  A new experiment to test for the time independence of the fine structure
constant, alpha, is proposed. The experiment utilizes 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 is
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 alpha^6 dependence. This
enables an enhancement of the sensitivity to temporal changes in a at microwave
frequencies.
