We report a fourfold improvement in the determination of nuclear magnetic moments for neutron-deficient isotopes of francium-207--213, reducing the uncertainties from 2% for most isotopes to 0.5%. These are found by comparing our high-precision calculations of hyperfine structure constants for the ground states with experimental values. In particular, we show the importance of a careful modeling of the Bohr-Weisskopf effect, which arises due to the finite nuclear magnetization distribution. This effect is particularly large in Fr and until now has not been modeled with sufficiently high accuracy. An improved understanding of the nuclear magnetic moments and Bohr-Weisskopf effect are crucial for benchmarking the atomic theory required in precision tests of the standard model, in particular atomic parity violation studies, that are underway in francium.
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 has been shown to be in good agreement with analytical expressions. It has been shown that the ratio of the anomalies for $s$ and $p_{1/2}$ 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.
We perform ab initio calculations of isotope shifts for isotopes of cesium (from A=123 to A=137) and francium (from A=207 to A=228). These calculations start from the relativistic Hartree-Fock method and make use of several techniques to include correlations. The field (volume) isotope shift is calculated by means of an all-order correlation potential method and within the singles-doubles partial triples linearized coupled-cluster approach. Many-body perturbation theory in two different formulations is used to calculate the specific mass shift. We discuss the strong points and shortcomings of …
We have measured the hyperfine splitting of the $7P_{1/2}$ state at the 100 ppm level in Fr isotopes ($^{206g,206m, 207, 209, 213, 221}$Fr) near the closed neutron shell ($N$ = 126 in $^{213}$Fr). The measurements in five isotopes and a nuclear isomeric state of francium, combined with previous determinations of the $7S_{1/2}$ splittings, reveal the spatial distribution of the nuclear magnetization, i.e. the Bohr-Weisskopf effect. We compare our results with a simple shell model consisting of unpaired single valence nucleons orbiting a spherical nucleus, and find good agreement over a range of…
The interaction of orbital electrons with the charge and magnetic moment of the nucleus polarizes it, and the detailed description requires a careful treatment of the nuclear vector polarizability. We present a complete and closed form expression for the resulting contribution to hyperfine splitting in light atomic systems.
Nuclear structure corrections of orders $Zα\, E_F$ and $(Zα)^2 E_F$ are calculated for the hyperfine splitting of the muonic deuterium. The obtained results disagree with previous calculations and lead to a $5\,σ$ disagreement with the current experimental value of the $2S$ hyperfine splitting in $μ$D.
Measuring the hyperfine structure (HFS) of long-lived $^3P_2$ states of divalent atoms may offer the opportunity of extracting relatively unexplored nuclear magnetic octupole and electric hexadecapole moments. Here, using relativistic many-body methods of atomic structure and the nuclear shell model, we evaluate the effect of these higher nuclear moments on the hyperfine structure. We find that the sensitivity of HFS interval measurements in $^{87}$Sr needed to reveal the perturbation caused by the nuclear octupole moment is on the order of kHz. Results of similar analyses for $^{9}$Be, $^{25}…
Relativistic and QED corrections were calculated for hyperfine splitting of the $2S_{1/2}$ ground state in $^{6,7}$Li atoms with an exact account for electronic correlations. The resulting theoretical predictions achieved such a precision level that, by comparison with experimental values, they enable determination of the nuclear properties. In particular, the obtained results show that the $^7$Li nucleus, having a charge radius smaller than $^6$Li, has about a 40% larger Zemach radius. Together with known differences in the electric quadrupole and magnetic dipole moments, this calls for a dee…
Hyperfine structures of $^{211}$Fr due to the interactions of magnetic dipole ($μ$), electric quadrupole ($Q$) and magnetic octupole ($Ω$) moments with the electrons are investigated using the relativistic coupled-cluster (RCC) theory with an approximation of singles, doubles and important valence triples excitations in the perturbative approach. Validity of our calculations are substantiated by comparing the results with their available experimental values. Its $Q$ value has also been elevated by combining the measured hyperfine structure constant of the $7p \ ^2P_{3/2}$ state with our improv…
Hyperfine structure (HFS) of atomic energy levels arises due to interactions of atomic electrons with a hierarchy of nuclear multipole moments, including magnetic dipole, electric quadrupole and higher rank moments. Recently, a determination of the magnetic octupole moment of the $^{173}\mathrm{Yb}$ nucleus was reported from HFS measurements in neutral ${}^{173}\mathrm{Yb}$ [PRA 87, 012512 (2013)], and is four orders of magnitude larger than the nuclear theory prediction. Considering this substantial discrepancy between the spectroscopically extracted value and nuclear theory, here we propose …
The hyperfine splitting of the ground and low-energy $3/2^+(7.8 \pm 0.5$ eV) levels in the $^{229}$Th nucleus in muonic atom ($μ^-_{1S_{1/2}}{}^{229}$Th$)^*$ has been calculated considering the distribution of the nuclear magnetization in the framework of collective nuclear model with the wave functions of the Nilsson model for the unpaired neutron. It is shown that (a) the hyperfine splitting of the $3/2^+$ isomeric state is anomalously weak, and the reduction of hyperfine structure in comparison with the model of a point nuclear magnetic dipole is close to 100\%, (b) partial inversion of lev…
Three models of the nuclear magnetization distribution are applied to predict the hyperfine structure of the hydrogenlike heavy ions and neutral thallium atoms: the uniformly magnetized ball model and single-particle models for the valence nucleon with the uniform distribution and the distribution determined by the Woods-Saxon potential. Results for the hydrogenlike ions are in excellent agreement with previous studies. The application of the Woods-Saxon model is now extended to the neutral systems with the explicit treatment of the electron correlation effects within the relativistic coupled …
We have investigated the correction to the hyperfine structure of heavy multicharged ions, which is connected with the nuclear-polarization effect caused by the unpaired bound electron. Numerical calculations are performed for hydrogenlike ions taking into account the dominant collective nuclear excitations. The correction defines the ultimate limit of precision in accurate theoretical predictions of the hyperfine-structure splittings.
Weak interactions within a nucleus generate a nuclear spin dependent parity violating electromagnetic moment; the anapole moment. In heavy nuclei, the anapole moment is the dominant contribution to spin-dependent atomic parity violation. We analyze a method to measure the nuclear anapole moment through the electric dipole transition it induces between hyperfine states of the ground level. The method requires tight confinement of the atoms to position them at the anti-node of a standing wave driving the anapole-induced E1 transiton. We explore the necessary limits in the number of atoms, excita…
We identify a potential means to extract the $^{229g}$Th $\rightarrow$ $^{229m}$Th nuclear excitation energy from precision microwave spectroscopy of the $5F_{5/2,7/2}$ hyperfine manifolds in the ion $^{229g}$Th$^{3+}$. The hyperfine interaction mixes this ground fine structure doublet with states of the nuclear isomer, introducing small but observable shifts to the hyperfine sub-levels. We demonstrate how accurate atomic structure calculations may be combined with measurement of the hyperfine intervals to quantify the effects of this mixing. Further knowledge of the magnetic dipole decay rate…
Recently the first laser spectroscopy measurement of the radioactive RaF molecule has been reported by Ruiz \textit{et al.} [Nature \textbf{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 i…
Calculations of the magnetic hyperfine structure rely on the input of nuclear properties -- nuclear magnetic moments and nuclear magnetization distributions -- as well as quantum electrodynamic (QED) radiative corrections for high-accuracy evaluation in heavy atoms. The uncertainties associated with assumed values of these properties limit the accuracy of hyperfine calculations. For example, for the heavy alkali-metal atoms Cs and Fr, these uncertainties may amount collectively to almost 1\% or 2\%, respectively. In this paper we propose a method for removing the dependence of hyperfine struct…
The hyperfine splitting in hydrogenlike $^{209}$Bi, $^{203}$Tl, and $^{205}$Tl is calculated with the nuclear magnetization determined from experimental data on the hyperfine splitting in the corresponding muonic atoms. The single-particle and configuration-mixing nuclear models are considered. The QED corrections are taken into account for both electronic and muonic atoms. The obtained results are compared with other calculations and with experiment.
The magnetic octupole moment of 87Rb is determined from hyperfine intervals in the 5p 2P[3/2] state measured by Ye et al. [Opt. Lett. 21, 1280 (1996)]. Hyperfine constants A = 84.7189(22) MHz, B = 12.4942(43) MHz, and C = -0.12(09) kHz are obtained from the published measurements. The existence of a significant value for C indicates the presence of a nuclear magnetic octupole moment Omega. Combining the hyperfine constants with atomic structure calculations, we obtain Omega = -0.58(39) (b muN). Second-order corrections arising from interaction with the nearby 5p 2P[1/2] state are found to be i…
The thorium nucleus with mass number $A=229$ has attracted much interest because its extremely low lying first excited isomeric state at about $8$eV opens the possibility for the development of a nuclear clock. However, neither the exact energy of this nuclear isomer nor properties, such as nuclear magnetic dipole and electric quadrupole moment are known to a high precision so far. The latter can be determined by investigating the hyperfine structure of thorium atoms or ions. Due to its electronic structure and the long lifetime of the nuclear isomeric state, Th$^{2+}$ is especially suitable f…
A theoretical model of hyperfine spectra of nuclear spin conversion of molecular isomers has been developed. The model takes into account the nuclear spin-spin and spin-rotation interactions, as well as the saturation of intramolecular mixing of molecular ortho and para states. The model has been applied to hyperfine spectra of nuclear spin conversion in ^{13}CH3F molecules subjected to an external electric field. Conditions under which the hyperfine structure in the spectra can be resolved have been determined.
The hyperfine structure of the long-lived $5D_{3/2}$ and $5D_{5/2}$ levels of Ba$^+$ ion is analyzed. A procedure for extracting relatively unexplored nuclear magnetic moments $Ω$ is presented. The relevant electronic matrix elements are computed in the framework of the ab initio relativistic many-body perturbation theory. Both the first- and the second-order (in the hyperfine interaction) corrections to the energy levels are analyzed. It is shown that a simultaneous measurement of the hyperfine structure of the entire $5D_J$ fine-structure manifold allows one to extract $Ω$ without contaminat…
The ground state hyperfine splitting values of high Z hydrogenlike ions are calculated. The relativistic, nuclear and QED corrections are taken into account. The nuclear magnetization distribution correction (the Bohr-Weisskopf effect) is evaluated within the single particle model with the g_{S}-factor chosen to yield the observed nuclear moment. An additional contribution caused by the nuclear spin-orbit interaction is included in the calculation of the Bohr-Weisskopf effect. It is found that the theoretical value of the wavelength of the transition between the hyperfine splitting components …
To search for the temporal variation of the fundamental constants one needs to know dependence of atomic transition frequencies on these constants. We study the dependence of the hyperfine structure of atomic $s$-levels on nuclear radius and, via radius, on quark masses. An analytical formula has been derived and tested by the numerical relativistic Hartree-Fock calculations for Rb, Cd$^+$, Cs, Yb$^+$ and Hg$^+$. The results of this work allow the use of the results of past and future atomic clock experiments and quasar spectra measurements to put constrains on time variation of the quark mass…
We report on our calculations of differential hyperfine anomalies in the nuclear single-particle model for a number of atoms and ions of interest for studies of fundamental symmetries violations. Comparison with available experimental data allows one to discriminate between different nuclear magnetization models, and this data supports the use of the nuclear single-particle model over the commonly-used uniform ball model. Accurate modelling of the nuclear magnetization distribution is important for testing atomic theory through hyperfine comparisons. The magnetization distribution must be adeq…
Using 201Hg+ contained within a linear quadrupole rf ion trap that is part of an atomic clock, we report the first high-resolution spectroscopy on the hyperfine structure of 201Hg+. We measure the absolute ground state hyperfine interval to be 29.954365821130(171)(62)(10) GHz, more than 8 orders of magnitude improvement over the previous measurement. The first error estimate in parentheses is the statistical error in the shifted line center measurement, the second is systematic uncertainty, and the third is calibration uncertainty in the hydrogen maser reference standard. By comparison to the …
Determination of nuclear moments for many nuclei relies on the computation of hyperfine constants, with theoretical uncertainties directly affecting the resulting uncertainties of the nuclear moments. In this work we improve the precision of such method by including for the first time an iterative solution of equations for the core triple cluster amplitudes into the relativistic coupled-cluster method, with large-scale complete basis sets. We carried out calculations of the energies and magnetic dipole and electric quadrupole hyperfine structure constants for the low-lying states of 229Th^(3+)…
The relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) and the non-relativistic multiconfiguration Hartree-Fock (MCHF) methods have been employed to calculate the magnetic dipole and electric quadrupole hyperfine structure constants of zinc. The calculated electric field gradients for the $ 4s 4p \,\, ^3 \! P^o_{1} $ and $ 4s 4p \,\, ^3 \! P^o_{2} $ states, together with experimental values of the electric quadrupole hyperfine structure constants, made it possible to extract a nuclear electric quadrupole moment $ Q(^{67}{\rm Zn}) = 0.122(10) $ b. The error bar has been evaluated in a q…
We have performed accurate calculations of the hyperfine structure of the $2\,^3\!P$ state in the $^9$Be atom with the help of highly optimized, explicitly correlated functions, accounting for the leading finite nuclear mass, radiative, nuclear structure and relativistic effects. By comparison with measurements, we have determined the $^9$Be nuclear quadrupole moment to be $Q_{\rm N} = 0.05350(14)$ barns, which is not only the most accurate result, but also disagrees with previous determinations.
We use relativistic Hartree-Fock and correlation potential methods to calculate nuclear spin-dependent parity non-conserving amplitudes (dominated by the nuclear anapole moment) between hyperfine structure components of the ground state of odd isotopes of K, Rb, Cs, Ba+, Yb+, Tl, Fr, and Ra+. The results are to be used for interpretation of current and future measurements.
Accurate calculations of the fine and hyperfine splitting of the 2P state in Li and Be$^+$ isotopes using the explicitly correlated Hylleraas basis set are presented. Theoretical predictions including the mixing of $P_{1/2}$ and $P_{3/2}$ states, relativistic and quantum electrodynamics effects on hyperfine interactions, are compared with experimental values. It is concluded that precise spectroscopic determination of the nuclear magnetic moments requires elimination of nuclear structure effects by combining measurements for two different states.
The hyperfine splitting of the ground state of H-, Li-, and B-like ions is investigated in details within the range of nuclear numbers Z = 7-28. The rigorous QED approach together with the large-scale configuration-interaction Dirac-Fock-Sturm method are employed for the evaluation of the interelectronic-interaction contributions of first and higher orders in 1/Z. The screened QED corrections are evaluated to all orders in (αZ) utilizing an effective potential approach. The influence of nuclear magnetization distribution is taken into account within the single-particle nuclear model. The speci…
By combining our theoretical calculation and recently measured electric quadrupole hyperfine structure constant of the $3d ^2D_{5/2}$ state in the singly ionized $^{43}$Ca, we determine its nuclear quadrupole moment to one percent accuracy. The obtained result, $-0.0444(6)b$, is about ten percent improvement over the considered standard value. We have employed the relativistic coupled-cluster theory at single and double excitations level to calculate the electronic wave functions. The accuracy of these wave functions are estimated by comparing our calculated magnetic dipole hyperfine constants…
Sturmian basis set for the Dirac equation with finite nuclear size: Application to polarizability, Zeeman and hyperfine splitting, and vacuum polarization
We measure hyperfine structure in the metastable ${^3P}_2$ state of $^{173}$Yb and extract the nuclear magnetic octupole moment. We populate the state using dipole-allowed transitions through the ${^3P}_1$ and ${^3S}_1$ states. We measure frequencies of hyperfine transitions of the ${^3P}_2 \rightarrow {^3S}_1$ line at 770 nm using a Rb-stabilized ring cavity resonator with a precision of 200 kHz. Second-order corrections due to perturbations from the nearby ${^3P}_1$ and ${^1P}_1$ states are below 30 kHz. We obtain the hyperfine coefficients as: $A=-742.11(2)$ MHz, $B=1339.2(2)$ MHz, which re…
The hyperfine splitting of the ground state of selected B-like ions within the range of nuclear charge numbers $Z=49-83$ is investigated in detail. The rigorous QED approach together with the large-scale configuration-interaction Dirac-Fock-Sturm method are employed for the evaluation of the interelectronic-interaction contributions of first and higher orders in $1/Z$. The screened QED corrections are evaluated to all orders in $αZ$ by using an effective potential. The influence of nuclear magnetization distribution is taken into account within the single-particle nuclear model.
High-resolution Zeeman spectroscopy of electronic-nuclear hyperfine levels of ${^5\mathrm{I}_8}\rightarrow{^5\mathrm{I}_7}$ transitions in Ho$^{3+}$:Y$_2$SiO$_5$ is reported. Crystal-field parameters determined for the two $C_1$ symmetry sites in Er$^{3+}$:Y$_2$SiO$_5$ are successfully used to model the Zeeman-hyperfine data, including the prediction of avoided crossings between hyperfine levels under the influence of an external magnetic field. The two six- and seven-coordinate substitutional sites may be distinguished by comparing the spectra with crystal-field calculations.
Parity violation experiment in Yb is made on the strongly forbidden M1 transition 6s^2 1S0 \to 5d6s 3D1. The hyperfine mixing of the 5d6s 3D1 and 5d6s 3D2 levels opens E2 channel, whose amplitude differs for F-sublevels of the 3D1 level. This effect may be important for the experimental search for the nuclear-spin-dependent parity violation effects predominantly caused by the nuclear anapole moment.
A large-scale configuration-interaction (CI) calculation is reported for the hyperfine splitting of the 2^2S and 3^2S states of ^7Li and ^9Be^+. The CI calculation based on the Dirac-Coulomb-Breit Hamiltonian is supplemented with a separate treatment of the QED, nuclear-size, nuclear-magnetization distribution, and recoil corrections. The nonrelativistic limit of the CI results is in excellent agreement with variational calculations. The theoretical values obtained for the hyperfine splitting are complete to the relative order of α^2 and improve upon results of previous studies.
$^{229}$Th is a promising candidate for developing a nuclear optical clock and searching the new physics beyond the standard model. Accurate knowledge of the nuclear properties of $^{229}$Th is very important. In this work, we calculate hyperfine-structure constants for the first four states of $^{229}$Th$^{3+}$ using the relativistic coupled-cluster method based on the Gauss basis set. The no-pair Dirac-Coulomb-Breit Hamiltonian with the lowest-order quantum electrodynamics (QED) correction is the starting point, together with all linear and non-linear terms of single and double excitations a…
A recent measurement of the hyperfine splitting in the ground state of Li-like $^{209}$Bi$^{80+}$ has established a "hyperfine puzzle" -- the experimental result exhibits a 7$σ$ deviation from the theoretical prediction [J. Ullmann et al., Nat. Commun. 8, 15484 (2017); J. P. Karr, Nat. Phys. 13, 533 (2017)]. We provide evidence that the discrepancy is caused by an inaccurate value of the tabulated nuclear magnetic moment ($μ_I$) of $^{209}$Bi. We perform relativistic density functional theory and relativistic coupled cluster calculations of the shielding constant that should be used to extract…
Theoretical investigations of the hyperfine structure of the hydrogen molecular ion (one electron and two protons) are discussed. The nuclear spin-rotation interaction has been found to be of the same sign as in the hydrogen molecule and the hyperfine transition frequencies can be accurately predicted. With measurements of the hyperfine structure of the deuterium molecular ion (or of HD+) it should be possible to obtain a value of the deuteron quadrupole moment that could be compared with the values obtained from the deuterium molecule and from nuclear theory.
