= Differential thermal-expansion uncertainty
\b[Differential thermal-expansion uncertainty] is the uncertainty in the unequal thermal length changes of two compared objects. In a <gauge-block thermal mismatch> correction, shared temperature and coefficient information determine which contributions cancel and which remain.
For approximately equal known nominal length $L$, coefficient difference $a=\alpha_X-\alpha_R$ and common temperature offset $\theta=T-T_0$, the differential correction is $d=La\theta$. Its first-order <sensitivity coefficients> are $\partial d/\partial a=L\theta$ and $\partial d/\partial\theta=La$. Hence
$$
u^2(d)\simeq L^2\left[\theta^2u^2(a)+a^2u^2(\theta)+2a\theta\operatorname{cov}(a,\theta)\right].
$$
Length uncertainty and other inputs require further terms when they are material to the result. General propagation must retain input <covariance>.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, sensitivity coefficients and correlated inputs]{ref}
The coefficient difference itself has
$$
u^2(a)=u^2(\alpha_X)+u^2(\alpha_R)-2\operatorname{cov}(\alpha_X,\alpha_R).
$$
Assuming two coefficient estimates independent merely because two blocks are present can therefore misstate the uncertainty. Similarly, a common thermometer contribution is not two unrelated temperature errors. NIST's handbook explicitly treats differences in block temperatures and expansion coefficients in comparisons.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§5.4.3.1, Temperature Effects, Eq. 5.8 and comparison discussion]{ref}
An original example takes $L=100\,\mathrm{mm}$, $a=6.5\times10^{-6}\,\mathrm{K^{-1}}$, $\theta=2\,\mathrm K$, independent $u(a)=0.5\times10^{-6}\,\mathrm{K^{-1}}$ and $u(\theta)=0.1\,\mathrm K$. The correction is $1.3\,\mathrm{\mu m}$. The two first-order standard-uncertainty contributions are 0.10 and $0.065\,\mathrm{\mu m}$, giving combined $u(d)\approx0.119\,\mathrm{\mu m}$. These are stipulated inputs, not a certificate.
A useful boundary case exposes <second-order thermal mismatch uncertainty>. If both central values $a$ and $\theta$ are zero, the linear sensitivities vanish even though both inputs remain uncertain. For independent inputs with those means, their product has exact variance $L^2u^2(a)u^2(\theta)$. With $L=100\,\mathrm{mm}$, $u(a)=10^{-6}\,\mathrm{K^{-1}}$ and $u(\theta)=0.1\,\mathrm K$, this gives 10 nm standard uncertainty, while first-order propagation gives zero. The GUM explicitly requires attention to higher-order terms when nonlinearity matters.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2, nonlinear functions and higher-order terms]{ref}
Specify the thermal model, shared inputs and approximation used. A small nominal correction is not automatically evidence that its uncertainty is negligible.
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