= Thermal expansion of a weakly anharmonic bond
{title2=$\alpha=-\gamma k_B/(2a\kappa^2)$}
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 $\langle x^2\rangle_0=k_BT/\kappa$; vanishing mean <force> then yields $\langle x\rangle=-\gamma k_BT/(2\kappa^2)$ 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.
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