Diffraction

1. Diffraction Analysis of crystal structure x-rays, neutrons and electrons Lett forkortet versjon av Anette Gunnes sin presentasjon MENA3100

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 MENA3100

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 = 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 MENA3100

4. Interference of waves Constructive interference Destructive interference 0 =(2n+1) =2n Sound, light, ripples in water etc etc Constructive and destructive interference MENA3100

5. Nature of light Newton: particles (corpuscles) Huygens: waves Thomas Young doubleslit experiment (1801) Path difference  phase difference Wave-particle duality MENA3100

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 MENA3100

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) MENA3100

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 MENA3100

9. Derivation of Bragg’s law θ θ dhkl θ x 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. MENA3100

10. Relationship between resiprocal vector and interplanar spacing θ Bragg’s law: Thus: MENA3100

11. Vector representation of Bragg law IkI=Ik0I=1/λ λx-rays>> λe k-k0 k = ghkl 2θ k0 (hkl) The limiting-sphere construction Diffracted beam Incident beam Reflecting sphere Limiting sphere MENA3100

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 MENA3100

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 MENA3100

14. 50 nm MENA3100

15. z wjc rj c a b vjb uja The intensity of a reflection is proportional to: y x Structure factors X-ray: 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) . 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. MENA3100

16. eiφ = cosφ + isinφ enπi = (-1)n eiφ+ e-iφ = 2cosφ Example: fcc 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(1+1+1+1)=4f If h, k, l are mixed integers (exs 112) then Fhkl=f(1+1-1-1)=0 (forbidden) MENA3100

17. The structure factor for fcc The reciprocal lattice of a FCC lattice is BCC What is the general condition for reflections for bcc? MENA3100

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? MENA3100