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. 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
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3. 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= nm
Scattered by atomic nuclei and electrons
Thickness less than ~200 nm
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4. Interference of waves Sound, light, ripples in water etc etc
Constructive and destructive interference
=2n
=(2n+1)
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5. 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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6. 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
Røntgen jobbet med en Crookes tube med sort papp rundt. Oppdaget at en fluoriserende skjerm i rommet lyste opp. Fant ut at strålingen gikk gjennom papp og bøker. Så når han holdt et objekt at han kunne se skjelettet sitt. Fotograferte sin kones hånd.
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7. 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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8. Laue conditions
Scattering from a periodic distribution of scatters along the a axis a ko 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
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9. The Laue equations
The Laue equations give three conditions for incident waves to be diffracted
by a crystal lattice
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) r k k0
k-k0
r*hkl
(hkl)
Δ=a.(k-ko)=h
Δ=b.(k-ko)=k
Δ=c.(k-ko)=l
If (k-k0) = r*, then φ= 2πn
r*= ha*+kb*+lc*
Δ=r . (k-k0)
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10. 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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11. Derivation of Bragg’s lawθ x
dhkl
Path difference Δ= 2x => phase shift
Constructive interference if Δ=nλ
This gives the criterion for constructive interference:
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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12. 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
The path difference: x-y
Y= x cos2θ and x sinθ=d
cos2θ= 1-2 sin2θ
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13. von Laue – Bragg equationθ
Vector normal to a plane θ
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14. The limiting-sphere construction
Vector representation of Bragg law
IkI=Ik0I=1/λ
λx-rays>> λe k
= ghkl
(hkl) k0
k-k0 2θ
Diffracted beam
Incident beam
Reflecting sphere
Limiting sphere
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15. The Ewald Sphere (’limiting sphere construction’)
Elastic scattering: 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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16. 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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17. 50 nm
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18. 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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19. 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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20. Example: 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 ]
What is the general condition
for reflections for fcc?
Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l))
What is the general condition
for reflections for bcc?
If h, k, l are all odd then:
Fhkl= f( )=4f
If h, k, l are mixed integers (exs 112) then
Fhkl=f( )=0 (forbidden)
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21. 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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22. 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?
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