Physics 361
Problem set 3
due in class October 21, 2009
P3.1 Anomalous density of states
A certain twodimensional simplesquare lattice of lat
tice constant
a
has the dispersion relation (
ω
L
(
k
))
2
=
c
2
L
k
2
, for longitudinal vibrations and (
ω
L
(
k
))
2
=
c
2
T
k
2
for transverse vibrations for
ka
1.
As explained in Chapter 23, the density of mode frequencies
g
(
ω
) determines the temper
ature dependence of the specific heat.
a) What is
g
(
ω
) for an
N

atom crystal at frequencies
ω
described by the dispersion
relation above? That is, what is the number of modes between
ω
and
ω
+
dω
?
Suppose that for some wavevector
~
k
0
, the
ω
L
(
k
) has an absolute maximum.
Near this
maximum
ω
L
(
k
) =
ω
0

A
(
~
k

~
k
0
)
2
. You can assume
ω
T
(
k
) always lies well below
ω
0
, so
that these modes don’t contribute to b).
b) Find the form of
g
(
ω
) when
ω
<
∼
ω
0
, and when
ω
>
∼
ω
0
.
P3.2 Poorman’s localization
In the alternatingspring system of Figure 22.9, there is a
gap in the density of states—
i.e.,
a frequency range where
there are no normal modes.
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 Spring '10
 Pucker
 Physics, Solid State Physics, Frequency, Wavelength, Fundamental physics concepts, Ω, twodimensional simplesquare lattice

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