Measurement of the $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-2{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ Energy Separation ($\Delta E-S$) in Hydrogen ($n=2\mathrm{}$)

We have measured the $2S_{\frac{1}{2}}-2P_{\frac{3}{2}}$ energy separation ($\Delta E-S$) in the $n=2\mathrm{}$ state of atomic hydrogen by an atomic-beam radio-frequency method. We used a beam of metastable atoms in the single hyperfine state ${\beta }_A(F,m_F=0,0)$. The electric dipole transitions ${\beta }_A-b(F,m_F=1,+1)$ and ${\beta }_A-d(F,m_F=1,-1)$ were observed at fixed frequencies by sweeping the Zeeman magnetic field parallel to the beam axis. Measurements of the frequencies of the rf field used to drive the transition and the magnetic fields at the line centers of the resonance curves, along with the theory of Zeeman splitting were used to determine the zero-field splitting ($\Delta E-S$). Our final value of ($\Delta E-S$) is 9911.250 ± 0.063 MHz, where the quoted uncertainty is one average deviation from the mean. Using the revised experimental value of the Lamb shift, $S_H=1057.90\mathrm{}\pm 0.10$ MHz, we obtain $\Delta E=10\mathrm{} 969.15+0.12$ MHz. The corresponding value for ${\alpha }^{-1\mathrm{}}$ is 137.0356 ± 0.0007.