  Ensembles of bismuth donor spins in silicon are promising storage elements
for microwave quantum memories due to their long coherence times which exceed
seconds. Operating an efficient quantum memory requires achieving critical
coupling between the spin ensemble and a suitable high-quality factor resonator
-- this in turn requires a thorough understanding of the lineshapes for the
relevant spin resonance transitions, particularly considering the influence of
the resonator itself on line broadening. Here, we present pulsed electron spin
resonance measurements of ensembles of bismuth donors in natural silicon, above
which niobium superconducting resonators have been patterned. By studying spin
transitions across a range of frequencies and fields we identify distinct line
broadening mechanisms, and in particular those which can be suppressed by
operating at magnetic-field-insensitive `clock transitions'. Given the donor
concentrations and resonator used here, we measure a cooperativity $C\sim 0.2$
and based on our findings we discuss a route to achieve unit cooperativity, as
required for a quantum memory.
