Classical Texture Texture Classical Analysis Analysis D. Chateigner D. Chateigner CRISMAT- -ENSICAEN; IUT ENSICAEN; IUT- -UCBN UCBN CRISMAT 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 -2 θ 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 (hk l ) 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: : aggregate of grains, different phases, sizes, Polycrystal 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, crystallites crystallites, , crystallographic crystallographic planes planes Grains, Friedel's law: I hk l = I -h-k- l using normal diffraction + or - directions not distinguished [hk l ] I + I - -[hk l ]
Texture effects effects on diffraction on diffraction diagrams diagrams Texture θ -2 θ scan: probes only parallel planes
Li 0.12 La 0.88 TiO 3 random bulk Li 0.12 La 0.88 TiO 3 /(100)-MgO Oriented film MgO 003 001 002
asymmetric scan: probes only inclined planes
mixed scan: probes specific planes for specific orientations 006 008 005
ω scan: probes orientation of only one plane type (fixed θ ), only for small ω - θ 2000 1500 Intensity (a.u) 1000 0.1° 500 0 17.5 18.0 18.5 ω (°)
limitations: available θ (or other) range diamond (Fd3m), 2.52 Å neutrons, up to 2 θ = 150° 111
limitations: 2 texture components same c-axes direction, but not same a p - p L = 0 ; p : random hk 0 l − 1 p 0 MgO 003 { } ∑ I 0 0 l i = = p = i 1 L { } ∑ 00 l I hk l hk l 001 002
limitations: 2 texture components, one inclined 2000 1500 Intensity (a.u) 1000 0.1° 500 0 17.5 18.0 18.5 ω (°) 2000 1500 Intensity (a.u) 1000 500 0 17.5 18.0 18.5 ω (°)
Representations of texture: pole figures of texture: pole figures Representations One crystallite oriented in the Pole sphere: - location of all [hk l ] ∈ unit sphere - dS = sin χ d χ d ϕ - ( χ , ϕ ): angles in the diffractometer space � Hard to visualise: needs pole figures
Stereographic projections: projections: equal equal angle angle Stereographic Poles: p(r', ϕ ): r’ = R tan( χ /2)
Lambert projections (equal equal area) area) Lambert projections ( Poles: p(r', ϕ ): r’ = 2R sin( χ /2)
Lambert/Schmidt stereographic 90 90 120 60 120 60 30 150 150 30 180 0 180 0 210 330 210 330 240 300 240 300 270 270 5° x 5° grid: 1368 points
Pole figures Pole figures {hk l }-Pole figure: location of distribution densities, for the {hk l } diffracting plane, defined in the crystallite frame K B , relative to the sample frame K A . Pole figures space: � , with y = ( ϑ y , ϕ y ) = [hk l ]* Direct Pole Figure: built on diffracted intensities I h ( y ), h = <hk l >* Normalised Pole Figure: built on distribution densities P h ( y ) Density unit: the "multiple of a random distribution", or "m.r.d."
Usual pole figure pole figure frames frames K K A Usual A Lineation direction RD G . . . TD M n F ND N Foliation plane metallurgy malacology geophysics Thin films: substrate directions … X A , Y A , Z A
Normalisation Normalisation π π 2 /2 ∫ ∫ 90 = ϑ ϕ ϑ ϑ ϕ total I I ( , ) sin d d 120 60 h y y y y y h ϕ = ϑ = 0 0 y y 150 30 π π 2 /2 ∫ ∫ random = total ϑ ϑ ϕ I I / sin d d 180 0 y y y h h ϕ = 0 ϑ = 0 y y 210 330 I ( ) y = P ( ) 240 300 h y 270 random h I h ϑ y ϕ I ( , ) h y - Only valid for complete pole figures: neutrons in symmetric geometry - Needs a refinement strategy to get I random for all h 's
Incompleteness and corrections of pole figures and corrections of pole figures Incompleteness Missing Bragg peaks Absorption + volume Defocusing (x-rays) Blind area Localisation
2θ -defocusing ω -defocusing χ -defocusing
Defocusing corrections: Intensity - Calibration on a random powder rand I χ Net intensities ω θ cor = meas 0 , , I I χ ω θ χ ω θ (point detector) , , , , rand I 0° 90° χ ω θ , , ⎡ ⎤ − bkg rand bkg I I I Peak maximum (point detector) χ ω θ ω θ ω θ = , , 0 , , 0 , , meas bkg I - I ⎢ ⎥ Integrated intensity (1D or 2D detector) χ ω θ ω θ , , 0 , , − bkg rand bkg I I I ⎢ ⎥ ⎣ ⎦ ω θ χ ω θ ω θ 0 , , , , 0 , , - Total integration of the peak (direct integration or fit)
Absorption/Volume corrections: χ Specific to each instrumental geometry Sample dependent (films, multilayers …) beam Modifies the defocusing curves Can be integrated in fitting procedures t Top film Intensity ( ( ) ) − − µ θ 1 exp 2 / sin T = χ I(0) I( ) i χ ( ( ) ) − − µ θ χ 1 exp 2 / sin cos T i 0° 90° ∑ − µ 2 T j j ( ( ) ) − − µ θ 1 exp 2 / sin exp( j ) T θ i sin Covered layer = χ I(0) I( ) i ∑ − µ 2 T j j ( ( ) ) − − µ θ χ 1 exp 2 / sin cos exp( j ) T θ χ i sin cos i
Single or multiple texture components, multiplicity multiplicity Single or multiple texture components, Double Single Tetragonal Cubic
Program convention ! convention ! Program
Outer aragonite layer Cypraea testudinaria Pnma space group real example example A real A
Texture types Texture types Random texture 3 degree of freedom All P h ( y ) homogeneous 1 m.r.d. density whatever y Planar texture 100 001 2 degree of freedom 1 [hk l ] at random in a plane
100 001 Fibre texture 1 degree of freedom 1 [hk l ] along 1 y direction Cyclic-Fibre texture c // Z A Cyclic-Planar texture ( a , b ) // (X A ,Y A )
Single crystal-like texture 100 110 0 degree of freedom 2 [hk l ]'s along 2 y directions Single-crystal and perfect 3D orientation not distinguished
Pole figure and Orientation spaces spaces Pole figure and Orientation dV( ) 1 y Pole figure expression: = P ( ) y dy π V 4 h dy = sin ϑ y d ϑ y d ϕ y π π 2 /2 ∫ 0 ∫ ϑ ϕ ϑ ϑ ϕ π P ( , ) sin d d = 4 h y y y y y ϕ = ϑ = 0 y y Orientation Distribution Function f(g): dV(g) 1 dg = sin( β )d β d α d γ = (g) dg 2 f π V 8 π π π 2 / 2 2 ∫ ∫ ∫ π 2 (g) dg = 8 f α = β = γ = 0 0 0
From Pole figures to Pole figures to the the ODF ODF From Pole figure: one direction fixed in K A Orientation: two directions fixed in K A 1 ~ ∫ = ϕ P ( ) (g) d Fundamental Equation of QTA h y f π 2 // h y Needs several pole figures to construct the f(g)
Pole figures from from g g Pole figures - Rotation of K A about the axis Z A through the angle α : [K A a K' A ]; associated rotation g 1 = { α ,0,0} - Rotation of K' A about the axis Y' A through the angle β : [K' A a K" A ]; associated rotation g 2 = {0, β ,0} - Rotation of K" A about the axis Z" A through the angle γ : [K" A a K"' A //K B ]; associated rotation g 3 = {0,0, γ } finally: g = g 1 g 2 g 3 = { α ,0,0} {0, β ,0} {0,0, γ } = { α , β , γ } Y' A Y' A X' A Z'' A Z'' A Y''' A g 2 = {45,45,0} g 3 = {45,55,45} g 1 = {45,0,0}
Euler angles conventions Euler angles conventions Matthies Roe Bunge Canova Kocks ϕ 1 = α + π /2 α Ψ ω = π/2 − α Ψ β Θ Φ Θ Θ γ Φ ϕ 2 = γ + 3 π /2 φ = 3π/2 − γ Φ = π − γ Bunge's convention Roe/Matthies's convention
From f(g) to f(g) to the the pole figures pole figures From
Deal with components in the ODF space Deal with components in the ODF space α γ β ODF γ -sections Pole figures Component: (Hexagonal system) g = {85,80,35}
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