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 of density of ions (or of conduction electrons) as $3/(4\pi R^3)$ , which defines R.

(a) Show that the frequency of a single ion set into oscillation is  $\omega = \sqrt{e^2/(MR^3)}$ .

(b) Estimate the value of this frequency for sodium, given that the atomic mass of sodium is 23g/mol, the metallic sodium concentration is $n = (2.65*10^2^2) cm^-3$ , and $e = (4.8 * 10^{-10}) esu$ .

(c) Estimate the order of magnitude of the sound velocity in metal.

(All answers please be as brief as possible but clearly written)

Answer 1

a) The number density of ions is 3/4πR³, the charge density will be 3e/4πR³
When an ion is displaced by a small distance r from its equilibrium position,the total charge contributing to the restoring force is:
q=3e/4πR³ * 4πr³/3=er³/R³
Md²r/dt²=-qe/r²=-e²r/R³
which gives an oscillation frequency of w=sqrt(e2/MR3)
b) R=cuberoot(3/8π)*a=cuberoot(3/8π)*429.06pm = 2.1126*10^-10m
M= 22.99*1:660*10^-27kg = 3:818*10^-26kg
e= 11*1.602*10^-19C = 1.762*10^-18C
w=sqrt(e²/4π*ε0*MR³)= 2.8*10¹⁴rad/s
c) The wave length to be of order of lattice constant, 1010 m, sound velocity will be
v=w/k= w*λ/2π = 10⁴ m/s