PROBLEM SET 2

  1. [Kittel, 4.7] Consider point ions of mass M and charge q 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 ¾ pR3, which defines R.

    a) Show that the frequency of a single ion set into oscillation is     w = (e2/MR3)1/2.

    b) Estimate the value of this frequency for sodium metal.

    c) From the above and some simple reasoning, estimate the velocity of sound in sodium. Find a literature value for the sound velocity in sodium for comparison.
  2. [Kittel, 5.1] From the dispersion relation for a mono-atomic linear lattice of N atoms with nearest-neighbor interactions, show that the density of modes is:

where wm is the maximum frequency.

  1. In GaAs the sound velocity is 4.7 x 105 cm/sec, and the atomic density is 4.42 x 1022 atoms/cm-3. Calculate the Debye frequency and Debye temperature.
  2. Consider a crystal of GaAs in which the sound velocity is 4.7 x 105 cm/sec and the optical phonon energy is 36 meV and is assumed to have no dispersion. Using the Debye model for the acoustic phonon energy and the Einstein model for the optical phonons, calculate the lattice vibration energy per cm3 at 77K, 300K, and 1000K. If the Ga-As bond energy is 1 eV, compare the phonon energy with the crystal binding energy per cm3.
  3. Due to the finite size of real crystals, the acoustic phonon spectrum is not continuous, but discrete. Therefore, as pointed out in class, the Debye model might break down at very low temperatures and go back over to some version of an Einstein model. Choose some practical values for the relevant parameters, and estimate the temperature range in which this effect would become important.
  4. Fill in the steps skipped over in the class discussion of the harmonic oscillator to show: