= Linear thermal expansion coefficient
{title2=$\alpha_L=L^{-1}(\partial L/\partial T)_P$}
The <linear thermal expansion coefficient> is the fractional change in a specified <length> per change in <temperature>, at fixed external <pressure> for a sample free to expand. For small changes, $\Delta L/L\simeq\alpha_L\Delta T$. An isotropic sample with the same response in all directions has volumetric <coefficient of thermal expansion> $\alpha=3\alpha_L$, since differentiating $V\propto L^3$ gives $dV/V=3\,dL/L$. Direction-dependent materials require a specified measurement direction. https://openstax.org/books/university-physics-volume-2/pages/1-3-thermal-expansion[OpenStax University Physics 2, §1.3: linear and volumetric expansion]{ref}.
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