Elastic Scattering and Kinematic Theory of Electron Diffraction

Dr. Hongzhou Zhang hozhang@tcd.ie

SNIAM 1.06 896 4655

Lecture 2

• Elements of the TEM

• Simple pictures of how to acquire images and diffraction patterns - Modes

• Alignment of the TEM

Reminder: Homework 1:Lecture 4 (due at lecture 5): Find a paper… please start looking for papers…

Content

• Signals in the TEM

• Elastic Scattering

– Quantitative description of the scattering process

– Atoms, Unit cells, Crystals

• Kinematic Theory of Electron Diffraction

– Geometrics of diffraction patterns

– Reciprocal lattice

Signals in a TEM

Specimen

Backscattered e

Secondary e Auger e

X-rays Cathodoluminescence

Incident e beam

Un-scattered e Inelastically Scattered e

Elastically Scattered e

Induced current

e, h pairs

Signals: Consequences of scattering processes

Scattering: Duality

• The natures of the scattering: interaction, energy exchange, …

• Scattering probability • Angular distribution of the

scattered waves/particles – Intensity – Amplitudes/phase

• Mean Free Path

Particle perspectives

Wave perspectives “Signals”

Atoms – Unit cells – Crystal

Coherent scattering:

Incoherent scattering:

j

j

r

rcoh

2

* j

j

r

rcohcohcohI

j

j

r

rincI

2

elastic inelastic

Coherent Diffraction, image contrast Neutron scattering – excitations (phonons, magnons)

Incoherent No sharp crystal-structure-related diffraction Broad angular dependence Compton scattering

Spectroscopies Absorption Analytical: EELS/ELNES/EXELFS

The incident – a precise phase relationship – The scattered

Scatterings: Different types

- Interactions: Single/plural/Multiple Single Scattering (Kinematic): not even for thin sample of ~nm Multiple-scattering: e-probe broadening

• Elastic: the solid remains in its original state - Total kinetic energy and momentum conserved - Coulomb atomic potential/screened (small angle scattering)

• Inelastic: excitation of energy states E = Ef-Ei

- Electron-electron interaction - Less localized processes

This is not all about using a coherent source

Differential cross section: Particle Perspectives

a: Impact parameter

Hyperbolic trajectories

2: scattering angle

a decreases

2 increases

Differential cross section:

I

I

IFlux density of the incident beam:

The electron flux scattered per solid angle:

I

d

d

I: # of electrons per unit time

• Anisotropic scattering:

beamincident theofdensity flux The

d angle solidper scatteredflux electron The section -cross lDifferenta

Angular distribution of scattered particles

• Total cross section:

s

dd

d

S

r

For a given , I

increases

d

dDecreases, a function of

Angular distribution

Scattering Potential

Scattering Length/amplitude: Wave Perspectives

rkiri

exp0

• Incident plane wave

• Scattered spherical wave

rrikrr

fikrr

frsc

expexp 00

f() ~ dimension of length

rrr si

• The total field

- Far from the scattering centre, r >> r’

Spherical wave ~ plane wave

rrkir

r

exp1

4

1

Cross-sections and scattering length

**

2m

hiej

2

00**

02

Ψm

ke

m

iej iiii

2

03

0**

2Ψf

r

r

m

ke

m

iej sssssc

rkiri

exp0Incident plane wave:

Scattered spherical wave: ikrr

frrikrr

frsc expexp 00

0

0

Charges passing through d :

Wave Perspectives :

2

f

d

d

djdrjsc 0

2 0

2

j

jr

d

d sc

Particle Perspectives: dS

dIj

d

dIj 0

djdI 0

The current, dI, scattered passing through solid angle d

dr

dI

dS

dIjsc 2

drjdI sc

2

f() ~ dimension of length

Current intensity: Charge conservation

ip̂

Scattering amplitude/length/factor: Wave perspectives

rErrVm

)(

2

22• The Schrodinger equation: rrV

mr

mE )(

2222

2

rrUrk )(2

0

2

A point source at r’ : rrikrr

rrG

exp1

4

1,

Satisfies the equation: '',2

0

2 rrrGk

• The amplitude of the total scattered wave • Sum of the amplitude of spherical waves from scattering centres • Scattering centres: Electrons of the atoms, atoms of the crystal

The weight of the spherical wave centred at r’: rrU )(

The total scattered amplitude: rdrrGrrUrs

3,)(

rdrrGrrUrs

3,)(

The asymptotic solution far from the nucleus (the detectors)

Spherical wave ~ plane wave rrkir

rrG

exp1

4

1,

rdrkirrVr

rkimrs

3

02exp)(

exp

2

The first Born approximation: single scattering rkir

00 exp

rdrkkirVr

rkimrs

3

002exp)(

exp

2

Coherent scattering

V(r): atomic, crystal,… )(

2)(

2rV

mrU

For each scattering, the wave before the scattering is the incident plane wave

The scattering factor

r

ikrfs

exp0

rdrkkirVr

ikrmrs

3

020 exp)(exp

2

• Scattered wave:

rdrqirVm

rdrkkirVm

f 3

2

3

02exp)(

2exp)(

2

0

0

22 2

2

2 EE

EE

E

m

• To work out f(), using the interaction potential:

2

4

0

3

0

3

0

2

2

expexp

42

q

j

j

fZ

jjj rdrr

rriqrdrqirZ

emf

x

sin20 kkkq

xfZh

mef

2

2

2

0 sin24

1

r

Zedr

rr

redrrrrVrV j

j

j

ij

22

0

1

)(

4

1...)...;(*)(

0kkq

Cross sections: Large angle ( fx = 0) Rutherford cross section:

2

2

2

0

2

22

sin

1

8

mv

Zef

d

d R

Small angle cross section: 4

2 1~

fd

d el

2 k0

k

Definition of the scattering factor:

For an atom:

Model potentials:

R

r

r

ZerV exp

4)(

0

2

3

1

ZaR H nm0529.0Ha

Wentzel model (screened Coulomb Potential)

Discussion on f()

Strongest in the forward direction, i.e. small scattering angle: coherent, constructive in the forward direction

Stronger for lower beam energy, i.e. longer

wavelength

f() is normally a complex number: additional phase

due to scattering

Length unit

Larger |f| for heavier atoms

xfZh

mef

2

2

2

0 sin24

1

• Born approximation: single scattering • Ignored: exchange effects, virtual inelastic scattering effects, spin, Bremsstrahlung energy losses, electron-induced damage, absorption

• Only can be measured for gas target: multiple scattering • WKB + muffin-tin model

• Atomic shell structure ignored Reimer, TEM, Springer

f(0): sensitive to the mean

inner potential

Total elastic cross section and mean-free-path length

dd

dd

d

del sin2

0

4

0

• The total elastic cross section per atom

2Rr

Billiard balls

• Total cross section (per atom) inelelt

• A layer of solid

• Scattering occurs when the electron strikes a portion of ds: dzdsNt

- The scattered fraction dj of j

• The current intensity of electrons remain unscatterred at z (direct beam) :

t

t

zjzNjj expexp 00

- The total mean-free-path length Nt

t

1

- Thickness, dz; Area, ds; volume: dzds

- # of Atoms: dsdzNA

dsdzN A

A

NN A

-123 mol1002.6 AN

A: the atomic weight

- Mass: dzds

dzNds

dzdsN

j

djt

t

Radiation Elastic mean free path (Å) Absorption length (Å) Probe size (Å)

Neutrons 108 109 107

X-rays 104 106 103

Electrons 102 103 1

2

f

d

d

Observable… … The strength of electron diffraction is 106 stronger than XRD

Carbon, organic materials, you

need 80kV beam

Energy Transfer in an Electron-Nucleus Collision

2

2

0' sin22

Mc

EEEEn

• The conservation of momentum

- Momentum of the nucleus: non-relativistic

- Momentum of the electrons: relativistic

nppp

nn EMp 2

2

021

EEEc

p 2

021

EEEc

p

• The conservation of energy nEEE

• The energy transferred

- Negligible for small

- For higher E and large

• Displacement energy: Ed 10-30 eV

Atoms being displaced from their lattice points: damage

• Knock-on processes: small cross section

• Threshold Ec: En=Ed, = 90o

dd

d elD

2/

min

sin

dn E

Mc

EEEE

min

2

2

0' sin22

d

ccn E

Mc

EEEE

2

022

ZoBelli, A. Gloter, C. P. Ewels, G. Seifert, C. Colliex, Electron knock-on cross section of carbon and boron nitride nanotubes, Physcial Review B 75, (2007). R. F. Egerton, R. McLeod, F. Wang, M. Malac, Basic questions related to electron-induced sputtering in the TEM. Ultramicroscopy 110, 991 (Jul, 2010). O. L. Krivanek et al., Gentle STEM: ADF imaging and EELS at low primary energies. Ultramicroscopy 110, 935 (Jul, 2010).

OK for elements on this side…

Diffraction in the TEM

• Selected Area Electron Diffraction (SAED) – >~100 nm

• Convergent-beam diffraction (CBED) – Each point: particular k

– Intensity across the disk ~ f(k)

– 1~10 nm

– Uniform: orientation and thickness

• Identity of Phase/orientation relationship

• Habit planes of precipitates, slip planes

• Exact crystallographic description of crystal defects

• Order/disorder, spinodal decomposition, magetic domains and similar phenomena

The one closest to that passes

the origin

Crystallography • You may want to review crystal structures …

• Chapter 1, Marc De Graef, Introduction to Conventional TEM, Cambridge • Chapter 1&2, Charles Kittel, Introduction to Solid State Physics, Wiley • Chapter 4-7, Ashcroft/Mermin, Solid State Physics, Brooks/Cole • Chapter 1-3, A, Kelly, et.al, Crystallography and Crystal Defects, Wiley • Gerald Burns/Am. Glazer, Space Group for Solid State Scientists, Academic

- Primitive cells

The origins of the unit cells: cobnamrg

The positions of atoms: czbyaxrk

0,0,01 r

2

1,

2

1,

2

12r

- Lattice Planes (miller indices): (hkl) The reciprocal intercepts in units of a, b, c

The intercepts in units of a, b, c

- Directions: [uvw]

Equivalent Directions: <uvw>

Equivalent planes: {uvw}

Braces/curly brackets

Angle brackets

Square brackets

Round brackets

The Assumptions of Kinematical Theory

• Monochromatic wave

• Plane waves (T, D)

• Free of distortion

• Negligible diffraction (all atoms receive an incident wave of the same amplitude)

• No interaction between T and Ds: Refractive index = 1

• No absorption

• No re-scattering of scattered waves

A geometrical Approach

• Diffraction of monochromatic light by a grating – Large distance R – In phase: path difference = n – → intensity : low

efficiency for large angle scattering

• Scattered wavelets – in phase in particular crystallographic directions – Scattering by an individual

atom f() – Scattering by a crystal

• Angular distribution – The Bragg law – The Laue Conditions – Structure &lattice

factors/amplitudes

The Bragg Law

d

Scattering angle: 2

T D

Bragg Law: simple method of visualising diffraction

• Crystal planes - Wavefront

• Path difference for strong diffraction

ndhkl sin2

The smallest angle: n =1 2

d

(1) Path difference: 2

sin

d

(1)?

(2)?

(2) Path difference: 2

332sinsin

ddd

(3)?

(3) Path difference: 243sinsin ddd

(4) n = 2

Diffraction along (1):

d 2

dd sin

Tilt the beam, and the diffraction tilts accordingly

Ap

pro

x.

Quantitative approach - Laue Conditions

nPr

• Scattering by individual atoms

• Incident/scattered waves: po, p (unit vectors)

• Path difference for constructive interference

Atoms: cwbvaur

0ppP

• Laue conditions for diffraction:

haP

2

2

kbP

2

2

lcP

2

2

The solution of the Laue conditions, k, satisfies the diffraction condition and defines diffracted beam k

po p

O B

A

A’ • Phase difference for constructive interference

nrP

2

2

hak 2

kbk 2

lck 2

0prOA

prAO

0pprAOOA

Path difference

00

2kp

kp

2

0kkk

ko k

k

b a

Laue Conditions: The solution • The solution

where ......;0;2;2;2 **** baccbbaa

Each point in reciprocal space: a set of particular planes in real space

Two vectors AB and BC in Plane (hkl)

0*****

bk

k

bah

h

aclbkah

k

b

h

aghkl

AB

0*****

bk

k

bcl

l

cclbkah

k

b

l

cghkl

BC

g

AB

)( plane hklghkl

g

BC

)( plane hklghkl

*** clbkahgk hkl

- Orthogonal axes (cubic, tetragonal, orthorhombic): Reciprocal Lattice coincide with Crystal lattice

cba

,*

k

b

OB

h

a

OA

l

c

OC

O

A

B

C

OAOB

OBOC

g d

aa 2*

- Properties

hkl

hkld

g2

gg

clbkah

h

a

g

gdhkl

2***

OA

c

a*

b*

c*

a

c

b

cba

cba

2*

hklgk

Diffraction

Laue Conditions and Diffraction

0k

k

gkkk

0

g O

L

G

• Construct the direction of diffraction beams

In the reciprocal space

- A line from L to O (the origin)

- in the direction of the incident beam

- Magnitude k0=2/

- A line from L to G (ghkl)

- Magnitude k=k0=2/

LG: The diffraction k due to the planes corresponding to ghkl

2

• The Bragg Law

hkl

hkld

gk

2

sin2 0

hkl

hkld

g

2sin

22

sin2 hkld

G2

hkl

hkld

ngnk

2

sin2 0 ndhkl sin2

nrg 2

Scattered by a Unit Cell – the Structure Factor

xfZh

mef

2

2

2

0 sin24

1

Atoms – Unit cells – Crystal

n

nnnn

n

nn lzkyhxififF 2expexp

• Unit cells

r

ikrfs

exp0

- Position of atom in the atomic plane czbyaxr nnnn

- Additional phase nnnnhklnn lzkyhxrgrP

2

2

• Atoms:

- Scattered amplitude factor by the cell:

Coherent Scattering, sum of the amplitude

0,0,01 r

2

1,

2

1,

2

12r

lkhiff

lzkyhxififF

AlNi

n

nnnn

n

nn

exp

expexp

- Intensity: 2FI

Nlkhff

Nlkhff

lkhfflkhff

FI

AlNi

AlNi

AlNiAlNi

2when

12when

2sincos

2

2

22

2

The same atoms

Nlkh

NlkhfFI

2when 0

12when 4 22

Systematic Absence

Systematic absence

d

T

D

A’ C’

For AA’-CC’ : sin2 )(hkld

Due to the addition scattering (200) A

C B

B’

For AA’-BB’ : 2sin

22

d

For BB’-CC’ : 2

sin2

2

d

No diffraction: 2

A’’ B’’ C’’

For AA’’-CC’’ :

243sinsin ddd

For AA’’-BB’’ :

For BB’’-CC’’ :

42

d

42

d

Diffraction beam: 4

1 2

3

(100) Systematic absence h+k+l =1 is an odd number

(200) Diffraction Occurs h+k+l =2 is an even number

Scattered by a Finite Crystal – Crystal Factor

Very thin samples: 100 nm ~ 500 unit cells

Atoms – Unit cells – Crystal

cnbnanr zyx

The origins of the unit cells:

- Additional phase

zgygxgnnn nznynxrgrP

2

2

zyx

zyx

NNN

nnn

zgygxg

n

n nznynxiFiFG

,,

,,

2expexp

- Scattered amplitude factor by the crystal:

X

XX

nN

n

nx

1

11

0

z

zz

g

yg

g

xg

iz

Niz

iy

Niy

ix

NixFG

2exp1

2exp1

2exp1

2exp1

2exp1

2exp1

- Intensity: 2GI

g

zg

g

yg

g

xg

z

Nz

y

Ny

x

NxFI

2

2

2

2

2

22

sin

sin

sin

sin

sin

sin

- Maximum: hxg kyg lzg hklgg

- Falls to zero: x

gN

hx1

;0sinsin ihihNNix xxg

Laue condition is relaxed!

Reciprocal Space

0k

k

g=ghkl+s

O

L

G

2

s

ghkl

g

zg

g

yg

g

xg

y

Ny

y

Ny

x

NxFI

2

2

2

2

2

22

sin

sin

sin

sin

sin

sin

sgg hkl

z

zz

y

yy

x

xx

s

Ns

s

Ns

s

NsFI

2

2

2

2

2

22

sin

sin

sin

sin

sin

sin

Perfect crystal: s varies slowly across the crystal

O

The Reciprocal lattice and Ewald Sphere

• Ewald sphere Construction

• Diffraction in 3D Specimen, Real space

Reciprocal space

- Choose a reciprocal lattice, O, as the origin of the lattice -A line from L to O (the origin)

- in the direction of the incident beam - Magnitude k0=2/

- A sphere centred at L, radius |k0| - The wavevectors k of all possible elastic diffracted waves emanating from L and terminated at the sphere

L

• The Ewald Sphere: – Radius of Ewald: 2/ ~ 17 nm-1 (100kV)

– Reciprocal lattice spacing: 0.3nm-1

– Flat surface: several streaks cut simultaneously

These g more or less perpendicular to the incident beam

Reciprocal Lattice: Laue Zones

nrg 2

*** clbkahg

cwbvaur

2* aa

nlwkvhu

0 lwkvhu rg

[uvw]

1g

2g

3g

• Zero-order Laue Zone:

The atomic planes (real space) have the zone axis as a common line of intersection

0rg

A plane in reciprocal space that is perpendicular to the beam

• high-order Laue Zone (HOLZ): 2rg

2

Beam direction r – real space

Selected Area Diffraction Optical Axis

k0 k

0k

k

Reciprocal Space

BFP of the OL

Image Plane of the DP R

hkl

hkld

g2

L

k

R

ghkl

LRdhkl

1212

LRdhkl

L : camera constant; L: camera length

...

8

31

2

L

R

R

Ldhkl

Projection lens imperfection makes is

unnecessary

Summary

• Atoms-Unit Cells-Crystal: Scattering factors

• Diffraction pattern: the spot array on the surface of the Ewald sphere

Lecture 4

• Indexing Diffraction Patterns

• Kikuchi Lines

Reciprocal – Real Spaces

Real Space Reciprocal Space

SC SC

Tetragonal Tetragonal

Hexagonal Hexagonal

FCC BCC

BCC FCC

Total Elastic/Inelastic Cross sections

Elastic Scattering Amplitude f() - WKB method

rkirirar ss

00 expexp

rdrV

EE

EE

Erdrnrs

0

0

2

21

2

Near Field: Modification of the incoming plane wave

1expexpexpexpexp 000000 rirkirkirkiri ss

The scattering amplitude: rdrkkiriikf s

3

0exp1exp

Wave-front behind the atom:

No absorption: 1ras

Phase shift due to the atom using the electron-refractive index:

Rewrite the exit wave:

Incident: i

rkiri

00 expIncident wave:

Scattered: sc

Far field, Fraunhofer diffraction, k: ikr

rfrsc exp0

Weak phase specimens (s <<1): rdrqirkf s

3exp

Using refractive index:

rdrqirV

EE

EE

Ef 3

0

0

2exp

2

2

0kkq

Coherent Scattering

Cross-section by a single electron: 2

j

j

r

rf

d

d

Cross-section by an atom (Z electrons):

d

dZfZf

d

d jr

j

j

j r

Z

r

ratom

22

2

2

~

Constructive coherent

Total Cross-section by an atom (Z electrons):

Z

r

atom

j

jr

d

d

d

d

d

d

dd

d

d

S

Z

rS

atom

j

jr

Different angular distribution due to

interference

Z

r SS

atomatom

j

jr

jr

Zdd

dd

d

d

Phase relationship

The bound electron is driven by the e-field of the wave

Oscillating dipole moment results in radiation