D. Chateigner, and the SOLSA Consortium Normandie Université

Microstructure and Texture Analysis: Theory and Case Studies

Trento, Italy, 26-30th Nov. 2018

Structure |Fh|2Φ,L

Texture f(g)Φ,L

Residual Stress <Cijkl(g)>Φ,L

Real samples Layered samples

thicknesses roughnesses ρ(z)…

Phase SΦ,L

defects (0D .. nD) broadening asymmetry

X-ray scattering “sees”

Phase ID

Raman, IR, XRF

Element ID

XRF

∫ϕ

ϕϕ=~

Sh~d)~,g(f)(P y

Rietveld: extended to lots of spectra

∑∑ ∑= =Φ

ΦΦΦΦΦΦ

ΦΦ

ηθηθηθΩθν

+ηθ=ηθLN

1i

N

1ihh

2hh

h2c

i0bc ),,(A),,(P),,(Fj)(Lp

VI),,(y),,(y SSSSS yyyyy

Texture: E-WIMV, components, Harmonics, Exp. Harmonics …

Strain-Stress:

( ) geogeo1

N

1m

1m

N

1mm

1N

1mm

1geo CSSSSS m

mm ====⎥⎥⎦

⎤

⎢⎢⎣

⎡= −

=

ν−

=

ν−−

=

ν− ∏∏∏

Geometric mean, Voigt, Reuss, Hill …

Layering:

AiΦ =ν iΦ sinθi sinθoµ i (sinθi + sinθo )

1− e−µ iτ iW{ } e−µkτ kWk<i∏

W =1

sinθi+

1sinθo

Stacks, coatings, multilayers …

Popa, Delft: Crystallite sizes, shapes, microstrains, distributions 0D-3D defects

Line Broadening:

X-Ray Reflectivity (specular):

Matrix, Parrat, DWBA, EDP …

X-Ray Fluorescence/GiXRF:

De Boer

Electron Diffraction Patterns:

2-waves Blackman

Microstructure (Line Broadening) Analysis Line Broadening causes Instrumental broadening Main contributions

Evolution of FWHM with x Extraction of the sample contribution Peak profiles Constant wavelengths

TOF neutrons Calibration of the instrumental contribution Back on diffraction expression Simple peak broadening characterization

Full-Width at Half-Maximum Integral breadth

Crystallite’s size effect Accounts for instrument in simple cases What is size?

Microstrains effect Williamson-Hall analysis Whole-pattern analysis Warren-Averbach-Bertaut analysis Anisotropic sizes and microstrains Examples

Line Broadening causes

-  Instrumental broadening

-  Finite size of the crystals acts like a Fourier truncation: size broadening

-  Imperfection of the periodicity due to dh variations inside crystals: microstrain broadening

-  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

∫+∞

∞−

−+=+⊗= dy)yx(g)y(f)x(b)x(b)x(g)x(f)x(h

)x(g)x(g)x(g g⊗= λ

Energy dispersion

Geometrical aberrations

Measured profile Sample contribution Background

0,0 0,2 0,4 0,6 0,8 1,0

0

20

40

60

80

100

Intensity

x

Extraction of the sample contribution Single peak

∑∑

=+

kj

)n(jkj

'kki)n(

i)1n(

ifg

hgff

f(x) obtained by deconvolution from h(x) [g(x) removal]

Stokes (1948): direct Fourier extraction Delhez et al. (1980): biases due to f(x) < g(x) and background

Richardson (1972): Bayesian deconvolution by iteration

Peak Profiles

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ θ−θ−

π=θ

2k

2ki

k

0i

H

)22(2ln4exp

H

2lnI2)2(GGauss:

Lorentz: m

2

k

kik

0i

H22

C1

1HCI

)2(L

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ θ−θ+

π=θ

m: Lorentzian order, [0,∞[ m = 1 "pure" Lorentzian function m = 1.5 "intermediate" Lorentzian function [Malmros et Thomas 1977] m = 2.0 "modified" Lorentzian function [Sonneveld et Visser 1975], Pearson VII

Constant wavelengths

V(x) = L(x) ⊗ G(x) Voigt:

Pseudo-Voigt: η)G(x)(1x)L(ηx)(pV −+=

η: mixing parameter

12

k

ki1

2

0k H

xxCAsAs11L

HQ)x(PVII

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠

⎞⎜⎝

⎛ ++=Split Pearson VII:

( )12

k

ki2

20

k HxxCAs11H

HQ)x(PVII

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ −++=

xi ≤ xk

xi > xk

12C 0L/11 −= 12C 0H/12 −=

k2

kk θsin/(3)Assinθ/(2)As(1)As)θ(As ++=

k2

0k00k0 θsin/(3)Lsinθ/(2)L(1)L)θ(L ++=

k2

0k00k0 θsin/(3)Hsinθ/(2)H(1)H)(θH ++=

Variable, parameterised Pseudo-Voigts, anisotropic …

TOF neutrons

∫+∞

∞−

−=⊗= t)E(t)dtG(xE(x)G(x)GE(x)h

βt-

hαt

tfor teβ)2(α

αβE(t)

tfor teβ)2(α

αβE(t)

>+

=

<+

=

Convolved Gaussian and rising and falling exponentials:

α and β: account for rising and falling exponential behaviours of the neutron pulse

∫+∞

∞−

−=⊗= t)E(t)dtpV(xE(x)pV(x)pVE(x)

Convolved pV and back-to-back exponentials:

0for teβα

αβE(t)

0for teβα

αβE(t)

βt-

αt

>+

=

≤+

=

Moderator pulse-shaped function (Ikeda et Carpenter 1985):

moderator theof constantsdecay :,

leakageneutron fast et2α

(t)S

leakageneutron slow Re(t)R)(1(t)R

with(t)R(t)S(t)I

αt23

k

βtk

kkk

βα

=

+δ−=

⊗=

−

−

Convolved pV and Ikeda-Carpenter …

Calibration of the instrument contribution

LaB6 NIST srm660b: flat-sample reflection geometry

FWHM (ω, χ, 2θ …) 2θ shift gaussianity asymmetry misalignments ...

0 20 40 60 80 100 1200.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4FW

HM

(2θ°

)

2θ(°)

D1B INEL CPS120

FWHM2 = U tan2θ + V tanθ + W (Thermal neutrons, Caglioti et al. 1958)

Evolution of the FWHM versus x: Laboratory X-rays and thermal neutrons

slits) ng(collimati W .const)2(

)dispersion(energy U tan2)2(

G

L

!

!

=θΔ

θλ

λΔ=θΔ (Lab. X-rays, Klug et Alexander 1974)

Accounts for instrument in simple cases Considering Gauss functions: )x(g)x(f)x(h ⊗=

)]x(g[FT)]x(f[FT)]x(h[FT =

]texp[)0(H)t(H ]/xexp[)0(h)x(h

]texp[)0(G)t(G ]/xexp[)0(g)x(g

]texp[)0(F)t(F ]/xexp[)0(f)x(f

2h

22h

2

2g

22g

2

2f

22f

2

βπ−=⇒βπ−=

βπ−=⇒βπ−=

βπ−=⇒βπ−=

spacedirect t ), (e.g. space reciprocal x ∈θ∈

]texp[)0(G]texp[)0(F ]texp[)0(H 2g

22f

22h

2 βπ−βπ−⇒βπ−

2g

2f

2h β+β=β

Lorentz functions: gfh ω+ω=ω

Gauss functions:

]h.csin[]h.c)1q(sin[

]h.bsin[]h.b)1p(sin[

]h.asin[]h.a)1n(sin[)h(T

)h(TFA

cba

cbahh

!!

!!!!!!

!!

!!!

!

!!!

!!!!!

π

+π

π

+π

π

+π=

=

directions c ,b ,a in the periods ofnumber :q p, ,n

function ceinterferen :)h(T

factor structure :F

amplitude scattered :A

cba

h

h

!!!

!!!!

!

!

Back on diffraction expression

0,0 0,2 0,4 0,6 0,8 1,0

0

20

40

60

80

100

H(α)

α

][sin])1n([sin)(H 2

2

πα

α+π=α

0,0 0,2 0,4 0,6 0,8 1,0

0

2000

4000

6000

8000

10000

H(α)

α

0,0 0,2 0,4 0,6 0,8 1,0

0

2

4

6

8

10

H(α)

α

n=9

n=2

n=99

(n+1)2

α+1/(n+1)

lh.ckh.bhh.a

:crystal infinite=

=

=

!!

!!!!

Simple peak broadening characterization

0,0 0,2 0,4 0,6 0,8 1,0

0

20

40

60

80

100

Intensity

θ(°)

ωω: Full-Width at Half-Maximum

θi θf

max2 I

d)(If

i

∫θ

θθ

θθ

=ββ: Integral Breadth (von Laue 1926)

θ0

Experimentally: β > ω

Crystallite’s size-shape effect

Scherrer formula (1918):

0h cos

KR

θω

λ=!

hR!

h!

K= 0.888 (Scherrer constant)

Depends on crystal shapes (Langford)

Since β > ω, Rh(β) < Rh(ω)

'Rh!

'h!

Scherrer analysis: Δθ

Δh

21

G0GIIIG ]H[H

1.8πε −=

]H[H1.8πε L0L

IIIL −=

Scherrer (1918)

Usually εL can be neglected (Delhez et al. 1993, Langford et al. 1993, Lutterotti et al. 1994)

cotanθβε

cotanθ4

βε

III

III

εIIImaxG,

εIIIG

><

><

=

= Stokes et Wilson 1944

If broadening only from microstrains

Williamson-Hall analysis

λsinθε4

R1

λcosθ

LIII

Lh

h

h +=ω Hall (1949), Lorentz-like

broadening

22

GIII

2G

2

λsinθε16

R

1λcosθβ

⎟⎠

⎞⎜⎝

⎛+=⎟⎠

⎞⎜⎝

⎛h

h

h Gauss-like broadening

λcosθhω

λsinθ

1LR −

h

LIIIε4 h

hkl 2h2k2l h’k’l’ 2h’2k’2l’

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

More elegant, mandatory for whole-pattern: Stokes deconvolution

Warren-Averbach-Bertaut analysis

Let construct a crystal as an integral of column-heights of the crystals:

( ) 1(R)dR with (R)dRmRh(m)0

mR=−= ∫∫

∞

≥DD

h(m): volume formed by all the columns of unit-base m: integer coordinate of the columns D(R): column-size distribution function

∫∞

=∂

∂

m(R)dR

mh(m)- D Fraction of columns of lengths R larger than m

(R)mh(m)- 2

2D=

∂

∂ Column-size distribution function

More elegant, mandatory for whole-pattern: Stokes deconvolution Bertaut-Warren-Averbach treatment, e.g. for a 00l peak:

( )∑ −π=

h

h

hhL

ssL2i-L 0eCΩ D

LSLL

RLL CCCCC ==

ε IIIhh

AS(L)

L

<R>

1

σD (L)

dL

AdL and (L)dL

Ad

R1

dLdA

Sv2

SL

2SA2

SL

2

F0L

SL

DD ==

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

→

What is size (coherent domain) ?

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+==

θβ

λ=

θ2

h

2R

hh

2h

h2h

R

σ1R

R

R

cosR h

!!

!

!

!! !

Apparent linear size (Bertaut 1949)

3h

4h

Vh2h

3h

Ah

2h2

h

hh

R

RR ;

R

RR

moment second :dS

dSRR

mean arithmetic :dS

dSRR

!

!!

!

!!

!!

!!

==

=

=

∫∫∫∫

If crystals with same sizes and shapes:

VhAhh R RR !!! ==

Recognised that β2θ overestimates Rh

∑ ∑Φ

=ΦΦΦ

=ΦΦΦΦ Ω+=

N

1iki

2k

K

Kkkkkibic AFPLpjSyy

1

Whole-Pattern (Rietveld) analysis

n

nn G

HF =

kiΦΩ Includes f(x) and g(x), but from Fourier deconvolution (Stokes 1948), extraction of f(x) is possible:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

πβ

β+

β

π

β

β=

G

L

GLi

xerfiRe)0(y)x(y

And, assuming Voigt profiles (Langford 1978, De Keijser et al. 1982, Balzar et Ledbetter

1993, Balzar 1999 ), separation of integral breadths are made is possible:

2gG

2hG

2fG

gLhLfL

β−β=β

β−β=β

Microstrains effect

λ=θsind2 h! 0dsin2cosd2 hh =∂θ+θ∂θ !!

h

hdd

tan !

!∂θ=θ∂ θε=ω tanIII

hh!!

IIIε

εIIε

Iε

r.m.s. d

d

h

hIIIh !

!!

∂=ε

Crystallite sizes, shapes, µstrains, distributions

X-rays

ω

(111)

(111)

<111>

X-rays

ω

(111)

(111)

<Rh> = R0 + R1P20(x) + R2P2

1(x)cosϕ + R3P21(x)sinϕ + R4P2

2(x)cos2ϕ + R5P22(x)sin2ϕ +

...<εh2>Eh4 = E1h4 + E2k4 + E3ℓ4 + 2E4h2k2 + 2E5ℓ2k2 + 2E6h2ℓ2 + 4E7h3k + 4E8h3ℓ + 4E9k3h +

4E10k3ℓ + 4E11ℓ3h + 4E12ℓ3k + 4E13h2kℓ + 4E14k2hℓ + 4E15ℓ2kh

Popa Line Broadening

•  Texture helps the "real" mean shape determination

∑∑==

ϕχ=ℓ

ℓℓℓ

"

0m

mmL

0h ),(KRR

Symmetrised spherical harmonics

)msin()(cosP)mcos()(cosP),(K mmm ϕχ+ϕχ=ϕχ ℓℓℓ

R0 R0, R1 < 0 R0, R1 > 0

R0, R6 > 0 R0, R2 and R6 > 0 R0, R6 < 0

R0, R4 > 0 R0, R1 > 0 R0, R1 < 0

m3m 6/m

1

Crystallite size (Å) along

Film thickness 10nm 15nm 20nm 25nm 35nm 40nm

[111] 176 153 725 254 343 379 [200] 64 103 457 173 321 386 [202] 148 140 658 234 337 381

10 nm 15 nm 20 nm

25 nm 35 nm 40 nm

Gold thin films

EMT nanocrystalline zeolite

Ng, Chateigner, Valtchev, Mintova: Science 335 (2012) 70

New active Li–Mn–O compound for high energy density Li-ion batteries

Freire, Kosovab, Jordy, Chateigner, Lebedev, Maignan, Pralong: Nature Mat. 15 (2016) 173

Rock-salt-type nanostructured material: shows a discharge capacity of 355 mAh/g