Answer to Question #242598 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.
The value of this frequency for sodium can be roughly estimated to be:

Answer 1

$R= (\frac{3}{8 \pi })^{1/3} \times a \\ = (\frac{3}{8 \pi})^{1/3} \times 429.06 \;pm \\ = 2.112 \times 10^{-10} \;m \\ M = 22.99 \times 1.66 \times 10^{-27} = 3.81 \times 10^{-26} \;kg \\ q = 11 \times 1.6 \times 10^{-19} \;C = 1.76 \times 10^{-18} \;C \\ ω= \sqrt{\frac{q^2}{4 \pi ε_0 MR^3} } = 2.8 \times 10^{14} \;rad/s$