Convolutional Multiple Whole Profile Fitting G abor Rib arik - PowerPoint PPT Presentation

Convolutional Multiple Whole Profile Fitting G´ abor Rib´ arik ribarik@renyi.hu Department of Materials Physics, Institute of Physics, E¨ otv¨ os University, Budapest, P .O.Box 32, H-1518, Hungary CMWP – p.1/70

Introduction: extracting microstructure using X-ray line profile analysis modeling size and strain broadening the MWP method the CMWP method CMWP application to ball milled Al-Mg alloys the CMWP program CMWP – p.2/70

(C)MWP-fit These methods are in fact: Whole Profile fitting, or Whole Powder Pattern fitting methods microstructural methods: the unit cell is NOT refined The aim is microstructure in terms of: size strain CMWP – p.3/70

Theory The Fourier coefficients of the line profiles (Warren & Averbach, 1952): A ( L ) = A S ( L ) A D ( L ) , this means that the observed profile is the convolution of size and strain profiles. If more physical effects and instrumental effects are simultaneously present: A ( L ) = A instr. ( L ) A size ( L ) A disl. ( L ) A pl.faults ( L ) · · · , I (2Θ) = I instr. (2Θ) ∗ I size (2Θ) ∗ I disl. (2Θ) ∗ I pl.faults (2Θ) ∗· · · CMWP – p.4/70

The size effect If we suppose: spherical crystallites lognormal f(x) size distribution density function: � x �� 2   � 1 1 log m f ( x ) = exp  −  ,   √ 2 σ 2 2 πσ x ( m and σ are the two parameters of the distribution). CMWP – p.5/70

The size effect The size intensity profile (Gubicza et al, 2000): � µ �   ∞ log µ sin 2 ( µ πs ) � m  d µ, I S ( s ) = √ erfc  ( πs ) 2 2 σ 0 where erfc is the complementary error function: ∞ 2 e − t 2 d t. � erfc( x ) = √ π x CMWP – p.6/70

The size effect If we suppose (Ribárik et al, 2001): ellipsoidal crystallite shape lognormal size distribution density function I S ( s ) has the same form than in the spherical case but m depends on the indices of reflection: m a m hkl = , � 1 � � cos 2 α 1 + ε 2 − 1 ( m a : the m parameter of the size distribution in the direction a , ε : ellipticity, α : the angle between the axis of revolution and the diffraction vector). CMWP – p.7/70

The size effect If the relative orientations of the crystallographic directions to the axis of revolution are known, cos α can be expressed by the indices of reflection. For cubic systems: l cos α = √ h 2 + k 2 + l 2 For hexagonal systems: l cos α = � c 2 4 a 2 ( h 2 + hk + k 2 ) + l 2 3 CMWP – p.8/70

The Size Fourier Transform: It can be expressed in an almost closed form which is suitable for fast numeric evaluation (Ribárik et al, 2001): � �   | L | √ � 9 m 3 exp 2 � log 4( 2 σ ) √ m − 3   A S ( L, m, σ ) = √ erfc 2 σ  −   2 3 2 σ  � �   | L | √ log 2 m 2 exp ( √ 2 σ ) m   √ | L | erfc − 2 σ  +   2 2 σ  � �   | L | log | L | 3 m   √ erfc  .   6 2 σ  CMWP – p.9/70

The strain effect The distortion Fourier coefficients (Warren & Averbach, 1952): − 2 π 2 g 2 L 2 � ε 2 A D ( L ) = exp � � L � , where g is the absolute value of the diffraction vector, � ε 2 L � is the mean square strain . The most important models for � ε 2 L � : Warren & Averbach (1952) Krivoglaz & Ryaboshapka (1963) Wilkens (1970) CMWP – p.10/70

The Wilkens dislocation theory Wilkens introduced the effective outer cut off radius of dislocations, R ∗ e , instead of the crystal diameter. Assuming infinitely long parallel screw dislocations with restrictedly random distribution (Wilkens, 1970): � b � L � 2 � � ε 2 πρ Cf ∗ L � = , 2 π R ∗ e where f ∗ is the Wilkens strain function (Wilkens, 1970). Kamminga and Delhez (2000) have shown that this strain function is also valid for edge- and curved dislocations. The distortion Fourier–transform in the Wilkens model: � L − πb 2 � �� A D ( L ) = exp 2 ( g 2 C ) ρL 2 f ∗ . R ∗ e CMWP – p.11/70

The Wilkens strain function � 7 + 512 1 � f ∗ ( η ) = − log η + 4 − log 2 η + 90 π � η arcsin V � 2 1 − 1 � d V − 4 η 2 π V 0 1 � 769 η + 41 1 90 η + 2 90 η 3 � � 1 − η 2 − 180 π � � 1 11 η 2 + 7 1 2 + 1 arcsin η + 1 3 η 2 6 η 2 , if η ≤ 1 , 12 π � 1 f ∗ ( η ) = 512 1 � 11 24 + 1 η − 4 log 2 η if η ≥ 1 , η 2 , 90 π � L � � = f ∗ ( η ) and η = 1 − 1 L � where f 2 exp e . 4 R ∗ R ∗ e CMWP – p.12/70

The Wilkens strain function 7 f * ( η ) 6 5 4 3 f * ( η ) 2 1 0 -1 -2 η 0 2 4 6 8 10 „ 7 « and 512 1 The Wilkens function and its approximations: − log η + 4 − log 2 η . 90 π CMWP – p.13/70

The dislocation arrangement parameter Wilkens introduced M ∗ , a dimensionless parameter: √ ρ M ∗ = R ∗ e The M ∗ parameter characterizes the dislocation arrangement: if the value of M ∗ is small, the correlation between the dislocations is strong if the value of M ∗ is large, the dislocations are distributed randomly in the crystallite CMWP – p.14/70

1 1 R ∗ e << R ∗ e >> √ ρ √ ρ M ∗ << 1 M ∗ >> 1 CMWP – p.15/70

The strain profile for fixed ρ and variable M ∗ values: CMWP – p.16/70

The shape of the strain profile for fixed ρ and variable M ∗ values: CMWP – p.17/70

Strain anisotropy According to (Ungár & Tichy, 1999), the average contrast factors of dislocations can be expressed in the following form for cubic crystals: C = C h 00 (1 − qH 2 ) , where h 2 k 2 + h 2 l 2 + k 2 l 2 H 2 = . ( h 2 + k 2 + l 2 ) 2 CMWP – p.18/70

For hexagonal crystals: C = C hk 0 (1 + a 1 H 2 1 + a 2 H 2 2 ) , where [ h 2 + k 2 + ( h + k ) 2 ] l 2 H 2 1 = c ) 2 l 2 ] 2 , [ h 2 + k 2 + ( h + k ) 2 + 3 2 ( a l 4 H 2 2 = c ) 2 l 2 ] 2 , [ h 2 + k 2 + ( h + k ) 2 + 3 2 ( a and a c is the ratio of the two lattice constants. CMWP – p.19/70

For orthorombic crystals: H 2 0 + a 1 H 2 1 + a 2 H 2 2 + a 3 H 2 3 + a 4 H 2 4 + a 5 H 2 � � C hkl = C h 00 , 5 where: h 4 H 2 a 4 0 = ” 2 “ h 2 a 2 + k 2 b 2 + l 2 c 2 k 4 H 2 b 4 1 = ” 2 “ h 2 a 2 + k 2 b 2 + l 2 c 2 l 4 H 2 c 4 2 = ” 2 “ h 2 a 2 + k 2 b 2 + l 2 c 2 CMWP – p.20/70

h 2 k 2 H 2 a 2 b 2 3 = ” 2 “ h 2 a 2 + k 2 b 2 + l 2 c 2 l 2 h 2 H 2 c 2 a 2 4 = ” 2 “ h 2 a 2 + k 2 b 2 + l 2 c 2 k 2 l 2 H 2 b 2 c 2 5 = ” 2 “ h 2 a 2 + k 2 b 2 + l 2 c 2 and a , b , c are the lattice constants. The constants C h 00 and C hk 0 are calculated from the elastic constants of the crystal (Ungár et al, 1999). CMWP – p.21/70

Planar and twin faults: The peak profile is the sum of a delta function and shifted and broadened Lorentzian profile functions the FWHM and shift value of the Lorentzians depend on the density of faults hkl -dependence: DIFFaX software (Treacy et al., Proc. Roy. Soc., 1991) The parameters were systematically calculated for each fundamental types of planar faults by Dr. Levente Balogh (see: L. Balogh, PhD thesis, Eötvös University, 2009). CMWP – p.22/70

Microstructural parameters (C)MWP-fit provides: size: m , σ , ε dislocations: ρ , M , q planar and twin faults: α , β CMWP – p.23/70

Multiple Whole Profile (MWP) fitting The method (Ribárik et al, 2001) is: a Whole Profile fitting method using ab-initio theoretical profile functions a Fourier method, which works on multiple profiles simultaneously The data must be prepared before applying the method: the profiles should be separated the instrumental broadening is corrected for by deconvolution using the Stokes method the separated and instrumental-free profiles are Fourier-transformed CMWP – p.24/70

MWP: profile separation M3377.dat 1 bg(x)+I(x) bg(x) p 0 +p 1 x I(x) I 1 (x) I 2 (x) 0.8 0.6 Intensity 0.4 100 101 0.2 004 0 -2 -1.5 -1 -0.5 0 0.5 1 ∆ K [1/nm] Example for the profile separation in the case of the strong overlapping peaks of a carbon black sample. CMWP – p.25/70

MWP: instrumental correction Example for the instrumental deconvolution in the case of an Al-6Mg sample. The A ( L ) Fourier transforms of the measured profile, the instrumental profile and the corrected profile are plotted as a function of L . CMWP – p.26/70

MWP: instrumental correction Example for the instrumental deconvolution in the case of an Al-6Mg sample. The raw measured and the corrrected intensity profile are plotted as a function of K . CMWP – p.27/70

MWP-fit (i) Multiple Whole Profile fitting of the Fourier–transforms. the measured intensity profiles are Fourier–transformed and normalized, they are fitted simultaneously by the normalized theoretical Fourier–transform: � L − πb 2 A ( L ) = A S ( L ) � �� 2 ( g 2 C ) ρL 2 f A S (0) exp , R ∗ e CMWP – p.28/70

MWP-fit (ii) Multiple Whole Profile fitting of the intensity profiles. In this procedure first the measured intensity profiles are normalized. Then all of them are fitted simultaneously by the normalized theoretical intensity function: F c ( s ) I ( s ) = , F c (0) where F c is the Cosine Fourier–transform of ( ?? ): ∞ � F c ( s ) = 2 A ( L ) cos(2 πLs ) d L. 0 In both cases [(i) and (ii)] all profiles are fitted simultaneously using a nonlinear least-squares algorithm. CMWP – p.29/70