Lamb Shift Contributions Research Articles

Lamb shift contribution of very light charged particles

The leading-order vacuum polarization contribution of very light millicharged fermions and scalar (spin-0) particles with charge $\ensuremath{\epsilon}e$ and mass $\ensuremath{\mu}$ to the Lamb shift of the hydrogen atom is shown to imply universal, i.e. $\ensuremath{\mu}$-independent, upper bounds on $\ensuremath{\epsilon}$: $\ensuremath{\epsilon}\ensuremath{\lesssim}{10}^{\ensuremath{-}4}$ for $\ensuremath{\mu}\ensuremath{\lesssim}1\text{ }\text{ }\mathrm{keV}$ in the case of fermions, and for scalars this bound is increased by a factor of 2. This is in contrast to expectations based on the commonly used approximation to the Uehling potential relevant only for conventionally large fermion (and scalar) masses.

Estimation of Lamb-shift effects for molecules: Application to the rotation-vibration spectra of water

A simple approach is presented for approximating Lamb shifts, the leading quantum electrodynamical (QED) effects, in molecules containing light elements. The Lamb-shift contributions to the electronic energies are estimated from scaled nuclear and electronic Darwin terms. QED effects on the rovibrational states of water are found to be significantly larger than current experimental uncertainties, and only one order of magnitude smaller than current computational uncertainties.

QED AND HYPERFINE SPLITTING IN THE 2S1/2-2P3/2 X-RAY TRANSITION IN LITHIUM-LIKE BI80+

The measurement of the Bi80+ 2s1/2−2p3/2 transition energy is discussed. The measurement resolved the hyperfine splitting of the (1s22s) F=4,5 ground configuration and allowed a determination of the Lamb shift contributions with a precision of about 9 × 10-4 providing the most accurate test of QED in a high-Z ion to date.

On the renormalization problem in non-relativistic QED: Formulation of an equivalent renormalization procedure for both non-relativistic and relativistic QED

The reasons for the wide-spread incorrect opinion that the non-relativistic QED is a non-renormalizable theory are discussed. It is pointed out that in both non-relativistic and relativistic QED the same energy renormalization condition, which, in the latter case, is reducible to the conventional mass renormalization, holds. Furthermore, by applying the energy renormalization in non-relativistic QED, for the first time, a general energy counter term, leading to finite bound-state Lamb-shift contributions in all orders of the interaction and for any initial radiation field, is derived unambiguously.

The magnetic trapping mode of an electron beam ion trap: New opportunities for highly charged ion research

Using x-ray spectroscopic techniques, we have investigated the properties of an electron beam ion trap (EBIT) after the electron beam is switched off. In the absence of the electron beam, bare, and hydrogenlike Kr35+ and Kr36+ ions remain trapped due to externally applied magnetic and electric fields for at least 5 s; xenon ions with an open L shell, i.e., Xe45+–Xe52+, remain trapped at least as long as 20 s. The ion storage time in this ‘‘magnetic trapping mode’’ depends on the pressure of background atoms as well as on the value of the externally applied trapping potential, and even longer ion storage times appear possible. The magnetic trapping mode enables a variety of new opportunities for atomic physics research involving highly charged ions, which include the study of charge transfer reactions, Doppler-shift-free measurements of the Lamb shift, measurements of radiative lifetimes of long-lived metastable levels, or ion-ion collision studies, by x-ray or laser spectroscopy, and mass spectrometry. Because the trap is filled in situ during the electron trapping phase, transfer losses associated with filling the trap from an external source are avoided. We present spectra of the K-shell emission from heliumlike and hydrogenlike Kr34+ and Kr35+ as well as Xe52+ and Xe53+ that are produced by charge transfer reactions in collisions between ions and neutral atoms. Marked differences with K-shell spectra produced by electron-impact excitation are evident. We use the measurements to infer the Lamb shift contribution to the energy of the 1s1/2–2p3/2 transition in hydrogenlike Xe53+ and determine it to be 31 276(12) eV. The measurement technique can be applied to any ion produced in an EBIT so that Doppler-shift-free Lamb shift measurements of hydrogenlike U91+ are within reach. We also illustrate the utility of the magnetic mode for lifetime determinations by measuring the 3.92(13) ms radiative decay of the 1s2s 3S1 level in heliumlike N5+.

Two-electron Lamb-shift calculations on heliumlike ions.

Complete second-order calculations of the two-electron contribution to the ground-state energy of heliumlike ions are presented and compared with recent experimental results. Calculations involve relativistic many-body effects and two-electron Lamb-shift (vacuum-polarization and self-energy) contributions as well as the recently presented nonradiative QED corrections. Agreement between experimental and theoretical results verifies the many-body part of the calculation. The accuracy of the results is not high enough to test the QED part of the calculations, but an improvement by half an order of magnitude would provide an experimental test of the QED effect for heavy ions beyond the first-order Lamb shift. {copyright} {ital 1996 The American Physical Society.}

ENERGY-LEVEL SCHEME AND TRANSITION PROBABILITIES OF S-LIKE IONS

Theoretical energy levels and transition probabilities are presented for 47 low-lying levels of sulfur-like ions for allZ≥ 18. The multiconfiguration Dirac–Fock technique is used to calculate energy levels and wave functions. The Breit interaction and Lamb shift contributions are calculated perturbatively as corrections to the Dirac–Fock energy. TheM1 andE2 transitions among the five levels of the ground-state configuration (3s23p4) and among four low-lying levels of the odd-parity configurations as well as theE1 transitions between excited and the ground levels are presented.

A measurement of the 1S–2S transition frequency in muonium

Abstract Doppler-free two-photon laser spectroscopy has been employed to measure the 1 2 S 1 2 −2 2 S 1 2 transition in the muonium atom (μ+e−). A value of 2 455 529 002(33) (46) MHz has been obtained, which agrees with QED calculations within two standard deviations. The Lamb shift contributions are tested to the level 8×10−3. The corresponding measurements in hydrogen and deuterium using the same apparatus and laser system provide a test of the applied systematic corrections and have verified the systematic error of 46 MHz quoted. The mass of the positive muon has been derived from the isotope shift in this transition and yields a value of 105.65880(29)(43) MeV c 2 .

Discrete-spectrum contribution to the Lamb-shift of the 2S and 2P hydrogenic states

A general formula due to Moses [Phys. Rev. A8, 1710 (1973)] for exact electric and magnetic multipole transition matrix elements in nonrelativistic hydrogenic atoms is reduced by carrying out additional integrations. By using the obtained reduced formula unrenormalized (finite) and renormalized Lamb-shift contributions stemming from individual discrete states as well as from the whole discrete spectrum are calculated in the case of the states 2S and 2P.

Calculation of the discrete-spectrum Lamb-shift contribution in atomic hydrogen

By applying the exact transition matrix elements, which were evaluated in our preceding paper, unrenormalized and renormalized Lamb-shift contributions stemming from individual discrete states as well as from the whole discrete spectrum are calculated in the case of the states 1S, 2S and 2P. For the first time to our knowledge, it is pointed out that the unrenormalized discrete spectrum contribution is completely finite and thus conceptual clarifications in the renormalization procedure are achieved.

Inclusion of retardation effects in the nonrelativistic treatment of the lamb shift

The retardation effects are included in the nonrelativistic calculation of the Lamb shift, by using a method which exploits theO(4) symmetry of the nonrelativistic hydrogen atom. This method has been previously applied by Lieber to the treatment of the Lamb shift. However, in contrast to his calculations, no kind of dipole approximation, leading to the introduction of an inconsistent cut-off frequency, is used in our treatment. Apart from obtaining cut-off-independent results, the inclusion of retardation effects makes also possible the calculation of the spin-radiation-field Lamb-shift contribution (which vanishes in the dipole approximation). Furthermore, our unambiguous renormalization, consisting of removing the unobservable free-electron self-energy contribution from our expressions, leads to finite analytical and numerical results for the energy level (Lamb) shifts. The comparison of these results with those obtained in a nonrelativistic treatment by other authors shows that our results are in better agreement with the experiment.

UNRENORMALIZED AND RENORMALIZED DISCRETE-SPECTRUM LAMB-SHIFT CONTRIBUTIONS FOR 1S AND 2S HYROGENIC STATES

A general formula due to Moses (Phys. Rev. A8, 1710 (1973)) for exact transition matrix elements in nonrelativistic hydrogenic atoms has been reduced by carrying out additional calculations. By using the obtained result, unrenormalized (finite) and renormalized Lamb-shift contributions stemming from individual discrete states as well as from the whole discrete spectrum are calculated.

Nonrelativistic Lamb shift for muonic molecules

A recently developed formula [R. Bukowski and B. Jeziorski, Phys. Rev. A46 (1992) 5437]. has been applied to estimate the soft-photon Lamb shift contribution to the energies of the muonic molecules ppμ, ddμ, ttμ, pdμ, ptμ and dtμ. The corresponding corrections to the dissociation energies for the excited P states of ddμ and dtμ have been found to be almost identical and equal to 0.048 meV. The magnitude of this stabilizing effect is too small to affect seriously the formation rates predictions.

Relativistic nuclear matter with composite nucleons.

Nuclear matter at high density requires a relativistic description. Recent two-loop calculations in local quantum field theory models at finite density yield disastrously large contributions. We argue that the composite structure of nucleons ought to soften these contributions. This argument is supported by a model calculation in which we find very strong reductions in the Lamb shift and vacuum fluctuation contributions. At nuclear matter densities and beyond, the total-loop result is perturbatively small in comparison to the one-loop calculation.

Two-photon laser spectroscopy of the muonium 1S?2S transition

The 12S1/2(F=1)−22S1/2(F=1) energy interval in muonium has been investigated by Doppler-free two-photon laser spectroscopy. The transition was observed by uniquely identifying and counting the muons released after the photoionization of the 2S state by the same laser field. The measured transition frequency of 2455 528016(58)(43) MHz is in good agreement with QED calculations. The experiment can be interpreted as a test of the Lamb shift contributions at the 1% level. The method is also well suited for a precise determination of the mass of the positive muon in a muoniumhydrogen isotope shift measurement.

Pure recoil corrections to the Lamb shift in hydrogenic atoms.

The recoil corrections of order (Z\ensuremath{\alpha}${)}^{6}$${\mathit{m}}^{2}$/M to the hydrogenic Lamb shift are reviewed, and additional terms previously thought to be small are calculated numerically. It is found that significant contributions arise from single-transverse--single-Coulomb interactions in which the bulk of the contribution comes from negative-energy electron intermediate states. This term is found to contain a lnZ\ensuremath{\alpha} dependence, which accounts for its large effect. Other smaller terms are also computed and combined with all previous Lamb-shift contributions. The current status of the Lamb shift in hydrogen is reviewed.

High precision variational calculations for the 1s 21S state of H − and the 1s 21s, 1s2s 1s and 1s2s 3s states of helium

High precision variational eigenvalues are presented for the 1s 21S state of H −, and the 1s 2 1S, 1s2s 1S and 1s2s 3S helium, together with relativistic, mass polarization and relativistic recoil corrections. In addition to the nonrelativistic energy, all corrections of order α 2, μ M ( μ M ) 2 and α 2 M are calculated to an accuracy of better than 10 −6 cm −1. Previous results of similar accuracy are available for the 1 sn d 1 D and 3 D states up to n = 8. A comparison with the high precision 1 s2 s 3 S 1−1 sn d 3 D 1 transition frequency measurements of Hlousek et al. indicates that the known quantum electrodynamic (Lamb shift) contribution of − 0.13488 cm −1 to the ionization energy of the 1s2s 3S 1 state is too small in magnitude by 0.00048(6) cm −1. A discrepancy of this size can reasonably be accounted for by as yet uncalculated contributions to the two-electron Lamb shift.

Quantum electrodynamics of many-electron atoms

The question of the validity of many body perturbation theory (MBPT) calculations that start from the Dirac-Coulomb Hamiltonian is addressed. An extension of the Furry representation of Quantum Electrodynamics (QED) is introduced in which the nuclear Coulomb field is replaced by a potential more relevant to many-electron atoms. QED perturbation theory for energies and matrix elements is set up and evaluated for exchanges of up to two photons. It is shown that in lowest and first order the expressions used in MBPT starting from the Dirac-Coulomb Hamiltonian are reproduced along with additional Lamb shift and Breit interaction contributions. In second order use of QED gives rise to slight differences with MBPT: however, simple rules pertaining to sums over negative energy states are given that eliminate possible difficulties corresponding to continuum dissolution.

Simultaneous observation of Lyman- alpha and Balmer- beta transitions in hydrogenic iron, Fe25+: A novel technique for 1s Lamb-shift measurement.

The $n=2$ to $n=4$ Balmer-$\ensuremath{\beta}$ transitions in hydrogenic ${\mathrm{Fe}}^{25+}$ have been observed in the first order of Bragg diffraction in a beam-foil light source. Simultaneous observation of the $1s\ensuremath{-}2p$ Lyman-$\ensuremath{\alpha}$ transitions from the same source in the fourth diffraction order leads to a new method for the measurement of the $1s$ ground-state Lamb shift which is less sensitive to uncertainties arising from Doppler shifts than previous work. With the use of the Balmer-$\ensuremath{\beta}$ transitions for calibration, the Lyman-$\ensuremath{\alpha}$ wavelengths are, for the $1s^{2}S_{\frac{1}{2}}\ensuremath{-}2p^{2}P_{\frac{3}{2}}$ transition, 1.778 15\ifmmode\pm\else\textpm\fi{}0.000 19 \AA{}, and for the $1s^{2}S_{\frac{1}{2}}\ensuremath{-}2p^{2}P_{\frac{1}{2}}$ transition, 1.783 64\ifmmode\pm\else\textpm\fi{}0.000 19 \AA{}. Values of the Lamb-shift contributions to the $1s\ensuremath{-}2p$ transitions, and the $2{p}_{\frac{1}{2},\frac{3}{2}}$ fine structure, are extracted.

Accurate wave-number measurements for theHe4i1s2p−1s3dtransitions and comparisons of several term separations with theory

We have measured the /sup 4/He 2/sup 3/P-3/sup 3/D (5875 A) and 2/sup 1/P-3/sup 1/D (6678 A) lines by Doppler-free intermodulated fluorescence spectroscopy, and also the spin-forbidden 2/sup 3/P/sub 1,2/-3/sup 1/D/sub 2/ (5874 A) and 2/sup 1/P/sub 1/-3/sup 3/D/sub 2/ (6679 A) lines as observed in Doppler-limited fluorescence spectra. The positive column of a low-pressure discharge was the source of the excited He atoms. The wave number of the tunable dye laser inducing the fluorescence was measured relative to an iodine-stabilized He-Ne laser by photographic Fabry-Perot interferometry. The results are sigma(2/sup 3/P/sub 0/-3/sup 3/D/sub 1/) = 17 013.762 08(9) cm/sup -1/, sigma(2/sup 3/P/sub 1/-3/sup 3/D/sub 2/) = 17 014.705 78(9) cm/sup -1/, sigma(2/sup 3/P/sub 1/-3/sup 1/D/sub 2/) = 17 018.123 8(4) cm/sup -1/, sigma(2/sup 1/P/sub 1/-3/sup 3/D/sub 2/) p = 14 966.652 04(10) cm/sup -1/, and sigma(2/sup 1/P/sub 1/-3/sup 1/D/sub 2/) = 14 970.069 84(10) cm/sup -1/. The 2/sup 3/P-3 /sup 3/D results are compared with two other recent determinations. A value of 1325.4(2.0) MHz obtained for the 3/sup 3/D/sub 1/-3/sup 3/D/sub 2/ interval by combination of our results with the accurately known 2/sup 3/P/sub 0/-2/sup 3/P/sub 1/ interval is consistent with several previous measurements. The measurements of the spin-forbiddenmore » lines give a more accurate connection between the low triplet and singlet terms of this spectrum. We have reevaluated several term separations (directly from our measurements or by combining them with other recent results) for comparison with available calculations. The 2/sup 3/S-2/sup 3/P separation in combination with previous calculations gives an experimental value of -0.177 6(5) cm/sup -1/ for the Lamb-shift contribution to this separation, as compared with Ermolaev's theoretical value of -0.173 cm/sup -1/.« less