High precision calculations of Fermi contact term for lithium-like ions

Abstract

The Fermi contact term is closely related to the atomic hyperfine constants. It often dominates the hyperfine splittings. The quality of the wave function near the origin and the correlation effect between electrons are two main factors which affect the numerical accuracy of the Fermi contact term. It is not an easy task to compute the Fermi contact term with high precision for a general atom. In the present paper, the Schrödinger equations of the <inline-formula><tex-math id="M4">\begin{document}$ {\text{1s2s3s}}{\;^{\text{4}}}{\text{S}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220923_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220923_M4.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ {\text{1s2s4s}}{\;^{\text{4}}}{\text{S}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220923_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220923_M5.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ {\text{1s2s2p}}{\;^{\text{4}}}{\text{P}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220923_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220923_M6.png"/></alternatives></inline-formula>states of lithium atom and lithium-like ions (<i>Z</i> = 4–10) are solved by using Rayleigh-Ritz variational method in Hylleraas coordinates. The variational energiesenergy converges to an accuracy of 10<sup>–13</sup>. Then the Fermi contact terms for these states are calculated based on the high precision variation wave functions. In particular, the Drachman global method are adopted in order to improve the convergence of the Fermi contact term. The effect of finite nuclear mass on Fermi contact term, i.e. the first-order mass polarization coefficient is also calculated. The Fermi contact term converges to an accuracy of 10<sup>–10</sup>, which is the most accurate result at present. Our results can be used as a reference for other theoretical methods or relevant experimental studies.