Principles & Practice of Electron Diffraction - CIME | EPFL

Principles & Practice of
Electron Diffraction

Duncan Alexander
EPFL-CIME

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 1

Contents

Introduction to electron diffraction 2
Elastic scattering theory

Basic crystallography & symmetry
Electron diffraction theory

Intensity in the electron diffraction pattern
Selected-area diffraction phenomena
Convergent beam electron diffraction

Recording & analysing selected-area diffraction patterns
Quantitative electron diffraction
References

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Introduction to
electron diffraction

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 3

Why use electron diffraction?

Diffraction: constructive and destructive interference of waves

! wavelength of fast moving electrons much smaller than spacing of atomic planes
=> diffraction from atomic planes (e.g. 200 kV e-, λ = 0.0025 nm)

! electrons interact very strongly with matter => strong diffraction intensity
(can take patterns in seconds, unlike X-ray diffraction)

! spatially-localized information
(≳ 200 nm for selected-area diffraction; 2 nm possible with convergent-beam electron diffraction)

! close relationship to diffraction contrast in imaging

! orientation information

! immediate in the TEM!

(" diffraction from only selected set of planes in one pattern - e.g. only 2D information)
(" limited accuracy of measurement - e.g. 2-3%)

(" intensity of reflections difficult to interpret because of dynamical effects)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 4

Image formation

Optical axis
Electron source

Condenser lens Selected area
aperture
Specimen
Objective lens
Back focal plane/
di raction plane

Intermediate
image 1

Intermediate lens

Projector lens BaTiO3 nanocrystals (Psaltis lab)
Image
Insert selected area aperture to choose
region of interest

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 5

Take selected-area diffraction pattern

Optical axis

Electron source

Condenser lens Selected area
aperture
Specimen
Objective lens
Back focal plane/
di raction plane

Intermediate
image 1

Intermediate lens

Projector lens Press “D” for diffraction on microscope console -
IDmi argaection alter strength of intermediate lens and focus
diffraction pattern on to screen

Find cubic BaTiO3 aligned on [0 0 1] zone axis

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 6

Elastic scattering
theory

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 7

Scattering theory - Atomic scattering factor

Consider coherent elastic scattering of electrons from atom

Differential elastic scattering Atomic scattering factor
cross section:

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 8

Scattering theory - Huygen’s principle

Periodic array of scattering centres (atoms)
Plane electron wave generates secondary wavelets

kk00

kDk2D1 k0 kD1 k0

Secondary wavelets interfere =>

strong direct beam and multiple orders of diffracted beams from constructive interference

Atoms closer together => scattering angles greater

=> Reciprocity!

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 9

Basic crystallography &
symmetry

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 10

Crystals: translational periodicity &
symmetry

Repetition of translated structure to infinity

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Crystallography: the unit cell

Unit cell is the smallest repeating unit of the crystal lattice
Has a lattice point on each corner (and perhaps more elsewhere)

Defined by lattice parameters a, b, c along axes x, y, z
and angles between crystallographic axes: α = b^c; β = a^c; γ = a^b

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 12

Building a crystal structure

Use example of CuZn brass
Choose the unit cell - for CuZn: primitive cubic (lattice point on each corner)

Choose the motif - Cu: 0, 0, 0; Zn: !,!,!

Structure = lattice +motif => Start applying motif to each lattice point

z

Motif: y Cu
Zn

x
z

y
x

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 13

Building a crystal structure

Use example of CuZn brass
Choose the unit cell - for CuZn: primitive cubic (lattice point on each corner)

Choose the motif - Cu: 0, 0, 0; Zn: !,!,!
Structure = lattice +motif => Start applying motif to each lattice point

Extend lattice further in to space

z

Motif: z y Cu
Zn

x

z
y

y

yy y

xxx x

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 14

Introduction to symmetry

As well as having translational symmetry, nearly all crystals obey other symmetries
- i.e. can reflect or rotate crystal and obtain exactly the same structure
Symmetry elements:

Mirror planes:

Rotation axes:

Centre of symmetry or
inversion centre:

Inversion axes: combination of rotation axis with centre of symmetry

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 15

Introduction to symmetry

Example - Tetragonal lattice: a = b ≠ c; α = β = γ = 90°
Anatase TiO2 (body-centred lattice) view down [0 0 1] (z-axis):

Identify mirror planes
Identify rotation axis: 4-fold = defining symmetry of tetragonal lattice!

y

Mirror plane y
Tetrad:
4-fold rotation
axis

z x x

O
Ti

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 16

More defining symmetry elements

Cubic crystal system: a = b = c; α = β = γ = 90°
View down body diagonal (i.e. [1 1 1] axis)

Choose Primitive cell (lattice point on each corner)
Identify rotation axis: 3-fold (triad)

Defining symmetry of cube: four 3-fold rotation axes (not 4-fold rotation axes!)

z

xy

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 17

(Cubic α-Al(Fe,Mn)Si: example of
primitive cubic with no 4-fold axis)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 18

More defining symmetry elements

Hexagonal crystal system: a = b ≠ c; α = β = 90°, γ = 120°
Primitive cell, lattice points on each corner; view down z-axis - i.e.[1 0 0]

Draw 2 x 2 unit cells
Identify rotation axis: 6-fold (hexad) - defining symmetry of hexagonal lattice

z a y
y
a a
z

120

a

x

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 19

The seven crystal systems

7 possible unit cell shapes with different symmetries that can be repeated by translation
in 3 dimensions

=> 7 crystal systems each defined by symmetry

Triclinic Monoclinic Orthorhombic Tetragonal Rhombohedral

Hexagonal Cubic

Diagrams from www.Wikipedia.org 20

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Four possible lattice centerings

P: Primitive - lattice points on cell corners

I: Body-centred - additional lattice point at cell centre

F: Face-centred - one additional lattice point at centre
of each face

A/B/C: Centred on a single face - one additional lattice 21
point centred on A, B or C face

Diagrams from www.Wikipedia.org

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

14 Bravais lattices

Combinations of crystal systems and lattice point centring that describe all possible crystals
- Equivalent system/centring combinations eliminated => 14 (not 7 x 4 = 28) possibilities

Diagrams from www.Wikipedia.org 22

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

14 Bravais lattices

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 23

Crystallography - lattice vectors

A lattice vector is a vector joining any two lattice points
Written as linear combination of unit cell vectors a, b, c:

t = Ua + Vb + Wc
Also written as: t = [U V W]

Examples: z z

z

yy y

x [1 0 0] x [0 3 2] x [1 2 1]

Important in diffraction because we “look” down the lattice vectors (“zone axes”)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 24

Crystallography - lattice planes

Lattice plane is a plane which passes through any 3 lattice points which are not in a straight line

Lattice planes are described using Miller indices (h k l) where the first plane away from the
origin intersects the x, y, z axes at distances:

a/h on the x axis
b/k on the y axis
c/l on the z axis

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 25

Crystallography - lattice planes

Sets of planes intersecting the unit cell - examples:

z

yz

x (1 0 0)

yz

x (0 2 2)

y

x (1 1 1)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 26

Lattice planes and symmetry

Lattice planes in a crystal related by the crystal symmetry

For example, in cubic lattices the 3-fold rotation axis on the [1 1 1] body diagonal
relates the planes (1 0 0), (0 1 0), (0 0 1):

z

xy
Set of planes {1 0 0} = (1 0 0), (0 1 0), (0 0 1), (-1 0 0), (0 -1 0), (0 0 -1)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 27

Weiss Zone Law

If the lattice vector [U V W] lies in the plane (h k l) then:
hU + kV + lW = 0

Electron diffraction:
Electron beam oriented parallel to lattice vector called the “zone axis”

Diffracting planes must be parallel to electron beam
- therefore they obey the Weiss Zone law*

(*at least for zero-order Laue zone)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 28

Electron diffraction
theory

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 29

Diffraction theory - Bragg law

Path difference between reflection from planes distance dhkl apart = 2dhklsinθ
22ddhhkklslsiinnθθ == λλ/=2->c-oBdnresastgtrrguulccattwiivv:ee iinntteerrffeerreennccee
nλ = 2dhklsinθ

+=

θθ
dhhkkll

Electron diffraction: λ ~ 0.001 nm 30
therefore: λ ≪ dhkl

=> small angle approximation: nλ ≈ 2dhklθ

∝Reciprocity: scattering angle θ dhkl-1

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Diffraction theory - 2-beam condition

θ kI

θ kD
θ

kI gd hkl G

000

2-beam condition: strong scattering from single set of planes

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 31

Multi-beam scattering condition

Electron beam parallel to low-index crystal orientation [U V W] = zone axis
Crystal “viewed down” zone axis is like diffraction grating with planes parallel to e-beam

In diffraction pattern obtain spots perpendicular to plane orientation
Example: primitive cubic with e-beam parallel to [0 0 1] zone axis

2 x 2 unit cells y

z

-1 0 0

0 -1 0 0 0 0 0 1 0

100 110

200 220

x 300

Note reciprocal relationship: smaller plane spacing => larger indices (h k l)
& greater scattering angle on diffraction pattern from (0 0 0) direct beam

Also note Weiss Zone Law obeyed in indexing (hU + kV + lW = 0)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 32

Scattering from non-orthogonal crystals

With scattering from the cubic crystal we can note that the diffracted beam for plane (1 0 0)
is parallel to the lattice vector [1 0 0]; makes life easy

However, not true in non-orthogonal systems - e.g. hexagonal:

z a y
a
(1 0 0) planes

120

x g1 0 0

[1 0 0]

=> care must be taken in reciprocal space!

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 33

The reciprocal lattice

In diffraction we are working in “reciprocal space”; useful to transform the crystal lattice in to
a “reciprocal lattice” that represents the crystal in reciprocal space:

Real lattice rn = n1a + n2b + n3c Reciprocal lattice r* = m1a* + m2b* + m3c*
vector: vector:

where:

a*.b = a*.c = b*.c = b*.a = c*.a = c*.b = 0

a*.a = b*.b = c*.c = 1

i.e. a* = (b ^ c)/VC VC: volume of unit cell

For scattering from plane (h k l) the diffraction vector:
ghkl = ha* + kb* + lc*

Plane spacing:

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 34

Fourier transforms for understanding
reciprocal space

Fourier transform: identifies frequency components of an object
- e.g. frequency components of wave forms

Each lattice plane has a frequency in the crystal lattice given by its plane spacing
- this frequency information is contained in its diffraction spot

The diffraction spot is part of the reciprocal lattice and, indeed the reciprocal lattice
is the Fourier transform of the real lattice

Can use this to understand diffraction patterns and reciprocal space more easily

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 35

The Ewald sphere

radius = 1/λ

kI: incident beam C
wave vector
kI kD
kD: diffracted
wave vector

0

Reciprocal space: sphere radius 1/λ represents possible scattering wave
vectors intersecting reciprocal space

Electron diffraction: radius of sphere very large compared to reciprocal lattice
=> sphere circumference almost flat

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 36

Ewald sphere in 2-beam condition

kI θ
θ
2θ θ kD

kI kD

0 0 0 ghkl G kI G

g
000

2-beam condition with one strong Bragg reflection corresponds to Ewald sphere
intersecting one reciprocal lattice point

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 37

Ewald sphere and multi-beam scattering

kI kD Assume reciprocal lattice points
are infinitely small
000
With crystal oriented on zone
axis, Ewald sphere may not

intersect reciprocal lattice points

However, we see strong diffraction
from many planes in this condition

Because reciprocal lattice points
have size and shape!

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 38

Fourier transforms and reciprocal lattice

Real lattice is not infinite, but is bound disc of material with diameter of
selected area aperture and thickness of specimen - i.e. thin disc of material

X

FT FT
“Relrod”
X

= 2 lengths scales in
reciprocal space!

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 39

Ewald sphere intersects Relrods

kI kD

000

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 40

Relrod shape

Shape (e.g. thickness) of sample is like a “top-hat” function
Therefore shape of Relrod is: sin(x)/x

Can compare to single-slit diffraction pattern with intensity:

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 41

Relrod shape

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 42

Intensity in the electron
diffraction pattern

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 43

Excitation error

Tilted slightly off Bragg condition, intensity of diffraction spot much lower

Introduce new vector s - “the excitation error” that measures deviation from
exact Bragg condition

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 44

Excitation error

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 45

Dynamical scattering

For interpretation of intensities in diffraction
pattern, single scattering would be ideal
- i.e.“kinematical” scattering

However, in electron diffraction there is
often multiple elastic scattering:
i.e.“dynamical” behaviour

This dynamical scattering has a high
probability because a Bragg-scattered beam
is at the perfect angle to be Bragg-scattered

again (and again...)

As a result, scattering of different beams
is not independent from each other

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 46

Dynamical scattering for 2-beam condition

For a 2-beam condition (i.e. strong scattering at ϴB) it can be derived that:

where:

and ξg is the “extinction distance” for the Bragg reflection:
Further:

i.e. the intensities of the direct and diffracted beams are complementary, and in
anti-phase, to each other. Both are periodic in t and seff

If the excitation distance s = 0 (i.e. perfect Bragg condition), then:

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 47

Extinction and thickness fringes

Dynamical scattering in the dark-field image –
=> Intensity zero for thicknesses t = nξg (integer n)

See effect as dark “thickness fringes” on wedge-shaped sample:

Composition changes in quantum wells 48
=> extinction at different thickness compared to substrate

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Dynamical scattering for 2-beam

2-beam condition: direct and diffracted beam
intensities beams π/2 out of phase:

Model with absorption using JEMS:

Bright-field image
showing modulation

with absorption:

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 49

2-beam: kinematical vs dynamical

Kinematical (weak interactions) Dynamical (strong interactions)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 50

Weak beam; kinematical approximation

Before we saw for 2-beam condition:
where:

Weak-beam imaging: make s large (~0.2 nm-1)
Now Ig is effectively independent of ξg -
“kinematical” conditions!

=> dark-field image intensity easier to interpret

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 51

Structure factor

Amplitude of a diffracted beam:

ri: position of each atom => ri: = xi a + yi b + zi c
K = g: K = h a* + k b* + l c*

Define structure factor:

Intensity of reflection:
Note fi is a function of s and (h k l)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 52

Forbidden reflections

Consider FCC lattice with lattice point coordinates:
0,0,0; !,!,0; !,0,!; 0,!,!

Calculate structure factor for (0 1 0) plane (assume single atom motif):

z

=>

y

x 53
Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Forbidden reflections

Cu3Au - like FCC Au but with Cu atoms on face-centred sites.
What happens to SADP if we gradually increase Z of Cu sites until that of Au (to obtain FCC Au)?

Diffraction pattern on [0 0 1] zone axis:

z

y

Au

x

Cu

Patterns simulated using JEMS

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 54

Forbidden reflections

Cu3Au - like FCC Au but with Cu atoms on face-centred sites.
What happens to SADP if we gradually increase Z of Cu sites until that of Au (to obtain FCC Au)?

Diffraction pattern on [0 0 1] zone axis:

z

y

Au

x

Cu

Patterns simulated using JEMS

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 55

Extinction rules

Face-centred cubic: reflections with mixed odd, even h, k, l absent:

Body-centred cubic: reflections with h + k + l = odd absent:

Reciprocal lattice of FCC is BCC and vice-versa

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 56

Selected-area
diffraction phenomena

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 57

Symmetry information

Zone axis SADPs have symmetry closely related to symmetry of crystal lattice

[0 0 1] Example: FCC aluminium

[1 1 0]

[1 1 1]

4-fold rotation axis

2-fold rotation axis 58

6-fold rotation axis - but [1 1 1] actually 3-fold axis
Need third dimension for true symmetry!

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Twinning in diffraction

Example: FCC twins
Stacking of close-packed {1 1 1} planes reversed at twin boundary:

A B CA B CA B CA B C
#A B CA B C BA C BA C

View on [1 1 0] zone axis:

{1 1 1} planes: B
A

1 -1 -1

1 -1 -1B
1 -1 1A

002

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 59

Twinning in diffraction

Example: Co-Ni-Al shape memory FCC twins observed on [1 1 0] zone axis
(1 1 1) close-packed twin planes overlap in SADP

Images provided by Barbora Bartová, CIME 60

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Epitaxy and orientation relationships

SADP excellent tool for studying orientation
relationships across interfaces

Example: Mn-doped ZnO on sapphire

Sapphire substrate Sapphire + film

Zone axes:
[1 -1 0]ZnO // [0 -1 0]sapphire

Planes:
c-planeZnO // c-planesapphire

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 61

Crystallographically-oriented precipitates

Co-Ni-Al shape memory alloy, austenitic with Co-rich precipitates

Bright-field image Dark-field image

!"#$%#&'#%()*+,&-./0
1&2'3)#.),2'+4'!"#"$"%!&'(%51167!89956617-:;:/:<'=>11>1?!899=116?-:;:/:
8,@'3)#.),2'+4'!"#"$"%!&'(%51167!89956617-:;:/:<%=>111?!899=116?-:;:/:

Images provided by Barbora Bartová, CIME

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 62

Double diffraction

Special type of multiple elastic scattering: diffracted beam
travelling through a crystal is rediffracted

Example 1: rediffraction in different crystal - NiO being reduced to Ni in-situ in TEM
Epitaxial relationship between the two FCC structures (NiO: a = 0.42 nm Ni: a = 0.37 nm)

Formation of satellite spots around Bragg reflections 63
Images by Quentin Jeangros, EPFL

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Double diffraction

Example 1: NiO being reduced to Ni in-situ in TEM movie

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 64

Double diffraction

Example 1I: rediffraction in the same crystal; appearance of forbidden reflections
Example of silicon; from symmetry of the structure {2 0 0} reflections should be absent

However, normally see them because of double diffraction

Simulate diffraction pattern
on [1 1 0] zone axis:

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 65

Ring diffraction patterns

If selected area aperture selects numerous, randomly-oriented nanocrystals,
SADP consists of rings sampling all possible diffracting planes
- like powder X-ray diffraction

Example: “needles” of contaminant cubic MnZnO3 - which XRD failed to observe!
Note: if scattering sufficiently kinematical, can compare intensities with those of X-ray PDF files

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 66

Ring diffraction patterns

Larger crystals => more “spotty” patterns

Example: ZnO nanocrystals ~20 nm in diameter

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 67

Ring diffraction patterns

“Texture” - i.e. preferential orientation - is seen as arcs of greater intensity
in the diffraction rings

Example: hydrozincite Zn5(CO3)2(OH)6 recrystallised to ZnO crystals 1-2 nm in diameter

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 68

Amorphous diffraction pattern

Crystals: short-range order and long-range order
Amorphous materials: no long-range order, but do have short-range order
(roughly uniform interatomic distances as atoms pack around each other)

Short-range order produces diffuse rings in diffraction pattern

Example:

Vitrified germanium
(M. H. Bhat et al. Nature 448 787 (2007)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 69

Kikuchi lines

Inelastic scattering event scatters electrons in all directions inside crystal
Some scattered electrons in correct orientation for Bragg scattering => cone of scattering

Cones have very large diameters => intersect diffraction plane as ~straight lines

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 70

Kikuchi lines

Position of the Kikuchi line pairs of (excess and deficient) very sensitive to specimen orientation

Can use to identify excitation vector; in particular s = 0 when diffracted beam coincides
exactly with excess Kikuchi line (and direct beam with deficient Kikuchi line)

Lower-index lattice planes => narrower pairs of lines

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 71

Kikuchi lines - “road map” to reciprocal space

Kikuchi lines traverse reciprocal space, converging on zone axes

- use them to navigate reciprocal space as you tilt the specimen!

Examples: Si simulations using JEMS

Si [1 1 0] Si [1 1 0] tilted off zone axis Si [2 2 3]

Obviously Kikuchi lines can be useful, but can be hard to see (e.g. from insufficient
thickness, diffuse lines from crystal bending, strain). Need an alternative method...

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 72

Convergent beam
electron diffraction

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 73

Convergent beam electron diffraction

Instead of parallel illumination with selected-area aperture, CBED uses
highly converged illumination to select a much smaller specimen region

Small illuminated area =>
no thickness and orientation variations

There is dynamical scattering, but it is useful!

Can obtain disc and line patterns
“packed” with information:

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 74

Convergent beam electron diffraction

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 75

Convergent beam electron diffraction

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 76

Convergent beam electron diffraction

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 77

Convergent beam electron diffraction

Kikuchi from: inelastic scattering convergent beam

“Kikuchi” lines much less
diffuse for CBED

=> use CBED to orientate
sample!

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 78

Convergent beam electron diffraction
– practical example

ZnO thin-film sample;
Conditions: convergent beam, large condenser aperture, diffraction mode

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 79

Convergent beam electron diffraction
– practical example

ZnO thin-film sample;
Conditions: convergent beam, large condenser aperture, diffraction mode

[1 1 0] zone axis

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 80

Recording & analysing
selected-area

diffraction patterns

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 81

Recording SADPs

Orientate your specimen by tilting
- focus the beam on specimen in image mode, select diffraction

mode and use “Kikuchi” lines to navigate reciprocal space
- or instead use contrast in image mode e.g. multi-beam zone axis corresponds

to strong diffraction contrast in the image

In image mode, insert chosen selected-area aperture;
spread illumination fully (or near fully) overfocus to obtain parallel beam

Select diffraction mode; focus diffraction spots using diffraction focus

Choose recording media:
- if CCD camera, insert beam stopper to cut out central, bright beam to avoid detector

saturation (unless you have very strong scattering to diffracted beams)
- if plate negatives, consider using 2 exposures: one short to record structure near central, bright

beam; one long (e.g. 60 s) to capture weak diffracted beams

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 82

Recording media:
plate negatives vs CCD camera

! no saturation damage ! immediate digital image
! high dynamic range ! linear dynamic range
! large field of view " small field of view

" need to develop, scan negative " care to avoid oversaturatation
" intensities not linear " reduced dynamic range

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 83

Calibrating your diffraction pattern

Plate negatives CCD camera

Record SADP from a known standard -
e.g. NiOx ring pattern

λL = dhklRhkl (D/2)C = dhkl-1

λ: e- wavelength (Å) D: diameter of ring (pixels)
L:“camera length” (mm) C: calibration (nm-1 per pixel)
dhkl-1: reciprocal plane spacing (nm-1)
dhkl: plane spacing (Å)
Rhkl: spot spacing on negative (mm)

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 84

CalibratingOpticalaxis rotation

UnlesEslecytoroun saoruerceusing rotation-corrected TEM (e.g. JEOL 2200FS), you must calibrate rotation
between image and diffraction pattern if you want to correlate orientation with image

Condenser lens Use specimen with clear shape orientation

Specimen Defocus diffraction pattern (diffraction focus/
Objective lens intermediate lens) to image pattern above BFP
Back focal plane/
di raction plane Diffraction spots now discs; in each disc there is an
image (BF in direct beam, DF in diffracted beams

Intermediate Selected area
image 1 aperture

Intermediate lens

Projector lens BF image (GaAs nanowire) Defocus SADP

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 85

Di raction

Analysing your diffraction pattern

Calculate planes spacings for
lower index reflections (measure

across a number and average)

Measure angles between planes

Compare plane spacings e.g. with
XRD data for expected crystals

Identify possible zone axes using
Weiss Zone Law

Simulate patterns e.g. using JEMS;
overlay simulation on recorded data

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 86

Indexing planes example

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 87

Indexing planes example

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 88

Indexing planes example

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 89

Quantitative electron
diffraction

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 90

Disadvantages of conventional SADP

" lose higher symmetry information
(projection effect;“2D” information; intensities not kinematical)

" dynamical intensity hard to interpret

" poor measurement accuracy of lattice parameters (2-3%)

Can solve with:

! higher order Laue zones: “3D” information

! advanced CBED: higher order symmetry, accurate lattice parameter
measurements, interpretable dynamical intensity

! electron precession:“kinematical” zone axis patterns

=> full symmetry/point group, space group determination; strain measurements;
polarity of non-centrosymmetric crystals; thickness determination; ...

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 91

Higher-order Laue Zones

ZOLZ: hU + kV + lW = 0 92
FOLZ: hU + kV + lW = 1
SOLZ: hU + kV + lW = 2

...

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Higher-order Laue Zones

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 93

Advanced CBED

Patterns from dynamical scattering in direct and diffraction discs allow determination of:
- polarity of non-centrosymmetric crystals

- sample thickness

JEMS simulation: GaN [1 -1 0 0] zone axis Simulation vs experiment:

t = 100 nm

t = 150 nm

t = 200 nm

t = 250 nm T. Mitate et al. Phys. Stat. Sol. (a)
192, 383 (2002)

000-2 0000 0002

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 94

HOLZ lines in CBED

Positions of “Kikuchi” HOLZ lines in direct CBED beam very sensitive to lattice parameters
=> use for lattice parameter determination with e.g. 0.1% accuracy, strain measurement

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 95

HOLZ lines in CBED

Because HOLZ lines contain 3D information, they also show true symmetry
e.g. three-fold {111} symmetry for cubic

- unlike apparent six-fold axis in SADP or from ZOLZ Kikuchi lines

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 96

HOLZ lines in CBED

Energy-filtered imaging mandatory for good quality CBED pattern
- e.g. Si [1 0 0] below taken with new JEOL 2200FS

Unfiltered Filtered

Images by Anas Mouti, CIME 97

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL

Precession electron diffraction

Tilt beam off zone axis, rotate => hollow-cone illumination
“Descan” to reconstruct “pointual” diffraction spots => spot pattern with moving beam

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 98

Precession electron diffraction

Because beam tilted off strong multi-beam axis, much less dynamical scattering

=> Multi-beam zone axis diffraction with “kinematical” intensity

Precession pattern shows higher order symmetry lost in conventional SADP

Precession pattern also much less sensitive to specimen tilt
- can try on the CM20 in CIME!

Images from www.nanomegas.com

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 99

Large angle CBED (LACBED)

Bragg and HOLZ lines superimposed on defocus image - use for:
- Burgers vector analysis: splitting of lines by dislocations

- orientation relationships: lines continuous/discontinuous across interfaces
- ...

Duncan Alexander: Principles & Practice of Electron Diffraction November 2010, EPFL 100