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Principal of EBSDGeneral View of EBSDEBSD Pattern GenerationCrystal systems and their symmetry

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Variety of materials are used around us ….

• Strength• stiffness• Density/weight• Hardness• Workability• Conductivity• Cost• Reliability• Safety• ・・・・• ・・・・• ・・・・

What is the criteria to select materials?

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What decide the property of material?

Composition/Distribution

Crystal structure/Phase

Grain size/shapeOrientationBoundary

Deformation / Heat treatments

Elements Texture

Element analysis?Texture measurement?Property measurement?・・・・

Property of materials are decided by combination

of these factors.

How we can analyze them????

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Strong Orientation

Weak Orientation

Different Property due to orientation

Because materials properties are “anisotropic”. We all know that the strength of wood varies with the direction of the grain.

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Before the fireFire at the tall building in Spain2005/02/12

Can we use this iron frame for reconstruction of this building?

Example of fire accident

After the fire

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Different property due to texture

If microstructure is different, the materials show different properties.

F

Rolled texture Re-crystallized texture

Isn’t the anisotropy averaged out in a polycrystal?

Not necessarily. A material will be isotropic if all of the grains have random orientations. But if the grains have similar orientations then the bulk material will exhibit anisotropy similar to the constituent crystals. The distribution of crystal orientations is called texture. Most forming processes produce materials with texture.

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Electron Beam DiffractionGeneration of EBSD Patterns

EBSD Pattern

Electron backscatter diffraction patterns (or EBSPs) are obtained in the SEM by focusing a stationary electron beam on a crystalline sample. The sample is tilted to approximately 70 degrees with respect to the horizontal. The diffraction pattern is imaged on a phosphor screen. The image is captured using a low-light CCD camera. The bands in the pattern represent reflecting planes in the diffracting crystal volume. Thus, the geometrical arrangement of the bands is a function of the orientation of the diffracting crystal lattice.

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Electron Diffraction

There are 4 ways to get Electron Diffraction patterns.

SEM TEM

ECP EBSP Spot Diffraction Kikuchi Pattern

Bulk Sample Thin film Sample

Bragg’s Law

d

nλ = 2d sin θB

λ

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

2212

cmeVeVm

h

oo

λ

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As shown in right figure, bands in EBSD pattern corresponds crystal planes.

Lattice plane

Weaker beam

Stronger beam

Crystal Plane

Bright line

Dark line

Intensity distribution of inelastic scattered electrons

Electrons come out from the sample surface

Band formation with Kikuchi patterns

Incident beam

Red arrows show the electrons come out from sample surface.

Sample surface

Generation of EBSD Patterns

Crystal planes Sample surface

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Diffracted electrons propagate as corn shape. Then the bands in EBSD pattern become hyperbola lines. EBSD patterns show the real lattice.

Generation of EBSD Patterns

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EBSD Patterns at different Acc. Voltage

Only band width in EBSD patterns change depending on Acc. Voltage.

10kV 40kV

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Kikuchi Pattern from sample surface

Inelastically scattered electrons (Noise)

1μm

Volume to generate EBSD pattern

High spatial resolutionSpatial resolution depends on SEM probe size

Shottky FE-SEM 10~15nmφLaB6 filament SEM 100~200nmφＷfilament S EM 200~300nm

Depth of Information is very shallowdepends on Acc. Voltage and samples

About 30-50 nm~ nearly one extinction distance

Generation of EBSD Patterns

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Crystal system and symmnetry

a

b

c

ab

c

Unit Cell – Lattice Parametersa, b & c – Vectors defining the unit cell (crystallographic axes)

a, b & c are the lengths of these vectors and α, β & γ are the angles between them. Together these form the set of lattice constants or parameters.

αβ

γ

Crystal plane: A set of lattice points that lie in one plane.

Crystal direction: A set of lattice points that lie along a line.

Crystal axes: The reference vectors that define the unit cell.

OA

ba

c2

a

b

c

OA 1, 1 & 1/2

uvw = [221][uvw] direction

<uvw> set of crystallographicallyequivalent directions

[110][100]

[101]

[001] [112] [111]

[021]

[010] a2

a1

a3

c

[1213]

[1100]

[1120]

Directions in the Crystal

Rule u v t w = indices = integers t = -(u+v) (All of a1, a2, a3 and c must be used.)

Indices in hexagonal and trigonal crystals

a2

a3

a1

<0, 1½, -1½, 0>

<0, 1, -1, 0, 0>

<-1 2 -1 0>

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(1 0 -1 0)<-1 2 -1 1>(1 0 -1 0)<-1 2 -1 3>

<0 0 0 1><0 0 0 1>

<-1 2 -1 3>

(1 0 -1 0)<-1 2 -1 1>

<-1 2 -1 1>

Indices in hexagonal and trigonal crystals

Lattice Planes(hkl) Miller indices of a plane

{hkl} set of symmetrically equivalent planes

(1210)

(010) (110)

(111) (112)

b/k

c/l

a/h

(1011) (1100)

Interplanar or d-Spacing

d010 d020

dhkl = a(h2 + k2 + l2)-1/2

(for cubics)

λ = 2dhklsinθ

dhkl = V[h2b2c2sin2α + k2a2c2sin2β + l2a2b2sin2γ+ 2hlab2c(cosαcosγ – cosβ)+ 2hkabc2(cosαcosβ – cosγ)+ 2kla2bc(cosβcosγ – cosα)]-1/2

V = abc[1 - cos2α - cos2β - cos2γ 2cosαcosβcosγ]1/2

Lines of lowest indices have the greatest spacing and density of lattice points

ab

(11)

(21)

(10)

(15)

2 dimensional examples

Band width is inversely proportional to d-spacing

111222

333

Color (hkl) d-spacing

——— 111 2.31

——— 200 2.00

——— 220 1.41

——— 311 1.21

——— 331 0.92

——— 042 0.89

Zone Ruleu = k1l2 – l1k2v = l1h2 – h1l2w = h1k2 – k1h2

Zone Axis – The line of intersection of two planes is the zone axis of the zone in which the two planes are located

Weiss Zone Lawhu + kv + lw = 0

Zone Axes

[002]

Structure Factor – Kinematic Calculation

Use space group information to locate atom positions in the unit cell.

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Structure Factor

In case of Cubic/FCC :Atom positions in Unit cell: (0,0,0),(½,½,0),( 0, ½,½), (½,0, ½)

)](cos)(cos)(cos1[ hllkkhfFhkl ++++++= πππ

When all h, k, l becomes even or odd, Fhkl has a value of 4 and make visible bands. All other cases, the intensity becomes 0 and bands are not visible. For example, FCC structure has (220)or (111) bands, but not (211).

In case of Cubic/BCC :Atom positions in Unit cell: (0,0,0),(½,½,½)

)](cos1[ lkhfFhkl +++= πWhen summation of h, k, l becomes even, Fhkl has a value of 2

and make visible bands. For example, BCC structure has (220)or (211) bands, but not (111) or (311) bands.

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Classification of Crystals

Crystal Lattice• Lattice :

An array of points in space arranged such that each point has identical surroundings

• Bravais Lattice:The unique lattice which can be build by translation and point symmetry in 3 di

mension. They are based on 7 crystal systems and its extension. Total number of Bravais lattice is 14 of 3 dimensional lattices.

• Point Group: Point group is defined by symmetry operation which holds consistency around

one point such as Rotation, Mirror and Inversion. Point group is classified into 32 groups.

• Space Group: Adding Screw and Glide operation to Point group makes whole 230 Space grou

p. Any crystals except quasi crystal are classified into one of these crystal groups.

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Crystal systems Rotation Lattice Parameter Relationships

Triclinic No rotation symmetry a ≠ b ≠ c α ≠ β ≠ γ

Monoclinic One ‘2 holds’ rotation a ≠ b ≠ c α = γ = 90° < β

OrthorhombicThree ‘2 holds’ rotations normal to each other

a ≠ b ≠ c α = β = γ = 90°

Tetragonal One ‘4 holds’ rotations a = b ≠ c α = β = γ = 90°

Rhombohedral One ‘3 holds’ rotations a = b = c α = β = γ < 120°⎯90 °

Hexagonal One ‘6 holds’ rotations a = b ≠ c α = β = 90°, γ = 120°

Cubic Four ‘3 holds’ rotations. a = b = c α = β = γ = 90

Seven Crystal Systems

Seven Crystal Systems are defined by rotation symmetry.

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ab

cαβ

γ

Triclinic

Only translation is available

No rotation symmetry

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Lattices with perpendicular axises

ab

c β

OrthorhombicMonoclinic

ab

c

a

c

TetragonalCubic

One ‘2 holds’ rotation Three‘2 holds’

rotations

One ‘4 holds’ rotation Four ‘3 holds’

rotations

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a

αα α

aa

Lattices with 3 holds rotation axis

RhombohedralHexagonal

Symmetries in EBSD Patterns

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Seven Crystal systems 32 Point Groups

14 Bravais Lattice

• Lattices delivered from simple lattice by combination of translation and point symmetry

• This Bravais lattices are reflected in the EBSD pattern as structure factors.

• Lattices delivered by rotation, inversion , mirror, improper rotation, etc.

• Symmetries we can see in the EBSD patterns

Symmetry Operation (Rotation)

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1 hold rotation(2π)

Symbol for this rotation operation is ‘Number’

2 holds rotation(π)

3 holds rotation(2π/3)

4 holds rotation(π/2)

6 holds rotation(π/3)

Symmetry Operation (Mirror plane)

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Green plane is a mirror plane

Symbol for this mirror operation : ‘m’

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Bring the points to opposite position with respect to the center.

(x, y, z)(–x, –y, –z)

Symbol for this Inversion operation : ‘-’

Symmetry Operation (Inversion)

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対称操作

反転および鏡映操作の拡張として次の操作も定義されている。

回反操作(n)•回転後に反転の操作

回映操作(n/m)•回転後に回転軸に垂直な面を鏡映面として鏡映の操作

_

回反操作 回映操作

Seven Crystal SystemsCrystal Systems (32)

Rotation AxesMirror Planes

Center of Inversion Lattice Parameter Relationships

2 3 4 6

Triclinic (2) - - - - - yes a ≠ b ≠ c α ≠ β ≠ γ

Monoclinic (3) 1 - - - 1 yes a ≠ b ≠ c α = γ = 90° < β

Orthorhombic (3) 3 - - - 3 yes a ≠ b ≠ c α = β = γ = 90°

Trigonal* (5) (Rhombohedral) 3 1 - - 3 yes a = b = c α = β = 90°, γ = 120°

Tetragonal (7) 4 - 1 - 5 yes a = b ≠ c α = β = γ = 90°

Hexagonal (7) 6 - - 1 7 yes a = b ≠ c α = β = 90°, γ = 120°

Cubic (5) 6 4 3 - 9 yes a = b = c α = β = γ = 90°

The conventional cell of the hexagonal system is frequently used for trigonal crystalsThe system with the highest symmetry in each class is shown (see webmineral.com)

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Symmetry in Material database

OIM-DC / Phase pageHerman Morgan expressionThis is Hexagonal, then

The 1st index: around <0001> axisThe 2nd index: around<10-10>axisThe 3rd index: around<1-210>axis

More explanations….

6/m : <0001> axis is 6holds rotation symmetry and the plane normal to this axis is mirror plane.

m : The plane normal to this <10-10> axis is mirror plane.

m : The plane normal to this <1 -2 1-10>axis is mirror plane.

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Symmetry axis for Herman-Morgan expression

Crystal Systems 1st index 2nd Index 3rd Index

Triclinic None

Monoclinic[010] in case of b-axis normal[001] in case of c-saix normal

Orthorhombic [100] [010] [001]

Tetragonal [001][100] [1-10][010] [110]

Trigonal* (Rhombohedral)

Hexagonal axis

[0001][10-10], [01-1-1]

[-1-120]Rhombohedral axis

[001][1-10], [01-1]

[-101]

Hexagonal [0001][10-10] [1-100][01-10] [12-30][-1-120] [-2-130]

Cubic [100] [111] [110]

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7つの結晶系で示された同じ結晶系を持つ場合でも異なる原子配列が可能な場合がある。例えば立方晶では次に示す３つの格子はすべてm3mという対称性を示すが、これらは見方を変える等によって同じ格子とみなすことはできない格子である。

このように並進対称性と点対称性のみの組合せで得られるユニークな独立した14種

類の格子が得られるがこれをブラベー格子という。ブラベー格子には上記のように同じ結晶系同じ点群に属するもので異なる原子配置を持つ格子を定義している。

Bravais Lattice

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Bravais Lattice

a b

c

Triclinic (P)

c

a b

SimpleOrthorhombic (P)

c

a b

Body-CenteredOrthorhombic (I)

c

a b

Base-CenteredOrthorhombic (C)

c

a b

Face-CenteredOrthorhombic (F)

aa

a

Face-CenteredCubic (F)

aa

a

SimpleCubic (P)

aa

a

Body-CenteredCubic (I)

aa

c

SimpleTetragonal (P)

c

a a

Body-CenteredTetragonal (I)

a a

a

ααα

Rhombohedral(R)

120

aa

c

Hexagonal(P)

ab

c

SimpleMonoclinic (P)

ab

c

Base-CenteredMonoclinic (C)

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Symmetry in EBSD Pattern

How OIM uses symmetry info of the crystal to index this pattern?

Mirror plane

4 holds rotation

2 holds rotation

OIM itself doesn’t consider the information of crystal symmetry when it index EBSD patterns. Information of crystal symmetry is considered when it build the material files for indexing the patterns.

After indexing the patterns, crystal symmetry information is also considered to calculate equivalent orientations.

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Set-up material file(s)(Symmetry, Lattice parameter and reflectors)

1. All equivalent crystal planes are listed. 2. Angles between crystal planes are calculated.

Capture EBSD Patterns.Detecting Bands

Indexing of detected bands List up inter planer angles and compare to look-up table which delivered from material file(s). Mirror indices of the band is decided by angular relations.

Calculate crystal orientation

Consider all equivalent crystal planes

OIM Analysis software calculate relations of equivalent crystal planes for orientation maps, plots and charts.

OIM-DC

OIM-Analysis

Indexing procedure of EBSD patterns

Unfortunately, crystal symmetry is not considered

in this process.

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Present situation of EBSD pattern indexing

Crystal symmetry information is used to calculate angular relations of equivalent crystal planes. When OIM indexes detected bands, crystal symmetry information is not considered. It just check the angular relations among bands.

Left example is the pattern from Ni-Sn alloy. Possible phases are listed as below.

Ni3Sn4 : Monocrinica = 12.2, b = 4.056, c = 5.21α = 90゜ β = 105.05゜γ = 90゜

NiSn : Hexagonala = 4.09, b = 4.09, c = 5.18α = 90゜ β = 90゜ γ = 120゜

Ni4Sn : Tetragonala = 5.11, b = 5.11, c = 4.88α = 90゜ β = 90゜ γ = 90゜

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Seeing the results of indexing, all three phase look reasonably OK. If we consider , Vote value, Fit value and CI value, NiSn (Hexagonal) looks most likely phase.

Ni3Sn4 NiSn Ni4Sn

Ni3Sn4 NiSn Ni4Sn

Vote 112 120 112Fit 0.71 0.8 0.91CI 0.000 0.075 0.075

Present situation of EBSD pattern indexing

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Let’s check detail of indexing when it is indexed as NiSn hexagonal. If this pattern is a hexagonal pattern, the green lines shown by red and yellow arrows are all equivalent crystal planes. So the intensity should be nearly same.

The band shown by yellow arrow is missing Then it is not appropriate to consider this pattern as hexagonal.

Present situation of EBSD pattern indexing

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Same as Hexagonal, if this pattern is indexed as Tetragonal, the bands shown by red and yellow arrows are equivalent bands. So the intensity of the bands should be nearly same. But it doesn’t look like that.

The bands shown by yellow arrow is much weaker .

Then it is difficult to consider it as Tetragonal patterns.

Present situation of EBSD pattern indexing

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Summary

EBSD Pattern shows real lattice, not reciprocal lattice. So we can see the symmerty of the crystal directly in the patterns.

We can distinguish 7 crystal systems and 14 Bravasis lattice from EBSD patterns.

It is effective to check intensity of equivalent bands to distinguish similarly indexed patterns.