Precision measurement of $\frac{g_I\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}_{}^{+}{}_{}{}^{}\right)}{g_J\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}\right)}$

A charge-exchange optical-pumping experiment has been performed to measure the ratio $\frac{g_I\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}_{}^{+}{}_{}{}^{}\right)}{g_J\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}\right)}$. The NMR frequency of free ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}_{}^{+}{}_{}{}^{}$ ions polarized by charge-exchange collisions with optically pumped ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ atoms was measured along with the ($F=2,M=-2\mathrm{}\leftrightarrow F=2,M=-1\mathrm{}$) Zeeman transition frequency of ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ in a magnetic field of ∼ 55 G. The magnetic field was produced in a shielded solenoid. The 30-${\mathrm{cm}}^3$ cylindrical optical-pumping cells were filled with a Ne buffer gas, and the ions were produced in a weak continuous electrodeless rf discharge which was run in a sidearm attached to the cells. The NMR frequency was ∼ 76 kHz and the charge-exchange and rf-broadened NMR linewidth was 60 Hz. The Zeeman frequency was ∼ 39 MHz and the Zeeman linewidth was 80 Hz due to the inhomogeneity of the magnetic field. Our result is $\frac{g_I\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}_{}^{+}{}_{}{}^{}\right)}{g_J\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}\right)}=-4.969\mathrm{}934\left(12\right)\mathrm{}\times {10}^{-4\mathrm{}}$, which can be compared with a previously measured value of $\frac{g_I\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}_{}^{+}{}_{}{}^{}\right)}{g_J\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}\right)}=-4.969\mathrm{}9147\left(44\right)\mathrm{}\times {10}^{-4\mathrm{}}$ to obtain the difference in the shielding constants of the atom and ion, $\sigma \left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}\right)-\sigma \left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}_{}^{+}{}_{}{}^{}\right)=(3.8\mathrm{}\pm 2.6)\times {10}^{-6\mathrm{}}$.