Linear thermal expansion coefficient (${\alpha }_L=L^{-1}(\partial L/\partial T{)}_P$)

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 $dV/V=3dL/L$. Direction-dependent materials require a specified measurement direction. ^[ref].

For a bond free of applied tension, use a local potential energy $V(a+x)=V(a)+\kappa x^2/2+\gamma x^3/6+\cdots$ with $\kappa >0$. Classical equipartition gives $⟨x^2{⟩}_0=k_{B}T/\kappa$; vanishing mean force then yields $⟨x⟩=-\gamma k_{B}T/(2{\kappa }^2)$ to first order in anharmonicity. Thus the linear thermal expansion coefficient is $\alpha =-\gamma k_B/(2a{\kappa }^2)$. For a Morse potential, $\gamma =-3\kappa /\xi$, giving positive expansion $3k_B/(2a\kappa \xi )$. This is a local bound-state approximation; the isolated cubic truncation is not a globally stable equilibrium potential.