Second-Order Doppler and Cavity Phase Dependent Frequency Shifts in Atomic Beam Frequency Standards

Theoretical expressions for the second-order Doppler and cavity phase dependent frequency shifts in atomic beam frequency standards are derived, and the concept of momenta of the velocity distribution function is introduced. The effects of frequency modulation of the microwave excitation are also described by analytical expressions. Momenta of the velocity distribution function which can be experimentally derived from the central part of the Ramsey pattern are given for the National Research Council of Canada frequency standards Cs III and Cs V. The values of the frequency shift arising from the second-order Doppler effect obtained for different operating conditions are in close agreement with those calculated from the truncated Maxwellian velocity distributions characteristic of the best fit with the experimental resonance patterns. The frequency shift arising from mistuning of the microwave cavity is also calculated. This shift is dependent on both the velocity distribution and the servo-system frequency modulation employed. The operating conditions required for cancellation of this shift are determined.