<i>Ab initio</i> calculation of hyperfine-structure constant <i>A</i> of Fr and evaluation of magnetic dipole moments of Fr isotopes
Authors: Lou Bing-Qiong , Li Fang , Wang Pei-Yan , Wang Li-Ming , Tang Yong-Bo ,   College of Physics and Materials Science, Henan Normal University, Xinxiang 453000, China,   Faculty of Arts and Sciences, Shenzhen Technology University, Shenzhen 518118, China
Published: December 2020

As the heaviest atom in alkali-metal elements, Fr atom has been regarded as a candidate for the search of the permanent electric dipole moment of the electron and of parity-nonconservation effects. Accurate knowledge of Fr atomic properties is of great interest. In this work, we use a relativistic coupled-cluster method to calculate the magnetic dipole hyperfine structure constants for <i>n</i>S (<i>n</i> = 7－12), <i>n</i>P (<i>n</i> = 7－12) and <i>n</i>D (<i>n</i> = 6－11) states of <sup>212</sup>Fr. A finite B-spline basis set is used to expand the Dirac radial function, including completely the single and double excitation in correlation calculation. Our results are compared with available theoretical and experimental values. The comparison shows that our method can offer accurate calculation of magnetic dipole hyperfine structure constant. For 7P state the differences between our results and experimental values are within 1%. The magnetic dipole hyperfine structure constants for 12S, <i>n</i>P (<i>n</i> = 9－12) and <i>n</i>D (<i>n</i> = 10－11) states are reported for the first time, which are very useful as benchmarks for experimental measurements and calculations by other theoretical methods of these quantities. In the relativistic coupled-cluster theoretical framework, we study the electron correlation effect on hyperfine-structure constant <i>A</i> for the S, P, and D states of Fr. We observe that the electron correlation effect is very important for hyperfine-structure constant properties. The D state has a considerable correlation effect. At the same time, we also investigate contribution trends of individual electron correlation effects involving direct, core-polarization and pair-correlation ones in S, P, and D Rydberg series. It is found that the dominant contributions for the S<sub>1/2</sub>, P<sub>1/2,3/2</sub> and <i>n</i>D<sub>3/2</sub> (<i>n</i> = 7－11) states are to from the direct effect; however, the dominant contributions for the 6D<sub>3/2</sub>, and <i>n</i>D<sub>5/2</sub> (<i>n</i> = 6－11) states are due to the pair-correlation and the core-polarization, respectively. For D<sub>5/2</sub> states, there is very strong cancellation among these individual correlation effects. The knowledge of these correlation trends is useful for studying the permanent electric dipole moment and parity-nonconservation effect of Fr in future. Moreover, the magnetic dipole moment <inline-formula><tex-math id="M2">\begin{document}$ {\mu}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M2.png"/></alternatives></inline-formula> for each of isotopes <sup>207−213,220−228</sup>Fr is determined by combining with experimental values for magnetic dipole hyperfine structure constant of 7P state. For each of isotope <sup>207−213</sup>Fr, our magnetic dipole moment <inline-formula><tex-math id="M3">\begin{document}$ {\mu}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M3.png"/></alternatives></inline-formula> is perfectly consistent with the experimental value, and our uncertainties are twice smaller than those in the experiments . For each of isotope <sup>220−228</sup>Fr, our magnetic dipole moment <inline-formula><tex-math id="M4">\begin{document}$ {\mu}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M4.png"/></alternatives></inline-formula> has a larger uncertainty, but is still in agreement with the experimental magnetic dipole moment <inline-formula><tex-math id="M5">\begin{document}$ {\mu}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20190113_M5.png"/></alternatives></inline-formula>.