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Problem set 3
due in class October 21, 2009
P3.1 Anomalous density of states
A certain twodimensional simplesquare lattice of lattice 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 temperature 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
.
Solution
a) The densityofstates integral in two dimensions is
1
(volume of crystal)
X
k
→
Z
BZ
d
2
k/
(2
π
)
2
For a linear dispersion
ω
=
ck
, the density of states is
g
(
ω
) =
Z
2
π k dk
(2
π
)
2
δ
(
ω

c

k

)
Using the substitution
x
=
ck
this gives
g
(
ω
) =
1
2
πc
2
Z
x dx δ
(
ω

x
) =
ω
2
πc
2
.
There is one transverse and one longitudinal mode, so the total density of states (per unit volume) is
g
(
ω
) =
ω
2
π
(
1
/c
L
2
+ 1
/c
T
2
)
.
b)
g
(
ω
) =
Z
d
2
k/
(2
π
)
2
δ
(
ω

ω
0

A
(
~
k

~
k
0
)
2
)
Deﬁning (
~
k

~
k
0
) as
~x
, we can ﬁnd the contribution from
~
k
near
~
k
0
.
g
(
ω
) =
Z
1
2
(2
π x dx
)
/
(2
π
)
2
δ
(
ω

ω
0

Ax
2
) =
π
(2
π
)
2
1
A
Z
d
(
Ax
2
)
δ
(
ω

ω
0

Ax
2
)
.
If
ω < ω
0
the integral is 1, so that
g
(
ω
) = (4
πA
)

1
If
ω > ω
0
, there are no states (in this region of
~
k
) so
g
(
ω
) = 0. Thus the density of states changes
discontinuously at
ω
0
from (4
πA
)

1
below to zero above. The speciﬁc heat
c
v
must thus have a negative
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This note was uploaded on 03/02/2010 for the course PHYSICS 336 taught by Professor Pucker during the Spring '10 term at King's College London.
 Spring '10
 Pucker
 Physics, Solid State Physics

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