Scattering and diffraction

1. Scattering and diffraction Basedonchapert 4 + somecrystallography

2. Repetition and continuation The probability of scattering is described in terms of either an “interaction cross-section” (σ)or a mean free path (λ). Differential scattering cross section (dσ/dΩ). i.e. the probability for scattering in a solid angle dΩ • 100keV: • σelastic = ~10-22 m2 • σinelastic = ~10-22 - 10-26 m2 • Is almost always the dominant component of the total scattering.

3. Scattering • Elastic Electron-nucleon High angle scattering Rutherford scattering Electron-electron Low angle scattering • Inelastic • Electron-nucleon • Bremsstrahlung • Electron-electron • X-rays • SE • Plasmons • Electron-atoms • Phonons

4. Inelastic scatteringelectron-nucleus interaction • Kramers cross section • To predict bremsstrahlung production N(E)=KZ(E0 – E)/E, N(E): number of bremsstrahlung photons, K: konstant E<~2 keV is absorbed in the specimen and detector

5. Ineleastic scatteringelectron-electron interaction σ (m2) Cross sections in Al assuming θ~0o 10-21 10-23 10-25 P E L K SE 100 200 300 400 Incident beam energy (keV)

6. Electron transitions and X-ray notation • K, L, M, N, O shells • Subshell L1, L2,… What is the effect of different Ionization cross sections? Given ”weights” within a family

7. Inelastic scattering → Ionization Total ionization cross section (Bethe -1930): ns: number of electrons in the ionized subshell bs and cs: constants for that shell • - The differential form show that the scattered • electron deviate through very small angles • (<~10 mrad). • The resultant characteristic X-ray is a spherical • wave emitted uniformly over 4π sr Ec: Critical ionization energy -Shell and Z dependent (Measured by EELS) -with relativistic correction (Williams -1933): Relativistic factor β=v/c

8. Critical ionization energy Ec is generally <20 keV

9. Difference between Ec and the X-ray energy

10. Cascade to ground • An ionized atom returns to ground state via a cascade of transitions.

11. Fluorescence yield, ωProbability of X-ray versus Auger electrons

12. Auger electrons E~few hundred eV- a few keV and strongly absorbed in the specimen.

13. Differential cross section for plasmon exitation a0: Bohr radius, θE=EP/2E0, Ep~15-20 eV σ→ 0, θ> 10 mrad

14. Phonon scattering • Scatter electrons to 5- 15 mrad • Diffuse background in diffraction pattern • Energy loss < 0.1 eV • Scattering increases with Z (~ Z3/2)

15. Beam damage Effect of HT? • Three principal forms • Radiolysis • Inelastic scattering breaks chemical bonds • Knock-on damage or sputtering • Displacement of atoms from the crystal lattice → point defects • Heating • Source of damage to polymers and biological tissue. Electron dose : Charge density (C/m2) hitting the specimen

16. Specimen heating • Depends on thermal conductivety of the specimen and beam current

17. Knock-on damage • Directly related to the incident beam energy • Primary way metals are damaged • Frenkel pair • Bond strength is a factor • Related to the displacement energy Threshold energy for dispacement of an atoms with atomic weight A:

18. Maximum transferable energy –Dispalcements threshold energy If more than the threshold energy is transfered to an atom it will dispalce from its site

19. Elastic scattering-Rutherford

20. Elastic scattering-Rutherford

21. Elastic scattering- small angles (<~3o) • Rutherford cross section can not be used • Scattering-factor approach is complementary • Wave nature of electrons Amplitudes: • Atomic scattering factor f(θ) • Structure factor F(θ)

22. Latticepropertiesofcrystals • The crystal structure is described by specifying a repeating element and its translational periodicity • The repeating element (usually consisting of many atoms) is replaced by a lattice point and all lattice points have the same atomic environments. Point lattice Repeating element in the example Lattice point Crystals have a periodic internal structure

23. Repeting element 1 2 3 What is the repeting element in example 1-3?

24. Repeting element 1 2 3

25. Enhetscellen: repetisjonsenheten 1 2 3 Valgfritt origo!

26. Point lattice repeting element unit cell Atoms and lattice points situated on corners, faces and edges are shared with neighbouring cells.

27. c α b β γ a Unit cell Elementary unit of volume! - Defined by three non planar lattice vectors: a, b and c -or by the length of the vectors a, b and c and the angles between them (alpha, beta, gamma). The origin of the unit cells can be described by a translation vector t: t=ua + vb + wc The atom position within the unit cell can be described by the vector r: r = xa + yb + zc

28. z c α β y b γ a x Axial systems The point lattices can be described by 7 axial systems (coordinate systems)

29. Bravais lattice The point lattices can be described by 14 different Bravais lattices Hermann and Mauguin symboler: P (primitiv) F (face centred) I (body centred) A, B, C (bace or end centred) R (rhombohedral)

30. z (001) (111) (110) (010) Z Z Z c/l b/k a/h (100) 0 y x Y Y Y X X X Lattice planes • Miller indexing system • Miller indices (hkl) of a plane is found from the interception of the plane with the unit cell axis (a/h, b/k, c/l). • The reciprocal of the interceptions are rationalized if necessary to avoid fraction numbers of (h k l) and 1/∞ = 0 • Planes are often described by their normal • (hkl) one single set of parallel planes • {hkl} equivalent planes

31. Hexagonal axial system a1=a2=a3 γ = 120o a2 a1 a3 (hkil) h + k + i = 0

32. The indices of directions (u, v and w) can be found from the components of the vector in the axial system a, b, c. The indices are scaled so that all are integers and as small as possible Notation [uvw] one single direction or zone axis <uvw> geometrical equivalent directions [hkl] is normal to the (hkl) plane in cubic axial systems z wc [uvw] c b a vb ua y x Directions

33. 27o 50 nm 15o 10o 0o Determination of the Bravais-lattice of an unknown crystalline phase Tilting series around common axis

34. 0o 50 nm 19o 25o 40o 52o Determination of the Bravais-lattice of an unknown crystalline phase Tilting series around a dens row of reflections in the reciprocal space Positions of the reflections in the reciprocal space

35. 011 111 001 101 6.04 Å 8.66 Å [101] [011] 7.94 Å 010 110 100 c b a Bravais-lattice and cell parameters [100] d = L λ / R From the tilt series we find that the unknown phase has a primitive orthorhombic Bravias-lattice with cell parameters: a= 6,04 Å, b= 7.94 Å og c=8.66 Å α= β= γ= 90o

36. Resiprocal lattice Important for interpretationof ED patterns Defined by the vectors a*, b* and c* which satisfy the relations: a*.a=b*.b=c*.c=1 and a*.b=b*.c=c*.a=a*.c=……..=0 Solution: a* is normal to the plane containing b and c etc. Unless a is normal to b and c, a* is not parallel to a. V: Volume of the unit cell V=a.(bxc)=b.(cxa)=c.(axb) Orthogonal axes: a* = 1/IaI, b*=1/IbI, c*=1/IcI

37. Reciprocal vectors, planar distances • The resiprocal vector • is normal to the plane (hkl). • and • the spacing between the (hkl) planes is given by Convince your self ! What is the dot product beteen the normal to a (hkl) plane with a vector In the (hkl) plane? • Planar distance (d-value) between planes {hkl} in a cubic crystal with lattice parameter a: Unit normal vector: n= ghkl/IghklI