Lamb Shift Calculations Research Articles

QEDMOD: Fortran program for calculating the model Lamb-shift operator

We present Fortran package QEDMOD for computing the model QED operator hQED that can be used to account for the Lamb shift in accurate atomic-structure calculations. The package routines calculate the matrix elements of hQED with the user-specified one-electron wave functions. The operator can be used to calculate Lamb shift in many-electron atomic systems with a typical accuracy of few percent, either by evaluating the matrix element of hQED with the many-electron wave function, or by adding hQED to the Dirac–Coulomb–Breit Hamiltonian. New version program summaryProgram Title: QEDMODProgram Files doi:http://dx.doi.org/10.17632/ct869v4gxn.1Licensing provisions: Apache License 2.0.Programming language: Fortran 77.Journal reference of previous version: Computer Physics Communications 189 (2015) 175-181, http://dx.doi.org/10.1016/j.cpc.2014.12.002.Does the new version supersede the previous version?: Yes.Nature of problem: Fortran code for computation of the model QED operator that approximates the Lamb shift in many-electron atoms.Solution method: The method of solution is based on the algorithm developed in Ref. [1].Reasons for the new version: Extension of the region of applicability; fixing of minor bugs.Summary of revisions: Region of supported nuclear charges is extended to Z∈[3,120]; a problem that appeared for the extended nuclear model for several selected values of Z is fixed. [1]V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Model operator approach to the Lamb shift calculations in relativistic many-electron atoms, Phys. Rev. A 88, 012513 (2013).

Model operator approach to the relativistic Lamb shift calculations in many-electron atoms and highly charged molecular ions

The quantum electrodynamics (QED) corrections to the alkali metal ionization potentials and to the ground-state energy of the U92+-U91+ one-electron quasi-molecule are evaluated with the use of the model operator approach. The QED contributions to the ionization energies in the alkali metals are obtained both by averaging the model operator with the valence electron wave functions and by incorporating the QED operator into the Dirac–Coulomb–Breit Hamiltonian. The two-center calculations of the vacuum-polarization and self-energy contributions to the Lamb shift in the one-electron quasi-molecule are performed for the different values of the internuclear distances (from the chemical distances up to the critical ones), the more simplified evaluation in the monopole approximation is also considered. The results of the calculations are compared with the theoretical predictions obtained previously in the framework of the ab initio approaches.

Model operator approach to the Lamb shift calculations in relativistic many-electron atoms

A model operator approach to calculations of the QED corrections to energy levels in relativistic many-electron atomic systems is developed. The model Lamb shift operator is represented by a sum of local and nonlocal potentials which are defined using the results of ab initio calculations of the diagonal and nondiagonal matrix elements of the one-loop QED operator with H-like wave functions. The model operator can be easily included in any calculations based on the Dirac-Coulomb-Breit Hamiltonian. The efficiency of the method is demonstrated by comparison of the model QED operator results for the Lamb shifts in many-electron atoms and ions with exact QED calculations.

Radiative Corrections in Highly Charged Ions and Tests of QED in Strong Electric and Magnetic Fields

To provide predictions of the Lamb shift of highly charged ions on the level of accuracy of about 10-6 has been achieved after exact results for the contributions of all two-photon self-energy diagrams have been performed. We report on the present status of our Lamb-shift calculations including all QED-corrections of first- and second-order in the finestructure constant α and all relevant nuclear effects. An excellent agreement between the most recent experimental data for Lamb shift of the 1s-ground state in hydrogenlike uranium can be stated. This can serve as a sensitive test of QED in the strongest electric fields accessible in nature. In a second part of this article we report about an all-order numerical evaluation of the one-photon selfenergy at low nuclear charge (Z = 1, ..., 5). A sensitive comparison of our numerical approach with analytical approach to the one-photon selfenergy confirms the consistency of these two different approaches (numerical and analytic) to very high precision.

Lamb-shift calculations for high-Z lithium-like ions.

We present ab initio calculations of the complete gauge invariant set of the self energy and vacuum polarization screening diagrams for the 2p\(2{\text{p}}_{{\text{1/2}}} - 2{\text{s}} \) transition in Li like ions. Various contributions to the transition energy are collected. The accuracy of theoretical predictions is discussed.

Measurement of QED and Hyperfine Splitting in the2s1/2-2p3/2X-Ray Transition in Li-likeB209i80+

A measurement of the ${2s}_{1/2}\ensuremath{-}{2p}_{3/2}$ x-ray transition in trapped Li-like ${\mathrm{Bi}}^{80+}$ ions was made that resolved the $0.820\ifmmode\pm\else\textpm\fi{}0.026\mathrm{eV}$ hyperfine splitting of the $({1s}^{2}2s{)}_{F\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4,5}$ ground configuration, providing the first such measurement in a multielectron highly charged ion. The intensity ratio of the two components is shown to be a new electron density diagnostic. The statistically averaged $2788.139\ifmmode\pm\else\textpm\fi{}0.039\mathrm{eV}$ energy of the ${2s}_{1/2}\ensuremath{-}{2p}_{3/2}$ transition provides the most accurate test of QED in a high- $Z$ ion to date, demonstrating the need for including higher-order terms in the Lamb shift calculations.

Nonrelativistic calculation of the Lamb shift to order α5

It is demonstrated, for the first time to our knowledge, that the Lamb-shift calculation for all states of hydrogenic atoms can be carried out to order α 5 by applying the methods of nonrelativistic quantum electrodynamics and without using the Dirac equation and the second quantization for the electron. The extremely small deviations from the standard relativistic results were calculated for different S- and non-S-states as well.

Complete Lamb-shift calculation to order α5 by applying the methods of non-relativistic quantum electrodynamics

It is shown, for the first time to our knowledge, that the Lamb shift for the states of hydrogenic atoms can be almost completely calculated by applying the methods of non-relativistic quantum electrodynamics and without using the Dirac equation and the second quantization for the electron. By taking into account the spin-orbit interaction, the complete Lamb shift to order α5 is calculated for different hydrogenic states.

Commutator expansion. II. Relativistic reduced Green’s functions and the Lamb shift calculation

This is a continuation of an earlier paper [J. Math. Phys. 34, 5509 (1993)]. A projection operator technique is introduced in the Lamb shift calculation in order to manage spurious infrared divergences. The method provides an efficient tool to exhibit factors of the field in the commutator expansion of the Lamb shift. Mass eigenfunction expansion concepts developed elsewhere provide the setting for the new technique. As a by-product of our method, relativistic reduced Green’s functions are found to appear in a natural way. The method may prove useful, since reduced Green’s functions are simpler than the full Green’s functions, and behave like constant operators as regards the parameter integrals.

Relativistic many‐body and QED calculations on atomic systems

A review is given of many-body perturbation methods, particularly in the all-order and coupled-cluster forms. Relativistic many-body schemes are analyzed in terms of one- and two-photon potentials, derived by means of QED. A complete second-order (nonradiative) calculation for He-like ions is presented, including repeated Breit interactions as well as the effects of retardation and of negative-energy states, but omitting the Lamb shift. Numerical results of some Lamb-shift calculations are also given. From the analysis, conclusions can be drawn concerning the accuracy of certain relativistic many-body approaches. © 1996 John Wiley & Sons, Inc.

Many-body calculation of the 2p1/2,3/2-2s1/2 transition energies in Li-like 238U.

We present a relativistic many-body calculation of the 2[ital s][sub 1/2] and 2[ital p][sub 1/2,3/2] ionization energies in Li-like [sup 238]U, based on the coupled-cluster singles and doubles approximation with the inclusion of the unretarded Breit interaction, using Dirac-Fock-Breit orbitals. A detailed comparison is made with a similar calculation using ordinary Dirac-Fock orbitals. The calculation yields the transition energies, and by comparison with the recently published experimental values we find the value for the remaining quantum-electrodynamical correction, which is compared with recent Lamb-shift calculations. The importance of an accurate nuclear model for evaluating the nuclear-size effect is also discussed.

A new tool for the lamb-shift calculation

We solve the Bethe-Salpeter equation for hydrogenic bound states by choosing an appropriate interaction kernel $K_c$. We want to use our solution to calculate up to a higher order the hydrogen Lamb-shift, and as a first application we present up to order $\left(\aa / \pi\right)(\za)^7$ the contribution of the lowest order self-energy graph, calculated {\it exactly}. The basic formalism is a natural extension to the hydrogenic bound states of the one previously presented by R. Barbieri and E. Remiddi and used in the case of positronium.

An improved Lamb shift calculation for a bound polaron in a purely two-dimensional quantum well

Using an improved variational wave function the 2s excited state energy of a Coulomb impurity bound polaron in a purely two-dimensional quantum well is calculated and the excitation energies and the phonon-induced Lamb shift corrections are obtained for several polar materials.

Interaction of a charged harmonic oscillator with a single quantized electromagnetic field mode.

The interaction of an atomic system with a single mode of a quantized electromagnetic field is investigated in the dipole approximation. Two approximations that are almost always made in this analysis are the neglect of the diamagnetic term in the Hamiltonian and the dropping of the rotating-wave approximation (RWA) terms that correspond to processes that do not conserve energy. Both approximations involve the neglect of terms that are of the same order of magnitude so that consistency demands that they be considered together. It is demonstrated that, in the dipole approximation, a minor modification of the usual application of quantum electrodynamics to any atomic system can exactly incorporate the effects of the diamagnetic term into a new frequency of the electromagnetic cavity mode; this frequency is slightly different from that of the empty cavity. This result is then applied to the specific case of a charged harmonic oscillator interacting with a single mode. For this simple model an analytic solution can then be found that includes the effects of the diamagnetic term and does not use the RWA. It is found that, for off resonance, the normal modes of the coupled system oscillate at shifted frequencies which are in agreement with the second-order perturbation-theory calculation of the Lamb shift. When the cavity mode is tuned to the atomic oscillator frequency, the perturbation-theory Lamb-shift calculation is no longer valid, and the exact solution derived in this paper predicts a resonant Lamb shift that is significantly larger than the off-resonant single-mode Lamb shift. Furthermore, the frequency spectrum is qualitatively changed from the single frequency of the free atomic system to a doublet spectrum. An analytic solution of this harmonic-oscillator problem can also be found when the RWA terms are dropped from the Hamiltonian. Comparison of the two solutions reveals that the RWA terms contribute a frequency shift of magnitude independent of the field intensity in the simple case of a charged harmonic oscillator.When trajectories of the exact and RWA solutions are compared in phase space, it is found that one effect of the RWA terms is to lift a twofold degeneracy of the phase-space orbits that occurs when the RWA terms are neglected. The harmonic-oscillator problem has several features in common with the interaction of a two-level atom with a single quantized field mode. Both Hamiltonians exhibit inversion symmetry and hence have parity as a good quantum number. Both Hamiltonians contain RWA terms. When these RWA terms are neglected, both Hamiltonians have a new constant of motion that can be identified with the total number of excitations in the combined system.

Lamb shift measurements in heavy ions

Recent experiments on the Lamb shift in hydrogenic heavy ions are compared with each other and with theory. The experiments are not yet good enough to test critically the latest Lamb-shift calculations.

Bethe sums for Lamb shift calculations in higher excited states

Bethe's average excitation energies have been systematically evaluated for all states with n ⪕ 8.

Two-Body Problem in Quantum Field Theory

We study an approach to the relativistic two-body problem that represents a considerable improvement of the Bethe-Salpeter equation. The main advantages are the improved properties of the simple ladder approximation. (a) It has the correct static limit and is in this sense equivalent to the sum of all crossed ladders of the Bethe-Salpeter equation, (b) in the case of massless exchange it is possible to solve analytically and obtain the wave functions and the T matrix in closed form, (c) the system is closed, that is, the two-particle T matrix is unitary. (d) The wave functions admit a complete quantum-mechanical interpretation without compromise of relativistic covariance, the current is conserved, and gauge invariance is respected. (e) The evaluation of physical amplitudes is greatly simplified, as illustrated by positronium decay and the Lamb shift. Both are carried out with full relativistic covariance, and the Lamb-shift calculation in particular is greatly simplified and clarified in comparison with other methods. Most of the results are applicable only to the case of spin-0 or spin-\textonehalf{} particles interacting through the exchange of a single scalar or vector boson. In particular, the calculations of positronium decay and the Lamb shift are carried out with spinless particles. It is shown, however, that the introduction of photon spin presents no problem and that the Lamb shift can be calculated in a fully gauge-invariant manner.

The Electromagnetic Shift of Landau Levels and the Magnetic Moment of the Electron

The change in the magnetic moment of the electron due to zero-point vibrations of the quantized radiation field is calculated from nonrelativistic quantum mechanics. Using ideas which underlie Bethe's calculation of the Lamb shift and Luttinger's calculation of the electron magnetic moment, but without using second quantization of the electron field, we obtain the familiar g = 2[1 + (e2/hc)]. In addition, the calculation gives the correct form for the magnetic-field-dependent radiative corrections to g, namely a first term proportional to B and a second term proportional to B lnB. These B-dependent radiative terms differ only by numerical factors of the order of unity from the relativistic quantum electrodynamics corrections derived by Gupta. The self-energy integrals are fairly sensitive to the Bethe cutoff, but choosing the high-frequency cutoff at μc2/ħ yields, simultaneously, the correct Lamb shift and the Schwinger g value.

The anomalous magnetic moment of the electron and the proton

The Schwebel-Sachs self-consistent field theory of quantum electrodynamics is used to calculate the anomalous magnetic moment of the electron without recourse to renormalization and perturbation techniques. It was shown by the originators of the theory that their basic postulate, that the « laws of nature must be described in terms of field variables which may be associated only with elementary interactions », led to an elimination of the self-interactions that plagued classical theory and to a new electromagnetic interaction, the dual Lorentz interaction, which has the form ϱH−(1/c)(j×E). It is shown that this dual Lorentz interaction gives rise to an additional magnetic interaction which may be interpreted as the anomalous magnetic moment. The anomaly is found to beA=2mcg/ħ whereg is the coupling constant of of the field-current interaction representing the dual Lorentz interaction and has the dimension of length. Using the value ofg obtained from the Lamb shift calculations of a previous paper (1), it is shown thatA=1.087·10−3. The small discrepancy between this value and that of the free relativistic electron measurement is attributed to approximations made in the Lamb shift calculations. Recalculation ofg on the basis of the experimental value of the anomaly yieldsg=2.24·10−14 cm. Calculation of the proton anomaly givesA=2.14 compared to the magnetic resonance value of 1.79. It would be of interest and value to determine the magnetic moment of the proton under the same experimental conditions which determine the magnetic moment of the « free » electron.

The anomalous magnetic moment of the electron and the proton

The Schwebel-Sachs self-consistent field theory of quantum electrodynamics is used to calculate the anomalous magnetic moment of the electron without recourse to renormalization and perturbation techniques. It was shown by the originators of the theory that their basic postulate, that the «laws of nature must be described in terms of field variables which may be associated only with elementary interactions», led to an elimination of the self-interactions that palgued classical theory and to a new electromagnetic interaction, the dual Lorentz interaction, which has the form ϱH−(1/c)(j×E). It is shown that this dual Lorentz interaction gives rise to an additional magnetic interaction which may be interpreted as the anomalous magnetic moment. The anomaly is found to beA=2mcg/ħ whereg is the coupling constant of of the field-current interaction representing the dual Lorentz interaction and has the dimension of length. Using the value ofg obtained from the Lamb shift calculations of a previous paper (1), it is shown thatA=1.087·10−3. The small discrepancy between this value and that of the free relativistic electron measurement is attributed to approximations made in the Lamb shift calculations. Recalculation ofg on the basis of the experimental value of the anomaly yieldsg2.24·10−14 cm. Calculation of the proton anomaly givesA-2.14 compared to the magnetic resonance value of 1.79. It would be of interest and value to determine the magnetic moment of the proton under the same experimental conditions which determine the magnetic moment of the «free» electron.