My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX A. THE QUASI-2D EWALD SUM
which reduces to
[ ( )
∑ πk
z 2 − k2 r‖
2 erfc
A 2k 2
k≠0
( σk
2
)
− 2√ πz 2
σA e−(σk/2)2 ]
+ O [ r 4] (A.26)
from a comparison of the contributions to the sum of all the k-vectors of a given
magnitude.
To proceed further, we use the following two-dimensional Fourier series:
with r = 0. This gives
∑
e −((r−R)/σ)2 = ∑
R
k
∑
e −(σk/2)2 =
k≠0
πσ 2
A e−(σk/2)2 e ik·r
A ∑
e −(R/σ)2 − 1
πσ 2
R
= A
πσ 2 − 1 + O [
e −L2 /σ 2]
(A.27)
(A.28)
which, when substituted back into equation (A.20), leads to
( ) (
v E (r) − ξ = 1 r + z 2 − r2 ‖ ∑
( ) )
πk σk
2 A erfc − 4
2 3σ 3√ + O [ r 4] [
+ O e −L2 /σ 2] .
π
k≠0
(A.29)
The remaining k-space sum may written in terms of the new variable β = 1/σ 2 :
∑
k≠0
( )
πk σk
A erfc = ∑ 2
k
= S(β).
( )
πk k
A erfc 2 √ β
(A.30)
Differentiating,
dS
dβ = ∑ k
= 2√ π √ β
A
= 2√ π √ β
A
√ π
2Aβ √ β k2 e −k2 /4β
d
dβ
d
dβ
( ∑
k
(
Aβ
π
e −k2 /4β
∑
R
)
e −R2 β
= 2√ β ∑
√ (1 − R 2 β)e −R2β .
π
R
)
(A.31)
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