 Gauge-block thermal mismatch

Gauge-block thermal mismatch is the unequal change in length of a reference and an unknown gauge block caused by different temperatures or thermal expansion coefficients. Bringing both blocks to the same temperature does not cancel the effect when their expansion coefficients differ and that common temperature differs from the reference temperature.

Let the lengths at $T_0=20{}^{\circ }\mathrm{C}$ be $L_{X,0}$ and $L_{R,0}$, and let their linear expansion coefficients be ${\alpha }_X,{\alpha }_R$. To first order,Thus subtract the final two terms from the observed difference to estimate the difference at $T_0$. NIST's handbook explicitly treats both temperature and coefficient differences in gauge-block comparisons.^[ref]

For approximately equal nominal length $L$ at common temperature $T$, the differential thermal contribution becomesIn an original example with $L=100\mathrm{mm}$, ${\alpha }_X=11.5\times 10^{-6}{\mathrm{K}}^{-1}$, ${\alpha }_R=5.0\times 10^{-6}{\mathrm{K}}^{-1}$ and $T-T_0=2\mathrm{K}$, the contribution is $1.3\mu \mathrm{m}$. Identical temperatures have not made the comparator reading a reference-temperature difference. These are stipulated material coefficients, not certificates for particular blocks.

Even equal coefficients leave an effect if the temperatures differ: at $11.5\times 10^{-6}{\mathrm{K}}^{-1}$ and 100 mm, a temperature difference 0.2 K contributes $0.23\mu \mathrm{m}$. Handling and incomplete equilibration can therefore matter before the readout is considered.

Differential thermal-expansion uncertainty must include the temperature measurement, the coefficients and their dependence. A common uncertain temperature is one shared input, not two unrelated thermometer errors. In the common-temperature model its sensitivity is $L({\alpha }_X-{\alpha }_R)$; if the coefficients agree, that particular common-temperature term cancels. Independent block-temperature differences need separate terms. General propagation retains the appropriate covariance.^[ref]

A controlled comparison measures or bounds block temperatures, allows equilibration, and records the coefficient assumptions. Comparing unlike materials without that information can turn a thermal contribution into an apparent calibration error.