Ø. Prytz

Introduction to diffraction

Øystein Prytz

January 21 2009

Ø. Prytz

Interference of waves

• Constructive and destructive interference

• Sound, light, ripples in water etc etc

)2

sin()(

)2

sin()(

2

1

xL

x

xL

x

=2n=(2n+1)

Ø. Prytz

Nature of light

• Newton: particles (corpuscles)

• Huygens: waves• Thomas Young double

slit experiment (1801)• Path difference phase

difference• Light consists of waves !• But remember blackbody

radiation and photoelectric effect !

Ø. Prytz

Discovery of X-rays

• Wilhelm Röntgen 1895/96• Nobel Prize in 1901• Particles or waves?• Not affected by magnetic

fields• No refraction, reflection or

intereference observed• If waves, λ10-9 m

Ø. Prytz

Max von Laue

• The periodicity and interatomic spacing of crystals had been deduced earlier (e.g. Auguste Bravais).

• von Laue realized that if X-rays were waves with short wavelength, interference phenomena should be observed like in Young’s double slit experiment.

• Experiment in 1912, Nobel Prize in 1914

Ø. Prytz

Bragg’s law

• William Henry and William Lawrence Bragg (father and son) found a simple interpretation of von Laue’s experiment• Consider a crystal as a periodic arrangement of atoms, this gives crystal planes• Assume that each crystal plane reflects radiation as a mirror • Analyze this situation for cases of constructive and destructive interference• Nobel prize in 1915

Ø. Prytz

Derivation of Bragg’s law

θ

θ

θ

x

)sin(

)sin(

hkl

hkl

dx

d

x

Path difference Δ= 2x => phase shiftConstructive interference if Δ=nλThis gives the criterion for constructive interference:

ndhkl )sin(2

dhkl

Bragg’s law tells you at which angle θB to expect maximum diffracted intensity for a particular family of crystal planes. For large crystals, all other angles give zero intensity.

But what happens if you place a plane in the middle?

Ø. Prytz

von Laue formulation

• Scattering angle related to the inverse plane spacing

• Waves often described using wave vectors

• The wave vector points in the direction of propogation, and its length inversely proportional to the wave length

hklB d

n

2)sin(

rkiAer 2)(

1

k

Ø. Prytz

von Laue formulation

k

'k

k

θ

1

'kk

'kkk

02

12

1

'2

')(

2

22

2

222

22

kkk

kkk

kkkkk

kkk

hkldg

gk

1

Vector normal to a plane

02 2 ggk

k

g

θ )sin()90cos( kgkggk

)sin(2

1)sin(

12

)sin(2 2

hkl

hkl

d

d

gkg

Ø. Prytz

The reciprocal lattice

• g is a vector normal to a set of planes, with length equal to the inverse spacing between them

• Reciprocal lattice vectors a*,b* and c*

• These vectors define the reciprocal lattice• All crystals have a real space lattice and a reciprocal

lattice• Diffraction techniques map the reciprocal lattice

*** clbkahg

)(*,

)(*,

)(*

bac

bac

acb

acb

cba

cba