Institute of Solid State PhysicsTechnische Universität Graz

Fourier Series and Fourier Transforms

Fourier series in 2-D and 3-D

Electrons in a crystal move in a 3-D periodic potential.

X-rays scatter from the periodic electron density.

Expanding a 1-d function in a Fourier series

xAny periodic function can be represented as a Fourier series.

f(x)

01

( ) cos(2 / ) sin(2 / )p pp

f x f c px a s px a

a

2 pGa

cos sin2 2

ix ix ix ixe e e ex xi

( ) iGxGG

f x f e

*

-G Gf fFor real functions:

2 2p p

G

c sf i

reciprocal lattice vector

Institute of Solid State Physics

Fourier series in 1-D, 2-D, or 3-D Technische Universität Graz

In two or three dimensions, a periodic function can be thought of as a pattern repeated on a Bravais lattice. It can be written as a Fourier series

( ) iG rGG

f r f e

Reciprocal lattice vectors (depend on the Bravais lattice)

Structure factors (complex numbers)

2 - ... 1,0,1,... G vb v ba

In 1-D:

Reciprocal space (Reziproker Raum)k-space (k-Raum)

p k

2k

k-space is the space of all wave-vectors.

A k-vector points in the direction a wave is propagating.

wavelength: momentum:

kx

kz

kyG

( ) iG rGG

f r f e

Any periodic function can be written as a Fourier series

Structure factor

Reciprocal lattice vector G

vi integers

1 1 2 2 3 3G b b b

2 3 3 1 1 2

1 2 31 2 3 1 2 3 1 2 3

2 , 2 , 2a a a a a ab b ba a a a a a a a a

kx

ky

Reciprocal lattice (Reziprokes Gitter)

2i j ija b

Determine the structure factors

Multiply by e-iG'r and integrate over a unit cell

( ) iG rGG

f r f e

'Gf V

-

1 ( ) iG rG cellf f r e drV

The structure factor is proportional to the Fourier transform of the pattern that gets repeated on the Bravais lattice, evaluated at that G-vector.

''

unit cell

( ) i G G riG r GGunit cell

f r e dr f e dr

Only G = G' is non zero.

Plane waves (Ebene Wellen)

exp expik r r ik r

Most functions can be expressed in terms of plane waves

A k-vector points in the direction a wave is propagating.

2k cos sinik re k r i k r

( ) ik rf r F k e dk

Fourier transforms

Most functions can be expressed in terms of plane waves

( ) ik rf r F k e dk

1 ( )2

ik rdF k f r e dr

This can be inverted for F(k)

http://lamp.tu-graz.ac.at/~hadley/ss1/crystaldiffraction/ft/ft.php

Fourier transform of f(r)

Fourier transforms

/ 2

/ 2

sin / 212

aikx

a

kaF k e dx

k

f x

Fourier transform:

sin / 2 ikxkaf x e dkk

Inverse transform:

0

0

1 10 02 2sin / 2 Si( ) Si( )

kikx

k

ka k x k xf x e dkk

Transmitted pulse:

Sine integral

a

Notations for Fourier Transforms

d = number of dimensions 1,2,3a,b = constants

Notations for Fourier Transforms

f(r) is built of plane waves

Notations for Fourier Transforms

Matlab

Notations for Fourier Transforms

Mathematica

Notations for Fourier Transforms

Engineering literature, usually on the 1-d case is considered.

Properties of Fourier transforms

Convolution (Faltung)

( )* ( ) ( ) ( )f r g r f r g r r dr

http://lamp.tu-graz.ac.at/~hadley/ss1/crystaldiffraction/ft/ft.php

http://lampx.tugraz.at/~hadley/num/ch3/3.3a.php

The reciprocal lattice is the Fourier transform of the real space lattice

crystal = Bravais_lattice(r) * unit_cell(r)

F(crystal) = F(Bravais_lattice(r))F(unit_cell(r))

k

a2

reciprocal

Cubes on a bcc lattice

( ) iG rGG

f r f e

Multiply by and integrate over a primitive unit cell.iG re

3

unit cell

( ) iG r Gf r e d r f V

http://lamp.tu-graz.ac.at/~hadley/ss1/crystaldiffraction/fourier.php

Cubes on a bcc lattice

31 ( ) expcellGf f r iG r d rV

fG is the Fourier transform of fcell evaluated at G.fcell is zero outside the primitive unit cell.

3

unit cell

( ) iG r Gf r e d r f V

V is the volume of the primitive unit cell.

4 4 4

33

4 4 4

1 2( )exp exp exp exp

a a a

cell x y zGa a a

Cf f r iG r d r iG x iG y iG z dxdydzV a

Volume of conventional u.c. a3. Two Bravais points per conventional u.c.

Cubes on a bcc lattice

The Fourier series for any rectangular cuboid with dimensions Lx×Ly×Lz repeated on any three-dimensional Bravais lattice is:

3

16 sin sin sin4 4 4

yx z

Gx y z

G aG a G aCf

a G G G

8 sin sin sin

2 2 2( ) exp

y yx x z z

G x y z

G LG L G LCf r iG r

VG G G

4 4 4

4 44

2sinexp cos sin 4exp

a a a x

x x xx

a aa x x x

G aiG x G x i G x

iG x dxiG iG G

http://lampx.tugraz.at/~hadley/ss1/crystaldiffraction/fourier.php

Spheres on an fcc lattice

( ) iG rGG

f r f e

3 3sphere

1 ( ) exp exp .cellGCf f r iG r d r iG r d r

V V

Integrate over

iG re

2

0 0

2

0 0

exp( ) sin

cos cos sin cos sin

R

G

R

Cf iG r r drd dV

C G r i G r r drd dV

20 0

2 cos cos sin cos sinR

G

Cf G r i G r r drdV

Multiply by and integrate over a primitive unit cell.

Spheres on an fcc lattice

cos cos sin cos sin and

sin cos cos cos sin ,

d G r G r G rdd G r G r G r

d

Integrate over

0

sin4 RG

G rCf rdrV G

20 0

2 cos cos sin cos sinR

G

Cf G r i G r r drdV

Both terms are perfect differentials

0 0

2 sin cos cos cosR

G

Cf G r i G r drV

0

Spheres on any lattice

Integrate over r

0

sin4 RG

G rCf rdrV G

34 sin cos .G Cf G R G R G RV G

The Fourier series for non-overlapping spheres on any three-dimensional Bravais lattice is:

3sin cos4( ) exp .

G

G R G R G RCf r iG rV G

Molecular orbital potential

2

0

1( )4

jr j

ZeU rr r

The Fourier series for any molecular orbital potential is:

Volume of the primitive unit cell

22

0

exp( )

G

iG rZeU rV G

position of atom j