Classical Texture Texture Classical Analysis Analysis D. - - PowerPoint PPT Presentation

Classical Classical Texture Texture Analysis Analysis

• D. Chateigner
• D. Chateigner

CRISMAT CRISMAT-

• ENSICAEN; IUT

ENSICAEN; IUT-

• UCBN

UCBN 6 bd. M. Juin 14050 Caen 6 bd. M. Juin 14050 Caen

Outline Outline

Qualitative aspects of crystallographic textures Grains, Crystallites and Crystallographic planes Normal diffraction Effects on diffraction diagrams, their limitations θ-2θ scans Asymmetric scans ω-scans (rocking curves) Representations of texture: pole figures Pole Sphere Stereographic projection Equal-area projection: Lambert/Schmidt projection Pole figures Localisation of crystallographic directions from pole figures Direct and normalised pole figures Normalisation Incompleteness and corrections of pole figures Single texture component Multiple texture components Pole figures and (hkl) multiplicity A real example

Pole figure types Random texture Planar textures Fibre textures Three-dimensional texture Pole Figures and Orientation spaces Mathematical expression of diffraction pole figures and ODF From pole figures to the ODF Orientations g and pole figures Euler angle conventions From f(g) to pole figures Deal with ODF in the space Plotting the ODF Inverse pole figures ODF refinement Generalised spherical harmonics WIMV Entropy modified WIMV and Entropy maximisation ADC, Vector and component methods ODF coverage Reliability and texture strength estimators Why needing Combined analysis

Qualitative aspects of texture Qualitative aspects of texture Polycrystal Polycrystal: : aggregate of grains, different phases, sizes, shapes, orientations … Diffraction: Diffraction:

• probes

probes lattice lattice planes: planes: crystallites crystallites, , not not grains grains

• x

x-

• rays

rays, neutrons or , neutrons or electrons electrons

SEM: SEM:

• grains,

grains, not not crystallites crystallites ( (coherent coherent, single , single crystal crystal domains domains) )

• shape

shape vs vs crystallographic crystallographic texture (EBSD) texture (EBSD)

Grains, Grains, crystallites crystallites, , crystallographic crystallographic planes planes Friedel's law: Ihkl = I-h-k-l using normal diffraction + or - directions not distinguished

[hkl]

• [hkl]

I+ I-

Texture Texture effects effects on diffraction

• n diffraction diagrams

diagrams

θ-2θ scan: probes only parallel planes

MgO

Li0.12La0.88TiO3 random bulk Oriented film Li0.12La0.88TiO3/(100)-MgO 001 002 003

asymmetric scan: probes only inclined planes

mixed scan: probes specific planes for specific orientations

005 006 008

ω scan: probes orientation of only one plane type (fixed θ), only for small ω-θ

17.5 18.0 18.5 500 1000 1500 2000 0.1° Intensity (a.u)

ω(°)

limitations: available θ (or other) range

diamond (Fd3m), 2.52 Å neutrons, up to 2θ = 150° 111

limitations: 2 texture components

same c-axes direction, but not same a

MgO

001 002 003

{ } { }

l l l

l l

00 hk i hk

L 1 hk I I = p : p ; p 1 p

• p

= L = = −

∑ ∑

i

random

limitations: 2 texture components, one inclined

17.5 18.0 18.5 500 1000 1500 2000 Intensity (a.u)

17.5 18.0 18.5 500 1000 1500 2000 0.1° Intensity (a.u)

Representations Representations of texture: pole figures

• f texture: pole figures

One crystallite oriented in the Pole sphere:

• location of all

[hkl] ∈ unit sphere

• dS = sinχ dχ dϕ
• (χ,ϕ): angles in the

diffractometer space Hard to visualise: needs pole figures

Stereographic Stereographic projections: projections: equal equal angle angle

Poles: p(r',ϕ): r’ = R tan(χ/2)

Lambert projections ( Lambert projections (equal equal area) area)

Poles: p(r',ϕ): r’ = 2R sin(χ/2)

30 60 90 120 150 180 210 240 270 300 330 30 60 90 120 150 180 210 240 270 300 330

stereographic Lambert/Schmidt 5° x 5° grid: 1368 points

Pole figures Pole figures

{hkl}-Pole figure: location of distribution densities, for the {hkl} diffracting plane, defined in the crystallite frame KB, relative to the sample frame KA. Pole figures space: , with y = (ϑy,ϕy) = [hkl]* Direct Pole Figure: built on diffracted intensities Ih(y), h = <hkl>* Normalised Pole Figure: built on distribution densities Ph(y) Density unit: the "multiple of a random distribution", or "m.r.d."

Usual Usual pole figure pole figure frames frames K KA

A

Lineation direction

.

ND RD

.

N G

.

nF TD M Foliation plane metallurgy malacology geophysics Thin films: substrate directions … XA, YA, ZA

Normalisation Normalisation

30 60 90 120 150 180 210 240 270 300 330

random

I ) ( I ) ( P

h

y y

h h

=

∫ ∫

= =

=

/2 2 total

d d sin ) , ( I I

π ϑ π ϕ

ϕ ϑ ϑ ϕ ϑ

y y

y y y y y h

h

∫ ∫

= =

=

/2 2 total random

d d sin / I I

π ϑ π ϕ

ϕ ϑ ϑ

y y

y y y

h h

) , ( I

y y ϕ

ϑ

h

• Only valid for complete pole figures:

neutrons in symmetric geometry

• Needs a refinement strategy to get Irandom for all h's

Incompleteness Incompleteness and corrections of pole figures and corrections of pole figures

Missing Bragg peaks Blind area Defocusing (x-rays) Localisation Absorption + volume

2θ-defocusing ω-defocusing χ-defocusing

Defocusing corrections:

• Calibration on a random powder
• Total integration of the peak (direct integration or fit)

90° 0° Intensity

Peak maximum (point detector) Integrated intensity (1D or 2D detector)

bkg rand bkg rand bkg bkg bkg meas rand rand meas cor

, , , , , , , , , , , , , , , , , , , , , , , ,

I I I I I I I

• I

I I I I

θ ω θ ω χ θ ω θ ω θ ω θ ω χ θ ω θ ω χ θ ω χ θ ω θ ω χ θ ω χ

− − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = =

Net intensities (point detector)

χ

t

χ

beam Specific to each instrumental geometry Sample dependent (films, multilayers …) Modifies the defocusing curves Can be integrated in fitting procedures

Absorption/Volume corrections:

( ) ( ) ( ) ( )

χ θ µ θ µ χ cos sin / 2 exp 1 sin / 2 exp 1 ) I( I(0)

i i

T T − − − − =

Top film

( ) ( ) ( ) ( )

) cos sin 2 exp( cos sin / 2 exp 1 ) sin 2 exp( sin / 2 exp 1 ) I( I(0) χ θ µ χ θ µ θ µ θ µ χ

i j j j i i j j j i

T T T T

∑ ∑

− − − − − − =

Covered layer

χ

0° 90° Intensity

Single or multiple texture components, Single or multiple texture components, multiplicity multiplicity

Double Single Tetragonal Cubic

Program Program convention ! convention !

A A real real example example

Cypraea testudinaria Outer aragonite layer Pnma space group

Texture types Texture types

Random texture 3 degree of freedom All Ph(y) homogeneous 1 m.r.d. density whatever y Planar texture 100 001 2 degree of freedom 1 [hkl] at random in a plane

100 001 Fibre texture 1 degree of freedom 1 [hkl] along 1 y direction Cyclic-Fibre texture c // ZA Cyclic-Planar texture (a,b) // (XA,YA)

Single crystal-like texture 0 degree of freedom 2 [hkl]'s along 2 y directions 100 110 Single-crystal and perfect 3D orientation not distinguished

Pole figure and Orientation Pole figure and Orientation spaces spaces

dy y y

h

) ( P 4 1 = V ) dV( π

Pole figure expression:

dy = sinϑy dϑy dϕy π ϕ ϑ ϑ ϕ ϑ

π ϑ π ϕ

4 = d d sin ) , ( P

/2 2 0 ∫

∫

= =

y y

y y y y y h

Orientation Distribution Function f(g): dg (g) 8 1 = V dV(g)

2 f

π

dg = sin(β)dβdαdγ

2 2 2 / 2

8 = dg (g) π

π γ π β π α

∫ ∫ ∫

= = =

f

From From Pole figures to Pole figures to the the ODF ODF

Pole figure: one direction fixed in KA Orientation: two directions fixed in KA

∫

=

y h y //

~ d (g) 2 1 ) ( P

h

ϕ π f

Fundamental Equation of QTA Needs several pole figures to construct the f(g)

Pole figures Pole figures from from g g

• Rotation of KA about the axis ZA through the angle α:

[KA a K'A]; associated rotation g1 = {α,0,0}

• Rotation of K'A about the axis Y'A through the angle β:

[K'A a K"A]; associated rotation g2 = {0,β,0}

• Rotation of K"A about the axis Z"A through the angle γ:

[K"A a K"'A//KB]; associated rotation g3 = {0,0,γ} finally: g = g1 g2 g3 = {α,0,0} {0,β,0} {0,0,γ} = {α,β,γ} X'A Y'A Y'A Z''A Z''A Y'''A

g2 = {45,45,0} g3 = {45,55,45} g1 = {45,0,0}

Matthies Roe Bunge Canova Kocks α Ψ ϕ1 = α + π/2 ω = π/2 − α Ψ β Θ Φ Θ Θ γ Φ ϕ2 = γ + 3π/2 φ = 3π/2 − γ Φ = π − γ

Euler angles conventions Euler angles conventions

Bunge's convention Roe/Matthies's convention

From From f(g) to f(g) to the the pole figures pole figures

Deal with components in the ODF space Deal with components in the ODF space

α β γ ODF γ-sections Pole figures Component: (Hexagonal system) g = {85,80,35}

Plotting f(g) Plotting f(g)

A 3D A 3D plotting plotting program program: ODF plot : ODF plot

γ = 0 γ = 10 γ = 20 γ = 30 γ = 40

ODF 3D ODF 3D-

• isometric

isometric view view ODF sections ( ODF sections (α α, , β β, or , or γ γ) )

Cartesian or Polar f(g) view Cartesian or Polar f(g) view

Polar Polar Cartesian Cartesian

β = 0: space deformation

Inverse pole figures Inverse pole figures

∫

=

y h y //

~ d (g) 2 1 ) ( P

h

ϕ π f

∫

=

h y h //

~ ~ d (g) 2 1 ) ( R

y

ϕ π f

Pole figures Inverse Pole figures 24 equivalent cubic sectors for the Inverse pole figure of a cubic system

ODF refinement ODF refinement

from Generalized Spherical Harmonics ( from Generalized Spherical Harmonics (Bunge Bunge): ):

( ) ( )

∑ ∑ ∑

− = − = ∞ =

Θ + =

l l m m n mn l l l n n l l

k C k l

h h h

y y φ

*

1 2 1 ) ( P

f (g) = Cl

mnTl mn(g) m,n=−l l

∑

l= 0 ∞

∑

∫

=

y h y //

~ d (g) 2 1 ) ( P

h

ϕ π f

One has to invert:

[ ]

∑ ∑

−

y h h h h

dy y y

2

) ( P N ) ( I

Least-squares Refinement procedure

∑ ∑

∞ = − =

=

) 2 ( ,

) ( ) (

λ λ λ λ λ n m mn mn e

g T C g f

But even orders are the only available parts:

from the WIMV iterative process (Williams from the WIMV iterative process (Williams-

• Imhof

Imhof-

• Matthies

Matthies-

• Vinel

Vinel): ):

h h

y

h h IM n M m n n n

P g f g f N g f

1 1 I 1 1

) ( ) ( ) ( ) ( ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∏ ∏

= = +

h h

y

h h IM M m

P N g f

1 exp 1 I 1

) ( ) ( ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∏ ∏

= =

and

h h h

y y

h h M w r M m n n n

n

P P g f g f

∏

= +

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

1 1

) ( ) ( ) ( ) (

with 0 < rn < 1, relaxation parameter, Mh number of division points of the integral around k, wh reflection weight

E-WIMV (Rietveld only): Entropy Entropy maximisation maximisation ( (Schaeben Schaeben): ):

h n h

y y

h h M r M m n n n

P P g f g f

∏

= +

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

1 1

) ( ) ( ) ( ) (

arbitrarily defined cells (ADC, arbitrarily defined cells (ADC, Pawlik Pawlik): ): Very similar to E-WIMV, with integrals along path tubes Vector method ( Vector method (Ruer Ruer, , Baro Baro, , Vadon Vadon): ):

Pi(h) = [σij(h)] fj

I linear equations for J unknown quantities:

Component method (Helming): Component method (Helming):

∑

+ =

c c c

g f I F g f ) ( ) (

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = =

2 2

~ exp 2 exp 1 2 ) ~ ( ) , ( ζ ζ ζ π g g f g g f

c

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 2 cos 1 2 ln ζ S ) ( ) ( 1 ) (

1

S I S I S N − =

Gaussian component:

Evaluation of the OD coverage Evaluation of the OD coverage

{100} pole figure, measured up to χ = 45°:

2

{100} + {110}, measured up to χ = 45°:

5 3 6 3

{100} + {110} + {111}, up to χ = 45°: Say 20 measured (5° x 5°) complete pole figures: = 20 x 1368 = 27360 experimental points ODF (5° x 5° x 5°, triclinic): 98496 points to refine

Estimators of Refinement Quality Estimators of Refinement Quality

Visual assessment

Helix pomatia (Burgundy land snail: Outer com. crossed lamellar layer) Bathymodiolus thermophilus (deep

• cean mussel: Outer Prismatic layer)

RP Factors: Individual pole figures:

( )

) (y P ~ x, ) (y P ~ ) (y P ~

• )

(y P ~ = ) (h RP

j

• h

J 1 j= j

• h

J 1 j= j c h j

• h

i x

i i i i

θ

∑ ∑

... 10 1, , x x for t x > for t 1 ) t , x ( ε = ⎩ ⎨ ⎧ ≤ = θ Averaged on all pole figures:

∑

I 1 = i i x x

) (h RP I 1 = RP

Bragg R-Factors:

[ ] ( )

) (y P ~ x, ) (y P ~ ) (y P ~

• )

(y P ~ = ) (h RB

j

• h

J 1 j= j 2

• h

J 1 j= 2 j c h j

• h

i x

i i i i

θ

∑ ∑

Weighted Rw-Factors:

[ ] ( )

) (y P ~ x, ) (y I w ) (y I w

• )

(y I w = ) (h Rw

j

• h

J 1 j= j 2 z h

• ij

J 1 j= 2 j c h c ij j

• h
• ij

i x

i i i i

θ

∑ ∑

) (y I 1 w

j

• h

ij

i

=

200 400 600 800 20 40 60 80 100 120 140

gRw0 gRw1 F2

RP RP1 F2

Texture strength estimators F2 ∈ ]1,∞[ > 1 m.r.d2 = 1: powder = ∞: single crystal

ODF Texture Index:

i i i

g g f F ∆ =

∑

) ( 8 1 ) (m.r.d.

2 2 2 2

π

Discrete OD

2 2 2

1 2 1 1

mn n m L

C F

λ λ λ λ λ λ

λ

∑ ∑ ∑

− = − = =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + =

Continuous ODF

Pole figures Texture Index:

( ) [ ]

i 2 i i

4 1 J y y

2 h

∆ =

∑

h

P π

S ∈ [0,-∞[ ≤ 0 = 0: powder = -∞: single crystal

Texture Entropy:

i i i i 2

g )] g ( f ln[ ) g ( f 8 1

• S

∆ π =

∑

S - F2:

• 7
• 6
• 5
• 4
• 3
• 2
• 1

50 100 150 200 250 300

Entropy F

Lower bound: Fon = 0

Fon + smooth texture component(s) Fon + Dirac-like texture component

012 104 006 110 113 202 024/108 116 211/122 1010

Belemnite rostrum ~ pure calcite

Why Why needing needing combined combined analysis analysis

• Solve the peak-overlap problems (intra- and inter-phases)

Resolved during ODF refinement

Polyphased Mylonite (Palm Canyon, CA)

10 20 30 40 50 60 70 1 2 3 4 5 6 7

Q102 Q110 P131 Q101 + B003 P201 + B111 B110+020 P111P111 B001

Intensity (x 106) 2Theta (°)

Using 0D detector hardly manageable

PC 82 mylonite Biotite Quartz Albite Anorthite K-spar Composition (weight %) 9.0 24.2 31.7 17.4 14.1

R3 Space group C2/m C-1

Textures & Microstructures 33, 1999, 35-43

Quartz Biotite foliation lineation Albite

Plasma-treated polypropylene films

10 15 20 25 30 35 40 45 20 40 60

• 102

220 150+060 131

• 131+041+"121"

111

• 111+130+031

040 110+011

Intensity (a.u.) 2θ(°)

Large broadening + overlaps + amorphous phase

• Don't want or can't powderise your sample:

. Rare: Ice from deep cores, meteorite rocks ... . Expensive: high-tech materials . Impossible: hard materials, polymers, thin structures ...

• Decreases instrument time:

. 5° x 5° grid = 1368 points / pole figure . ODF: needs as much pole figures as possible

• Access to other parameters:

. crystal sizes, micro-strains, stacking faults + twins (QMA) . residual strains and stresses (QSA) . Structure determination . Phase proportions (QPA) . Thicknesses, roughnesses (XRR)

• Avoid false minima due to parameter correlation:

. phase and texture . Structure and texture . Structure and strains . Thickness and phase …

• Benefit of these correlation to access "true" values

Textured materials: between powder and single-crystal, angular discrimination

• Easier to practice !