Consider 1 mole of Al which has the Debye temperature given
in the table in Kittel. Calculate the energy in J required to raise the
temperature from (a) T=0K to T=10K and (b) T=0K to T=300K . In each case
give the result if you use i) the Dulong-Petit classical specific heat and
ii) the quantum expresion for the specific heat
in the Debye approximation. (Explain how you carry out the calculations.
If you have difficulty in finding an analytic
expression for the energy in the Debye approximation, you may use an approximate
integration of the curve given in Kittel chapter 5, figure 7.)
Using the Debye temperatures for diamond carbon and silicon given in
Kittel, for each material compute the average sound velocity that appears
in the expression for the Debye temperature. Compare the values to
the longitudinal sound velocity in the (100) direction that can
be found from the elastic constants in Chapter 3.
What energy phonons contribute most to the heat capacity at
temperature T which is much below the Debye temperature?
(Hint: examine the form of the integrand in Eq. 30 of Chapter 5.)
Estimate the root mean square displacememt Delta xrms
for an atom using the formulas
given in class (lecture notes 9 - not given in Kittel -
the main formulas are repeated here). Use the mass of a Al atom and the
estimate C = 100 eV/nm2 (the same as was given in problem set 4).
(a) Find an estimate of the T=0 zero point Delta xrms. The analysis
in the lecture notes led to the estimate (hbar^2/CM)1/4.
(b) Find an estimate for T=1000K using the high temperature estimate
(2kBT/C)1/2.
(c) Compare the values with the nearest neighbor distance in Al.
Kittel, Problem 5-1.
Consider two crystals of identical size and shape but Debye temperatures
differing by a factor of 2. In the low temperature regime, where the thermal
conductivity is limited by the surface scattering, what is the ratio of the
thermal conductivities of the two crystals.