1. Analysis of crystal structure x-rays, neutrons and electrons
Diffraction
Analysis of crystal structure
x-rays, neutrons and electrons
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2. 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= 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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MENA3100

3. gives the Laue equations:
Two lattice points separated by a vector r
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) k0 r k
k-k0
r*hkl
(hkl)
r = a, b or c and IkI=Ik0I=λ
gives the Laue equations:
Δ=hλ Δ=kλ Δ=lλ
If (k-k0) = r*, then φ= 2πn
r*= ha*+kb*+lc*
Δ=r . (k-k0)
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MENA3100

4. The path difference: x-y
Bragg’s law d θ y x
nλ = 2dsinθ
Planes of atoms responsible for a diffraction peak behave as a mirror
1/d2=(h/a)2+(k/b)2+(l/c)2
Orthorhombic lattice
The path difference: x-y
Y= x cos2θ and x sinθ=d
cos2θ= 1-2 sin2θ
k-k0
r*hkl
(hkl)
ghkl = r*hkl
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MENA3100

5. The limiting-sphere construction
Vector representation of Bragg law
IkI=Ik0I=λ
λx-rays>> λe k
= ghkl
(hkl) k0
k-k0 2θ
Diffracted beam
Incident beam
Reflecting sphere
Limiting sphere
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MENA3100

6. 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.
Forbidden reflections y’ y x θ x’ d θ
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).
From the structure factor of the unit cell one can determine if the hkl reflection it is allowed or forbidden.
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MENA3100

7. Structure factors 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.
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) . rj
uja a b x z c y
vjb
wjc
The intensity of a reflection is proportional to:
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8. Example: Cu, fcc eiφ = cosφ + isinφ enπi = (-1)n eix + e-ix = 2cosx
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?
If h, k, l are all odd then:
Fhkl= f( )=4f
What is the general condition
for reflections for bcc?
If h, k, l are mixed integers (exs 112) then
Fhkl=f( )=0 (forbidden)
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MENA3100