The dependence of phonon and paramagnetic resonances on the fine structure constant in sapphire and the possibility of a test of time dependence

A new experiment to test for the time independence of the fine structure constant, α, is proposed. The experiment is based on a similar proposal at optical frequencies. We propose to utilize orthogonally polarized transverse electric and transverse magnetic whispering gallery modes in a single sapphire resonator tuned to similar frequencies. When configured as a dual mode sapphire clock, we show that the anisotropy of sapphire makes it possible to undertake a sensitive measurement from the beat frequency between the two modes. At infrared frequencies this is possible due to the different effect of the lowest phonon frequency on the two orthogonally polarized modes. At microwave frequencies we show that the phonon effect is too small. We show that the electron spin resonance of paramagnetic impurities (such as Cr³⁺) in the lattice effects only one polarization with an α⁶ dependence. This enables an enhancement of the sensitivity to temporal changes in α at microwave frequencies. In both cases degradation of cavity Q is experienced due to Kramers Kronig relations. The effect on both cases is assessed.