Accurate calculation of hyperfine-induced 5d6s <sup>3</sup>D<sub>1,3</sub>→6s<sup>2</sup> <sup>1</sup>S<sub>0</sub> E2 transitions and hyperfine constants of ytterbium atoms
Authors: Zhao Guo-Dong , Cao Jin , Liang Ting , Feng Min , Lu Ben-Quan , Chang Hong ,   National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China,   Key Laboratory of Time Reference and Applications, Xi’an 710600, China,   School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 101408, China,   Hefei National Laboratory, Hefei 230026, China
Published: March 2024

The parity violation effects via the <inline-formula><tex-math id="M14">\begin{document}$ {\mathrm{5d6s\; {^3D_1} \to 6s^2 \; {^1S_0}}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M14.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M14.png"/></alternatives></inline-formula> transition have been extensively investigated in ytterbium atoms. However, the M1 transition between the excitation state <inline-formula><tex-math id="M15">\begin{document}$ {\mathrm{5d6s\; {^3D_1}}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M15.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M15.png"/></alternatives></inline-formula> and the ground state <inline-formula><tex-math id="M16">\begin{document}$ {\mathrm{6s^2 \; {^1S_0}}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M16.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M16.png"/></alternatives></inline-formula>, as well as the hyperfine-induced E2 transition, significantly affects the detection of parity violation signal. Therefore, it is imperative to obtain the accurate transition probabilities for the M1 and hyperfine-induced E2 transitions between the excitation state <inline-formula><tex-math id="M17">\begin{document}${\mathrm{ 5d6s\; {^3D_1} }}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M17.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M17.png"/></alternatives></inline-formula> and the ground state <inline-formula><tex-math id="M18">\begin{document}$ {\mathrm{6s^2\; {^1S_0}}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M18.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M18.png"/></alternatives></inline-formula>. In this work, we use the multi-configuration Dirac-Hartree-Fock theory to precisely calculate the transition probabilities for the <inline-formula><tex-math id="M19">\begin{document}${\mathrm{ 5d6s \; {^3D_1} \to 6s^2 \; {^1S_0} }}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M19.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M19.png"/></alternatives></inline-formula> M1 and hyperfine-induced <inline-formula><tex-math id="M20">\begin{document}${\mathrm{ 5d6s \; ^3D_{1,3} \to 6s^2 \; {^1S_0} }}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M20.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M20.png"/></alternatives></inline-formula> E2 transitions. We extensively analyze the influences of electronic correlation effects on the transition probabilities according to our calculations. Furthermore, we analyze the influences of different perturbing states and various hyperfine interactions on the transition probabilities. The calculated hyperfine constants of the e <inline-formula><tex-math id="M21">\begin{document}$ {\mathrm{^3D_{1,2,3}}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M21.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M21.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M22">\begin{document}${\mathrm{ ^1D_2}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M22.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M22.png"/></alternatives></inline-formula> states accord well with experimental measurements, validating the rationality of our computational model. By combining experimentally measured hyperfine constants with the theoretically derived electric field gradient of the extra nuclear electrons at the nucleus, we reevaluate the nuclear quadrupole moment of the <inline-formula><tex-math id="M23">\begin{document}$ ^{173} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M23.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M23.png"/></alternatives></inline-formula>Yb nucleus as <inline-formula><tex-math id="M24">\begin{document}$ Q = 2. 89(5) \;\rm {b} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M24.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240028_M24.png"/></alternatives></inline-formula>, showing that our result is in excellent agreement with the presently recommended value.