Nanopowder crystallite sizes and shapes from diffraction experiments D. Chateigner, L. Lutterotti Normandie Université, IUT-UCN, CRISMAT-ENSICAEN Univ. Trento Nanodays, Caen, 2 nd Feb. 2017
Diffraction “sees” Texture Structure f(g) Φ ,L |F h | 2 Φ ,L Real Layered samples thicknesses roughnesses Residual Stress Phase ρ (z)… <C ijkl (g)> Φ ,L S Φ ,L defects (0D .. nD) broadening asymmetry
Line Broadening causes - Instrumental broadening - Finite size of the crystals acts like a Fourier truncation: size broadening - Imperfection of the periodicity due to d h variations inside crystals: microstrain effect - Generally: 0D, 1D, 2D, 3D defects - All quantities are average values over the probed volume electrons, x-rays, neutrons: complementary distributions: mean values depend on distributions’ shapes
Irradiated Fluorapatites
Instrumental broadening 100 g ( x ) g ( x ) g ( x ) = ⊗ 80 g λ 60 Intensity 40 20 Energy dispersion 0 0,0 0,2 0,4 0,6 0,8 1,0 Geometrical aberrations x + ∞ h ( x ) f ( x ) g ( x ) b ( x ) b ( x ) f ( y ) g ( x y ) dy = ⊗ + = + ∫ − − ∞ Sample contribution Background Measured profile
Back on diffraction expression � � � � A F T � � ( h ) = h h a b c � � � � � � � sin[ ( n 1 ) a . h ] sin[ ( p 1 ) b . h ] sin[ ( q 1 ) c . h ] π + π + π + � � � � � T ( h ) � � = � � a b c sin[ a . h ] sin[ b . h ] sin[ c . h ] π π π � A : scattered amplitude h � F : structure factor h � � T � � ( h ) : interferen ce function a b c � � � n , p, q : number of periods in the a , b , c directions
2 sin [ ( n 1 ) ] 10000 π + α n=99 H ( ) α = 8000 2 sin [ ] πα 6000 H( α ) 4000 (n+1) 2 100 2000 n=9 80 0 0,0 0,2 0,4 0,6 0,8 1,0 60 α H( α ) 40 α +1/(n+1) 20 10 n=2 0 8 0,0 0,2 0,4 0,6 0,8 1,0 α 6 � � H( α ) 4 a . h h = � � 2 infinite crystal : b . h k = � 0 � c . h l = 0,0 0,2 0,4 0,6 0,8 1,0 α
Crystallite’s size-shape effect Scherrer analysis: Δθ Scherrer formula (1918): K λ � R = Δ h h � cos R � ω θ 0 h h � � R h ' h ' K= 0.888 (Scherrer constant) Depends on crystal shapes (Langford) Since β > ω , R h ( β ) < R h ( ω )
After Scherrer analysis … Williamson-Hall (1949) Warren-Averback-Bertaut (1952) Whole-Pattern analysis: Langford (1978), de Keijser (1982), Balzar et Ledbetter (1982) … But deconvolution of contributions (Stokes 1948) ! Rietveld (1969): convolution ! More infos: http://www.ecole.ensicaen.fr/~chateign/ formation/course/Classical_Microstructure.pdf
Rietveld: extended to lots of spectra N N L ν 2 Φ i Φ y ( y , , ) y ( y , , ) I Lp ( ) j F ( y , , ) P ( y , , ) A ( y , , ) ∑ ∑ ∑ θ η = θ η + θ Ω θ η θ η θ η c S b S 0 h h h S h S i S Φ Φ Φ Φ Φ 2 V i 1 1 c h = Φ = Φ ~ ~ P ( y ) f ( g , ) d = ϕ ϕ ∫ Texture: h S E-WIMV, components … ~ ϕ 1 − N N N Strain-Stress: ⎡ ⎤ ( ) ν 1 1 1 − ν − ν − − m S S S S S C ∏ ∏ ∏ = = = = = ⎢ m ⎥ m m m m geo geo geo ⎢ ⎥ m 1 m 1 m 1 ⎣ ⎦ = = = Geometric mean, Voigt, Reuss, Hill … Layering: top film ( ( ) ) ( ( ) ) C g 1 exp Tg / cos / 1 exp 2 T / sin cos = − − µ χ − − µ ω χ 1 2 χ XRR: Parrat, DWBA, EDP … XRF, PDF …
Popa Line Broadening model Crystallite sizes, shapes, µ strains, distributions <111> X-rays (111) X-rays (111) (111) (111) ω ω • Texture helps the "real" mean shape determination Symetrised spherical harmonics L ℓ m m � R R K ( , ) ∑ ∑ = χ ϕ ℓ ℓ h m m m K ( , ) P (cos ) cos( m ) P (cos ) sin( m ) χ ϕ = χ ϕ + χ ϕ ℓ ℓ ℓ ℓ 0 m 0 = = 0 (x) + R 2 P 2 1 (x)cos ϕ + R 3 P 2 1 (x)sin ϕ + R 4 P 2 2 (x)cos2 ϕ + R 5 P 2 2 (x)sin2 ϕ + <R h > = R 0 + R 1 P 2 ... 2 >E h 4 = E 1 h 4 + E 2 k 4 + E 3 ℓ 4 + 2E 4 h 2 k 2 + 2E 5 ℓ 2 k 2 + 2E 6 h 2 ℓ 2 + 4E 7 h 3 k + 4E 8 h 3 ℓ + 4E 9 k 3 h + < ε h 4E 10 k 3 ℓ + 4E 11 ℓ 3 h + 4E 12 ℓ 3 k + 4E 13 h 2 k ℓ + 4E 14 k 2 h ℓ + 4E 15 ℓ 2 kh
R 0 R 0 , R 1 < 0 R 0 , R 1 > 0 1 R 0 , R 0 , R 6 > 0 R 0 , R 6 < 0 R 2 and R 6 > 0 6/m m3m R 0 , R 4 > 0 R 0 , R 1 < 0 R 0 , R 1 > 0
Gold thin films Film thickness Crystallite 10nm 15nm 20nm 25nm 35nm 40nm size (Å) along [111] 176 153 725 254 343 379 [200] 64 103 457 173 321 386 [202] 148 140 658 234 337 381 20 nm 10 nm 15 nm 25 nm 35 nm 40 nm
EMT nanocrystalline zeolite Ng, Chateigner, Valtchev, Mintova: Science 335 (2012) 70
Combined Analysis approach Extracted Intensities X-Ray specular Reflectivity Roughness, electron Density WIMV, E-WIMV & EDP, Harmonics, components, ADC Thickness Fresnel, pole figures Matrix (Parrat), Orientation Distribution Function inverse pole figures DWBA Structural parameters Rietveld atomic positions, substitutions, vibrations cell parameters Structure Multiphased, layered samples: + Thickness, Microstructure Le Bail Anisotropic Sizes + and µ - strains ( Popa ), Voigt, phase % Stacking faults ( Warren ), Reuss, Distributions, Turbostratism (Ufer) Geometric Popa-Balzar, mean sin 2 ψ Phase ratio (amorphous + crystalline) Le Bail Rietveld TEM, XRF, PDF Residual stresses Strain Distribution Function
Why not more ? Magnetic Nuclear (isotopic) scattering SANS, n-Tomography, PDF � p σ , T ∇ , T ij � � E neutrons Structure Macroscale H Local environment Magnetic structure Texture Magnetic Texture � Residual Stresses Magnetic roughness Phases Vacancies µ h Thickness Atomic scale ν MAUD, Jana Roughness Porosity Fullprof … Size and shape Amorphization electrons photons muons (X, γ , IR … ) Open Databases Composition Interfaces Nanoscales Misorientations Dislocations Twins, Faults
Combined Analysis Workshop in Caen: 3 rd – 7 th July 2017 ! www.ecole.ensicaen.fr/~chateign/formation/ Thanks !
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