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




g  ha *  kb * lc *
• Reciprocal lattice vectors a*,b* and c*
 
 
 



b c
c a
a b
a*     , b *     , c *    
a  (b  c )
b  (c  a )
c  (a  b )
• These vectors define the reciprocal lattice
• All crystals have a real space lattice and a reciprocal lattice
• Diffraction techniques map the reciprocal lattice
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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
=2n
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2
1 ( x)  sin(
x)
L
2
2 ( x)  sin(
x )
L
=(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, λ10-9 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


(r )  Ae2ik r
k
1

Scattering from a periodic distribution of scatters along the a axis
ko
a
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
g hkl  h a *  k b*  l c *
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The Laue equations
The Laue equations give three conditions for incident waves to be diffracted
by a crystal lattice
•
•
•
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)
If (k-k0) = r*, then φ= 2πn
r*= ha*+kb*+lc*
Δ=r . (k-k0)
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Two lattice
points separated
by a vector r
k0
r
k
Δ=a.(k-ko)=h
(hkl)
r*hkl
Δ=b.(k-ko)=k
Δ=c.(k-ko)=l
k-k0
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(  ) 
x
θ
d hkl
 x  d hkl sin(  )
θ
θ
dhkl
x
Path difference Δ= 2x => phase shift
Constructive interference if Δ=nλ
This gives the criterion for constructive interference:
   2d hkl sin(  )  n
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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Bragg’s law
y
• nλ = 2dsinθ
θ
– Planes of atoms responsible
for a diffraction peak behave
as a mirror
d
x
θ
The path difference: x-y
Y= x cos2θ and x sinθ=d
cos2θ= 1-2 sin2θ
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von Laue – Bragg equation

k0
θ

k
 
k  g
1
g
d hkl
Vector normal to a plane
 
2ko  g  g 2  0

g

k
ko  k 
1

 θ
ko
 
ko  g  ko g cos(90   )  ko g sin(  )

 
ko  k  k



( k o  k ) 2  k 2


2
k o  2k o  k  k 2  k 2


1
1
 2k o  k  k 2  2
2




2k o  k  k 2  0
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 2ko g sin(  )   g 2
1
1
2 sin(  ) 

d hkl
2d hkl sin(  )  
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The limiting-sphere construction
•
Vector representation of
Bragg law
•
IkI=Ik0I=1/λ
– λx-rays>> λe
= ghkl
Incident beam
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The Ewald Sphere (’limiting sphere
construction’)
Elastic scattering:
k  k'
k

k’
The observed diffraction pattern is
the part of the reciprocal lattice that
is intersected by the Ewald sphere
g
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The Ewald Sphere is flat (almost)
Cu Kalpha X-ray:  = 150 pm => small k
Electrons at 200 kV:  = 2.5 pm => large k
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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.
y’y
θ
d
x’
x
θ
– Forbidden reflections
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).
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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
N
X-ray:
Fg  Fhkl   f j( x ) exp( 2i (hu j  kv j  lw j ))
j 1
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.
wjc
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) .
z
rj
c
a b
v jb
uj a
y
x
The intensity of a reflection is
Fg Fg
proportional to:
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Example: fcc
•
•
•
eiφ = cosφ + isinφ
enπi = (-1)n
eix + e-ix = 2cosx
N
Fg  Fhkl   f j exp( 2 i (hu j  kv j  lw j ))
j 1
Atomic positions in the unit cell:
[000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ]
What is the general condition
for reflections for fcc?
What is the general condition
for reflections for bcc?
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) then
Fhkl=f(1+1-1-1)=0 (forbidden)
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The structure factor for fcc
The reciprocal lattice of a FCC lattice is BCC
What is the general condition
for reflections for 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
Fg  Fhkl   f j exp( 2 i (hu j  kv j  lw j ))
j 1
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