= Differential length comparator
An instrument for comparing two <lengths> through their difference. Suppose a test rod and reference both have length $L_0$ at <temperature> $T_0$, and have constant <linear thermal expansion coefficients> $\alpha_s$ and $\alpha_r$ over a small interval. Their first-order models give
$$
L_s(T)=L_0[1+\alpha_s(T-T_0)],\qquad L_r(T)=L_0[1+\alpha_r(T-T_0)].
$$
For the observed difference $D(T)=L_s(T)-L_r(T)$, the <derivative> $b=dD/dT$ therefore determines
$$
\boxed{\alpha_s=\alpha_r+\frac{b}{L_0}}.
$$
The reference's independently calibrated <linear thermal expansion coefficient> is essential: the difference alone identifies $\alpha_s-\alpha_r$, not either coefficient separately.
For an original numerical example, stipulate $L_0=1.00\,\mathrm m$, $\alpha_s=1.0\times10^{-6}\,\mathrm{K^{-1}}$ and $\alpha_r=9.0\times10^{-6}\,\mathrm{K^{-1}}$. Heating both by $10\,\mathrm K$ gives sample and reference increases of $10\,\mu\mathrm m$ and $90\,\mu\mathrm m$. Thus \b[the sample expands even though its measured difference decreases] by $80\,\mu\mathrm m$. These are chosen model parameters, not measured values for a historical <Invar> specimen or reference.
If $\alpha_r$ and $b$ have independent <standard uncertainties> $0.2\times10^{-6}\,\mathrm{K^{-1}}$ and $0.3\,\mu\mathrm{m\,K^{-1}}$, and uncertainty in $L_0$ is negligible, <variance additivity for independent random variables> gives
$$
u(\alpha_s)=\sqrt{u(\alpha_r)^2+\frac{u(b)^2}{L_0^2}}
\simeq0.36\times10^{-6}\,\mathrm{K^{-1}}.
$$
Repeating the difference measurement can reduce its random contribution; it does not remove uncertainty in the reference. A <temperature gradient> or unequal rod <temperatures> also breaks the common-temperature model and can introduce a <systematic error>.
\Image[/differential-expansion-and-abba-drift.png]
{title=Chosen rod expansion and instrument drift models: a negative length difference need not imply contraction}
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