Relativistic Recoil Corrections Research Articles

2D energy-momentum tensor distributions of nucleon in a large- Nc quark model from ultrarelativistic to nonrelativistic limit

Form factors of the energy-momentum tensor (EMT) can be interpreted in certain frames in terms of spatial distributions of energy, stress, linear and angular momentum, based on 2D or 3D Fourier transforms. This interpretation is in general subject to ``relativistic recoil corrections,'' except when the nucleon moves at the speed of light like e.g. in the infinite-momentum frame. We show that it is possible to formulate a large-${N}_{c}$ limit in which the probabilistic interpretation of the nucleon EMT distributions holds also in other frames. We use the bag model formulated in the large-${N}_{c}$ limit as an internally consistent quark model framework to visualize the information content associated with the 2D EMT distributions. In order to provide more intuition, we present results in the physical situation and in three different limits: by considering a heavy-quark limit, a large system-size limit and a constituent-quark limit. The visualizations of the distributions in these extreme limits will help to interpret the results from experiments, lattice QCD, and other models or effective theories.

Theoretical prediction of the fine and hyperfine structure of heavy muonic atoms

Precision calculations of the fine and hyperfine structure of muonic atoms are performed in a relativistic approach and results for muonic 205 Bi, 147 Sm, and 89 Zr are presented. The hyperfine structure due to magnetic dipole and electric quadrupole splitting is calculated in first order perturbation theory, using extended nuclear charge and current distributions. The leading correction from quantum electrodynamics, namely vacuum polarization in Uehling approximation, is included as a potential directly in the Dirac equation. Also, an effective screening potential due to the surrounding electrons is calculated, and the leading relativistic recoil correction is estimated.

Theoretical study of the hyperfine-interaction constants and the isotope-shift factors for the 3s21S0–3s3p3,1P1o transitions in Al+

We calculated the magnetic dipole and the electric quadrupole hyperfine interaction constants of 3s3p $^{3,1}P^o_1$ states and the isotope shift, including mass and field shift, factors for transitions from these two states to the ground state 3s$^2~^1S_0$ in Al$^+$ ions using the multiconfiguration Dirac-Hartree-Fock method. The effects of the electron correlations and the Breit interaction on these physical quantities were investigated in detail based on the active space approach. It is found that the CC and the higher-order correlations are considerable for evaluating the uncertainties of the atomic parameters concerned. The uncertainties of the hyperfine interaction constants in this work are less than 1.5\%. Although the isotope shift factors are highly sensitive to the electron correlations, reasonable uncertainties were obtained by exploring the effects of the electron correlations. Moreover, we found that the relativistic nuclear recoil corrections to the mass shift factors are very small and insensitive to the electron correlations for Al$^{+}$. These atomic parameters present in this work are valuable for extracting the nuclear electric quadrupole moments and the mean-square charge radii of Al isotopes.

Relativistic corrections for the ground electronic state of molecular hydrogen

We recalculate the leading relativistic corrections for the ground electronic state of the hydrogen molecule using variational method with explicitly correlated functions which satisfy the interelectronic cusp condition. The new computational approach allowed for the control of the numerical precision which reached about 8 significant digits. More importantly, the updated theoretical energies became discrepant with the known experimental values and we conclude that the yet unknown relativistic recoil corrections might be larger than previously anticipated.

Relativistic recoil effects on energy levels in a muonic atom: A Grotch-type calculation of the second-order vacuum-polarization contributions

Adjusting a previously developed Grotch-type approach to a perturbative calculation of the electronic vacuum-polarization effects in muonic atoms, we find here the two-loop vacuum polarization relativistic recoil correction of order $\alpha^2(Z\alpha)^4m^2/M$ in light muonic atoms. The result is in perfect agreement with the one previously obtained within the Breit-type approach. We also discuss here simple approximations of the irreducible part of the two-loop vacuum-polarization dispersion density, which was applied to test our calculations and can be useful for other evaluations with an uncertainty better than 1%.

Relativistic recoil effects for energy levels in a muonic atom within a Grotch-type approach. I. General approach

Recently we calculated relativistic recoil corrections to the energy levels of the low-lying states in muonic hydrogen induced by electron vacuum polarization effects. The results were obtained by Breit-type and Grotch-type calculations. The former were described in our previous papers in detail, and here we present the latter. The Grotch equation was originally developed for pure Coulomb systems and allowed to express the relativistic recoil correction to order ${(Z\ensuremath{\alpha})}^{4}{m}^{2}/M$ in terms of the relativistic nonrecoil contribution ${(Z\ensuremath{\alpha})}^{4}m$. Certain attempts to adjust the method to electronic vacuum polarization took place in the past, however, the consideration was incomplete and the results were incorrect. Here we present a Grotch-type approach to the problem and in a series of papers consider relativistic recoil effects in order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{4}{m}^{2}/M$ and ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{4}{m}^{2}/M$. That is the first paper of the series and it presents a general approach, while two other papers present results of calculations of the $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{4}{m}^{2}/M$ and ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{4}{m}^{2}/M$ contributions in detail. In contrast to our previous calculation, we address now a variety of states in muonic atoms with a certain range of the nuclear charge $Z$.

Relativistic recoil corrections to the electron-vacuum-polarization contribution in light muonic atoms

The relativistic recoil contributions to the Uehling corrections are revisited. We consider a controversy in recent calculations based on different approaches including Breit-type and Grotch-type calculations. We have found that calculations of those authors were in fact done in different gauges and in some of those gauges contributions the retardation and two-photon-exchange effects were missed. We have evaluated such effects and obtained a consistent result from those approaches. We present a correct expression for the Grotch-type approach which produces a correct gauge-invariant result. We also consider a finite-nuclear-size correction for the Uehling term. The results are presented for muonic hydrogen and deuterium atoms and for muonic helium-3 and helium-4 ions.

Relativistic reduced-mass and recoil corrections to vacuum polarization in muonic hydrogen, muonic deuterium, and muonic helium ions

The reduced-mass dependence of relativistic and radiative effects in simple muonic bound systems is investigated. The spin-dependent nuclear recoil correction of order $(Z\ensuremath{\alpha}){}^{4} \ensuremath{\mu}{}^{3}/{m}_{N}^{2}$ is evaluated for muonic hydrogen and deuterium and muonic helium ions ($\ensuremath{\mu}$ is the reduced mass and ${m}_{N}$ is the nuclear mass). Relativistic corrections to vacuum polarization of order $\ensuremath{\alpha}(Z\ensuremath{\alpha}){}^{4}\ensuremath{\mu}$ are calculated, with a full account of the reduced-mass dependence. The results shift theoretical predictions. The radiative-recoil correction to vacuum polarization of order $\ensuremath{\alpha}(Z\ensuremath{\alpha}){}^{5} \mathrm{ln}{}^{2}(Z\ensuremath{\alpha})\ensuremath{\mu}{}^{2}/{m}_{N}$ is obtained in leading logarithmic approximation. The results emphasize the need for a unified treatment of relativistic corrections to vacuum polarization in muonic hydrogen, muonic deuterium, and muonic helium ions, where the mass ratio of the orbiting particle to the nuclear mass is larger than the fine-structure constant.

Universality of leading relativistic corrections to bound state gyromagnetic ratiosThis paper was presented at the International Conference on Precision Physics of Simple Atomic Systems, held at École de Physique, les Houches, France, 30 May – 4 June, 2010.

We discuss the leading relativistic (nonrecoil and recoil) corrections to bound state g-factors of particles with arbitrary spin. These corrections are universal for any spin and depend only on the free particle gyromagnetic ratios. We explain the physical reasons behind this universality.

QED calculation of the2p1∕2−2sand2p3∕2−2stransition energies and the ground-state hyperfine splitting in lithiumlike scandium

We present the most accurate up-to-date theoretical values of the $2{p}_{1∕2}\text{\ensuremath{-}}2s$ and $2{p}_{3∕2}\text{\ensuremath{-}}2s$ transition energies and the ground-state hyperfine splitting in ${\mathrm{Sc}}^{18+}$. All two- and three-electron contributions to the energy values up to the two-photon level are treated in the framework of bound-state QED without $\ensuremath{\alpha}Z$ expansion. The interelectronic interaction beyond the two-photon level is taken into account by means of the large-scale configuration-interaction Dirac-Fock-Sturm (CI-DFS) method. The relativistic recoil correction is calculated with many-electron wave functions in order to take into account the electron-correlation effect. The accuracy of the transition energy values is improved by a factor of 5 compared to the previous calculations. The CI-DFS calculation of interelectronic-interaction effects and the evaluation of the QED correction in an effective screening potential provide significant improvement for the $2s$ hyperfine splitting. The results obtained are in good agreement with recently published experimental data.

Correlation and quantum electrodynamic effects on the radiative lifetime and relativistic nuclear recoil in Ar13+ and Ar14+ ions

The radiative lifetime and mass isotope shift of the 1s22s22p 2P3/2 - 2P1/2 M1 transition in Ar13+ ions have been determined with high accuracies using the Heidelberg electron beam ion trap. This fundamentally relativistic transition provides unique possibilities for performing precise studies of correlation and quantum electrodynamic effects in many-electron systems. The lifetime corresponding to the transition has been measured with an accuracy of the order of one per thousand. Theoretical calculations predict a lifetime that is in significant disagreement with this high-precision experimental value. Our mass shift calculations, based on a fully relativistic formulation of the nuclear recoil operator, are in excellent agreement with the experimental results and confirm the absolute necessity to include relativistic recoil corrections when evaluating mass shift contributions even in medium-Z ions.

Relativistic Nuclear Recoil Corrections to the Energy Levels of Hydrogenlike Ions

Results of a calculation of the relativistic nuclear recoil corrections to the 1s and 2s state energies of hydrogenlike ions are presented for extended nuclei in the range Z = 10–92.

Recoil correction to the ground-state energy of hydrogenlike atoms

The recoil correction to the ground state energy of hydrogenlike atoms is calculated to all orders in \alpha Z in the range Z = 1-110. The nuclear size corrections to the recoil effect are partially taken into account. In the case of hydrogen, the relativistic recoil correction beyond the Salpeter contribution and the nonrelativistic nuclear size correction to the recoil effect, amounts to -7.2(2) kHz. The total recoil correction to the ground state energy in hydrogenlike uranium (^{238}U^{91+}) constitutes 0.46 eV.

Electro-magnetic nucleon form factors and their spectral functions in soliton models

It is demonstrated that in simple soliton models essential features of the electro-magnetic nucleon form factors observed over three orders of magnitude in momentum transfer $t$ are naturally reproduced. The analysis shows that three basic ingredients are required: an extended object, partial coupling to vector mesons, and relativistic recoil corrections. We use for the extended object the standard skyrmion, one vector meson propagator for both isospin channels, and the relativistic boost to the Breit frame. Continuation to timelike $t$ leads to quite stable results for the spectral functions in the regime from the 2- or 3-pion threshold to about two rho masses. Especially the onset of the continuous part of the spectral functions at threshold can be reliably determined and there are strong analogies to the results imposed on dispersion theoretic approaches by the unitarity constraint.

Nuclear recoil corrections to the 2p3/2 state energy of hydrogen-like and high-Z lithium-like atoms in all orders in alpha Z

The relativistic nuclear recoil corrections to the energy of the 2p3/2 state of hydrogen-like and the (1s)22p3/2 state of high-Z lithium-like atoms in all orders in alpha Z are calculated. The calculations are carried out using the B-spline method for the Dirac equation. For low Z the results of the calculation are in good agreement with the alpha Z-expansion results. It is found that the total nuclear recoil contribution to the energy of the (1s)22p3/2-(1s)22s transition in lithium-like uranium constitutes -0.09 eV.

Relativistic nuclear recoil corrections to the energy levels of hydrogenlike and high-Z lithiumlike atoms in all orders in alpha Z.

The relativistic nuclear recoil corrections to the energy levels of low-lying states of hydrogenlike and high-Z lithiumlike atoms in all orders in \ensuremath{\alpha}Z are calculated. The calculations are carried out using the B-spline method for the Dirac equation. For low Z the results of the calculation are in good agreement with the \ensuremath{\alpha}Z-expansion results. It is found that the nuclear recoil contribution, in addition to Salpeter's contribution, to the Lamb shift (n=2) of hydrogen is -1.32(6) kHz. The total nuclear recoil correction to the energy of the (1s${)}^{2}$${\mathit{p}}_{1/2\mathrm{\ensuremath{-}}}$(1s${)}^{2}$2s transition in lithiumlike uranium constitutes -0.07 eV and is largely made up of QED contributions.

Relativistic nuclear recoil corrections to the energy levels of multicharged ions

The relativistic nuclear recoil corrections of order ( alpha Z)4 m2/M to the energy levels of two- and three-electron multicharged ions are calculated. The calculations are made with the hydrogen-like Dirac wavefunctions and are intended for the ions in the range 10<or approximately=2<or approximately=40. Comparison of the calculated corrections with the uncertainties of experimental values of the transition energies is given.

Energies and relativistic corrections for the Rydberg states of helium: Variational results and asymptotic analysis.

The results of an extended series of high-precision variational calculations for all states of helium up to n=10 and L=7 (excluding S states above n=2) are presented. Convergence of the nonrelativistic eigenvalues ranges from five parts in ${10}^{15}$ for the 2P states to four parts in ${10}^{19}$ for the 10K states. Relativistic and quantum electrodynamic corrections of order ${\mathrm{\ensuremath{\alpha}}}^{2}$, ${\mathrm{\ensuremath{\alpha}}}^{3}$, ${\mathrm{\ensuremath{\alpha}}}^{2}$\ensuremath{\mu}/M, ${\mathrm{\ensuremath{\alpha}}}^{2}$(\ensuremath{\mu}/M${)}^{2}$, and ${\mathrm{\ensuremath{\alpha}}}^{3}$\ensuremath{\mu}/M are included and the required matrix elements listed for each state. For the 1s2p $^{3}$${\mathit{P}}_{\mathit{J}}$ states, the lowest-order spin-dependent matrix elements of the Breit interaction are determined to an accuracy of three parts in ${10}^{9}$, which, together with higher-order corrections, would be sufficient to allow an improved measurement of the fine-structure constant. Methods of asymptotic analysis are extended to provide improved precision for the relativistic and relativistic-recoil corrections. A comparison with the variational results for the high-angular-momentum states shows that the ``standard-atomic-theory'' and ``long-range-interaction'' pictures discussed by Hessels et al. [Phys. Rev. Lett. 65, 2765 (1990)] come into agreement, thereby resolving what appeared to be a discrepancy. The comparison shows that the asymptotic expansions for the total energies are accurate to better than \ifmmode\pm\else\textpm\fi{}100 Hz for L&gt;7, and results are presented for the 9L, 10L, and 10M states (i.e., angular momentum L=8 and 9). Significant discrepancies with experiment persist for transitions among the n=10 states, which cannot be easily accommodated by supposed higher-order corrections or additional terms. Finally, the asymptotic analysis indicates that a revision to the quantum-defect method is required for the analysis of high-precision data.

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.

New variational techniques for the 1snd states of helium.

New variational techniques are described which yield a factor of 1000 improvement in accuracy for the energies of the 1snd $^{1}\mathrm{D}$ and $^{3}\mathrm{D}$ states of helium up to n=8. Convergence to better than \ifmmode\pm\else\textpm\fi{}10 kHz is obtained, making possible high-precision comparisons with experiment for fine structure and singlet-triplet splittings. The comparisons are sensitive to QED, relativistic recoil, and second-order mass-polarization corrections.