Answer to Question #242599 in Optics for Reena Biswas

Question

Consider point ions of mass 'M' and charge 'e' immersed in a uniform sea of conduction electrons.The ions are imagined to be in stable equilibrium when at regular lattice points. If one ion is displaced a small distance 'r' from its equilibrium position, the restoring force is largely due to the electric charge within the sphere of radius 'r' centered at the equilibrium position. Take the number density of ions (or of conduction electrons) as 34πR3, which defines R.
Estimate the order of magnitude of the velocity of sound in the metal?

Answer 1

The oscillation frequency, according to the provided data, is

$$\omega=\sqrt{\frac{e^2}{4\pi\epsilon_0MR^3}}=2.8·10^{14}\text{ s}^{-1}.$$

The wavelength has the order of magnitude close to the lattice constant (10^-10 m), so, the

$$v=\frac{\omega\lambda}{2\pi}=\frac{2.8·10^{14}·10^{-10}}{2\pi}=4456\text{ m/s},$$

the order of magnitude is 10³.