Question

The Drude model can be used to estimate the force constants,
vibrational frequencies, and related properties in metals. We use a
crude model to describe the vibration of a single ion in a monovalent metal:
The ion is assumed to be a positive point charge in a spherical unit cell that is
filled with electrons of the appropriate density n. All the rest of the crystal is
ignored. (a) Consider a small displacement of the ion from the center of the
unit cell. Show that the restoring force is proportional to the magnitude of the displacement. Hint: Use Gauss’ law to calculate the force. (b) What is the
vibrational frequency for a single ion of sodium? (c) Performa crude estimate
of the speed of sound in sodium, and compare your result to the experimental
value of 3200 ms−1.

Answer 1

consider a single ion in a monovalent atom
the ion is a point +ve charge q, and -q charge is distributed in a sphere with electron density n
now
q = nVe ( where V is volume of the sphere and e is charge on electron)

a. lets say the charge is moved by distance x from the center of the sphere of radius R such that
V = 4*pi*R^3/3

then from gauss law, electric field at the radius x is E(x)
E(x)*4*pi*x^2 = q'/epsilon
here
q' = -nV'*e
V' = 4*pi*x^3/3
hence

E(x)*4*pi*x^2 = -n*e*4*pi*x^3/3*epsilon
E(x) = -n*e*x/3*epsilon

hence restoring force on the charge is
F(x) = q*E(x) = -4n^2*pi*R^3*e^2*x/9*epsilon

hence we can see the restoring force is proportional to the displacement from the center

b. hence vibrational frequency = f
   also
   w = 2*pi*f
   and
   from simple harmonic motion definition
   w = sqrt(k/m) = sqrt()
   when
   m*a = -kx = restoring force
   hence
   mass of sodium ion = m
   m*a = -4n^2*pi*R^3*e^2*x/9*epsilon
   hence
   w = sqrt(4n^2*pi*R^3*e^2/9*epsilon*m)
   f = w/2*pi = sqrt(4n^2*pi*R^3*e^2/9*epsilon*m)/2*pi

c. speed of sound in sodium , c = lambda*f
   where lambda is interatomic distance for sodium, lambda = 0.19 nm
   c = 0.19*10^-9 * sqrt(4n^2*pi*R^3*e^2/9*epsilon*m)/2*pi
   putting in vakues of n ( electronic density), R = lambda = 0.19 nm, epsilon = permittivity of free space, e charge of electron we can find approximate speed of sound in the metal