Developing a field independent frequency reference by averaging over transitions to multiple hyperfine levels

We show that by averaging over transitions to multiple hyperfine levels, quadrupole shifts and dominant Zeeman effects exactly cancel whenever the nuclear spin, I, is at least as large as the total electronic angular momentum, J. The average frequency thus defines a frequency reference which is inherently independent of external magnetic fields and electric field gradients. We use Lu+ to illustrate the method although the approach could be readily adapted to other atomic species. This approach practically eliminates the quadrupole and Zeeman shift considerations for many clock candidates.