  The Y2SiO5 (YSO) crystal is a dielectric material with biaxial anisotropy
with known values of refractive index at optical frequencies. It is a
well-known rare-earth (RE) host material for optical research and more recently
has shown promising performance for quantum-engineered devices. In this paper,
we report the first microwave characterization of the real permittivity tensor
of a bulk YSO sample, as well as an investigation of the temperature dependence
of the tensor components from 296 K down to 6 K. Estimated uncertainties were
below 0.26%, limited by the precision of machining the cylindrical dielectric.
Also, the electrical Q-factors of a few electromagnetic modes were recorded as
a way to provide some information about the crystal losses over the temperature
range. To solve the tensor components necessary for a biaxial crystal, we
developed the multi-mode technique, which uses simultaneous measurement of low
order Whispering Gallery Modes. Knowledge of the permittivity tensor offers
important data, essential for the design of technologies involving YSO, such as
microwave coupling to electron and hyperfine transitions in RE doped samples at
low temperatures.
