Nuclear Size Corrections Research Articles

Nuclear mass and size corrections to the magnetic shielding

Nuclear mass and size corrections to the magnetic shielding

Muonic x-ray spectroscopy on implanted targets

Muonic x-ray spectroscopy uses muons to obtain information about the structure of the atom and the nucleus. In muonic atoms, the energy levels of atomic orbitals are significantly more sensitive to the finite size correction. By probing these orbitals using x-ray spectroscopy, the nuclear size correction can be extracted, providing valuable input for laser spectroscopy in the form of absolute charge radii with a relative precision better than 10−3. Continuing on developments that allowed measurements on target quantities of about 5 μg, we showed the feasibility of using implanted targets. In the future, this will allow the measurement of absolute charge radii of long-lived radioactive isotopes that are not available in sufficient enrichment or large quantities. In this contribution, we shall report on the target preparation, involving high-fluence implantation, and on the preliminary results of the muX experimental campaign.

First-principles calculation of the frequency-dependent dipole polarizability of argon

In this work we report state-of-the-art theoretical calculations of the dipole polarizability of the argon atom. Frequency dependence of the polarizability is taken into account by means of the dispersion coefficients (Cauchy coefficients), which is sufficient for experimentally relevant wavelengths below the first resonant frequency. In the proposed theoretical framework, all known physical effects including the relativistic, quantum electrodynamics, finite nuclear mass, and finite nuclear size corrections are accounted for. We obtained ${\ensuremath{\alpha}}_{0}=11.0775(19)$ for the static polarizability and ${\ensuremath{\alpha}}_{2}=27.976(15)$ and ${\ensuremath{\alpha}}_{4}=95.02(11)$ for the second and fourth dispersion coefficients, respectively. The result obtained for the static polarizability agrees (within the estimated uncertainty) with the most recent experimental data [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)] but is less accurate. The dispersion coefficients determined in this work appear to be the most accurate in the literature, improving by more than an order of magnitude upon previous estimates. By combining the experimentally determined value of the static polarizability with the dispersion coefficients from our calculations, the polarizability of argon can be calculated with accuracy of around $10\phantom{\rule{0.16em}{0ex}}\mathrm{ppm}$ for wavelengths above roughly $450\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$. This result is important from the point of view of quantum metrology, especially for a new pressure standard based on thermophysical properties of gaseous argon. Additionally, in this work we calculate the static magnetic susceptibility of argon, which relates the refractive index of dilute argon gas with its pressure. While our results for this quantity are less accurate than in the case of the polarizability, they can provide, via the Lorenz-Lorentz formula, the best available theoretical estimate of the refractive index of argon.

Investigation of two-photon 2s→1s decay in one-electron and one-muon ions

We study the radiative decay of the $2s$ state of one-electron and one-muon ions, where the two-photon mechanism plays an important role. Due to the nuclear size corrections, the radiative decay of the $2s$ state in the electron and muon ions is qualitatively different. Based on the accurate relativistic calculation, we introduce a two-parameter approximation, which makes it possible to describe the two-photon angular-differential transition probability for the polarized emitted photons with high accuracy. The emission of photons with linear and circular polarizations is studied separately. We also investigate the transition probabilities for the polarized initial and final states. The investigation is performed for ions with atomic numbers $1\ensuremath{\le}Z\ensuremath{\le}120$.

Exploring the x-ray photoabsorption spectrum of sodium in the energy range 1070–1090 eV by elucidating the inner shell excitation transition probabilities

The researchers report on a new study of the x-ray photoabsorption spectrum of sodium near the K-edge, involving the calculations of the excitation energies and the excitation transition probabilities for K-shell and M-shell electron excitations transitions in atomic sodium. The calculations were carried out by employing the ‘multi configuration Dirac–Fock method’, Breit interactions and ‘finite nuclear size corrections’ are included in the calculations, the transition probabilities were convoluted into ‘Breit–Wigner’ line shapes, and single 1s photoionization was also taken into account. In addition, the [1s]np and the [1s3s]ns n′p singlet–triplet energy splitting was calculated. Results of the calculations were compared with previous measurements of the photoabsorption spectrum of sodium near the K-edge, and a very good agreement was observed.

Vacuum polarization and finite-nuclear-size effects in the two-photon decay of hydrogenlike ions

The total two-photon decay rate of hydrogen-like ions is studied using relativistic quantum electrodynamics. In particular, we analyse how finite nuclear size and QED vacuum polarization corrections affect the decay rate. To calculate these corrections, a finite basis set method based on $B$-splines is used for the generation of quasi-complete atomic spectra and, hence, of the relativistic Green's function. By making use of this $B$-spline approach, high precision calculations have been performed for the $2s_{1/2} \to 1s_{1/2} + 2\gamma$ and $2p_{1/2} \to 1s_{1/2} + 2\gamma$ decay of hydrogen-like ions along the entire isoelectronic sequence. The results of these calculations show that both, QED and finite nuclear size effects, are comparatively weak for the $2s_{1/2} \to 1s_{1/2} + 2\gamma$ transition. In contrast, they are much more pronounced for the $2p_{1/2}\to 1s_{1/2} + 2\gamma$ decay, where, for hydrogen-like Uranium, the decay rate is reduced by 0.484% due to the finite nuclear size and enhanced by 0.239% if the vacuum polarization is taken into account.

Skyrme-type nuclear interaction as a tool for calculating the finite-nuclear-size correction to atomic energy levels and the bound-electron g factor

A state-of-the-art approach for calculating the finite nuclear size correction to atomic energy levels and the bound-electron $g$ factor is introduced and demonstrated for a series of highly charged hydrogen-like ions. Firstly, self-consistent mean-field calculations based on the Skyrme-type nuclear interaction are employed in order to produce a realistic nuclear proton distribution. In the second step, the obtained nuclear charge density is used to construct the potential of an extended nucleus, and the Dirac equation is solved numerically. The ambiguity in the choice of a Skyrme parametrization is supressed by fine-tuning of only one parameter of the Skyrme force in order to accurately reproduce the experimental values of nuclear radii in each particular case. The homogeneously charged sphere approximation, the two-parameter Fermi distribution and experimental nuclear charge distributions are used for comparison with our approach, and the uncertainties of the presented calculations are estimated. In addition, suppression of the finite nuclear size effect for the specific differences of $g$ factors is demonstrated.

Analysis of QED and non-adiabaticity effects on the rovibrational spectrum of H3 + using geometry-dependent effective nuclear masses.

The influence of QED effects (including one- and two-electron Lamb-shift, Araki-Sucher term, one-loop self-energy, and finite nuclear size correction) together with non-adiabatic effects on the rovibrational bound states of H3 + has been investigated. Non-adiabaticity is modeled by using geometry-dependent effective nuclear masses together with only one single potential energy surface. In conclusion, for rovibrational states below 20 000 cm-1, QED and relativistic effects do nearly compensate, and a potential energy surface based on Born-Oppenheimer energies and diagonal adiabatic corrections has nearly the same quality as the one including relativity with QED; the deviations between the two approaches for individual rovibrational states are mostly below 0.02 cm-1. The inclusion of non-adiabatic effects is important, and it reduces deviations from experiments mostly below 0.1 cm-1.

Nuclear Size Corrections to the Energy Levels of Single-Electron Atoms

A study is made of nuclear size corrections to the energy levels of single-electron atoms for the ground state of hydrogen like atoms. We consider Fermi charge distribution to the nucleus and calculate atomic energy level shift due to the finite size of the nucleus in the context of perturbation theory. The exact relativistic correction based upon the available analytical calculations is compared to the result of first-order relativistic perturbation theory and the non-relativistic approximation. We find small discrepancies between our perturbative results and those obtained from exact relativistic calculation even for large nuclear charge number Z.

Weighted difference ofgfactors of light Li-like and H-like ions for an improved determination of the fine-structure constant

A weighted difference of the $g$-factors of the Li- and H-like ion of the same element is studied and optimized in order to maximize the cancellation of nuclear effects. To this end, a detailed theoretical investigation is performed for the finite nuclear size correction to the one-electron $g$-factor, the one- and two-photon exchange effects, and the QED effects. The coefficients of the $Z\alpha$ expansion of these corrections are determined, which allows us to set up the optimal definition of the weighted difference. It is demonstrated that, for moderately light elements, such weighted difference is nearly free from uncertainties associated with nuclear effects and can be utilized to extract the fine-structure constant from bound-electron $g$-factor experiments with an accuracy competitive with or better than its current literature value.

Finite nuclear size corrections to the recoil effect in hydrogenlike ions

The finite nuclear size corrections to the relativistic recoil effect in H-like ions are calculated within the Breit approximation. The calculations are performed for the 1s, 2s, and 2p1/2 states in the range Z = 1–110. The obtained results are compared with previous evaluations of this effect. It is found that for heavy ions the previously neglected corrections amount to about 20% of the total nuclear size contribution to the recoil effect calculated within the Breit approximation.

Radiative nonrecoil nuclear finite size corrections of order α(Zα)5 to the hyperfine splitting of S-states in muonic hydrogen

On the basis of quasipotential method in quantum electrodynamics we calculate nuclear finite size radiative corrections of order α(Zα)5 to the hyperfine structure of S-wave energy levels in muonic hydrogen and muonic deuterium. For the construction of the particle interaction operator we employ the projection operators on the particle bound states with definite spins. The calculation is performed in the infrared safe Fried–Yennie gauge. Modern experimental data on the electromagnetic form factors of the proton and deuteron are used.

Differential energy measurement between He- and Li-like uranium intra-shell transitions

We present the first clear identification and highly accurate measurement of the intra-shell transition 1s2p 3P2→1s2s 3S1 of He-like uranium performed via x-ray spectroscopy. The present experiment was conducted at the gas-jet target of the ESR storage ring in GSI (Darmstadt, Germany), where a Bragg spectrometer, with a bent germanium crystal, and a Ge(i) detector were mounted. Using the ESR deceleration capabilities, we performed a differential measurement between the 1s2p 3P2→1s2s 3S1 He-like U transition energy, at 4510 eV, and the 1s22p 2P3/2→1s22s 2S1/2 Li-like U transition energy, at 4460 eV. By a proper choice of the ion velocities, the x-ray energies from the He- and Li-like ions could be measured, in the laboratory frame, at the same photon energy. This allowed for a drastic reduction of experimental systematic uncertainties, principally due to the Doppler effect, and for a comparison with theory without the uncertainties arising from one-photon quantum electrodynamics predictions and nuclear size corrections.

Experimental efforts at NIST towards one-electron ions in circular Rydberg states

Experimental effort is underway at NIST to enable tests of theory with one-electron ions synthesized in circular Rydberg states from captured bare nuclei. Problematic effects that limit the accuracy of predicted energy levels for low-lying states are vanishingly small for high-angular-momentum (high-L) states; in particular, the nuclear size correction for high-L states is completely negligible for any foreseeable improvement of measurement precision. As an initial step towards realizing such states, highly charged ions are extracted from the NIST electron beam ion trap (EBIT) and steered through the electrodes of a Penning trap. The goal is to capture bare nuclei in the Penning trap for experiments to make one-electron atoms in circular Rydberg states with dipole (E1) transitions in the optical domain accessible to a frequency comb.

Nuclear-size correction to the Lamb shift of one-electron atoms

The nuclear-size effect on the one-loop self-energy and vacuum polarization is evaluated for the $1s$, $2s$, $3s$, $2{p}_{1/2}$, and $2{p}_{3/2}$ states of hydrogen-like ions. The calculation is performed to all orders in the nuclear binding strength parameter $Z\ensuremath{\alpha}$. Detailed comparison is made with previous all-order calculations and calculations based on the expansion in the parameter $Z\ensuremath{\alpha}$. Extrapolation of the all-order numerical results obtained toward $Z=1$ provides results for the radiative nuclear-size effect on the hydrogen Lamb shift.

Finite nuclear size correction to vacuum polarization in hydrogen-like ions

The finite nuclear size corrections to the Dirac binding energy of an electron and to the vacuum polarization in hydrogen-like ions are considered. These corrections were calculated for the 1s1/2, 2s1/2, 2p1/2, and 2p3/2 states in a wide range of the nuclear charge Z = 1–100. The results obtained are compared with the available analytical data.

Proposal for the determination of nuclear masses by high-precision spectroscopy of Rydberg states

The theoretical treatment of Rydberg states in one-electron ions is facilitated by the virtual absence of the nuclear-size correction, and fundamental constants like the Rydberg constant may be in the reach of planned high-precision spectroscopic experiments. The dominant nuclear effect that shifts transition energies among Rydberg states therefore is due to the nuclear mass. As a consequence, spectroscopic measurements of Rydberg transitions can be used in order to precisely deduce nuclear masses. A possible application of this approach to hydrogen and deuterium, and hydrogen-like lithium and carbon is explored in detail. In order to complete the analysis, numerical and analytic calculations of the quantum electrodynamic self-energy remainder function for states with principal quantum number n = 5, …, 8 and with angular momentum ℓ = n − 1 and ℓ = n − 2 are described .

Fundamental constants and tests of theory in Rydberg states of one-electron ions

The nature of the theory of circular Rydberg states of hydrogen-like ions allows highly accurate predictions to be made for energy levels. In particular, uncertainties arising from the problematic nuclear size correction which beset low angular-momentum states are negligibly small for the high angular-momentum states. The largest remaining source of uncertainty can be addressed with the help of quantum electrodynamics calculations, including a new nonperturbative result reported here. More stringent tests of theory and an improved determination of the Rydberg constant may be possible if predictions can be compared with precision frequency measurements in this regime. The diversity of information can be increased by utilizing a variety of combinations of ions and Ryberg states to determine fundamental constants and test theory.

Rovibrational levels of HD

The dissociation energies of all rotation-vibrational states of the molecular HD in the ground electronic state are calculated to a high accuracy by including nonadiabatic, relativistic alpha(2), and quantum electrodynamic alpha(3) effects, with approximate treatment of small higher order alpha(4), and finite nuclear size corrections. The obtained result for the ground molecular state of 36 405.7828(10) cm(-1) is in a small disagreement with the latest most precise experimental value.

Theoretical transition probabilities for inner shell excitations in rubidium near the K edge

Abstract We were able to reproduce the near K edge photoabsorption spectrum of rubidium in the energy range 15,200–15,235 eV. This was done by calculating the transition probabilities for inner shell excitations to singly and doubly excited states. The calculations have been carried out using multi-configuration Dirac–Fock wave functions with the inclusion of the Breit interaction, quantum electrodynamics contributions and finite nuclear size corrections. The transition probabilities were then convoluted into Lorentzian line shapes. A rough estimate of the effect of photoionization was also taken into account. Good overall agreement was attained when the results of the calculation is compared with previous measurements of the K edge photoabsorption spectrum of rubidium.