Transcript Diffraction

Diffraction
Analysis of crystal structure
x-rays, neutrons and electrons
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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
– Electron radiation (200kV): λ=0.00251 nm
• Scattered by atomic nuclei and electrons
• Thickness less than ~200 nm
– Neutron radiation λ~0.1nm
• Scattered by atomic nuclei
• Several cm thick samples
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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)
If (k-k0) = r*, then φ= 2πn
r*= ha*+kb*+lc*
Δ=r . (k-k0)
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k0
Two lattice
points separated
by a vector r
r
k
(hkl)
r*hkl
k-k0
r = a, b or c and IkI=Ik0I=λ
gives the Laue equations:
Δ=hλ Δ=kλ Δ=lλ
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
• 1/d2=(h/a)2+(k/b)2+(l/c)2
Y= x cos2θ and x sinθ=d
cos2θ= 1-2 sin2θ
– Orthorhombic lattice
d hkl  1 / g hkl
g hkl  h a *  k b*  l c *
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(hkl)
ghkl = r*hkl
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r*hkl
k-k0
The limiting-sphere construction
•
Vector representation of
Bragg law
•
IkI=Ik0I=λ
– λx-rays>> λe
= ghkl
Incident beam
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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: Cu, 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 ]
Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l))
What is the general condition
for reflections for fcc?
What is the general condition
for reflections for bcc?
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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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