydrogen Lamb shift of the 1S and 2S states. PACS numbers: 06.20.Jr, 31.30.jf, 21.10.Ft The hydrogen atom is one of the simplest and the most accurately measured atomic systems in physics. The highest precision has been achieved for the 1S-2S transition, whose frequency was measured [1] with the relative uncertainty of 4.2 × 10 −15 or 10 Hz. In future, it should be possible to increase the experimental accuracy even further, closely approaching the 1.3 Hz natural linewidth of the 2S level. A combination of measured transition frequencies in hydrogen with sophisticated theoretical calculations provides the method for determination of the Rydberg constant [2], which is currently the most accurately known fundamental constant. A coproduct of the determination procedure of the Rydberg constant is the proton charge radius.The proton charge radius has recently received much attention, after the experiments on the muonic hydrogen [3,4] reported a large (7σ) unexplained difference of the proton charge radius values extracted from the muonic hydrogen and the usual (electronic) hydrogen. One of the possible explanation of this puzzle might be a yet undiscovered problem in the theory of the electronic hydrogen. For this reason, investigations of possible inconsistencies in the hydrogen theory are of particular importance today. In the present Letter we report calculations that remove the second-largest theoretical uncertainty in the hydrogen theory and resolve a long-standing discrepancy between the analytical and the numerical approach to the nuclear recoil effect.General expressions for the nuclear recoil effect to first order in the electron-nucleus mass ratio m/M and to all orders in the nuclear binding strength parameter Zα (Z is the nuclear charge number and α is the fine-structure constant) were obtained by one of us [5,6] (see also [7]) and later rederived by other authors [8][9][10]. Numerical calculations to all orders in Zα were reported in Refs. [11][12][13]. The results of these calculations agreed well with the first terms of the Zα expansion [9,14,15]. However, a significant disagreement was later found for the higher-order Zα expansion terms. Specifically, the Zα-expansion result for the (Zα) 7 m/M contribution for the 1s state obtained within the leading logarithmic approximation [16, 17] yielded −(18/π) (Zα) 7 m 2 /M ...

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“…[22][23][24]. The numerical accuracy of these studies, however, was not sufficient for making any conclusions about the higher-order fns recoil effect for hydrogen.…”

mentioning

confidence: 97%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Nuclear Recoil Effect in the Lamb Shift of Light Hydrogenlike Atoms

Yerokhin

Shabaev

2015

Phys. Rev. Lett.

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“…The derivative with respect to κ in equation (29) must be taken at a fixed n r . Using the formulas for B 0 , C 0 , B −1 , and C −1 given above and reccurence equations (23)- (26), we can calculate the integrals A s , B s , and C s for any integer s. Explicit formulas for these calculations were derived in [7].…”

Section: Application Of the Virial Relations For Evaluation Of The Avmentioning

confidence: 99%

“…In [14], formulas (58) and (60) were employed to calculate the interelectronicinteraction corrections to the hyperfine splitting in Li-like ions. In [29], virial relations (16)- (19) with V (r) = −αZ/r were used to evaluate the recoil correction to the Lamb shift for an extended nucleus. As an example, let us demonstrate how the virial relations with V (r) = −αZ/r can be employed to evaluate the nuclear-size correction of the lowest order in mR (R is the nuclear charge radius) to the integral…”

Section: Other Applications Of the Virial Relationsmentioning

confidence: 99%

Virial Relations for the Dirac Equation and Their Applications to Calculations of Hydrogen-Like Atoms

Shabaev

2003

Lecture Notes in Physics

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Virial relations for the Dirac equation in a central field and their applications to calculations of H-like atoms are considered. It is demonstrated that using these relations allows one to evaluate various average values for a hydrogenlike atom. The corresponding relations for non-diagonal matrix elements provide an effective method for analytical evaluations of infinite sums that occur in calculations based on using the reduced Coulomb-Green function. In particular, this method can be used for calculations of higher-order corrections to the hyperfine splitting and to the g factor in hydrogenlike atoms.

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Numerical electronic structure calculations for atoms. II. Generalized variable transformation and relativistic calculations

Andrae

Hinze

2000

Int. J. Quantum Chem.

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ABSTRACT:The pairs of radial functions P i and Q i , which are part of the four-component single-particle spinors in the relativistic description of the electronic structure of bound states of atoms, are usually determined as solutions of eigenvalue problems. The latter constitute two-point boundary value problems which involve coupled pairs of first-order ordinary radial differential equations. To introduce a suitable notation, the theory involved in relativistic electronic structure calculations for atoms is briefly reviewed, including a general treatment of arbitrary transformations for the radial variable. Such variable transformations must be specified then, either explicitly (by a transformation function) or implicitly (by the solution method itself), for any actual calculation, though initially the variable transformation is almost completely independent of the numerical method to be applied. We consider suitable transformation functions out of which an actual choice can be made within wide limits. In our approach, the resulting transformed radial differential equations are discretized then with finite-difference methods. Since the accuracy and efficiency of these methods is increased when a constant step size h between contiguous points (in the transformed variable) is used, it is only here where the careful choice of the variable transformation is important. The resulting system of linear equations is solved for the radial functions with standard linear algebra methods rather than "shooting" methods. The present work extends the general study of variable transformations, given recently for nonrelativistic electronic structure calculations in Part I, to the relativistic case. Important results following from the present work are (i) a discretization scheme for first-order ordinary differential equations similar to the ANDRAE, REIHER, AND HINZE well-known standard Numerov scheme used within the nonrelativistic framework, (ii) a consistent numerical algorithm with an overall truncation error of order h 4 , (iii) a method for handling effective potentials behaving singularly like r −1 at the origin (as encountered, e.g., when the atomic nucleus is represented by a pointlike charge density distribution), and, in connection with this last point, (iv) the avoidance of nonanalytic short-range behavior of the solutions to be obtained from the differential equations.

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Recoil correction to the ground-state energy of hydrogenlike atoms

Cited by 72 publications

References 23 publications

Nuclear Recoil Effect in the Lamb Shift of Light Hydrogenlike Atoms

Nuclear Recoil Effect in the Lamb Shift of Light Hydrogenlike Atoms

Virial Relations for the Dirac Equation and Their Applications to Calculations of Hydrogen-Like Atoms

Numerical electronic structure calculations for atoms. II. Generalized variable transformation and relativistic calculations

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