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Diffraction

Analysis of crystal structure

x-rays, neutrons and electrons

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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

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Radiation: x-rays, neutrons and electrons

• Elastic scattering of radiation– No energy is lost

• The wave length of the scattered wave remains unchanged

• Regular arrays of atoms interact elastically with radiation of sufficient short wavelength – CuKα x-ray radiation: λ=0.154 nm

• Scattered by electrons• ~from sub mm regions

– Neutron radiation λ~0.1nm• Scattered by atomic nuclei• Several cm thick samples

– Electron radiation (200kV): λ=0.00251 nm• Scattered by atomic nuclei and electrons• Thickness less than ~200 nm

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Interference of waves

• Sound, light, ripples in water etc etc

• Constructive and destructive interference )

2sin()(

)2

sin()(

2

1

xL

x

xL

x

=2n =(2n+1)

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Nature of light

• Newton: particles (corpuscles)

• Huygens: waves• Thomas Young double

slit experiment (1801)• Path difference phase

difference• Light consists of waves !• Wave-particle duality

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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

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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

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Laue conditions

rkiAer 2)(

1

k

Scattering from a periodic distribution of scatters along the a axis

a

ko

k

The scattered wave will be in phase and constructive interference will occur if the phase difference is 2π.

Φ=2πa.(k-ko)=2πa.g= 2πh, similar for b and c

*** clbkahg hkl

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The Laue equations

• Waves scattered from two lattice points separated by a vector r will have a path difference in a given direction.

• The scattered waves will be in phase and constructive interference will occur if the phase difference is 2π.

• The path difference is the difference between the projection of r on k and the projection of r on k0, φ= 2πr.(k-k0)

Two lattice points separated by a vector r

rk

k0

If (k-k0) = r*, then φ= 2πnr*= ha*+kb*+lc*

Δ=r . (k-k0)k-k0

r*hkl(hkl)Δ=a.(k-ko)=h

Δ=b.(k-ko)=k

Δ=c.(k-ko)=l

The Laue equations give three conditions for incident waves to be diffractedby a crystal lattice

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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

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Derivation of Bragg’s law

)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

θ

θ

θ

x

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.

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• nλ = 2dsinθ– Planes of atoms responsible

for a diffraction peak behave as a mirror

Bragg’s law

d

θ

θ

y

x

The path difference: x-y

Y= x cos2θ and x sinθ=dcos2θ= 1-2 sin2θ

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von Laue – Bragg equation

0k

k

k

θ

1

kko

kkko

02

12

1

2

)(

2

22

2

222

22

kkk

kkk

kkkkk

kkk

o

o

oo

o

hkldg

gk

1

Vector normal to a plane

02 2 ggko

ok

g

θ )sin()90cos( gkgkgk ooo

)sin(2

1)sin(

12

)sin(2 2

hkl

hkl

o

d

d

ggk

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The limiting-sphere construction

• Vector representation of Bragg law

• IkI=Ik0I=1/λ

– λx-rays>> λe k= ghkl

(hkl)

k0

k-k0

2θIncident beamDiffr

acte

d be

am

Limiting sphereReflecting sphere

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The Ewald Sphere (’limiting sphere construction’)

1

'kk

Elastic scattering:

k k’

g

The observed diffraction pattern is the part of the reciprocal lattice that is intersected by the Ewald sphere

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Cu Kalpha X-ray: = 150 pm => small kElectrons at 200 kV: = 2.5 pm => large k

The Ewald Sphere is flat (almost)

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50 nm

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Allowed and forbidden reflections

• Bravais lattices with centering (F, I, A, B, C) have planes of lattice points that give rise to destructive interference for some orders of reflections.

– Forbidden reflections

d

θ

θ

y

xx’

y’

In most crystals the lattice point corresponds to a set of atoms.

Different atomic species scatter more or less strongly (different atomic scattering factors, fzθ).

From the structure factor of the unit cell one can determine if the hkl reflection it is allowed or forbidden.

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Structure factors

The structure factors for x-ray, neutron and electron diffraction are similar. For neutrons and electrons we need only to replace by fj

(n) or fj(e) .

N

j

xjhklg fFF

1

)( 2exp( ))( jjj lwkvhui X-ray:

The coordinate of atom j within the crystal unit cell is given rj=uja+vjb+wjc. h, k and l are the miller indices of the Bragg reflection g. N is the number of atoms within the crystal unit cell. fj(n) is the x-ray scattering factor, or x-ray scattering amplitude, for atom j.

rj

ujaa b

x

z

c

y

vjb

wjc

The intensity of a reflection is proportional to:

ggFF

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Example: fcc

• eiφ = cosφ + isinφ

• enπi = (-1)n

• eix + e-ix = 2cosx

N

jjhklg fFF

1

2exp( ))( jjj lwkvhui

Atomic positions in the unit cell: [000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ]

Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l))

If h, k, l are all odd then:Fhkl= f(1+1+1+1)=4f

If h, k, l are mixed integers (exs 112) thenFhkl=f(1+1-1-1)=0 (forbidden)

What is the general condition for reflections for fcc?

What is the general condition for reflections for bcc?

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The structure factor for fcc

What is the general condition for reflections for bcc?

The reciprocal lattice of a FCC lattice is BCC

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The reciprocal lattice of bcc

• Body centered cubic lattice • One atom per lattice point, [000] relative to the lattice point• What is the reciprocal lattice?

N

jjhklg fFF

1

2exp( ))( jjj lwkvhui