 Differential thermal-expansion uncertainty

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.