Micro Diffraction

$: Micro Diffraction$
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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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] [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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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] [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.

Micro Diffraction

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