Homework 5
PHZ 5156
Due Thursday, December 4
1. Consider a periodic charge density
ρ
(
~
r
) defined such that
ρ
(
~
r
+
~
R
n
1
n
2
n
3
) =
ρ
(
~
r
)
.
(1)
We define the vector
~
R
n
1
n
2
n
3
to be
~
R
n
1
n
2
n
3
=
n
1
~
a
1
+
n
2
~
a
2
+
n
3
~
a
3
(2)
where
n
1
,
n
2
, and
n
3
are integers and the primitive lattice vectors are defined in
terms of a Cartesian system by
~
a
1
=
L
ˆ
x
,
~
a
2
=
L
ˆ
y
, and
~
a
3
=
L
ˆ
z
.
a) Determine the appropriate reciprocal lattice vectors
~
G
such that we can write the
periodic charge density
ρ
(
~
r
) as a summation
ρ
(
~
r
) =
X
~
G
ρ
~
G
exp(
i
~
G
·
~
r
)
(3)
where the summation is over all possible vectors
~
G
.
b) Show that the Coulomb potential due to this charge density is also periodic and
given by (up to a constant),
V
(
~
r
) =
X
~
G
V
~
G
exp(
i
~
G
·
~
r
)
(4)
which can be written in terms of the
ρ
~
G
as,
V
(
~
r
) =
X
~
G
4
πρ
~
G
G
2
exp(
i
~
G
·
~
r
)
(5)
with
G
2
=
~
G
·
~
G
. Hint: Consider the Poisson equation.
c) Show that for a point charge at
~
r
0
and its periodic images, that
ρ
~
G
=
1
Ω
exp(

i
~
G~
r
0
)
with Ω =
L
3
.
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 Fall '08
 Johnson,M
 Electrostatics, Electric charge, Fundamental physics concepts, Energy density, periodic charge density

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