Magnetic Moments of Rhenium-186 and Rhenium-188 and Analysis of the Rhenium Hyperfine Structure

The nuclear moments of rhenium-186 and rhenium-188 have been measured by the method of triple resonance in an atomic beam. They are found to be ${\mu }_I\mathrm{}\left(186\right)=+1.728\mathrm{}\left(.003\right)\mathrm{}$ nm and ${\mu }_I\mathrm{}\left(188\right)=+1.777\mathrm{}\left(.005\right)\mathrm{}$ nm, including the correction for diamagnetic shielding. With these values, an analysis based on the wave function of Trees has been undertaken of the hyperfine structure (hfs) in the ${}_{}{}^6{}_{}{}^{}S_{\frac{5}{2}}^{}{}_{}{}^{}$ ground state arising from the half-filled $5d$ shell. Calculations are made of contributions arising from the breakdown of Russell-Saunders coupling within the configuration ${\mathrm{}\left(5d\right)\mathrm{}}^5{\mathrm{}\left(6s\right)\mathrm{}}^2$ and configuration mixing by ${\mathrm{}\left(5d\right)\mathrm{}}^6\mathrm{}\left(6s\right)\mathrm{}$. It is shown that the principal contribution to the magnetic dipole hfs comes from relativistic effects in ${}_{}{}^6{}_{}{}^{}S_{\frac{5}{2}}^{}{}_{}{}^{}$. This is consistent with the small value obtained for the hyperfine anomaly ${}_{}{}^{186}{}_{}{}^{}\Delta _{}^{188}{}_{}{}^{}=0.1\mathrm{}\left(0.4\right)\mathrm{}$%. It is found that if the radial wave functions of Cohen for $5d$ electrons are used to calculate relativistic corrections, then good agreement is obtained with the measured $A$ constant. The correction factors of Casimir are shown to be too small by a factor of four. The quadrupole hyperfine structure is found to arise primarily from the cancellation of two off-diagonal matrix elements: a nonrelativistic element $〈{}_{}{}^4{}_{}{}^{}P_{\frac{5}{2}}^{}{}_{}{}^{}\left|q_J\right|{}_{}{}^4{}_{}{}^{}D_{\frac{5}{2}}^{}{}_{}{}^{}〉$ and a relativistic element $〈{}_{}{}^6{}_{}{}^{}S_{\frac{5}{2}}^{}{}_{}{}^{}\left|q_J\right|{}_{}{}^4{}_{}{}^{}P_{\frac{5}{2}}^{}{}_{}{}^{}〉$. Unfortunately, the accuracy is insufficient to permit a reliable determination of even the sign of the quadrupole moment. The measured nuclear moments are compared with the predictions of the Nilsson model. Good agreement is found if free-nucleon $g$ factors are used, and excellent agreement is obtained with quenched $g$ factors.