Scattering and diffraction

Based on chapert 4 + some crystallography

Repetition and continuationThe 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.

Scattering

• ElasticElectron-nucleon

High angle scatteringRutherford scattering

Electron-electron Low angle scattering

• InelasticElectron-nucleon

Bremsstrahlung

Electron-electron X-rays SEPlasmons

Electron-atomsPhonons

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 absorbedin the specimen and detector

Ineleastic scatteringelectron-electron interaction

100 200 300 400 Incident beam energy (keV)

σ (m2)

10-21

10-23

10-25

Cross sections in Alassuming θ~0o

P

EL

K

SE

Electron transitions and X-ray notation

• K, L, M, N, O shells– Subshell L1, L2,…

What is the effect of differentIonization cross sections?

Given ”weights” within a family

Inelastic scattering → IonizationTotal ionization cross section (Bethe -1930):

ns: number of electrons in the ionized subshellbs 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

-with relativistic correction (Williams -1933):

Relativistic factor β=v/c

Ec: Critical ionization energy -Shell and Z dependent (Measured by EELS)

Critical ionization energy

Ec is generally <20 keV

Difference between Ec and the X-ray energy

Cascade to ground

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

Fluorescence yield, ωProbability of X-ray versus Auger electrons

Auger electrons

E~few hundred eV- a few keVand strongly absorbed in the specimen.

Differential cross section for plasmon exitation

a0: Bohr radius, θE=EP/2E0, Ep~15-20 eV σ→ 0, θ> 10 mrad

Phonon scattering

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

Beam damage

• 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.

Effect of HT?

Electron dose : Charge density (C/m2) hitting the specimen

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

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:

Maximum transferable energy –Dispalcements threshold energy

If more than the threshold energy is transfered to an atom it will dispalce from its site

Elastic scattering-Rutherford

Elastic scattering-Rutherford

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)

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.

Repeating element in the example

Crystals have a periodic internal structure

Lattice point

Point lattice

Repeting element 1 2 3

What is the repeting element in example 1-3?

Repeting element

1 2 3

Enhetscellen: repetisjonsenheten 1 2 3

Valgfritt origo!

Point lattice repeting element unit cell

Atoms and lattice points situated on corners, faces and edges are shared with neighbouring cells.

Unit cell

- 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).

Elementary unit of volume!

a

c

bα

βγ

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

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 a≠b≠c α=γ=90o ≠ β

Orthorombic a≠b≠c α= β=γ=90o

Tetragonal a=b≠c α= β=γ=90o

Cubic a=b=c α= β=γ=90o

Hexagonal a1=a2=a3≠c α= β=90o

γ=120o

Rhombohedral a=b=c α= β=γ ≠ 90o

Bravais lattice

The point lattices can be describedby 14 different Bravais lattices The point lattices can be describedby 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)

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

Z

Y

X

(010)

(001)

(100)

Z

Y

X

(110)

(111)

Z

Y

X

y

z

x

c/l

0a/h b/k

Hexagonal axial system

a1

a2

a3

a1=a2=a3

γ = 120o

(hkil)h + k + i = 0

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

uaa b

x

z

c

y

vb

wc

[uvw]

Determination of the Bravais-lattice of an unknown crystalline phase

Tilting series around common axis

0o

10o

15o

27o

50 nm

50 nm

Tilting series around a dens row of reflections in the reciprocal space

0o

19o

25o

40o

52o

Positions of the reflections in the reciprocal space

Determination of the Bravais-lattice of an unknown crystalline phase

Bravais-lattice and cell parameters

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

6.0

4 Å

7.94 Å8.66 Å

a

bc

100

110

111

010

011

001 101

[011] [100] [101]

d = L λ / R

Resiprocal latticeImportant for interpretation of ED patternsImportant for interpretation of 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:

Vbac

Vacb

Vcba

/)(

/)(

/)(

*

*

*

V: Volume of the unit cell

V=a.(bxc)=b.(cxa)=c.(axb)

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.

Orthogonal axes:a* = 1/IaI, b*=1/IbI, c*=1/IcI

Reciprocal vectors, planar distances

• Planar distance (d-value) between planes {hkl} in a cubic crystal with lattice parameter a:

222 lkh

ad hkl

–The resiprocal vector

is normal to the plane (hkl).

andthe spacing between the (hkl) planes is given by

*** clbkahg hkl

hklhkl gd /1

Convince your self !

What is the dot product beteen the normal to a (hkl) plane with a vector In the (hkl) plane?

Unit normal vector: n= ghkl/IghklI