Marder M.P. Condensed Matter Physics
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APPENDICES
895
Condensed Matter
Physics,
Second Edition
by Michael P. Marder
Copyright © 2010 John Wiley & Sons, Inc.
A. Lattice Sums and Fourier Transforms
A.l One-Dimensional Sum
The following sum arises in Section 3.2.4.
1=0
N-\
,ilaq
There are N terms in this sum.
i(l+\)aq
(A.l)
(A.2)
/=o
N-\
:
\ e 1 — 1 -\- g
lNa
1 Replacing / by / +
1
is the same as removing (A.3)
^-^' the first term in the sum, and adding an extra
term on at the end.
/=0
iZ
q
-\+e
iNaq
e
iNaq _ j
giaq _ \
e
iNaq/2
gin Nag
/
2
e
iaq/2
sm
aq
j2
Solving Eq. (A.2) for £,.
Using s\n{x) = (e'
x
—
e~'
x
)/2i.
2
sm
Naq/2
'
S
?l ~ T7I2
sin aq/2
The result is plotted in Figure 3.3.
(A.4)
(A.5)
(A.6)
(A.7)
A.2 Area Under Peaks
Because Figure 3.3 looks like a sum of Dirac delta functions, it is useful to find the
area under each of the peaks. In fact, both T,
q
and
\T,
q
\
2
can be regarded as sums of
delta functions. The area under a peak of
T,
q
is obtained by integrating from —
IT/a
to ir/a and is
n/a giNaq _ j
dq —: —
dz Z
N
1
iaz z
—
1
2-7T 2TT
— =N—,
a L
The limits of integration go from midway be- (A.8)
tween one set of peaks to midway between
the next.
Defining z = exp[m</] and taking the unit cir- (A.9)
cle as the integration contour.
The residue of the pole at z = 0 is 1. Recall (A. 10)
that L = aN is the length of the system.
897
Condensed Matter
Physics,
Second Edition
by Michael P. Marder
Copyright © 2010 John Wiley & Sons, Inc.
898
Appendix A. Lattice Sums and Fourier Transforms
and
.2vr
S, = J2 N—ô(q-2nl'/a) = J2 2ir6(qa-2irl'). (AM)
V
/'
|2
The area under a peak of |S
a
| is found similarly.
I
f
■it a „iNaq _ i
j I |2
"<7
— H
dzz?-lz-
N
-l
iaz
z — 1
z~ '
—
1
ia z^l-z)
2
dz 1
iaz^l-z)
2
27rJV
„ AT
=
2-KN—
a L
There
is no pole in the two discarded
terms.
Using
the residue theorem with aid of
the
Tay-,
lor
expansion
1
/
(
1
—
z)
2
= ^
;
^
0
(' +
1
)z'
■
(A.
12)
(A.13)
(A.
14)
(A.15)
(A.
16)
So the area under a peak of |£
ö
|
2
is N times the area under a peak of S
9
.
A.3 Three-Dimensional Sum
In the three-dimensional case, let ai ... 03 be primitive vectors, and consider the
collection of lattice points of the form
R=^2
l
a
a
a
with 0 < l
a
< M.
Then
E*
=
£«**
R
3
n
Q=l
3
n'
a=l
3
V^
e
il
a
q-a
a
JMäa
q _
Y
e
ia
a
•<?
— 1
Using
Eq. (3.13)
IJ(27r)
J2ô{q-a
a
-2Trm'
a
).
SeeEq.(A.ll).
(A. 17)
(A. 18)
(A.
19)
(A.20)
(A.21)
Q=l
The values of q at which the delta functions peak are the reciprocal lattice vectors
K, obeying Eq. (3.18). If one integrates any of the delta functions by dq, what is
the result? Define the variables Q
a
= q-â
a
. Then
■a
a
-2Trm'
a
)
(A.22)
Discrete Case
899
dQ
dQ
Y[6(Q
a
-27rm'
a
) (A.23)
dq
\ pj The matrix of
derivatives
of
Q
by q is a matrix
whose
rows are the Cartesian components of (A.24)
|ö
I
• (fl2 X ^3)! "^ the three primitive vectors
3.
The determinant
of
this matrix is the volume of the primitive
cell
defined by the three primitive
vectors.
Therefore
?« = V N(
2ir
^2e
i
«
R
= Y
/
N
( —
)
3o
(q-
K
)-
withL
3
=
v.
(A.25)
A.4 Discrete Case
Rather than choosing to evaluate E
ö
for all values of q, one often has reason to
restrict q to the discrete values
2vr/
n
= , With / an integer. (A.26)
Na
It is clear from Eq. (A.l) that E
q
= N when q = 0, q = 2-n/a, q = 4ir/a, . . .,
and it is clear from Eq. (A.5) that S
0
vanishes otherwise. So with q restricted by
Eq. (A.26) one has
^
=
N
£ <W/
a
(A.27)
= N/ SQK- Because 2irl/a are just the reciprocal lattice (A.28)
' ^ \/f*rtnrv in nni* rlim^neir^n
K
In three dimensions, (A.28) generalizes naturally to
E
e
lc/
= N N ö^p-
N
'
s tne tota
l number of elements in the sum (A 29)
^ 1
K
'
over«.
R
K
Equation (6.12) provided a correspondence between discrete and continuous delta
functions, which says to multiply the discrete delta function by
(2TT)
3
/V.
Using
this rule turns Eq. (A.29) into
$><^ = (27r)
3
^X>(<7-^ (A.30)
R k
which is identical to Eq. (A.25).
An additional result in one dimension follows from computing
Y,J>
r
=
Y,<?*
ar
'
Na
(A.31)
q
l=\
e
2nir/a
_ 1
=
iTÙr/Na _
]
=
aN
J2 Kr-INO) =
L8{r).
Sincere [0, L}. (A.32)
e l
t
Similarly, if q lies in the first Brillouin zone, then
900 Appendix A. Lattice Sums and Fourier Transforms
£ e
r
^ = VÖ(r).
q£B.Z.
A.5 Convolution
The convolution of two functions A(r) and B(r) is
A*B(r)
= f
<rf
A{?)B(
Fourier transforms of convolutions give products:
(A.33)
7-7)
f dre^
T
A*B(f)
= f drdVe^A^Bir-r')
= f drdi" e
i
^
(7
-
7
'^
+
^A(r')B(r-r')
fdr'e
i
^A(r') f dr
e^B(r)
= A(q)B(q).
Similarly, Fourier transforms of products give convolutions:
f dre^A{r)B{r) = f drd?6{r-r')e
i
ï
7
A(r')B(r)
-I
drdr'dq'-
2TT
-e^A^Bij)
[
2~i [
d7>e
^'
1
"
A
(
7
')} [
j
dré^-^Bir)
A*B(q)
2TT '
(A.34)
(A.35)
(A.36)
(A.37)
(A.38)
(A.39)
(A.40)
(A.41)
(A.42)
A.6 Using the Fast Fourier Transform
The fast
Fourier
transform (FFT) is a rapid numerical algorithm for evaluating the
sum
N
^ = £
e
27rilm/N
G,
1=0
(A.43)
Press et
al.
(1992) describe the algorithm and provide source code; better and more
elaborate routines can be obtained at http://www.netlib.org. Routines
computing multidimensional generalizations of (A.43) are available as well.
There are many subtleties involved in relating a discrete sum such as (A.43)
to a continuous integral; some are discussed by Brigham (1988) and Nussbaumer
(1982).
It is risky to use the discrete transform blindly, but in cases where it is
appropriate, here is a prescription for relating it to continuous Fourier transforms.
Using the Fast Fourier Transform 901
Suppose that the goal is to use a discrete transform with N points to describe
the Fourier transform of a continuous function G(x), with the interval covered by
the discrete transform corresponding to the interval
[—x
max
,
x
max
\.
Define
dx = 2
X
-^ (A.44)
N
and
dk = ^-. (A.45)
Ndx
Also define a function / that shuffles the index of a Fourier transform in a conven-
tional way:
function 1
(
j)
if ( j>(N+l)/2 )
{
return (j-N-1)
} else {
return (j-1)
Next, fill an array F
(
j
)
with values of the function G(x) according to
for j=l through N
{
x=l(j)*dx
F
(
j)=G(x)*dx
}
After invoking a fast Fourier transform routine on the array F
(
j ), the original
F
(
j
)
will be overwritten with its discrete Fourier transform, and the value of
F
(
j
)
will be the Fourier transform G{k), where
k=l(j)*dk
If one begins instead with G(k) rather than G{x), one has instead
for j=l through N
{
k=l(j)*dk
F(j)=G(k)*dk/(2*pi)
}
and uses an appropriate flag in the Fourier transform routine to obtain the inverse
transform.
Fast Fourier transform routines exist also for multidimensional sums, and they
obey conventions naturally generalized from those for the one-dimensional sums.
For example, suppose one wants to carry out the sum in Eq. (7.33),
Y^VMq-K).
(A.46)
902
Appendix A. Lattice Sums and Fourier Transforms
This sum is in the form of a convolution, so by far the fastest way to carry it out is
to take the Fourier transform of U^, the transform of
tj),
multiply the two, and then
invert the transform. Put another way, the potential energy operator is diagonal in
real space, so it should be computed there. The bookkeeping needed to compute
K\ i
,12,13,
and load Fourier components of
U
and
ip
into
N
x
N x N
arrays is indicated
by'
'
for jl=l through N
{
for j2=l through N
{
for
j 3=1
through N
{
for m=l through 3
{
K(m)=l
(
jl)*bl(m)+l(j2)*b2(m)+1(j3)*b3 (m)
Builds reciprocal lattice vector
K
kp
(m)
=q
(m)
-K
(m)
}
,,,,/-;i -;o AI\ —ri iv\ Loads Fourier components of potential into
UU(J±, JZ, U3)-U(K) arrayuu.
P
s
\ J
-L r
J^-r J
~>
I ~ pSl
(
Kp
)
Loads Fourier components of
ip
into array p s
References
E. O. Brigham (1988), The Fast Fourier Transform and its Applications, Prentice Hall, Englewood
Cliffs,
NJ.
H. J. Nussbaumer (1982), Fast Fourier Transform and Convolution Algorithms, Springer-Verlag,
Berlin.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (1992), Numerical Recipes in C,
2nd ed., Cambridge University Press, Cambridge.