Kittel, Problem 4-2. To answer this, show that the expressions
for an atomic vibration involving differences go over to derivative expressions
in the limit of long waves.
Kittel, Problem 4-3
Kittel, Problem 4-5. Give your results in terms of C and the mass M.
Calculate the speed of sound for a longitudinal wave propagating in
a [100] direction for two cases: diamond carbon and gold. Use the elastic constants
any any other needed information given in the text.
Consider a simple cubic crystal with lattice constant a and
one atom per cell of mass M. Assume the atoms interact with nearest-neighbor
forces phi(R) with second derivative C = phi''.
Answer the questions below in terms of a, M, and C.
A. Give expression for the elastic constant C11 and the bulk
modulus B.
B. Give the expression for the longitudinal sound velocity v
in the [100] direction in terms of the appropriate elastic constant.
C. Give the expression for the dispersion curve omega(k) for longitudinal motion
as a function of
wavevector k in the [100] direction. Show that this leads to a velocity
of sound in agreement with part B and give the expression for the
frequency omegaBZ for k at the boundary of the Brillouin zone.
D. Find values of each of the quantities C11, B, v, and omegaBZ,
for the case where M = mass of the Al atom, a = 0.286 nm (the nearest-neigh. distance
in Al given in Kittel), and an estimate of C = 100 eV/nm2
(This is a very crude estimate of phi'' based upon the idea that
displacement of an atom by 0.1 nm should change the energy by of order 1 eV.)
E. Even though Al does not form the simple cubic structure and the value of C is
a crude estimate, the results should be of the same order of magnitude as
in real Al. Compare C11 and B with the actual values
for Al given in Kittel.
In Figure 11 (chapter 4) of Kittel, are shown the measured
dispersion curves of Na, which has the bcc structure. It is a good
approximation to assume the interaction phi(R) acts only between nearest neighbors.
A.
From the value of the longitudinal frequency at the zone
boundary in the [100] direction, find the value of the second derivative
phi''. (Note that you must treat Na as bcc and the neighbors are not
oriented along the [100] direction.)
B. Give the expression for the dispersion curve for transverse motion (k in the
[100] direction, displacement in the [010] direction) using
the value of phi'' from part A. What is the value of
the frequency at the Brillouin zone boundary?