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Does the material choice and material thickness for the hot and cold
plates effect the performance of a LTD Stirling. Would copper work
better than aluminum of the same thickness? For the same material is
thinner better assuming it is sufficiently strong and stiff?

Material selection for building a single LTD SE would depend more on availability than what would be best for mass production. Hard grade machinable aluminum in reasonable 1/8" to 1/4" thicknesses would be useable.

My guess is 3/16" would be optimum. Thicker H & C plates would take longer to come to steady state running temperatures, and the engine would run longer when the heat source is removed.

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