= Gauge-block thermal mismatch
\b[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,
$$
D_{\rm obs}=L_{X,0}-L_{R,0}
+\alpha_XL_{X,0}(T_X-T_0)-\alpha_RL_{R,0}(T_R-T_0).
$$
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.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§5.4.3.1, Temperature Effects, Eq. 5.8 and following comparison discussion]{ref}
For approximately equal nominal length $L$ at common temperature $T$, the differential thermal contribution becomes
$$
\delta D_{\rm th}\simeq L(\alpha_X-\alpha_R)(T-T_0).
$$
In an original example with $L=100\,\mathrm{mm}$, $\alpha_X=11.5\times10^{-6}\,\mathrm{K^{-1}}$, $\alpha_R=5.0\times10^{-6}\,\mathrm{K^{-1}}$ and $T-T_0=2\,\mathrm K$, the contribution is $1.3\,\mathrm{\mu m}$. 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 $11.5\times10^{-6}\,\mathrm{K^{-1}}$ and 100 mm, a temperature difference 0.2 K contributes $0.23\,\mathrm{\mu m}$. 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>.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, sensitivity coefficients and correlated inputs]{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.
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