Microdiffraction
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Microdiffraction
Patricia Muñoz Alonso
Master in materials engineering
Structural Characterization of Materials I: Microscopy and Diffraction
Microdiffraction
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‐ Introduction
Electron diffraction is a mighty method for studying the structure of materials. In a transmission
electron microscope (TEM), electrons penetrate a thin specimen and it is therefore possible to
form a transmission electron diffraction pattern from electrons that have passed through a thin
specimen.
The diffracted electrons are focused by means of electromagnetic lenses into a regular disposal
of diffraction spots (the electron diffraction pattern).
The diffraction patterns are formed in the reciprocal space, while the image plane is formed at
the real space. The transformation from the real space to the reciprocal space is given by the
Fourier transform, so a diffraction pattern is a Fourier transform of the periodic crystal lattice,
giving us information on the periodicities in the lattice and atomic positions
If a selected area aperture is inserted and the parallel incident beam illumination is used, a
diffraction pattern from a specific area as small as 100 nm in diameter is obtained. This mode is
called selected area diffraction, SAED.
The minimum area is limited owing to the spherical aberration of the objective lens with
selected area diffraction technique.
There is other diffraction mode, Convergent beam electro diffraction, CBED, where the area for
diffraction is chosen by focusing the incident beam into a very fine spot (2nm) on the region of
interest. There is another mode of electron diffraction called microdiffraction, in which the
angle of incidence is in between that of SAD and CBED.
The difference between microdiffraction and CBED is the convergence angle. In
microdiffraction the angle is very small (<0.01º).
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The angle of convergence, α, is proportional to the diameter of the diffraction disc in the
diffraction pattern so SAD diffraction pattern consists of a set of spots, while microdiffraction
gives a set of small discs and the CBED pattern consists of a set of discs bigger than
microdiffraction.
Condensorlens
Specimen
Objective lens
Back focal plane
Image plane
Intermediatelens
Intermediatelens
Beamconvergenceα about 0.01º
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The diffraction event can be described in reciprocal space by the Ewald sphere creation. A
sphere with radius 1/λ is drawn through the origin of the reciprocal lattice. Now, for each
reciprocal lattice point that is located on the Ewald sphere of reflection, the Bragg condition is
satisfied and diffraction occurs. The observed diffraction pattern is the part of the reciprocal
lattice that is intersected by the Ewald sphere. The zone axis of a given diffraction pattern is the
reciprocal space vector normal to any of the reciprocal lattice vectors in the pattern.
We can control the sphere because the radius is connected to the wavelength of the electron
beam, controlled by the energy we put into the beam (kV). At a zone-axis orientation, the
reflections in the diffraction pattern break up into zones called Laue zones. The central zone is
called the zero-order Laue zone.
SAED Microdiffraction CBED
Specimen
Objective lens
Diffractionpattern
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ZOLZ: Zero Order Laue Zone
FOLZ: First Order Laue Zone
SOLZ: Second Order Laue Zone
When the selected area diffraction method is chosen, the angular view of the back focal plane of
the objective lens is usually restricted to the ZOLZ. We can see the points that intersect in the
other layers if the collection angle is large. These layers produce outer rings known as higher
order Laue zone rings (HOLZ).
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Microdiffraction patterns
The information present on microdiffraction patterns is used to identify the crystal system, the
orientation of the crystal with respect to the electron beam, the Bravais lattice and the glide
planes of the structure.
When poor CBED patterns are obtained because the samples are composed of small particles or
crystal or large lattice parameter, it is necessary to improve the angular resolution and reduce
the diffuse scattering in the diffraction pattern. We can do that by using microdiffraction
technique.
Microdiffraction pattern gives the "net" symmetry of the zero-order Laue zone (ZOLZ) and the
"net" symmetry of the whole pattern (WP) ZOLZ + HOLZ patterns, this allows the
determination of the Laue class and in consequence of the crystal system.
In addition, the possible shift between the ZOLZ and the FOLZ patterns is connected with the
Bravais modes, the possible periodicity difference between the ZOLZ and the FOLZ pattern is
connected with the presence of glide planes.
These crystallographic features are simply and reliably obtained, in a methodical manner, from
a few patterns by means of tables and theoretical patterns established for each crystal system.
-Identification of the crystal system
The "'net" symmetry of the reciprocal lattice depends on the crystal system so the "net"
symmetry of microdiffraction patterns is used to identify the crystal system.
The net symmetry only takes into account the position of the reflections on the pattern (the
intensity is not considered). The identification is made looking the microdiffraction pattern with
the highest "net" symmetry. There are the next types of symmetry: 1 2 3 4 6 m. 2mm 3m 4mm
6mm.
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The "net" symmetries of the microdiffraction patterns are directly connected to the crystal
system as indicated in table below and the corresponding zone axis by means of the next Table.
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"Net symmetry
whole
pattern ZOLZ
6mm (6mm) [0001]
3m (6mm) <111> [0001]
4mm (4mm) [001] <001>
2mm (2mm) [001] <100> <110> <11⎯20>
[010] <110>
<001> for
Pa3 <1⎯100>
[100]
m (2mm) [u0w] [u0w] <u0w> <uv0> [u⎯u 0 w] [uvt0]
[0vw] [uv0] <uvw> <1 1⎯2 0> [u⎯u 0 w]
[uv0] [uvw] [u u⎯2⎯u w]
2 (2)2 [010]
1 (2)2 [uvw] [uvw] [uvw] [uvw] [uvw] [uvtw] [uvtw]
Crystal system Tri. Mono Ortho Tetra Cubic hR rHombohedral hP hexagonal
Bravais lattice Bravais lattice
Trigonal Hexagonal
-Identification of the Bravais lattice and identification of the glide planes.
There is a comparison between the experimental patterns equivalent to these specific zone axes
with theoretical patterns drawn for all the possible Bravais lattices and for all the possible glide
planes found in the 230 space groups. The Bravais lattice, the nature and the orientation of the
glide plane and a partial extinction symbol are indicated on each theoretical drawing.
Bravais lattices with centering (F, I, A, B, C) have planes of lattice points that give rise to
destructive interference for some orders of reflections as a result, produce typical shifts between
the ZOLZ and the FOLZ reflection nets for the for the specific principal zone axes given in
table below.
The relative spacing of reflections in the HOLZ as compared with those in the ZOLZ which
provides information about the presence of glide planes
Kinematical forbidden reflections are produce by glide planes produce, except for on the
particular zone axes which are exactly perpendicular to the glide planes. When the forbidden
reflections are really absent, they can be without a doubt distinguished from the permitted
reflections. We observe a typical periodicity difference between the ZOLZ and the FOLZ
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reflection nets on the microdiffraction patterns. The smallest rectangle or square with sides
parallel to the "net" mirrors is drawn in the ZOLZ and in the FOLZ to see this difference.
These rectangles give information about the ZOLZ/FOLZ periodicity difference connected with
the presence of glide planes.
The zone axes which allow one to observe the ZOLZ/FOLZ periodicity differences are given in
table
Crystal system Mono.
Unique axis b ortho. Tetragonal Cubic Hexagonal Trigonal
Zone axes
required for
identification of
the Bravais
lattice
[0 1 0] [1 0 0] [0 0 1] <001> P only P and R only
or or and
[0⎯1 0] [0 1 0] <110>
or for
[0 0 1] cI and cP
Zone axes
required for
simultaneous
identification of
the Bravais
lattice and glide
planes
[0 1 0] [1 0 0] [0 0 1] <001> <1 1⎯2 0> Rhombohedral Hexagonal
or or and and and Bravais Bravais
[0⎯1 0] [0 1 0] <100> <110> <1⎯1 0 0> lattice lattice
or and <1 1⎯2 0> <1 1⎯2 0>
[0 0 1] <110> and
<1⎯1 0 0>
‐ Identification of the partial extinction symbol.
If we want to identify of the Bravais lattices and the glide planes, it is necessary the observation
of the same zone axes. For each of the crystal systems, the theoretical microdiffraction patterns
for all possible Bravais lattices and for all possible glide planes are drawn. Additionally a
comparison between experimental and theoretical patterns is made and each theoretical pattern
gives us an individual partial extinction symbol introduced by Buerger.
Depending on the crystal system, one, two or three required zone axes leads to the partial
extinction symbol. The resulting symbol is in agreement with a few possible space groups listed
in table 3.2 of the International Tables for Crystallography.
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‐ Determination of the point group and final deduction of possible space groups
The "ideal" symmetry of a microdiffraction pattern is connected with the point group and a
strategy to identify the point group from microdiffraction pattern is proposed in tables.
Example
Identification of the nitride γ′-Fe4N
It is possible to get a similar structure to the perlite in Fe-C in Fe4N. To do that Fe–N binary
specimens have to heat at 840ºC in nitrogen atmosphere and then cool slowly to obtain the α-
ferrite + γ′-Fe4N pearlitic microstructure. To identify the crystal structure of the γ′-Fe4N nitride
electron microdiffraction technique has been used. The TEM used operated at 120 kV.
Steps:
‐ Determine the "'net" symmetry from the ZOLZ and HOLZ at principal axes to deduce
the crystal system.
‐ Investigate the ZOLZ/FOLZ shift and periodicity differences to get the Bravais lattice
and to reveal the glide planes.
‐ Use the "ideal" symmetry to identify the point group.
‐ Deduce the space group or a set of space groups with the information we have. The
method requires a very limited number of patterns, and the crystallographic data are
identified comparing with the theoretical patterns and tables given for each crystal
system.
Procedure:
The “net” symmetries for the Zero Order Laue Zones (ZOLZ) recorded along 〈001〉 and
〈111〉 zone axes are (4 mm) and (6 mm) respectively. These “net” symmetries correspond to
a cubic system;
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(4mm) (6mm)
〈001〉and 〈111〉 electron microdiffraction ZAPs for the γ′-Fe4N nitride
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Crystal system Mono.
Unique axis b ortho. Tetragonal Cubic Hexagonal Trigonal
Zone axes
required for
identification of
the Bravais
lattice
[0 1 0] [1 0 0] [0 0 1] <001> P only P and R only
or or and
[0⎯1 0] [0 1 0] <110>
or for
[0 0 1] cI and cP
Zone axes
required for
simultaneous
identification of
the Bravais
lattice and glide
planes
[0 1 0] [1 0 0] [0 0 1] <001> <1 1⎯2 0> Rhombohedral Hexagonal
or or and and and Bravais Bravais
[0⎯1 0] [0 1 0] <100> <110> <1⎯1 0 0> lattice lattice
or and <1 1⎯2 0> <1 1⎯2 0>
[0 0 1] <110> and
<1⎯1 0 0>
The shift and the periodicity difference between the ZOLZ and FOLZ (First Order Laue Zone)
reflection nets along slightly tilted 〈001〉 and 〈011〉 zone axes are related to the P– – –
extinction symbol.
Electron microdiffraction patterns along 〈001〉 and 〈011〉 showing ZOLZ and HOLZ areas
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The “ideal” ZOLZ symmetries recorded along 〈001〉, 〈011〉 and 〈111〉 zone axes are
(4 mm), (2 mm) and (6 mm), respectively. These “ideal” symmetries lead to the m⎯3 m point
group.
the space group for the present nitride is P m⎯3 m
Example 2
χ Phase present in a duplex austenitic-ferritic stainless steel as small particles with an average
size of about 0.1 /μm is observed with the electron microscope at 40 V potential with a small
convergence angle.
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[001] zone axis microdiffraction pattern with (4mm), 4mm "net" and (4mm), 2mm "ideal" symmetries.
Zone axis microdiffraction pattern with (6mm), 3m "net" symmetries.
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[011] zone axis microdiffraction pattern with (2mm), 2mm "net" symmetries.
The highest "net" symmetries observed for this phase are (4mm), 4mm and (6mm), 3m which,
according to Table correspond to a cubic system.
The specific ZAPs to observe for Bravais lattice and glide plane identifications are :
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Crystal system Mono.
Unique axis b Ortho. Tetragonal Cubic Hexagonal Trigonal
Zone axes
required for
identification of
the Bravais
lattice
[0 1 0] [1 0 0] [0 0 1] <001> P only P and R only
or or and
[0⎯1 0] [0 1 0] <110>
or for
[0 0 1] cI and cP
Zone axes
required for
simultaneous
identification of
the Bravais
lattice and glide
planes
[0 1 0] [1 0 0] [0 0 1] <001> <1 1⎯2 0> Rhombohedral Hexagonal
or or and and and Bravais Bravais
[0⎯1 0] [0 1 0] <100> <110> <1⎯1 0 0> lattice lattice
or and <1 1⎯2 0> <1 1⎯2 0>
[0 0 1] <110> and
<1⎯1 0 0>
On the (001) ZAP, two sets of perpendicular "net" mirrors ml, m 2 and m'1, m' 2 are recognized.
The smallest squares drawn in the ZOLZ and in the FOLZ have their sides parallel to the m1,
m2 mirrors and they are equal. FOLZ reflections are present on the m'1, m'2 mirrors but absent
on ml, m2 mirrors. The resultant partial extinction symbols are I -.o or F--.
The (110) ZAP shows (2mm), 2mm "net" symmetries. There are not FOLZ reflections the two
perpendicular "net" m1 and m2 mirrors and the two rectangles with sides parallel to the mirrors
drawn in the ZOLZ and in the FOLZ are identical. The individual partial extinction symbol I -
So the partial extinction symbols leads to I---
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The point group is identified from observation of the (001) ZAP "ideal" symmetry as indicated
in Table. This (001) pattern shows a (4mm) "ideal" ZOLZ symmetry and WP "ideal" symmetry
is 2mm. Looking to Table the point group matching to (4mm), 2mm "ideal" symmetries is 43m.
The space group of this χ phase is I⎯4 3 m.