Question #43486

Consider a crystalline solid where the interatomic potential energy $U(r)$ between two adjacent atoms separated by a distance $r$ is modeled as: $U(r) = U_0 + \frac{1}{2}c(r-r_0)^2 - g(r-r_0)^3$ where $r_0$ is the equilibrium separation at absolute zero temperature. The constants $c$ (in $\text{J}\cdot\text{m}^{-2}$) and $g$ (in $\text{J}\cdot\text{m}^{-3}$) are positive. The term $-g(r-r_0)^3$ represents the anharmonicity of the potential. Assume the anharmonic term is small compared to the thermal energy $kT$, where $k$ is the Boltzmann constant (in $\text{J}\cdot\text{K}^{-1}$) and $T$ is the absolute temperature (in $\text{K}$). Using classical statistical mechanics, determine the coefficient of linear thermal expansion $\alpha$ of the solid. (Hint: The average displacement $\langle x \rangle = \langle r - r_0 \rangle$ can be evaluated using the Boltzmann factor $e^{-U(x)/kT}$ and the approximation $e^{gx^3/kT} \approx 1 + \frac{gx^3}{kT}$. Standard integral: $\int_{-\infty}^{\infty} x^4 e^{-ax^2} dx = \frac{3}{4a^2}\sqrt{\frac{\pi}{a}}$)