  Many searches for axion cold dark matter rely on the use of tunable
electromagnetic resonators. Current detectors operate at or near microwave
frequencies and use cylindrical cavities with cylindrical tuning rods. The
cavity performance strongly impacts the signal power of the detector, which is
expected to be very small even under optimal conditions. There is strong
motivation to characterize these microwave cavities and improve their
performance in order to maximize the achievable signal power. We present the
results of a study characterizing the HAYSTAC (Haloscope At Yale Sensitive to
Axion Cold dark matter) cavity using bead perturbation measurements and
detailed 3D electromagnetic simulations. This is the first use of bead
perturbation methods to characterize an axion haloscope cavity. In this study,
we measured impacts of misalignments on the order of 0.001 in and demonstrated
that the same impacts can be predicted using electromagnetic simulations. We
also performed a detailed study of mode crossings and hybridization between the
TM$_{010}$ mode used in operation and other cavity modes. This mixing limits
the tuning range of the cavity that can be used during an axion search. By
characterizing each mode crossing in detail, we show that some mode crossings
are benign and are potentially still useful for data collection. The level of
observed agreement between measurements and simulations demonstrates that
finite element modeling can capture non-ideal cavity behavior and the impacts
of very small imperfections. 3D electromagnetic simulations and bead
perturbation measurements are standard tools in the microwave engineering
community, but they have been underutilized in axion cavity design. This work
demonstrates their potential to improve understanding of existing cavities and
to optimize future designs.
