ppm is short for 10
-6 unit/unit, so a 1-meter long diamond with a CTE of 1 ppm/
oC would expand 1 micron for each 1
oC increase in T.
However, the CTE of tourmaline (
is midway between diamond & gold,
so the thermal stress would be less. An average CTE of tourmaline is about 6 x10
-6/
oC, so for a 1 cm length and [Δ]T = -53
oC,
[Δ]length (tourmaline) = 3.18 [μ]m,
[Δ]length (gold) = 7.42[μ]m
If the stone is a solid chunk & the gold alloy is thin, the [Δ]CTE x [Δ]T gives a strain of 0.04% in the gold alloy. Don't know the elastic limit of your alloy, but it's maybe 0.2%. In this case, only get the plastic deformation necessary for the loosening effect if the gold alloy has been heavily work hardened or is already under considerable tensile stress. Some pre-existing tensile stress is likely; I suspect jewelers use it keep the stone tight.
One needs the stress-strain curves for tourmaline & 18-kt gold, and any initial cold work & pre-existing stress in the 18-kt gold, in order to do the calculations.