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Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Principles & Practice of Electron Diffraction
1
Duncan AlexanderEPFL-CIME
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Contents
2
Introduction to electron diffraction
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 CIME, EPFL
Introduction to electron diffraction
3
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Diffraction: constructive and destructive interference of waves
! electrons interact very strongly with matter => strong diffraction intensity (can take patterns in seconds, unlike X-ray diffraction)
" diffraction from only selected set of planes in one pattern - e.g. only 2D information
! 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)
! spatially-localized information(≳ 200 nm for selected-area diffraction; 2 nm possible with convergent-beam electron diffraction)
! orientation information
! close relationship to diffraction contrast in imaging
! immediate in the TEM!
" limited accuracy of measurement - e.g. 2-3%
" intensity of reflections difficult to interpret because of dynamical effects
Why use electron diffraction?
4
(" 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 CIME, EPFL
Insert selected area aperture to choose region of interest
Optical axis
Electron source
Condenser lens
Specimen
Objective lens
Back focal plane/di!raction plane
Intermediate lens
Projector lens
Image
Intermediate image 1
Optical axis
Electron source
Condenser lens
Specimen
Objective lens
Back focal plane/di!raction plane
Intermediate lens
Projector lens
Image
Intermediate image 1
Selected areaaperture
BaTiO3 nanocrystals (Psaltis lab)
Image formation
5
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Press “D” for diffraction on microscope console - alter strength of intermediate lens and focus
diffraction pattern on to screen
Optical axis
Electron source
Condenser lens
Specimen
Objective lens
Back focal plane/di!raction plane
Intermediate lens
Projector lens
Image
Intermediate image 1
Selected areaaperture
Optical axis
Electron source
Condenser lens
Specimen
Objective lens
Back focal plane/di!raction plane
Intermediate lens
Projector lens
Di!raction
Intermediate image 1
Selected areaaperture
Find cubic BaTiO3 aligned on [0 0 1] zone axis
Take selected-area diffraction pattern
6
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Elastic scattering theory
7
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Consider coherent elastic scattering of electrons from atom
Differential elastic scatteringcross section:
Atomic scattering factor
Scattering theory - Atomic scattering factor
8
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Atoms closer together => scattering angles greater
=> Reciprocity!
Periodic array of scattering centres (atoms)
k0k0k0k0k0k0k0k0k0
k0
k0
kD1
k0
kD2
k0
kD2k0kD1
Plane electron wave generates secondary wavelets
Secondary wavelets interfere => strong direct beam and multiple orders of diffracted beams from constructive interference
Scattering theory - Huygen’s principle
9
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Basic crystallography & symmetry
10
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Repetition of translated structure to infinity
Crystals: translational periodicity & symmetry
11
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Unit cell is the smallest repeating unit of the crystal latticeHas a lattice point on each corner (and perhaps more elsewhere)
Defined by lattice parameters a, b, c along axes x, y, zand angles between crystallographic axes: α = b^c; β = a^c; γ = a^b
Crystallography: the unit cell
12
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Use example of CuZn brassChoose the unit cell - for CuZn: primitive cubic (lattice point on each corner)
x
y
z
Choose the motif - Cu: 0, 0, 0; Zn: !,!,!
x
y
z
Motif:Cu
Znx
y
z
Structure = lattice +motif => Start applying motif to each lattice point
x
y
z
Motif:Cu
Znx
y
z
x
y
z
Motif:Cu
Znx
y
z
Building a crystal structure
13
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
x
y
z
Motif:Cu
Znx
y
z
Use example of CuZn brassChoose 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
x
y
z
Cu
Zn
x
y
z
Cu
Zn
x
y
z
Cu
Zn
y
x
y
zCu
Zn
y
y
x
y
zCu
Zn
y
y
y
x
y
zCu
Zn
y
y
y
y
Building a crystal structure
14
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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:
Inversion axes: combination of rotation axis with centre of symmetry
Centre of symmetry orinversion centre:
Introduction to symmetry
15
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Example - Tetragonal lattice: a = b ≠ c; α = β = γ = 90°
Anatase TiO2 (body-centred lattice) view down [0 0 1] (z-axis):
x
y
z
O
Ti
x
y
z
O
Ti
Identify mirror planes
x
y
z
O
Ti
x
y
z
O
Ti
x
y
z
O
Ti
x
y
z
O
Ti
x
y
z
O
Ti
x
y
z
O
Ti
Tetrad:4-fold rotationaxis
Mirror plane
Identify rotation axis: 4-fold = defining symmetry of tetragonal lattice!
Introduction to symmetry
16
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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!)
x y
z
x y
z
x y
z
More defining symmetry elements
17
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
(Cubic α-Al(Fe,Mn)Si: example of primitive cubic with no 4-fold axis)
18
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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]
x
yz
120
a
a
Draw 2 x 2 unit cells
Identify rotation axis: 6-fold (hexad) - defining symmetry of hexagonal lattice
x
yz
120
a
a
x
yz
120
a
a
More defining symmetry elements
19
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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
The seven crystal systems
20
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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 latticepoint centred on A, B or C face
Diagrams from www.Wikipedia.org
Four possible lattice centerings
21
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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
14 Bravais lattices
22
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
14 Bravais lattices
23
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
A lattice vector is a vector joining any two lattice pointsWritten as linear combination of unit cell vectors a, b, c:
t = Ua + Vb + WcAlso written as: t = [U V W]
Examples:z
x
y
z
x
y
z
x
y
[1 0 0] [0 3 2] [1 2 1]
Important in diffraction because we “look” down the lattice vectors (“zone axes”)
Crystallography - lattice vectors
24
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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 axisb/k on the y axisc/l on the z axis
Crystallography - lattice planes
25
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Sets of planes intersecting the unit cell - examples:
x
z
y
x
z
y
x
z
y
(1 0 0)
(0 2 2)
(1 1 1)
Crystallography - lattice planes
26
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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 diagonalrelates the planes (1 0 0), (0 1 0), (0 0 1):
x y
z
x y
z
x y
z
x y
z
x y
z
Set of planes {1 0 0} = (1 0 0), (0 1 0), (0 0 1), (-1 0 0), (0 -1 0), (0 0 -1)
Lattice planes and symmetry
27
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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)
Weiss Zone Law
28
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Electron diffraction theory
29
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Path difference between reflection from planes distance dhkl apart = 2dhklsinθ
Electron diffraction: λ ~ 0.001 nmtherefore: λ ≪ dhkl
=> small angle approximation: nλ ≈ 2dhklθReciprocity: scattering angle θ ∝ dhkl
-1
dhkld!
hkld! !
hkld! !
hkld! !
hkld! !
+ =
hkld! !
+ =
hkl
2dhklsinθ = λ/2 - destructive interference2dhklsinθ = λ - constructive interference=> Bragg law:nλ = 2dhklsinθ
Diffraction theory - Bragg law
30
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
d
!
!
hkl
!
!
!
!!
k I
!
!!
k I
k D
k I!
!
0 0 0 Gg
!
k I
k D
k I
2-beam condition: strong scattering from single set of planes
Diffraction theory - 2-beam condition
31
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
x
yz
x
yz
0 0 00 0 0
1 0 0
x
yz
0 0 0
1 0 0
0 1 0
x
yz
0 0 0
1 0 0
0 1 0
1 1 0
x
yz
0 0 0
1 0 0
0 1 0
1 1 0
2 0 0
x
yz
0 0 0
1 0 0
0 1 0
1 1 0
2 0 0 2 2 0
x
yz
Electron beam parallel to low-index crystal orientation [U V W] = zone axisCrystal “viewed down” zone axis is like diffraction grating with planes parallel to e-beam
In diffraction pattern obtain spots perpendicular to plane orientationExample: primitive cubic with e-beam parallel to [0 0 1] zone axis
Note reciprocal relationship: smaller plane spacing => larger indices (h k l)& greater scattering angle on diffraction pattern from (0 0 0) direct beam
2 x 2 unit cells
0 0 0
1 0 0
0 1 0
1 1 0
2 0 0 2 2 0
3 0 0
0 0 0
1 0 0
0 1 0
1 1 0
2 0 0 2 2 0
3 0 0
-1 0 0
0 -1 0
Also note Weiss Zone Law obeyed in indexing (hU + kV + lW = 0)
Multi-beam scattering condition
32
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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:
x
yz
120
a
a
(1 0 0) planes
yz
120
a
a
[1 0 0]
(1 0 0) planes
yz
120
a
a
[1 0 0] g1 0 0
(1 0 0) planes
=> care must be taken in reciprocal space!
Scattering from non-orthogonal crystals
33
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
For scattering from plane (h k l) the diffraction vector: ghkl = ha* + kb* + lc*
rn = n1a + n2b + n3cReal lattice
vector:
In diffraction we are working in “reciprocal space”; useful to transform the crystal lattice in toa “reciprocal lattice” that represents the crystal in reciprocal space:
r* = m1a* + m2b* + m3c*Reciprocal lattice
vector:
a*.b = a*.c = b*.c = b*.a = c*.a = c*.b = 0
a*.a = b*.b = c*.c = 1
a* = (b ^ c)/VC
where:
i.e. VC: volume of unit cell
Plane spacing:
The reciprocal lattice
34
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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 latticeis the Fourier transform of the real lattice
Can use this to understand diffraction patterns and reciprocal space more easily
Fourier transforms for understanding reciprocal space
35
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
radius = 1/!
0
kIkD
C
Reciprocal space: sphere radius 1/λ represents possible scattering wave vectors intersecting reciprocal space
kI: incident beam wave vector
kD: diffracted wave vector
Electron diffraction: radius of sphere very large compared to reciprocal lattice=> sphere circumference almost flat
The Ewald sphere
36
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
kIkD
ghkl
kk
G0 0 0
2!
!
!
0 0 0 Gg
!
k I
k D
k I
2-beam condition with one strong Bragg reflection corresponds to Ewald sphereintersecting one reciprocal lattice point
Ewald sphere in 2-beam condition
37
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
0 0 0
kIkDkk
With crystal oriented on zone axis, Ewald sphere may not
intersect reciprocal lattice points
However, we see strong diffractionfrom many planes in this condition
Assume reciprocal lattice pointsare infinitely small
Because reciprocal lattice pointshave size and shape!
Ewald sphere and multi-beam scattering
38
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Real lattice is not infinite, but is bound disc of material with diameter ofselected area aperture and thickness of specimen - i.e. thin disc of material
X
FT FT
X “Relrod”
= 2 lengths scales inreciprocal space!
Fourier transforms and reciprocal lattice
39
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
0 0 0
kIkDkk
0 0 0
kIkDkk
Ewald sphere intersects Relrods
40
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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:
Relrod shape
41
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Relrod shape
42
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Intensity in the electron diffraction pattern
43
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Tilted slightly off Bragg condition, intensity of diffraction spot much lower
Introduce new vector s - “the excitation error” that measures deviation fromexact Bragg condition
Excitation error
44
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Excitation error
45
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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 beamis at the perfect angle to be Bragg-scattered
again (and again...)
As a result, scattering of different beams is not independent from each other
Dynamical scattering
46
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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:
If the excitation distance s = 0 (i.e. perfect Bragg condition), then:
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
Dynamical scattering for 2-beam condition
47
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL 48
Dynamical scattering in the dark-field image –
=> Intensity zero for thicknesses t = nξg (integer n)
Composition changes in quantum wells=> extinction at different thickness compared to substrate
See effect as dark “thickness fringes” on wedge-shaped sample:
Extinction and thickness fringes
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Dynamical scattering for 2-beam
49
2-beam condition: direct and diffracted beam intensities beams π/2 out of phase:
Bright-field image showing modulation
with absorption:
Model with absorption using JEMS:
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
2-beam: kinematical vs dynamical
50
Kinematical (weak interactions) Dynamical (strong interactions)
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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
Weak beam; kinematical approximation
51
where:
Before we saw for 2-beam condition:
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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:
Structure factor
52
Note fi is a function of s and (h k l)
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Consider FCC lattice with lattice point coordinates:0,0,0; !,!,0; !,0,!; 0,!,!
x
z
y
Calculate structure factor for (0 1 0) plane (assume single atom motif):
x
z
y
=>
x
z
y
x
z
y
Forbidden reflections
53
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
x
z
y
Au
Cu
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)?
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
Patterns simulated using JEMS
Diffraction pattern on [0 0 1] zone axis:
x
z
y
Forbidden reflections
54
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
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)?
x
z
y
Au
CuPatterns simulated using JEMS
Diffraction pattern on [0 0 1] zone axis:
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
Forbidden reflections
55
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Reciprocal lattice of FCC is BCC and vice-versa
Extinction rules
56
Face-centred cubic: reflections with mixed odd, even h, k, l absent:
Body-centred cubic: reflections with h + k + l = odd absent:
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Selected-area diffraction phenomena
57
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Symmetry information
58
Zone axis SADPs have symmetry closely related to symmetry of crystal lattice
Example: FCC aluminium[0 0 1]
[1 1 0]
[1 1 1]
4-fold rotation axis
2-fold rotation axis
6-fold rotation axis - but [1 1 1] actually 3-fold axisNeed third dimension for true symmetry!
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Twinning in diffraction
59
Example: FCC twinsStacking of close-packed {1 1 1} planes reversed at twin boundary:
A B C A B C A B C A B C# A B C A B C B A C B A C
Example: FCC twinsStacking of close-packed {1 1 1} planes reversed at twin boundary:
A B C A B C A B C A B C
View on [1 1 0] zone axis:
{1 1 1} planes:
1 -1 1
0 0 2
1 -1 -1
{1 1 1} planes:{1 1 1} planes:
1 -1 1A
1 -1 -1B
A B
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Twinning in diffraction
60
Example: Co-Ni-Al shape memory FCC twins observed on [1 1 0] zone axis
Images provided by Barbora Bartová, CIME
(1 1 1) close-packed twin planes overlap in SADP
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Epitaxy and orientation relationships
61
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 CIME, EPFL
Crystallographically-oriented precipitates
62
Images provided by Barbora Bartová, CIME
Bright-field image Dark-field image
!"#$%#&'#%()*+,&-./01&2'3)#.),2'+4'!"#"$"%!&'(%51167!89956617-:;:/:<'=>11>1?!899=116?-:;:/:8,@'3)#.),2'+4'!"#"$"%!&'(%51167!89956617-:;:/:<%=>111?!899=116?-:;:/:
Co-Ni-Al shape memory alloy, austenitic with Co-rich precipitates
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Double diffraction
63
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
Images by Quentin Jeangros, EPFL
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Double diffraction
64
Example 1: NiO being reduced to Ni in-situ in TEM movie
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Double diffraction
65
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 patternon [1 1 0] zone axis:
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Ring diffraction patterns
66
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 CIME, EPFL
Ring diffraction patterns
67
Larger crystals => more “spotty” patterns
Example: ZnO nanocrystals ~20 nm in diameter
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Ring diffraction patterns
68
Example: hydrozincite Zn5(CO3)2(OH)6 recrystallised to ZnO crystals 1-2 nm in diameter
“Texture” - i.e. preferential orientation - is seen as arcs of greater intensityin the diffraction rings
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Amorphous diffraction pattern
69
Crystals: short-range order and long-range order
Vitrified germanium (M. H. Bhat et al. Nature 448 787 (2007)
Example:
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
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Inelastic scattering event scatters electrons in all directions inside crystal
Cones have very large diameters => intersect diffraction plane as ~straight lines
Kikuchi lines
70
Some scattered electrons in correct orientation for Bragg scattering => cone of scattering
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Kikuchi lines
71
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 coincidesexactly 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 CIME, EPFL
Kikuchi lines - “road map” to reciprocal space
72
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 CIME, EPFL
Convergent beam electron diffraction
73
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Instead of parallel illumination with selected-area aperture, CBED useshighly converged illumination to select a much smaller specimen region
Convergent beam electron diffraction
74
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 CIME, EPFL
Convergent beam electron diffraction
75
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Convergent beam electron diffraction
76
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Convergent beam electron diffraction
77
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Convergent beam electron diffraction
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Kikuchi from: inelastic scattering convergent beam
“Kikuchi” lines much lessdiffuse for CBED
=> use CBED to orientatesample!
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Convergent beam electron diffraction– practical example
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ZnO thin-film sample;Conditions: convergent beam, large condenser aperture, diffraction mode
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Convergent beam electron diffraction– practical example
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ZnO thin-film sample;Conditions: convergent beam, large condenser aperture, diffraction mode
[1 1 0] zone axis
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Recording & analysing selected-area
diffraction patterns
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Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Recording SADPs
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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 CIME, EPFL
! no saturation damage! high dynamic range and sensitivity
! linear dynamic range! large field of view
" time consuming loading, scanning
Recording media:image plates vs CCD camera
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! immediate digital image! linear dynamic range" small field of view
" care to avoid oversaturatation" reduced dynamic range
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Calibrating your diffraction pattern
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Plate negatives
λL = dhklRhkl
λ: e- wavelength (Å)L: “camera length” (mm)
dhkl: plane spacing (Å)Rhkl: spot spacing on negative (mm)
CCD camera
(D/2)C = dhkl-1
D: diameter of ring (pixels)C: calibration (nm-1 per pixel)
dhkl-1: reciprocal plane spacing (nm-1)
Record SADP from a known standard -e.g. NiOx ring pattern
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Calibrating rotation
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Unless you are using rotation-corrected TEM (e.g. JEOL 2200FS), you must calibrate rotationbetween image and diffraction pattern if you want to correlate orientation with image
Optical axis
Electron source
Condenser lens
Specimen
Objective lens
Back focal plane/di!raction plane
Intermediate lens
Projector lens
Di!raction
Intermediate image 1
Selected areaaperture
Use specimen with clear shape orientation
Defocus diffraction pattern (diffraction focus/intermediate lens) to image pattern above BFP
Diffraction spots now discs; in each disc there is an image (BF in direct beam, DF in diffracted beams
BF image (GaAs nanowire) Defocus SADP
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Analysing your diffraction pattern
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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
Simulate patterns e.g. using JEMS;overlay simulation on recorded data
Identify possible zone axes using Weiss Zone Law
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Indexing planes example
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Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Indexing planes example
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Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Indexing planes example
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Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Quantitative electron diffraction
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Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Disadvantages of conventional SADP
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" 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 CIME, EPFL
Higher-order Laue Zones
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ZOLZ: hU + kV + lW = 0FOLZ: hU + kV + lW = 1SOLZ: hU + kV + lW = 2
...
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Higher-order Laue Zones
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Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Advanced CBED
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Patterns from dynamical scattering in direct and diffraction discs allow determination of:- polarity of non-centrosymmetric crystals
- sample thickness
T. Mitate et al. Phys. Stat. Sol. (a)192, 383 (2002)
Simulation vs experiment:JEMS simulation: GaN [1 -1 0 0] zone axis
t = 100 nm
t = 150 nm
t = 200 nm
t = 250 nm
000-2 0000 0002
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
HOLZ lines in CBED
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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 CIME, EPFL
HOLZ lines in CBED
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Because HOLZ lines contain 3D information, they also show true symmetrye.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 CIME, EPFL
HOLZ lines in CBED
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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
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Precession electron diffraction
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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 CIME, EPFL
Precession electron diffraction
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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 CIME, EPFL
Large angle CBED (LACBED)
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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 CIME, EPFL
Nano-area electron diffraction
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Image the condenser aperture using a third condenser lens
=> nanometer-sized beam with parallel illumination
Zuo et al. Microscopy Research and Technique 64 347 (2004)
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Nano-area electron diffraction
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Electron diffraction pattern from single double-walled carbon nanotube- can determine chirality
Method developed for nano-objects where no dynamical scattering problem but phase is required - therefore need coherent illumination that you do not obtain with CBED
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
Phase/orientation mapping
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NanoMEGAS ASTAR: phase and orientation mapping in TEM– similar to EBSD in SEM but e.g. much higher spatial resolution (~5 nm possible)
Record diffraction patterns as electron probe moved across sample
Analyse diffraction patterns by “template matching”– i.e. correlate to ~2000 patterns simulated at different orientations
Combine with precession and can achieve angular resolution of < 1°
Orientation map for nanocrystalline Cu:Phase map showing local martensitic structure of steel at stacking faults:
Images from NanoMEGAS company literature
Duncan Alexander: Principles & Practice of Electron Diffraction CIME, EPFL
“Transmission Electron Microscopy”, Williams & Carter, Plenum Press
http://www.doitpoms.ac.uk
“Transmission Electron Microscopy: Physics of Image Formation and Microanalysis (Springer Series in Optical Sciences)”, Reimer, Springer Publishing
“Large-Angle Convergent-Beam Electron Diffraction Applications to Crystal Defects”, Morniroli, Taylor & Francis Publishing
References
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http://crystals.ethz.ch/icsd - access to crystal structure file databaseCan download CIF file and import to JEMS
Web-based Electron Microscopy APplication Software (WebEMAPS)http://emaps.mrl.uiuc.edu/
JEMS Electron Microscopy Software Java versionhttp://cimewww.epfl.ch/people/stadelmann/jemsWebSite/jems.html
“Electron diffraction in the electron microscope”, J. W. Edington, Macmillan Publishers Ltd
http://escher.epfl.ch/eCrystallography