Physics 573: Problem set 3
-
A one-dimensional crystal comprises atoms of alternating masses
m1 and m2 with equal spacing and equal
effective spring constants between the atoms. Calculate the phonon dispersion relations,
and discuss the limit as m1 approaches m2.
Discuss the ratio of the amplitudes for the two different modes near the Brillouin zone
boundary.
-
The effective spring constant between planes of atoms separated by na in metals is
frequently of a form proportional to
sin{k0na}/na, where k0 is a constant. If you
combine this with the dispersion relation for phonons and take the derivative
∂ω2/∂k you should find a peculiar result. This
slope of the dispersion blows up near k = k0. This effect is
called the Kohn anomaly.
-
(a) Use the dispersion relation for modes in a one-dimensional monatomic lattice to
obtain the phonon density of modes per energy (dk/dω) for
that
lattice. Show that it is proportional to
[ωmax2 - ω2]-1/2,
where ωmax is the largest allowed frequency.
(b) Near k = 0, the dispersion ω(k) must be parabolic by
symmetry, so for optical phonons it will look like
ω(k) = ωmax - Ak2. Calculate
the density of phonon states for ω > ωmax and for
ω < ωmax.
-
(a) In a layered dielectric material (like graphite), the phonons are mainly confined to
the
planes of strongly coupled atoms and there is not much coupling between the layers.
Calculate the
heat capacity for a layer in the Debye approximation. How does it depend on temperature
as T → 0?
(b) If the layers are coupled together discuss the T → 0 temperature dependence
for weak coupling and for coupling between layers similar in strength to the in-plane
coupling.
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