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![Page 1: 7/3-11MENA3100 Diffraction Analysis of crystal structure x-rays, neutrons and electrons Lett forkortet versjon av Anette Gunnes sin presentasjon.](https://reader030.fdocuments.us/reader030/viewer/2022032523/56649d6f5503460f94a51195/html5/thumbnails/1.jpg)
7/3-11 MENA3100
Diffraction
Analysis of crystal structure
x-rays, neutrons and electrons
Lett forkortet versjon av Anette Gunnes sin presentasjon
![Page 2: 7/3-11MENA3100 Diffraction Analysis of crystal structure x-rays, neutrons and electrons Lett forkortet versjon av Anette Gunnes sin presentasjon.](https://reader030.fdocuments.us/reader030/viewer/2022032523/56649d6f5503460f94a51195/html5/thumbnails/2.jpg)
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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MENA3100
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
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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
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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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MENA3100
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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MENA3100
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)
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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
![Page 9: 7/3-11MENA3100 Diffraction Analysis of crystal structure x-rays, neutrons and electrons Lett forkortet versjon av Anette Gunnes sin presentasjon.](https://reader030.fdocuments.us/reader030/viewer/2022032523/56649d6f5503460f94a51195/html5/thumbnails/9.jpg)
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.
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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:
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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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MENA3100
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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MENA3100
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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MENA3100
50 nm
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MENA3100
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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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?
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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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MENA3100
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