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.
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