1. Scattering and diffraction
Based on chapert 4 +
some crystallography

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 = ~ m2
Is almost always the dominant component of the total scattering.

3. Scattering Elastic Inelastic Electron-nucleon 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 scattering electron-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 scattering electron-electron interaction
σ (m2)
Cross sections in Al
assuming θ~0o
10-21
10-23
10-25 P E L K SE
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, ω Probability 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. Lattice properties of crystals
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 What is the repeting element in example 1-3? 1 2 3
What is the repeting element in example 1-3?
Hvordan kan vi beskrive den periodiske strukturen til krystaller?
-ved å beskrive den minste rep.enheten.
-rep.enhet. kalles enhetscellen
For eks. 1-3 hva er enhetscellen?
-Eks. viser utsnitt fra tenkte kryst. i tre dim.
- ant. atomslag
-atom i forkant eksisterer også i bakkant
-det er ingen ”grønne” atomer i senter av utsnitt

24. Repeting element 1 2 3 Hvor mange atomer er det i en enhetscellen?
Enkel!
Hvor mange atomer er det i en enhetscellen?
Brukes til å finne tetthet til strukturen når V kjent
V=a*a*a (Kubisk), a lik lengden av en sideflate
Hva er enhetscellen/Hvor stor er den i eks. 2?

25. Enhetscellen: repetisjonsenheten
Valg av origo er fritt!
Det er like riktig å beskrive strukturen ved å ta utg.pkt. i et annet origo.
Først rødt atom i origo, nå blått.
For eks. 3 kan det se slik ut…..
Valgfritt origo!

26. Point lattice repeting element unit cell
Dette danner et gitter.
Eksempel
Gitterpunkt deles med nærliggende enhetsceller på samme måte som
atomene.
Kittre med et gitterpunkt pr. enhetscelle kalles primitivt.
Eks. på to Primitivt kubbisk gitter, m et og to atomer pr. gitterpunkt.
Flatesentrert kubisk gitter
Eksempler på hva vi kaller for Baravais gitre
Forslag på gruppe i enhetscelle i eksempel 3 til neste gruppetime. Vet aksesystem, og at det er 16 atomer i enhetscellen.
Atoms and lattice points situated on corners, faces and edges
are shared with neighbouring cells.

27. Unit cell Elementary unit of volume!b α β γ
- 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. Axial systems
The point lattices can be described by 7 axial systems (coordinate systems) x y z a b c α γ β
Axial system
Axes
Angles
Triclinic
a≠b≠c
α≠β≠γ≠90o
Monoclinic
α=γ=90o ≠ β
Orthorombic
α= β=γ=90o
Tetragonal
a=b≠c
Cubic
a=b=c
Hexagonal
a1=a2=a3≠c
α= β=90o
γ=120o
Rhombohedral
α= β=γ ≠ 90o
Enhetscellen defineres av tre ikke planære vektorer a,b og c som er aksene i enhetscellen.
-kan også beskrives ved lengdene til a, b og c samt vinklene mellom dem.
Vektorene definerer et aksesystem. I eksempel 1 var…, 2 var… Dette er eksempler på hva vi kaller kubisk aksesystem
I eks. 3 var var aksene a=b men ikke lik c, vinkl. 90 gr.…Dette er et eksempel på hva vi kaller et tetragonalt aksesystem. (Kommer tilbake til andre mulige aksesyst.)
Gjentakende like grupper av atomer i enhetscellen kan erstattes med gitterpunkt. 1 eller flere atomer pr. pkt.

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. Lattice planes Miller indexing systemz x
c/l
a/h
b/k
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 Z Y X
(110)
Crystals are described in the axial system of their unit cell
Når vi skal beskrive posisjonene til gitterpunkter og atomer bruker vi posisjonsvektorer av typen-…….
Hvor a, b og c er enhetsvektorene I de ulike krystall systmene, u, v, w er lengden langs hver av disse vektorene.
Hva ville uvw være for et atom på en sideflate?
Plan I enhetcelle kan angiv ved normalen til planet eller ved Miller-indeksene hkl Z Y X
(010)
(001)
(100)
(111) Z Y X

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

32. Directions 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 ua a b x z c y vb wc
[uvw]

33. Determination of the Bravais-lattice of an unknown crystalline phase
50 nm
Tilting series around common axis

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

35. Bravais-lattice and cell parameters
100
110
111
010
011
001
101
[011]
[100]
[101]
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
6.04 Å
7.94 Å
8.66 Å

36. Resiprocal lattice Important for interpretation of ED patterns
Defined by the vectors a*, b* and c* which satisfy the relations:
a*.a=b*.b=c*.c= 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