hw5 - Homework 5 PHZ 5156 Due Thursday December 4 1...

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
.