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Diamagnetic correction to the $\bm{^9}$Be$\bm{^+}$ ground-state hyperfine constant

We report an experimental determination of the diamagnetic correction to the $^9$Be$^+$ ground state hyperfine constant $A$. We measured $A$ = $-625\,008\,837.371(11)$ Hz at a magnetic field $B$ of 4.4609 T. Comparison with previous results, obtained at lower values of $B$ (0.68 T and 0.82 T), yields the diamagnetic shift coefficient $k$ = $2.63(18) \times 10^{-11}$ T$^{-2}$, where $A(B)=A_0\times (1+k B^2)$. The zero-field hyperfine constant $A_0$ is determined to be $-625\,008\,837.044(12)$ Hz. The $g$-factor ratio ${g_I}^\prime/g_J$ is determined to be $2.134\,779\,852\,7(10) \times 10^{-4}$, which is equal to the value measured at lower $B$ to within experimental error. Upper limits are placed on some other corrections to the Breit-Rabi formula. The measured value of $k$ agrees with theoretical estimates.