Question #43527

A hypothetical crystalline solid has a simple cubic lattice with a lattice constant $a$ at $0 \text{ K}$. The interatomic potential energy $U(x)$ between adjacent atoms is modeled as: $U(x) = \frac{1}{2} c x^2 - \frac{1}{3} g x^3$ where $x = r - a$ is the displacement from the $0 \text{ K}$ equilibrium separation, and $c$ and $g$ are positive constants. The solid is heated to a temperature $T$ (such that $k_B T \ll c a^2$, where $k_B$ is the Boltzmann constant) and allowed to expand freely. The effective Young's modulus of the solid at temperature $T$ is defined as $Y(T) = \frac{1}{a} \left. \frac{d^2 U}{dx^2} \right|_{x = \langle x \rangle}$, where $\langle x \rangle$ is the mean interatomic displacement at temperature $T$. Assuming classical statistical mechanics applies, which of the following expressions correctly represents $Y(T)$? (Given: $\int_{-\infty}^{\infty} x^4 e^{-\alpha x^2} dx = \frac{3}{4\alpha^2} \sqrt{\frac{\pi}{\alpha}}$ and $e^z \approx 1 + z$ for small $z$)