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Diffraction
Analysis of crystal structure
x-rays, neutrons and electrons
Lett forkortet versjon av Anette Gunnes sin presentasjon
7/3-11
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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 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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Interference of waves
• Sound, light, ripples in water etc
etc
• Constructive and destructive
interference
2
1 ( x)  sin(
x)
L
2
2 ( x)  sin(
x )
L
Constructive interference
Destructive interference
0
=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
• 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 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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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
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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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Relationship between resiprocal vector and interplanar spacing
ko  k 
θ

k0

k

g  2k sin  
Bragg’s law:

g
1
Thus:
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2 sin 

1 2 sin 

d

g
1
d
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
1

k’
The observed diffraction pattern is
the part of the reciprocal space that
is intersected by the Ewald sphere
g
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The Ewald Sphere is almost flat when 1/
becomes large
Cu Ka 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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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
proportional to:
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Fg Fg
Example: fcc
•
•
•
eiφ = cosφ + isinφ
enπi = (-1)n
eiφ + e-iφ = 2cosφ
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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