1. Analysis of crystal structure x-rays, neutrons and electrons
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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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 wavelength of the scattered wave remains unchanged
Regular arrays of atoms interact elastically with radiation of sufficient short wavelength
CuKα x-ray radiation: λ = 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 = nm
Scattered by atomic nuclei and electrons
Thickness less than ~200 nm
Sample volume down to (10 nm)3
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4. Interference of waves Sound, light, ripples in water etc etc
Constructive and destructive interference
Constructive interference
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
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 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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8. 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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9. 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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10. Relationship between resiprocal vector and interplanar spacingθ
Bragg’s law:
Thus:
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11. 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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12. The Ewald Sphere (’limiting sphere construction’)
Elastic scattering: k k’
The observed diffraction pattern is the part of the reciprocal space that is intersected by the Ewald sphere g
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13. The Ewald Sphere is almost flat when 1/l becomes large
Cu Ka X-ray:  = 150 pm => small k
Electrons at 200 kV:  = 2.5 pm => large k
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14. 50 nm
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15. 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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16. Example: fcc eiφ = cosφ + isinφ enπi = (-1)n eiφ + e-iφ = 2cosφ
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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17. 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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18. 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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