Title:Revisiting the hyperfine interval for the $2s2p$ $^3\!P_{J}$ state in $^9$Be

Abstract:Using relativistic multiconfiguration Dirac-Hartree-Fock method, we calculate the hyperfine-structure properties of the $2s2p$ $^3\!P_{J}$ state in $^9$Be. The hyperfine-structure properties encompass first-order hyperfine-structure parameters, as well as second-order and third-order corrections arising from the hyperfine mixing of different $2s2p$ $^3\!P_{J}$ levels. Based on our theoretical results, we reanalyze the previously reported measurement of the hyperfine interval for the $2s2p$ $^3\!P$ state in $^9$Be [A. G. Blachman and A. Lurio, Phys. Rev. 153, 164(1967)], yielding updated hyperfine-structure constants. Our results show that the hyperfine-structure constant $B$ of $2s2p$ $^3\!P_{1}$ is notably sensitive to second-order correction. Conversely, accurately determining the hyperfine-structure constant $B$ of $2s2p$ $^3\!P_{2}$ necessitates consideration of the hyperfine-structure constant $C$ in the first-order hyperfine interaction equation. The updated hyperfine-structure constant $B$ of the $2s2p$ $^3\!P_{2}$ state is found to be $1.4542(67)$~MHz, which is approximately $1.7\%$ larger than the previous value of $1.427(9)$~MHz.
By combining our theoretical results with the updated hyperfine-structure constant for the $2s2p$ $^3\!P_{2}$ state, we extract the electric quadrupole moment $Q$ of $^9$Be nucleus to be $0.05320(50)$~b. This value is consistent with the most recent determination using the few-body precision calculation method. Additional, we also discuss the reasons for the discrepancy between the $Q$ values obtained through few-body and previous many-body calculations.