Direct measurement of the ${}^3$He${}^{+}$ magnetic moments

Abstract

Helium-3 has nowadays become one of the most important candidates for studies in fundamental physics [1, 2, 3], nuclear and atomic structure [4, 5], magnetometry and metrology [6] as well as chemistry and medicine [7, 8]. In particular, ${}^3$He nuclear magnetic resonance (NMR) probes have been proposed as a new standard for absolute magnetometry [6, 9]. This requires a high-accuracy value for the ${}^3$He nuclear magnetic moment, which, however, has so far been determined only indirectly and with a relative precision of $12$ parts per billon (p.p.b.) [10,11]. Here we investigate the ${}^3$He${}^{+}$ ground-state hyperfine structure in a Penning trap to directly measure the nuclear $g$-factor of ${}^3$He${}^{+}$ $g_I^{\prime }=-4.2550996069(30{)}_{stat}(17{)}_{sys}$, the zero-field hyperfine splitting $E_{\mathrm{HFS}}^{\exp }=-8665649865.77(26{)}_{stat}(1{)}_{sys}$ Hz and the bound electron $g$-factor $g_e^{\text{exp}}=-2.00217741579(34{)}_{stat}(30{)}_{sys}$. The latter is consistent with our theoretical value $g_e^{\text{theo}}=-2.00217741625223(39)$ based on parameters and fundamental constants from [12]. Our measured value for the ${}^3$He${}^{+}$ nuclear $g$-factor allows for the determination of the $g$-factor of the bare nucleus $g_I=-4.2552506997(30{)}_{stat}(17{)}_{sys}(1{)}_{theo}$ via our accurate calculation of the diamagnetic shielding constant [13] $\ddot{I}{ƒ}_{{}^{3}H{e}^{+}}=0.00003550738(3)$. This constitutes the first direct calibration for ${}^3$He NMR probes and an improvement of the precision by one order of magnitude compared to previous indirect results. The measured zero-field hyperfine splitting improves the precision by two orders of magnitude compared to the previous most precise value [14] and enables us to determine the Zemach radius [15] to $r_Z=2.608(24)$ fm.