Ground state hyperfine splitting in 6,7Li atoms and the nuclear structure

Precision spectroscopy on 9Be overcomes limitations from nuclear structure

Many powerful tests of the standard model of particle physics and searches for new physics with precision atomic spectroscopy are hindered by our lack of knowledge of nuclear properties. Ideally, these properties may be derived from precise measurements of the most sensitive and theoretically best-understood observables, often found in hydrogen-like systems. Although these measurements are abundant for the electric properties of nuclei, they are scarce for the magnetic properties, and precise experimental results are limited to the lightest of nuclei1-4. Here we focus on 9Be, which offers the unique possibility to use comparisons between different charge states available for high-precision spectroscopy in Penning traps to test theoretical calculations typically obscured by nuclear structure. In particular, we perform high-precision spectroscopy of the 1s hyperfine and Zeeman structure in hydrogen-like 9Be3+. We determine the effective Zemach radius with an uncertainty of 500 ppm, and the bare nuclear magnetic moment with an uncertainty of 0.6 parts per billion- uncertainties unmatched beyond hydrogen. Moreover, we compare our measurements with the measurements conducted on the three-electron charge state 9Be+ (ref. 5), which enables testing the calculation of multi-electron diamagnetic shielding effects of the nuclear magnetic moment at the parts per billion level. Furthermore, we test the quantum electrodynamics methods used for the calculation of the hyperfine splitting. Our results serve as a crucial benchmark for transferring high-precision results of nuclear magnetic properties across different electronic configurations.

Higher-Order QED Corrections to the Hyperfine Splitting in ^{3}He

We present a calculation of the hyperfine splitting of the 2^{3}S state in the ^{3}He atom with inclusion of all QED effects up to α^{3}E_{F}, where E_{F} is the Fermi splitting. Using the experimental value of the 1S hyperfine splitting in ^{3}He^{+}, we eliminate uncertainties from the nuclear structure and obtain the theoretical prediction for ^{3}He of ν_{hfs}=-6 739 701 181(41)  Hz, which is in perfect agreement with the experimental value -6 739 701 177(16)  Hz [S. D. Rosner and F. M. Pipkin, Phys. Rev. A 1, 571 (1970)PLRAAN0556-279110.1103/PhysRevA.1.571]. This result constitutes a 40-fold improvement in precision as compared to the previous value and is the most accurate theoretical prediction ever obtained for a nonhydrogenic system.

Measurement of Hyperfine Structure and the Zemach Radius in ^{6}Li^{+} Using Optical Ramsey Technique

We investigate the 2^{3}S_{1}-2^{3}P_{J} (J=0, 1, 2) transitions in ^{6}Li^{+} using the optical Ramsey technique and achieve the most precise values of the hyperfine splittings of the 2^{3}S_{1} and 2^{3}P_{J} states, with smallest uncertainty of about 10 kHz. The present results reduce the uncertainties of previous experiments by a factor of 5 for the 2^{3}S_{1} state and a factor of 50 for the 2^{3}P_{J} states, and are in better agreement with theoretical values. Combining our measured hyperfine intervals of the 2^{3}S_{1} state with the latest quantum electrodynamic (QED) calculations, the improved Zemach radius of the ^{6}Li nucleus is determined to be 2.44(2) fm, with the uncertainty entirely due to the uncalculated QED effects of order mα^{7}. The result is in sharp disagreement with the value 3.71(16) fm determined from simple models of the nuclear charge and magnetization distribution. We call for a more definitive nuclear physics value of the ^{6}Li Zemach radius.

Independently Optimized Orbital Sets in GRASP—The Case of Hyperfine Structure in Li I

In multiconfiguration Dirac–Hartree–Fock (MCDHF) calculations, there is a strong coupling between the localization of the orbital set and the configuration state function (CSF) expansion used to determine it. Furthermore, it is well known that an orbital set resulting from calculations, including CSFs describing core–core correlation and other effects, which aims to lower the weighted energies of a number of targeted states as much as possible, may be inadequate for building CSFs that account for correlation effects that are energetically unimportant but decisive for computed properties, e.g., hyperfine structures or transition rates. This inadequacy can be traced in irregular or oscillating convergence patterns of the computed properties as functions of the increasing orbital set. In order to alleviate the above problems, we propose a procedure in which the orbital set is obtained by merging several separately optimized, and mutually non-orthogonal, orbital sets. This computational strategy preserves the advantages of capturing electron correlation on the total energy through the variational MCDHF method and allows to target efficiently the correlation effects on the considered property. The orbital sets that are merged are successively orthogonalized against each other to retain orthonormality. The merged orbital set is used to build CSFs that efficiently lower the energy and also adequately account for the correlation effects that are important for the property. We apply the procedure to compute the hyperfine structure constants for the 1s22s2S1/2 and 1s22p2P1/2,3/2o states in 7Li and show that it leads to considerably improved convergence patterns with respect to the increasing orbital set compared to standard calculations based on a single orbital set, energy-optimized in the variational procedure. The perspectives of the new procedure are discussed in a broader context in the summary.

Deformed explicitly correlated Gaussians

Deformed explicitly correlated Gaussian (DECG) basis functions are introduced, and their matrix elements are calculated. All matrix elements can be calculated analytically in a closed form, except the Coulomb one, which has to be approximated by a Gaussian expansion. The DECG basis functions can be used to solve problems with nonspherical potentials. One example of such potential is the dipole self-interaction term in the Pauli-Fierz Hamiltonian. Examples are presented showing the accuracy and necessity of deformed Gaussian basis functions to accurately solve light-matter coupled systems in cavity QED.

Hyperfine Structure of the First Rotational Level in H_{2}, D_{2} and HD Molecules and the Deuteron Quadrupole Moment

We perform the four-body calculation of the hyperfine structure in the first rotational state J=1 of the H_{2}, D_{2}, and HD molecules and determine the accurate value for the deuteron electric quadrupole moment Q_{d}=0.285 699(15)(18)  fm^{2} in significant disagreement with former spectroscopic determinations. Our results for the hyperfine parameters agree very well with the currently most accurate molecular-beam magnetic resonance measurement performed several decades ago by N.F. Ramsey and coworkers. They also indicate the significance of previously neglected nonadiabatic effects. Moreover, a very good agreement with the recent calculation of Q_{d} based on the chiral effective field theory, although much less accurate, indicates the importance of the spin dependence of nucleon interactions in the accurate description of nuclei.

Precision Calculation of Hyperfine Structure and the Zemach Radii of ^{6,7}Li^{+} Ions

The hyperfine structures of the 2^{3}S_{1} states of the ^{6}Li^{+} and ^{7}Li^{+} ions are investigated theoretically to extract the Zemach radii of the ^{6}Li and ^{7}Li nuclei by comparing with precision measurements. The obtained Zemach radii are larger than the previous values of Puchalski and Pachucki [Phys. Rev. Lett. 111, 243001 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.243001] and disagree with them by about 1.5 and 2.2 standard deviations for ^{6}Li and ^{7}Li, respectively. Furthermore, our Zemach radius of ^{6}Li differs significantly from the nuclear physics value, derived from the nuclear charge and magnetic radii [Phys. Rev. A 78, 012513 (2008)PLRAAN1050-294710.1103/PhysRevA.78.012513] by more than 6σ, indicating an anomalous nuclear structure for ^{6}Li. The conclusion that the Zemach radius of ^{7}Li is about 40% larger than that of ^{6}Li is confirmed. The obtained Zemach radii are used to calculate the hyperfine splittings of the 2^{3}P_{J} states of  ^{6,7}Li^{+}, where an order of magnitude improvement over the previous theory has been achieved for ^{7}Li^{+}.

Matrix elements of explicitly correlated Gaussian basis functions with arbitrary angular momentum

A new algorithm for calculating the Hamiltonian matrix elements with all-electron explicitly correlated Gaussian functions for quantum-mechanical calculations of atoms with arbitrary angular momentum is presented. The calculations are checked on several excited states of three and four electron systems. The presented formalism can be used as unified framework for high accuracy calculations of properties of small atoms and molecules.