  The microwave cavity experiment is the most sensitive way of looking for
axions in the 0.1-10 GHz range, corresponding to masses of 0.5 - 40 $\mu$eV.
The particular challenge for frequencies greater than 5 GHz is designing a
cavity with a large volume that contains a resonant mode that has a high form
factor, a high quality factor, a wide dynamic range, and is free from intruder
modes. For HAYSTAC, we have designed and constructed an optimized high
frequency cavity with a tuning mechanism that preserves a high degree of
rotational symmetry, critical to maximizing its figure of merit. This cavity
covers an important frequency range according to recent theoretical estimates
for the axion mass, 5.5 - 7.4 GHz, and the design appears extendable to higher
frequencies as well. This paper will discuss key design and construction
details of the cavity, present a summary of the design evolution, and alert
practitioners of potentially unfruitful avenues for future work.
