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Charles-Édouard Guillaume investigated iron-nickel alloys whose thermal changes are unusually small. Invar has low thermal expansion over a useful temperature range, making it valuable for dimensional references. Elinvar addresses a related metrological problem through a small temperature dependence of elasticity, useful when an instrument's behavior depends on an elastic element. These materials show why composition and magnetic behavior can matter as much as geometrical precision when building a stable measuring instrument.^[ref]

A physicist.^[ref]

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 giveFor the observed difference $D(T)=L_s(T)-L_r(T)$, the derivative $b=dD/dT$ therefore determinesThe 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 $L_0=1.00\mathrm{m}$, ${\alpha }_s=1.0\times 10^{-6}{\mathrm{K}}^{-1}$ and ${\alpha }_r=9.0\times 10^{-6}{\mathrm{K}}^{-1}$. Heating both by $10\mathrm{K}$ gives sample and reference increases of $10\mu \mathrm{m}$ and $90\mu \mathrm{m}$. Thus 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 $0.2\times 10^{-6}{\mathrm{K}}^{-1}$ and $0.3\mu \mathrm{m}{\mathrm{K}}^{-1}$, and uncertainty in $L_0$ is negligible, variance additivity for independent random variables givesRepeating 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.

Elinvar names a family of alloys designed to have a small temperature coefficient of elastic stiffness over a useful operating range. This addresses a different problem from Invar: low thermal expansion does not automatically imply temperature-independent Young's modulus.^[ref]

For a simple elastic oscillator, $f\propto \sqrt{k/m}$, so a small stiffness change gives $\Delta f/f\simeq \frac{1}{2}\Delta k/k$ at fixed mass. An alloy with suitably compensated stiffness can reduce thermal drift in a mechanical frequency standard. Its composition, heat treatment and operating range still matter; the name does not mean exact invariance at every temperature.