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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 $\partial d/\partial \theta =La$. HenceLength uncertainty and other inputs require further terms when they are material to the result. General propagation must retain input covariance.^[ref]

The coefficient difference itself hasAssuming 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.^[ref]

An original example takes $L=100\mathrm{mm}$, $a=6.5\times 10^{-6}{\mathrm{K}}^{-1}$, $\theta =2\mathrm{K}$, independent $u(a)=0.5\times 10^{-6}{\mathrm{K}}^{-1}$ and $u(\theta )=0.1\mathrm{K}$. The correction is $1.3\mu \mathrm{m}$. The two first-order standard-uncertainty contributions are 0.10 and $0.065\mu \mathrm{m}$, giving combined $u(d)\approx 0.119\mu \mathrm{m}$. 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 $L^2{u}^2(a)u^2(\theta )$. 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.^[ref]

Specify the thermal model, shared inputs and approximation used. A small nominal correction is not automatically evidence that its uncertainty is negligible.

Ted Doiron, identified as Theodore D. Doiron in NIST's official publication record, is a dimensional metrology researcher and coauthor with John S. Beers of The Gauge Block Handbook by Ted Doiron and John S. Beers. The preface uses Ted Doiron and attributes substantial changes to the handbook's measurement-assurance program to the author abbreviated TD.^[ref]

The scientific challenge is to obtain a reproducible length from a material artifact under changing environmental and measurement conditions. Thermal expansion is a particularly clear example. For a small temperature change,where $\alpha$ is the linear expansion coefficient. An original example with $L(T_0)=100\mathrm{mm}$, stipulated $\alpha =11.5\times 10^{-6}{\mathrm{K}}^{-1}$ and $T-T_0=0.1\mathrm{K}$ gives a length increase $0.115\mu \mathrm{m}$. A small temperature difference can therefore matter when comparing precision gauge blocks.

Even a shared temperature does not cancel every thermal effect. For two equal nominal lengths with different expansion coefficients, the first-order differential change isUsing $L=100\mathrm{mm}$, coefficient difference $1\times 10^{-6}{\mathrm{K}}^{-1}$ and common offset $0.2\mathrm{K}$ gives $20\mathrm{nm}$. This gauge-block thermal mismatch survives a simple subtraction even though both blocks have reached the same temperature. A temperature difference between the blocks introduces another contribution.

The handbook treats temperature corrections and their control alongside comparison procedures and measurement assurance.^[ref] A useful experiment compares the corrected result over controlled environmental changes and checks an independent measurement standard, rather than equating a stable display with an accurate physical length. The examples above are illustrative calculations, not an attributed material dataset or a claim about current NIST instrument performance.