Bohr Weisskopf Research Articles

Bohr-Weisskopf effect in the potassium isotopes

The magnetic hyperfine structure constants have been calculated for low-lying levels in neutral potassium atom taking into account the Bohr--Weisskopf (BW) and Breit--Rosenthal (BR) effects. According to our results the $4p_{1/2}$ state of K~I is free from both BR and BW corrections on the level of the current theoretical uncertainties. Using this finding and the measured values of the $A(4p_{1/2})$ constants, we corrected the nuclear magnetic moments for several short-lived potassium isotopes. The BW correction is represented as a product of atomic and nuclear factors. We calculated the atomic factor for the ground state of K I, which allowed us to extract nuclear factors for potassium $I^\pi = 3/2^+$ isotopes from the experimental data. In this way the application range of the single-particle nuclear model for nuclear-factor calculation in these isotopes has been clarified.

Empirical Determination of the Bohr-Weisskopf Effect in Cesium and Improved Tests of Precision Atomic Theory in Searches for New Physics.

The finite distribution of the nuclear magnetic moment across the nucleus gives a contribution to the hyperfine structure known as the Bohr-Weisskopf (BW) effect. We have obtained an empirical value of -0.24(18)% for this effect in the ground and excited s states of atomic ^{133}Cs. This value is found from historical muonic-atom measurements in combination with our muonic-atom and atomic many-body calculations. The effect differs by 0.5% in the hyperfine structure from the value found using the uniform magnetization distribution, which has been commonly employed in the precision heavy-atom community over the last several decades. We also deduce accurate values for the BW effect in other isotopes and states of cesium. These results enable cesium atomic wave functions to be tested in the nuclear region at an unprecedented 0.2% level, and are needed for the development of precision atomic many-body methods. This is important for increasing the discovery potential of precision atomic searches for new physics, in particular for atomic parity violation in cesium.

Bohr-Weisskopf effect: From hydrogenlike-ion experiments to heavy-atom calculations of the hyperfine structure

In this paper we study the influence of electron screening on the Bohr-Weisskopf (BW) effect in many-electron atoms. The BW effect gives the finite-nucleus magnetization contribution to the hyperfine structure. Relativistic atomic many-body calculations are performed for s and p_1/2 states of several systems of interest for studies of atomic parity violation and time-reversal-violating electric dipole moments -- Rb, Cs, Fr, Ba+, Ra+, and Tl. For s states, electron screening effects are small, and the relative BW correction for hydrogenlike ions and neutral atoms is approximately the same. We relate the ground-state BW effect in H-like ions, which may be cleanly extracted from experiments, to the BW effect in s and p_1/2 states of neutral and near neutral atoms through an electronic screening factor. This allows the BW effect extracted from measurements with H-like ions to be used, with screening factors, in atomic calculations without recourse to modelled nuclear structure input. It opens the way for unprecedented accuracy in accounting for the BW effect in heavy atoms. The efficacy of this approach is demonstrated using available experimental data for H-like and neutral Tl-203 and Tl-205.

Comparison of Methods for Eliminating the Bohr–Weisskopf Effect in Atomic Spectra of $${}^{{209}}$$Bi Heavy Ions

Comparison of Methods for Eliminating the Bohr–Weisskopf Effect in Atomic Spectra of $${}^{{209}}$$Bi Heavy Ions

Atomic calculations of the hyperfine-structure anomaly in gold

The magnetic hyperfine structure constants have been calculated for low-lying levels in neutral gold atom and gold-like ion of mercury taking into account Bohr--Weisskopf (BW) effect. BW effect is represented as a product of atomic and nuclear ($d_\mathrm{nuc}$) factors. We have calculated the atomic factors, which enable one to extract BW-correction values for far from stability gold nuclei from the experimental data. The possible uncertainty of our atomic calculations have been estimated by the comparison with the available experimental data. It has been shown that the standard single-particle approach in $d_\mathrm{nuc}$ calculation reasonably well describes experimental data for $11/2^-$ gold isomers and $3/2^+$ ground state of $\rm ^{199}Au$. At the same time, it fails to describe the hyperfine constant in $^{197}\mathrm{Au}$. This indicates the more pronounced configuration mixing in $\rm ^{197}Au$ than in $\rm ^{199}Au$.

The Bohr–Weisskopf Effect in the Atomic Spectra of Heavy Ions of 209Bi

Two ways of eliminating the Bohr–Weisskopf effect in the theory of hyperfine splitting (HFS) are compared: the ratio and specific-difference (SD) approaches to the differences between the HFS values in different shells. A comparison reveals a number of disadvantages in the SD approach. Conclusions based on results obtained in regard to further refining the magnetic moment of the 209Bi nucleus with heavy ions in the storage rings of GSI Darmstadt and IMP Lanzhou are considered.

Nuclear magnetization distribution effect in molecules: Ra+ and RaF hyperfine structure.

Recently, the first laser spectroscopy measurement of the radioactive RaF molecule has been reported by Ruiz et al. [Nature 581, 396 (2020)]. This and similar molecules are considered to search for the new physics effects. The radium nucleus is of interest as it is octupole-deformed and has close levels of opposite parity. The preparation of such experiments can be simplified if there are reliable theoretical predictions. It is shown that the accurate prediction of the hyperfine structure of the RaF molecule requires to take into account the finite magnetization distribution inside the radium nucleus. For atoms, this effect is known as the Bohr-Weisskopf (BW) effect. Its magnitude depends on the model of the nuclear magnetization distribution which is usually not well known. We show that it is possible to express the nuclear magnetization distribution contribution to the hyperfine structure constant in terms of one magnetization distribution dependent parameter: BW matrix element for 1s-state of the corresponding hydrogen-like ion. This parameter can be extracted from the accurate experimental and theoretical electronic structure data for an ion, atom, or molecule without the explicit treatment of any nuclear magnetization distribution model. This approach can be applied to predict the hyperfine structure of atoms and molecules and allows one to separate the nuclear and electronic correlation problems. It is employed to calculate the finite nuclear magnetization distribution contribution to the hyperfine structure of the 225Ra+ cation and 225RaF molecule. For the ground state of the 225RaF molecule, this contribution achieves 4%.

Hyperfine Anomaly in Eu Isotopes and the Universiability of the Moskowitz–Lombardi Formula

A method for determining the hyperfine anomaly, without using the nuclear magnetic moments, is used on a series of unstable isotopes of Eu. The large number of experimental data in Eu makes it possible to extract the hyperfine anomaly for a number of unstable isotopes. Calculations of the Bohr–Weisskopf effect and hence the hyperfine anomaly are performed using the particle-rotor formalism. The result from the calculations and experiments is compared with other theoretical calculations and the empirical Moskowitz–Lombardi formula. The results show that the Moskowitz–Lombardi formula is not universal.

Calculation of Francium Hyperfine Anomaly

The Dirac–Hartree–Fock plus many-body perturbation theory (DHF + MBPT) method has been used to calculate hyperfine structure constants for Fr. Calculated hyperfine structure anomaly for hydrogen-like ion is in good agreement with analytical expressions. It has been shown that the ratio of the anomalies for s and p1/2 states is weakly dependent on the principal quantum number. Finally, we estimate Bohr–Weisskopf corrections for several Fr isotopes. Our results may be used to improve experimental accuracy for the nuclear g factors of short-lived isotopes.

The theory of the Bohr–Weisskopf effect in the hyperfine structure

Description of the Bohr–Wesskopf effect in the hyperfine structure of few-electron heavy ions is a challenging problem, which can be used as a test of both QED and atomic calculations. However, for twenty years the research has actually been going in a wrong direction, aimed at fighting the Bohr–Weisskopf effect through its cancellation in specific differences. Alternatively, we propose the constructive model-independent way, which enables the nuclear radii and their momenta to be retrieved from the hyper-fine splitting (HFS). The way is based on analogy of HFS to internal conversion coefficients, and the Bohr–Weisskopf effect – to the anomalies in the internal conversion coefficients. It is shown that the parameters which can be extracted from the data are the even nuclear momenta of the magnetization distribution. The radii R2 and – for the first time – R4 are obtained in this way by analysis of the experimental HFS values for the H- and Li-like ions of 209Bi. The critical prediction concerning the HFS for the 2p12 state is made. The present analysis shows high sensitivity of the method to the QED effects, which offers a way of precision test of QED. Experimental recommendations are given, which are aimed at retrieving data on the HFS values for a set of a few-electron configurations of each atom.

Comparison of hyperfine anomalies in the [formula omitted] and [formula omitted] levels of 85Rb and 87Rb

We observe a hyperfine anomaly in the measurement of the hyperfine splitting of the 6S1/2 excited level in rubidium. We perform two step spectroscopy using the 5S1/2→5P1/2→6S1/2 excitation sequence. We measure the splitting of the 6S1/2 level and obtain for the magnetic dipole constants of 85Rb and 87Rb A=239.18(4)MHz and A=807.66(8)MHz, respectively. The hyperfine anomaly difference of δ8587=−0.0036(2) comes from the Bohr Weisskopf effect: a correction to the point interaction between the finite nuclear magnetization and the electrons, and agrees with that obtained in the 5S1/2 ground state.

Multi-configuration Dirac–Fock calculations of the hyperfine structure constants A and B of neutral Cu, Ag and Au

The multi-configuration Dirac–Fock method has been employed to calculate the magnetic dipole hyperfine structure constant A and the electric quadrupole hyperfine structure constant B of neutral Cu, Ag and Au. The contributions from Breit interaction, quantum electrodynamics (QED) corrections, core–valence correlations, core–core correlations and Bohr–Weisskopf correction have been taken into account. Through the use of the active space method, the calculated values of the magnetic dipole hyperfine structure constant A of the ground state are compared with the other available data and are found to be in good agreement with them. In this paper we give the magnetic dipole hyperfine structure constant and the electric quadrupole hyperfine structure constant of 4s 2S1/2, 4p 2P1/2 and 2P3/2 of Cu, 5s 2S1/2, 5p 2P1/2 and 2P3/2 of Ag, 6s 2S1/2, 6p 2P1/2 and 2P3/2 of Au. The hyperfine splits of these ground states are also calculated. These data provide reference values for a variety of scientific applications, especially for an atomic clock.

Precision hfs of 126Cs( [formula omitted]) by ABMR

The hfs separation Δ ν of 126Cs( T 1 / 2 = 1.63 m ) in the 6 s S 1 / 2 2 ground state was obtained in a precision measurement near zero magnetic field by means of atomic beam magnetic resonance with laser optical pumping on-line with the CERN-PSB-ISOLDE mass separator. The result, Δ ν = 3629.514 (0.001) MHz, corrects significantly a previous published value from a high-field experiment. With our result, the precision of the nuclear magnetic moment, μ ( Cs 126 ) ≈ 0.776 μ N , is now limited by the influence of extended nuclear structure on the hfs (the Bohr–Weisskopf effect).

Hyperfine splitting of hydrogenlike atoms based on relativistic mean field theory

We evaluate the hyperfine splitting of hydrogenlike $^{209}\mathrm{Bi}$ and $^{207}\mathrm{Pb}$ atoms based on a relativistic method for both the electron system and the nucleon system. The Bohr-Weisskopf (BW) effect is calculated with Lorentz covariant current. It is shown that the BW correction to the hyperfine splitting (HFS) is $0.58%--0.67%$ for $^{209}\mathrm{Bi}^{82+}$ and $3.79%--4.00%$ for $^{207}\mathrm{Pb}^{81+}$. It is also concluded that relativistic mean field theory reproduces the observed values of the HFS within the accuracy of $5%$ in $^{209}\mathrm{Bi}^{82+}$ and $13%$ in $^{207}\mathrm{Pb}^{81+}$.

Hyperfine structure and \(\user1{g} \) factor measurements on Ba+ and Eu+ isotopes

Paul and Penning ion traps have been used to determine hyperfine structure coupling constants as well as nuclear \(\user1{g} \) factors of Ba+ ions with great precision. Attempts are under way to measure both quantities for a long chain of Eu isotopes in order to determine differential hyperfine anomalies for a better understanding of the Bohr–Weisskopf effect.

Bohr–Weisskopf effect: influence of the distributed nuclear magnetization on hfs

Nuclear magnetic moments provide a sensitive test of nuclear wave functions, in particular those of neutrons, which are not readily obtainable from other nuclear data. These are taking added importance by recent proposals to study parity non-conservation (PNC) effects in alkali atoms in isotopic series. By taking ratios of the PNC effects in pairs of isotopes, uncertainties in the atomic wave functions are largely cancelled out at the cost of knowledge of the change in the neutron wave function. The Bohr–Weisskopf effect (B–W) in the hyperfine structure interaction of atoms measures the influence of the spatial distribution of the nuclear magnetization, and thereby provides an additional constraint on the determination of the neutron wave function. The added great importance of B–W in the determination of QED effects from the hfs in hydrogen-like ions of heavy elements, as measured recently at GSI, is noted. The B–W experiments require precision measurements of the hfs interactions and, independently, of the nuclear magnetic moments. A novel atomic beam magnetic resonance (ABMR) method, combining rf and laser excitation, has been developed for a systematic study and initially applied to stable isotopes. Difficulties in adapting the experiment to the ISOLDE radioactive ion beam, which have now been surmounted, are discussed. A first radioactive beam measurement for this study, the precision hfs of 126Cs, has been obtained recently. The result is 3629.515(∼0.001) MHz. The ability of ABMR to determine with high precision nuclear magnetic moments in free atoms is a desideratum for the extraction of QED effects from the hfs of the hydrogen-like ions. We also point out manifestations of B–W in condensed matter and atomic physics.

Some analytic results on the Uehling correction to hyperfine splitting in a muonic atom

We analyze the Uehling correction to the hyperfine structure of the ground state in hydrogenlike electronic and muonic atoms with a pointlike nucleus. The results are obtained analytically without any expansion over ~k. PACS No.: 31.20D Resume : Nous analysons la correction d’Uehling a la structure hyperfine du fondamental d’atomes hydrogenoides, electroniques ou muoniques, avec un noyau ponctuel, les resultats sont analytiques, sans recours a une expansion en puissances de ~k. [Traduit par la redaction] This work is a direct continuation of a previous paper [1] devoted to the Uehling correction to the Lamb shift (Fig. 1a) in a hydrogenlike atom with a pointlike nucleus. Here we calculate the spin-dependent part of the correction (Figs. 1b and 1c). In ref. 1, we examined the kind of integral for the complete Dirac wave function and the Uehling potential as a perturbation. More details will be found in a later paper1 where the method will be developed for any single-photon potential diagram including the one in Fig. 1b. The hyperfine structure (hfs) is more sensitive to the nuclear structure than the Lamb shift. As is well known, the theoretical uncertainty of the hyperfine splitting value for a hydrogenlike atom comes mainly from estimates of the nuclear magnetization distribution correction (the Bohr–Weisskopf effect). That is correct both for muonic and electronic atoms, but in the first case this correction is significantly larger and investigations of the hyperfine structure of muonic atoms can be used as a way to measure that effect and to apply the result to an electronic atom with the same nucleus. The hyperfine structure in a muonic atom being of the order +] ,p @ps is more important 1 S.G. Karshenboim. Manuscript in preparation. Received June 8, 1998. Accepted July 2, 1998. S.G. Karshenboim. D.I. Mendeleev Institute for Metrology, 198005 St. Petersburg, Russia. Telephone: 7-812-2591051; FAX: 7-812-1130114; e-mail: sgk@onti.vniim.spb.su V.G. Ivanov. Pulkovo Observatory, 196140, St. Petersburg, Russia. e-mail: ivanovv@gao.pnpi.spb.ru V.M. Shabaev. St. Petersburg State University, 198904, St. Petersburg, Russia. e-mail: shabaev@pobox.spbu.ru Can. J. Phys. 76: 503–506 (1998) © 1998 NRC Canada 504 Can. J. Phys. Vol. 76, 1998 Fig. 1. The Uehling corrections to the energy. The orbiting particle with mass 6 is either a muon or an electron and a particle in vacuum polarization loop is an electron.