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¨

• tv¨
• s 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) = AS(L)AD(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) = Ainstr.(L)Asize(L)Adisl.(L)Apl.faults(L) · · · , I(2Θ) = Iinstr.(2Θ)∗ Isize(2Θ)∗ Idisl.(2Θ)∗ Ipl.faults(2Θ)∗· · ·

CMWP – p.4/70

The size effect

If we suppose: spherical crystallites lognormal f(x) size distribution density function:

f(x) = 1 √ 2πσ 1 x exp   −

• log

x m 2 2σ2    ,

(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):

IS(s) =

∞

• µ sin2(µ πs)

(πs)2 erfc   log µ m

• √

2σ   dµ,

where erfc is the complementary error function:

erfc(x) = 2 √π

∞

• x

e−t2 dt.

CMWP – p.6/70

The size effect

If we suppose (Ribárik et al, 2001): ellipsoidal crystallite shape lognormal size distribution density function

IS(s) has the same form than in the spherical case but m

depends on the indices of reflection:

mhkl = ma

• 1 +

1 ε2 − 1

• cos2 α

,

(ma: 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:

cos α = l √ h2 + k2 + l2

For hexagonal systems:

cos α = l

• 4

3 c2 a2 (h2 + hk + k2) + l2

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):

AS(L, m, σ) = m3 exp 9 4( √ 2σ)

2

3 erfc     log

• |L|

m

• √

2σ − 3 2 √ 2σ     − m2 exp ( √ 2σ)

2

2 |L| erfc     log

• |L|

m

• √

2σ − √ 2σ     + |L|3 6 erfc     log

• |L|

m

• √

2σ     .

CMWP – p.9/70

The strain effect

The distortion Fourier coefficients (Warren & Averbach, 1952):

AD(L) = exp

• −2π2g2L2ε2

L

• ,

where

g is the absolute value of the diffraction vector, ε2

L is the mean square strain.

The most important models for ε2

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):

ε2

L =

b 2π 2 πρ Cf∗ L 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:

AD(L) = exp

• −πb2

2 (g2C)ρL2f∗ L R∗

e

• .

CMWP – p.11/70

The Wilkens strain function

f∗(η) = − log η + 7 4 − log 2

• + 512

90π 1 η+ 2 π

• 1 − 1

4η2 η

• arcsin V

V dV − 1 π 769 180 1 η + 41 90η + 2 90η3 1 − η2− 1 π

• 11

12 1 η2 + 7 2 + 1 3η2

• arcsin η + 1

6η2, if η ≤ 1, f∗(η) = 512 90π 1 η − 11 24 + 1 4 log 2η 1 η2, if η ≥ 1,

where f

• L

R∗

e

• = f∗(η) and η = 1

2 exp

• −1

4 L R∗

e .

CMWP – p.12/70

The Wilkens strain function

• 2
• 1

1 2 3 4 5 6 7 2 4 6 8 10

f*(η) η f*(η)

The Wilkens function and its approximations: − log η + „7 4 − log 2 « and 512 90π 1 η .

CMWP – p.13/70

The dislocation arrangement parameter

Wilkens introduced M∗, a dimensionless parameter:

M∗ = R∗

e

√ρ

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

R∗

e << 1 √ρ

R∗

e >> 1 √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 = Ch00(1 − qH2),

where

H2 = h2k2 + h2l2 + k2l2 (h2 + k2 + l2)2 .

CMWP – p.18/70

For hexagonal crystals:

C = Chk0(1 + a1H2

1 + a2H2 2),

where

H2

1 =

[h2 + k2 + (h + k)2] l2 [h2 + k2 + (h + k)2 + 3

2(a c)2l2]2,

H2

2 =

l4 [h2 + k2 + (h + k)2 + 3

2(a c)2l2]2,

and a

c is the ratio of the two lattice constants.

CMWP – p.19/70

For orthorombic crystals:

Chkl = Ch00

• H2

0 + a1H2 1 + a2H2 2 + a3H2 3 + a4H2 4 + a5H2 5

• ,

where:

H2

0 =

h4 a4

“

h2 a2 + k2 b2 + l2 c2

”2

H2

1 =

k4 b4

“

h2 a2 + k2 b2 + l2 c2

”2

H2

2 =

l4 c4

“

h2 a2 + k2 b2 + l2 c2

”2

CMWP – p.20/70

H2

3 =

h2k2 a2b2

“

h2 a2 + k2 b2 + l2 c2

”2

H2

4 =

l2h2 c2a2

“

h2 a2 + k2 b2 + l2 c2

”2

H2

5 =

k2l2 b2c2

“

h2 a2 + k2 b2 + l2 c2

”2

and a, b, c are the lattice constants. The constants Ch00 and Chk0 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

0.2 0.4 0.6 0.8 1

• 2
• 1.5
• 1
• 0.5

0.5 1 Intensity ∆K [1/nm] 100 101 004 M3377.dat bg(x)+I(x) bg(x) p0+p1x I(x) I1(x) I2(x)

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:

A(L) = AS(L) AS(0) exp

• −πb2

2 (g2C)ρL2f L 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:

I(s) = Fc(s) Fc(0) ,

where Fc is the Cosine Fourier–transform of (??):

Fc(s) = 2

∞

• A(L) cos(2πLs) dL.

In both cases [(i) and (ii)] all profiles are fitted simultaneously using a nonlinear least-squares algorithm.

CMWP – p.29/70

MWP application to Cu sample

• 0.2

0.2 0.4 0.6 0.8 1 1.2 200 400 600 800 1000 Normalized Amplitude Frequency [nm] 111 200 220 311 222 400 measured data theoretical curve

Results of the MWP fit:

m = 62nm

σ = 0.53 ρ = 1.7 · 1015 m−2

M = Re√ρ = 1.7 q = 1.84

The measured (solid lines) and theoretical fitted (dashed lines) Fourier transforms for copper sample deformed by ECAP (equal-channel angular pressing) as a function of the Fourier Frequency, L. The difference plot is also given.

CMWP – p.30/70

MWP application to PbS sample

• 0.2

0.2 0.4 0.6 0.8 1 1.2 200 400 600 800 1000 1200 1400 1600 1800 Normalized Amplitude Frequency [nm] 111 200 220 311 222 400 331 420 422 measured data theoretical curve

The measured (solid lines) and theoretical fitted (dashed lines) Fourier–transforms for PbS sample as a function of the Fourier variable (Frequency), L, plotted by the program

• evaluate. The difference plot is also given in the bottom of the figure.

CMWP – p.31/70

MWP application to PbS sample

• 0.2

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 1.4 Normalized Intensity ∆K [1/nm] 111 200 220 311 222 400 331 420 422 measured data theoretical curve

The measured (solid lines) and theoretical fitted (dashed lines) intensity profiles for PbS sample plotted by the program evaluate. The difference plot is also given in the bottom of the figure. The indices of reflections are also indicated.

CMWP – p.32/70

Comparison with TEM

0.002 0.004 0.006 0.008 0.01 0.012 0.014 50 100 150 200 250 size distribution crystallite size [nm] m=62, σ=0.53 TEM data

The size distribution density function corresponding to the parameters of the MWP fit and the size distribution obtained by TEM for an ECA pressed copper sample.

CMWP – p.33/70

Comparison with TEM

The TEM micrograph (a) and the size distribution functions (b) measured by TEM and X-ray line profile analysis for nanoncrystalline Si3N4 particles.

CMWP – p.34/70

Comparison with TEM

(a) High resolution TEM image of nanocrystalline titanium sample (b) Fourier-filtered image from the white frame in (a), showing the dislocation arrangement in the grain boundary.

CMWP – p.35/70

Convolutional MWP (CMWP) fitting

When the MWP fitting procedure is used: the deconvolution of the physical and the instrumental profiles introduces noise, the selection of the analytical function used in the peak separation influences the shape of the individual profile. When the CMWP method (Ribárik et al, 2004) is used: the whole measured powder diffraction pattern is fitted by the sum of a background polinom and profile functions. the profile functions are calculated as the convolution of the theoretical functions for physical broadening and the instrumental profiles. Therefore neither the separation of the peaks nor the deconvolution is needed.

CMWP – p.36/70

The theoretical intensity pattern

Itheoretical = BG(2Θ) +

• hkl

Ihkl

MA XIhkl(2Θ − 2Θhkl 0 ),

where:

Ihkl = Ihkl

• instr. ∗ Ihkl

size ∗ Ihkl

• disl. ∗ Ihkl

pl.faults,

Ihkl

instr.: measured instumental profile which is directly used

The measured and theoretical patterns are compared using a nonlinear least-squares algorithm, the fitted parameters are the microstructural parameters (no individual profile parameters are used).

CMWP – p.37/70

Instrumental pattern of LaB6

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 20 40 60 80 100 120 140 160 Intensity 2Theta LaB6

CMWP – p.38/70

Instrumental pattern of LaB6

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 25 30 35 40 Intensity 2Theta LaB6

CMWP – p.39/70

Al-3Mg ball milled 3 h.

10000 20000 30000 40000 50000 60000 40 60 80 100 120 140 Intensity 2θ [o] 111 200 220 311 222 400 331 420 422 measured data theoretical curve difference

100 1000 10000 100000 40 60 80 100 120 140 Intensity Counts 2θ [o] 111 200 220 311 222 400 331 420 422 measured data theoretical curve

CMWP – p.40/70

Al-6Mg ball milled 6 h.

10000 20000 30000 40000 50000 40 60 80 100 120 140 Intensity 2θ [o] 111 200 220 311 222 400 331 420 422 measured data theoretical curve difference

100 1000 10000 100000 40 60 80 100 120 140 Intensity Counts 2θ [o] 111 200 220 311 222 400 331 420 422 measured data theoretical curve

Results of the CMWP fit:

m = 21nm

σ = 0.36 ρ = 1016 m−2

M = Re√ρ = 1.3 q = 1.3

CMWP – p.41/70

Al-20Mg ball milled 32 h.

• 2000

2000 4000 6000 8000 10000 12000 14000 16000 30 40 50 60 70 80 90 100 110 120 Intensity 2θ [o] 111 200 220 311 222 400 331 420 measured data theoretical curve difference

1000 10000 100 Intensity 2θ [o] 111 200 220 311 222 400 331 420 measured data theoretical curve

CMWP – p.42/70

Microstructure of ball milled Al-20Mg

1 2 3 4 5 6 5 10 15 20 25 30 35 ρ [1015 1/m2] milling period [h] "Al-Mg-t-rho-spline.dat" using 1:2

(Révész et al., J. of Metast. and Nanocr. Mat., 2005)

CMWP – p.43/70

Microstructure of ball milled Al-20Mg

10 20 30 40 50 60 70 80 90 5 10 15 20 25 30 35 <x>area [nm] milling period [h] "Al-Mg-t-Xarea-spline.dat" using 1:2

(Révész et al., J. of Metast. and Nanocr. Mat., 2005)

CMWP – p.44/70

CMWP features

supports cubic, hexagonal or orthorombic crystal system it supports (practically) unlimited number of phases the lognormal size distribution and spherical or ellipsoidal crystallite shape can be used planar faults effect (intrinsic, extrinsic faults or twins) can be inlcuded the Wilkens or the Groma-Csikor strain function can be used it supports the fitting of the contrast factor parameters

• r individual contrast factors can be used

it is using the measured instrumental profiles directly the peak positions and intensities can be refined

CMWP – p.45/70

CMWP features

each parameter value can be fixed, d · e can be fixed too weighting and parameter scaling is implemented it has a command line interface: evaluate which can be used interactively from a terminal window or by setting the parameter values and options in the ini files it can be automatically invoked without interaction (e.g. using it in a cycle or running it as a cron/at job) it has a graphical JAVA interface: every parameter value and option can be set in a JAVA panel and all the functions of the program can be reached it can be run using the WWW frontend: in this case only a working Web browser is needed, the input files can be uploaded to the server and the parameter values and

• ptions can be set by the frontend

CMWP – p.46/70

File formats

ASCII input files:

powder pattern file instrumental profiles indexing file background spline’s base points file ini files

CMWP – p.47/70

Powder pattern file

2-column file, contains: 2Θ, I.

Example: . . . 100.7800 335 100.7950 331 100.8100 335 100.8250 331 100.8400 342 100.8550 335 . . .

CMWP – p.48/70

Instrumental profiles

2-column files, containing: K − K0, I.

K = 2sin(Θ)

λ

K0: the K value at the peak center.

Example: . . .

• 0.00223

0.945234

• 0.00112

0.955681 0.985527 0.00111 0.983823 0.00222 0.97676 . . .

CMWP – p.49/70

Indexing file

3-column file, contains: 2Θ0, IMA

X, hkl, phase.

Example: 38.2887 13826 111 44.4726 5828 200 64.6108 2544 220 77.5747 2143 311 81.7108 579 222 98.1789 205 400 110.714 566 331 115.119 488 420 . . .

CMWP – p.50/70

Spline background

2-column file, contains: 2Θ, I.

100 200 300 400 500 600 700 800 30 40 50 60 70 80 90 Counts 2Theta spline base points interpolated spline function

Example: 30 120 40 420 60 320 65 520 70 580 93 670

CMWP – p.51/70

The CMWP webpage

Available at:

http://www.renyi.hu/cmwp

After an easy Registration you get access to: the WWW frontend of the CMWP and MWP programs: you can evaluate online your samples, only a Web browser is needed the complete program packages: you can download and install it on your own Linux computer Documentation page: http://www.renyi.hu/cmwp/doc

CMWP – p.52/70

The JAVA frontend of CMWP

CMWP – p.53/70

The MKSPLINE program of CMWP

CMWP – p.54/70

The peak indexing program of CMWP

CMWP – p.55/70

Setting the individual C values

CMWP – p.56/70

The WWW frontend of CMWP

CMWP – p.57/70

The CMWP upload page

CMWP – p.58/70

The CMWP instrumental upload page

CMWP – p.59/70

The CMWP sample listing page

CMWP – p.60/70

The CMWP evaluation page I.

CMWP – p.61/70

The CMWP evaluation page II.

CMWP – p.62/70

The CMWP evaluation page III.

CMWP – p.63/70

The CMWP evaluation page IV.

CMWP – p.64/70

The CMWP evaluation page V.

CMWP – p.65/70

The CMWP evaluation page VI.

CMWP – p.66/70

The CMWP results page

CMWP – p.67/70

Example of a solution file

CMWP – p.68/70

Example of a statistics file

CMWP – p.69/70

Summary

The MWP method works on the separated profiles or their Fourier transforms and it’s a powerful method to extract microstructural parameters from X-ray peak profiles. The CMWP method works in the direct space on the whole pattern, the instrumental effect is included in the model based pattern by using convolution and therefore neither the separation of the peaks nor the deconvolution is needed. Using these methods several microstructural parameters can be determined for size, dislocations, and planar faults.

CMWP – p.70/70