Question #43541

In a 1D model of a crystalline solid, an atom of mass $M$ is bound in an anharmonic potential given by $U(x) = \frac{1}{2} k x^2 - g x^3$, where $x$ is the displacement from its equilibrium position at $0\text{ K}$, and $k$ and $g$ are positive constants. At a thermodynamic temperature $T$, the time-averaged mean displacement of the atom due to thermal agitation is observed to be $\langle x \rangle = \alpha T$, where $\alpha$ is a positive constant. The atom undergoes small oscillations about this new mean position. Assuming $k_B$ is the Boltzmann constant and the thermal energy is small enough that $\frac{1}{2}k\langle x^2 \rangle \approx \frac{1}{2}k_B T$, which of the following expressions represents the time period of these small oscillations at temperature $T$?