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

multiply by cos(2p'x/a) and integrate over a period.

0 0

( ) cos 2 / cos 2 / cos 2 /2

a ap

p

acf x px a dx c px a px a dx

0

2 ( ) cos 2 /a

pc f x px a dxa

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

( ) iGxG

G

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 rG

G

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

ky

G

( ) iG rG

G

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)

Determine the structure factors in 1-D

Multiply by e-iG'x and integrate over a period a

( ) iGxG

G

f x f e

'

cos(( ') ) sin( ') )G G GG unit cell

f G G x if G G x dx f a

-

1 ( ) iGxG cellf f x e dx

a

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

( ) iG x i G G xG

Gunit cell

f x e dx f e dx

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 dk

k

Inverse transform:

0

0

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

ikx

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.

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

http://lampx.tugraz.at/~hadley/ss2/linearresponse/dft/dft0.php

Properties of Fourier transforms

Convolution (Faltung)

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

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 rG

G

f r f e

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

3

unit cell

( ) iG rGf 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 rGf 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

Spheres on an fcc lattice

( ) iG rG

G

f r f e

3 3

sphere

1 ( ) exp exp .cellG

Cf f r iG r d r iG r d rV 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

2

0 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 R

G

G rCf rdrV G

2

0 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 R

G

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:

3

sin 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

2

20

exp( )

G

iG rZeU rV G

position of atom j

Institute of Solid State PhysicsTechnische Universität Graz

The quantization of the electromagnetic field

Wave nature and the particle nature of light

Unification of the laws for electricity and magnetism (described by Maxwell's equations) and light

Quantization of the harmonic oscillator

Planck's radiation law

Serves as a template for the quantization of phonons, magnons, plasmons, electrons, spinons, holons and other quantum particles that inhabit solids.

Maxwell's equations

0

0 0 0

0

E

B

BEt

EB Jt

In vacuum the source terms J and are zero.

The vector potential

Maxwell's equations in terms of A Coulomb gauge 0A

B A

AE Vt

2

0 0 2

0

0

At

A

A At t

AAt

The wave equation

normal mode solutions have the form:

Using the identity 2( )A A A

22 2

2 .Ac At

2

0 0 2

AAt

( , ) cos( )A r t A k r t

Normal mode solutions

put the solution into the wave equation

22 2

2

Ac At

( , ) cos( )A r t A k r t

c k

cf

2 2 2c k A A

dispersion relation

wave equation:

normal modesolution:

Normalschwingungen oder Normalmoden

Dispersion relation

c k

k

cf

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EM waves propagating in the x direction

The electric and magnetic fields are

0 ˆcos( )xA A k x t z

0

0

ˆsin( )

ˆsin( )

x

x x

AE A k x t zt

B A k A k x t y

Quantization (using a trick)

The wave equation for a single mode.

The equation for a single mode is mathematically equivalent to:

2

2 2 2 22

( , )( , )x y zA k tc k k k A k t

t

2

2

xx mt

c2k2 , m 1

Quantization

Classical mathematical equivalence quantum mathematical equivalence

Rewriting this in terms of the electromagnetic field variables:

c k

Dispersion relation

12 0,1,2E j j

m

c2k2 , m 1

12 0,1,2E j j

j is the number of photons in that mode

periodic boundary conditions

Boundary conditions

fixed boundary conditions

2 nkL

2 2nk

L