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The microwave cavity experiment is the most sensitive way of looking foraxions 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 acavity with a large volume that contains a resonant mode that has a high formfactor, a high quality factor, a wide dynamic range, and is free from intrudermodes. For HAYSTAC, we have designed and constructed an optimized highfrequency cavity with a tuning mechanism that preserves a high degree ofrotational symmetry, critical to maximizing its figure of merit. This cavitycovers an important frequency range according to recent theoretical estimatesfor the axion mass, 5.5 - 7.4 GHz, and the design appears extendable to higherfrequencies as well. This paper will discuss key design and constructiondetails of the cavity, present a summary of the design evolution, and alertpractitioners of potentially unfruitful avenues for future work.

A Symmetric Multi-rod Tunable Microwave Cavity for the HAYSTAC Dark Matter Axion Search