7/3-11 MENA3100

Diffraction

Analysis of crystal structure

x-rays, neutrons and electrons

Lett forkortet versjon av Anette Gunnes sin presentasjon

MENA3100

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 wavelength 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 sample volume of the order of (0.1 mm)3

– Neutron radiation λ ~ 0.1nm• Scattered by atomic nuclei• From sample volume of the order of (10 mm)3

– Electron radiation (200 kV): λ = 0.00251 nm• Scattered by atomic nuclei and electrons• Thickness less than ~200 nm• Sample volume down to (10 nm)3

MENA3100

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)

Constructive interference Destructive interference

0

MENA3100

Nature of light

• Newton: particles (corpuscles)

• Huygens: waves• Thomas Young double

slit experiment (1801)• Path difference phase

difference• 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 within 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 (Friedrich, Knipping and von Laue), Nobel Prize in 1914 (von Laue)

MENA3100

Bragg’s law

•William Lawrence Bragg 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 semitransparent mirror • Analyze this situation for cases of constructive and destructive interference• Nobel prize together with his father in 1915 for solving the first crystal structures

MENA3100

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.

MENA3100

Relationship between resiprocal vector and interplanar spacing

0k

k

g

θ 1

kko

dg

1

sin2

sin2 kg

Bragg’s law:

sin21

d

Thus:

MENA3100

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 space that is intersected by the Ewald sphere

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

The Ewald Sphere is almost flat when 1/ becomes large

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50 nm

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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

MENA3100

Example: fcc

• eiφ = cosφ + isinφ

• enπi = (-1)n

• eiφ + e-iφ = 2cosφ

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?

MENA3100

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