Relativistic doppler effect in199Hg+ stored ions atomic frequency standard

RF traps are widely used nowadays in high resolution hyperfine spectroscopy. The spectrum of the microwave hyperfine transition at 40.5 GHz of the fundamental level2S1/2 of199Hg+ is phase modulated at the secular frequency of the stored ions and it consists of a narrow central line and lateral bands broadened by the ion trajectory and velocity distribution. The central line itself is shifted by the second-order Doppler effect which is the most important systematic error of stored ions frequency standards. In this paper the relativistic Doppler effect in the case of199Hg+ stored ions is deduced applying a mathematical formalism and using physical parameters that we have measured experimentally.