Powder Diffraction & Synchrotron Radiation

$: Powder Diffraction & Synchrotron Radiation$
•Prof. Paolo Scardi, Università di Trento

•Diffrazione da Materiali Policristallini •1

Powder Diffraction & Synchrotron Radiation

P. ScardiDepartment of Materials Engineering and Industrial Technologies

University of Trento

International Doctoral School in International Doctoral School in Materials Science & EngineeringMaterials Science & Engineering

About 15 new positions per year

University of Trento

Information and applications:

http://portale.unitn.it/drmse/

next deadline: September 14th

[email protected]



3

P. Scardi – Powder Diffraction & Synchrotron RadiationX SILS School - Duino, 11.09.2009

PRESENTATION OUTLINE

• Main applications of XRPD and RS-XRPD

PART I

• Some advantages and peculiaritiesof synchrotron radiation X-ray powderdiffraction (SR-XRPD)

PART II

• Diffraction from nanocrystalline andhighly deformed materials

4


single crystal100

110

020120220

powder(bulk polycrystalline)100

110

020120220

100

110

020120220

DIFFRACTION: SINGLE CRYSTAL AND POWDER



5


DIFFRACTION: SINGLE CRYSTAL AND POWDER

6


DEBYE-SCHERRER GEOMETRY

POWDER



7


SRXRD POWDER GEOMETRY: A TYPICAL EXAMPLE

ID31 Goniometer andnine-crystal analyzer

X-r

ayde

tect

or

Parallel beam geometry at ID31 (ESRF)capillary holder / high temperature blower

8


SRXRD POWDER GEOMETRY: A TYPICAL EXAMPLEParallel beam geometry of MCX (Elettra)



9



10



BEAMTIME APPLICATIONS:DEADLINE IS SEPTEMBER 15th



11


TYPICAL LAB GEOMETRY: BRAGG-BRENTANO (POWDER)

BULK orPOWDER

source

detector

20 40 60 80 100 120 1400

10002000

3000

4000

5000

6000

7000

Inte

nsity

(co

unts

)

2θ (degrees)

12


(why) do we need synchrotron radiation?

LAB vs SR XRD

… don’t use a cannonto kill a fly !

quoted by G. Artioli



13


1) High brillance, much better counting statistics / shorter data collectiontime (à fast kinetics, in situ studies)

40 60 80 100 120 140

10

100

1000

Inte

nsity

(co

unts

)

2θ (degrees) 10 2 0 30 40 50 60 70 80 9 0 100

10

100

Inte

nsity

(x10

3 cou

nts)

2θ (degrees)

SOME ADVANTAGES OF SRXRD

CuKα λ=0.15406 nm ESRF ID31 λ=0.0632 nmiron powder (ball milled)

14


2) With proper selection of optics, very narrow instrumental profile:increased resolution and accuracy in the measurement of peakposition, intensity and profile width/shape.

Lab instrument: ID31 @ESRF: FWHM≈0.05-0.1° FWHM≈0.003-0.004°




15


3) Extending the accessible region of reciprocal space well beyond whattraditional lab instruments can make

λ1 λ2<λ1


16


10 20 30 40 50 60 70 80 90 100

10

100 (110

)

(200

)

(2

11)

(220

)

(3

10)

(222

)

(3

21)

(400

)

(3

30),

(411

)(4

20)

(332

)

(4

22)

(431

), (5

10)

(521

)

(4

40)

(433

), (5

30)

(600

), (4

42)

(532

), (6

11)

(620

)

(5

41)

(622

)

(6

31)

(444

)

Inte

nsity

(x10

3 cou

nts)

2θ (degrees)

CuKα λ=0.15406 nm ESRF ID31 λ=0.0632 nm


40 60 80 100 120 140

10

100

1000

Inte

nsity

(co

unts

)

2θ (degrees)

(110

)

(200

)

(211

)

(220

)

(310

)

(222

)

3) Extending the accessible region of reciprocal space well beyond whattraditional lab instruments can make



17


4) Tuning the energy according to adsorption edges. Resonant scattering, control of fluerescence emission and depth of analysis.


4 6 8 10 12 14 16 18 200

200

400

600

µ/ρ

(cm

2 /g)

X-ray energy (keV)

Absorption edge of Fe

CuK

α

0

tI I e

µ ρρ

− =

18


X-RAY POWDER DIFFRACTION

• Crystal structure determination(Powder diffraction structure solution and refinement)

• Phase Identification – pure crystalline phases or mixtures(Search-Match procedures)

• Quantitative Phase Analysis (QPA)

• Crystalline domain size/shape and lattice defect analysis(Line Profile Analysis - LPA)

• Determination of residual stress field (Residual Stress Analysis)

• Amorphous phase analysis (radial distribution function)

most frequent applications

• Determination of preferred orientations (Texture Analysis)



19


Structure solution of heptamethylene-1,7-bis(diphenylphosphane oxide)

Structural formulaPh2P(O)(CH2)7P(O)Ph2

B.M. Kariuki , P. Calcagno, K. D. M. Harris, D. Phi lp and R.L. Johnston, Angew. Chem. Int. Ed. 1999, 38, No. 6, 831-835.

STRUCTURE SOLUTION: WHY POWDER ?

20


Structure solution/refinement of a complex triclinic organic compound (C24H16O7)

STRUCTURE SOLUTION & REFINEMENT: SRXRD

K. D. Knudsen et al., Angew. Chem. Int. Ed., 37 (1998) 2340

• Narrow peak profiles• Large number of measurable peaks• Accurate peak position/intensity• X-ray energy tuning to adsorption edges



21


Site occupancy in battery electrode material LaNi3.55Mn0.4Al0.3Co0.75)J.-M. Joubert et al., J. Appl. Cryst. 31 (1998) 327



?????????

22






23





24


Solving Larger Molecular Crystal Structures from Powder Diffraction Data by Exploiting Anisotropic Thermal Expansion, M. Brunelli et al., Angew. Chem. Int. Ed. 42, 2029, (2003)

90 K

130 K

160 K

• Narrow peak profiles• Large number of measurable peaks• Accurate peak position/intensity• X-ray energy tuning to adsorption edges• Anisotropic thermal expansion




25


à in research & in industry (production, quality control and diagnostics)Each crystalline phase has its own pattern that can be used as a ‘fingerprint’

‘Fingerprints’ of unknown substances can be compared with those of known crystalline phases of a database à Search-Match procedures

PHASE IDENTIFICATION

26


ICDD datababes(International Centre for Diffraction Data – www.icdd.com)

PDF-2Peak pos/int

PDF-4

full structuralinformation




27


Automatic search-match procedures based on peak position / intensity


28


Intensity of i-th data-point in the pattern

, , ,ci j k j k j k j bij ky S I P yφ= ⋅ ⋅ +∑ ∑

Integrated Intensityk-th peak of j-th phase

∝ |F|2

Scale factorof j-th phase Profile function

Background term

THE RIETVELD METHOD

PreferredOrientation

F(hkl) = (1/V) Σj fj(S) e2πi(hxj + kyj + lzj)e-Bj sin2θ/λ2

From the lecture of G. Zanotti



29


In a polyphasic mixture: weight fraction xj of the phase j

j j jj

l l ll

S vx

S vρ

ρ=

∑

THE RIETVELD METHOD

, , ,ci j k j k j k j bij ky S I P yφ= ⋅ ⋅ +∑ ∑

Integrated Intensityk-th peak of j-th phase

∝ |F|2

Scale factorof j-th phase Profile function

Background termPreferredOrientation

à normalization condition (Σ phase fraction = 1)

Intensity of i-th data-point in the pattern

30


Example: mixture of mineral phases in a ligand

RIETVELD-BASED QPA

2Th Degrees6560555045403530252015

Sqr

t(Cou

nts)

140130120110100908070605040302010

0-10-20-30

Lime CaCO3 26.28 %Dolomite CaMg(CO3)2 9.49 %Quartz 2.39 %Gypsum 0.69 %Bassanite 30.67 %Anhydrite 22.34 %Belite C2S 8.13 %



31


Structural and electronic properties of noncubic fullerides A’40C60 (A’=Ba,Sr)

STRUCTURE SOLUTION IN MULTIPHASE SAMPLES


C.M. Brown et al., Phys. Rev. Let. 83 (1999) 2258

ESRF BM161 λ=0.084884 nm

32


Structural and electronic properties of noncubic fullerides A’40C60 (A’=Ba,Sr)


C.M. Brown et al., Phys. Rev. Let. 83 (1999) 2258

STRUCTURE SOLUTION IN MULTIPHASE SAMPLES



33


Crystalline structure

↓long-range order

AMORPHOUS PHASE ANALYSIS

Amorphous phase

↓short-range order

34


Amorphous specimen of volume V (N atoms with scattering factor f, ):

AMORPHOUS PHASE ANALYSIS

( ) ( ) ( )20

11 4 2

2π ρ ρ π

π

≅ + −

∫V

I s Nf r r Sin sr drs

( ) ( ) ( )0 20

4 4 8 1 2I s

r r r s Sin sr dsNf

π ρ π ρ π π∞

≅ + −

∫By Fourier inversion:



35


Structure of nanocrystalline materials using atomic Pair Distribution Function (PDF) analysis: study of LiMoS2.

V. Petkov et al., Phys. Rev. B 65 (2002) 092105

PAIR DISTRIBUTION FUNCTION: USE OF SRXRD

( ) ( ) 0: 4π ρ ρ = − reduced PDF G r r r

36


Structure of nanocrystalline materials using atomic Pair Distribution Function (PDF) analysis: study of LiMoS2.

V. Petkov et al., Phys. Rev. B 65 (2002) 092105

PAIR DISTRIBUTION FUNCTION: USE OF SRXRD

( ) ( ) 0: 4π ρ ρ = − reduced PDF G r r r



37


LINE PROFILE ANALYSIS: SRXRDXerogel obtained by vacuum-drying: broad diffraction lines of nanocrystalline fcc phase

10 20 30 40 50 60 70 80 90 100

0

10000

20000

30000

40000

50000

60000

70000

80000

Inte

nsity

(co

unts

)

2θ (degrees)

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0<D>=2.25(10) nm

p(D

) (a

.u.)

D (nm)

P. Scardi & Leoni , ECS Transactions, 3(9) (2006) 125.

75 peaks

ESRF ID31 - glass capillary, λ=0.06325 nm

0.8 hour

38


QUESTIONS ??



39


Scattering from a small crystal of Cu (fcc)SCATTERING FROM NANOCRYSTALS

( ) ( )* *2 2m ni d r i d rm n

m n

f e f eπ π⋅ − ⋅

= ∑ ∑

mr

nrO

*c m n

m n

I A A∝ ∑ ∑

2 *I A AA∝ =

One scattering centre (electron, atom, unit cell)0s λ

s λ*d

40


mr

nrO

( ) ( )* *2 2m ni d r i d ruc m n

m nI f e f e

π π⋅ − ⋅∝ ∑ ∑

SCATTERING FROM NANOCRYSTALSScattering from a unit cell of Cu (fcc)



41


mr

nrO

mn m nr r r= −

( ) ( )* *2 2m ni d r i d ruc m n

m nI f e f e

π π⋅ − ⋅∝ ∑ ∑

SCATTERING FROM NANOCRYSTALSScattering from two atoms in a Cu (fcc) unit cell

( )*2 mni d rm n

m nf f e

π ⋅= ∑∑

42


( ) ( )* *2 2m ni d r i d ruc m n

m nI f e f e

π π⋅ − ⋅∝ ∑ ∑

SCATTERING FROM NANOCRYSTALSScattering from two atoms in a Cu (fcc) unit cell

( )*2 mni d rm n

m nf f e

π ⋅= ∑∑

mnr

0s λ

s λ

*d0* 2 sins s

d θλ λ−

= =

2θ



43


Two possible approaches

1. Reciprocal space approach(Laue – Wilson)

2. Direct space (or Real space) approach(Debye)

SCATTERING FROM A POWDER

44


SCATTERING FROM NANOCRYSTALS

mnr

0s λ

s λ

*d

2θnn n nu a vr b w c= + +

( ) ( )* *2 2π π⋅ − ⋅∝ ∑ ∑m ni d r i d r

uc m nm n

I f e f e ( )2

2

1

π + +

=

= ∑ n n nN

i u h v k w ln

n

f e

0s s ha lcd kbλ

∗ ∗∗ ∗−= = + +

2F=

Two possible approaches - #1 reciprocal space1. Factorize the contribution from a

unit cell (|F|2 – F, structure factor )



45


Then build the diffraction signalfor a small crystal (unit cell volume, Vuc)

SCATTERING FROM A POWDERTwo possible approaches - #1 reciprocal space

1. Factorize the contribution from aunit cell (|F|2 – F, structure factor )

( )2

2 2

1

n n nN

i u h v k w luc n

nI F f e π + +

=

∝ = ∑

(Interference function) à see ZANOTTI’s lecture

( ) ( ) ( )2 2 2 2

2 2 2 2

sin sin sinsin ( ) sin ( ) sin ( )sc

uc

F Nh Nk NlI

V h k lπ π ππ π π

∝ L=Na

a

( ) ( ) ( )2 2 2 2

2 2 2 2 2 2 2' ' '

sin sin sin( ') ( ') ( ')sc

h k luc

F Nh Nk NlI

V h h k k l lπ π π

π π π

∞ ∞ ∞

=−∞ =−∞ =−∞

∝− − −∑ ∑ ∑

46




(Interference function)

( ) ( ) ( )2 2 2 2

2 2 2 2

sin sin sinsin ( ) sin ( ) sin ( )sc

uc

F Nh Nk NlI

V h k lπ π ππ π π

∝

( ) ( ) ( )2 2 2 2

2 2 2 2 2 2 2' ' '


h k luc

F Nh Nk NlI


π π π

∞ ∞ ∞

=−∞ =−∞ =−∞

∝− − −∑ ∑ ∑

010 110 210

000 100 200

010 110 210

2θ

0ν λ

ν λ

Inte

nsity

h = 2h = 10




47



(Interference function)

( ) ( ) ( )2 2 2 2

2 2 2 2 2 2 2' ' '


h k luc

F Nh Nk NlI


π π π

∞ ∞ ∞

=−∞ =−∞ =−∞

∝− − −∑ ∑ ∑

010 110 210

000 100 200

010 110 210

2θ

0ν λ

ν λ

( )2100

2 2

sin( 1)

ππ

∝−scNh

Ih

Example:(100) point

Inte

nsity

h = 2h = 10

Two possible approaches - #1 reciprocal space1. Factorize the contribution from a

unit cell (|F|2 – F, structure factor )


48


-2 -1 0 1 2

-2

-1

0

1

2

2 λ

Powder Diffractionsphere

NANOCRYSTAL à POWDERTwo possible approaches - #1 reciprocal space



49


-2 -1 0 1 2

-2

-1

0

1

2

2 λ

NANOCRYSTAL à POWDERTwo possible approaches - #1 reciprocal space

50


Example: (001) peak, powder made of cubic crystallites, cube edge L

001

002

0s sθ θ

*001d

h - [100]

k - [010]

( ) ( ) ( ) ( )

2 2 2 22 2

2 2 2 2

sin ( ) sin ( ) sin ( ) sin ( )Nh Nk Nl NlI F dh dk Fh k l lπ π π π

π π π π∝ ⋅ →∫∫

1=l

l

I(l)

l - [001]

NANOCRYSTAL à POWDER

( )2

2sin ( )1

(0)

π

π

∞

−∞ =∫

Nl

ldl

I N

L=Na

a

Two possible approaches - #1 reciprocal space



51




Then build the diffractionsignal for a small crystal,

and integrate over thepowder diffractionsphere for calculatingthe signal from alldomains in the powder

100

110

020120220

100

110

020120220

( )2 * ,PDI F d D∝ ΦL

standard Powder Diffraction approach

( ) ( ) ( )2 2 2

2 2 2

sin sin sinsin ( ) sin ( ) sin ( )

π π ππ π πNh Nk Nlh k l

52



mr

nrO

mn m nr r r= −

( ) ( )* *2 2m ni d r i d ruc m n

m nI f e f e

π π⋅ − ⋅∝ ∑ ∑ ( )*2 mni d r

m nm n

f f eπ ⋅

= ∑∑

Two possible approaches - #2 real space2. Average over all possible orientations of

rmn in space



53


( ) ( )* *2 2π π⋅ − ⋅∝ ∑ ∑m ni d r i d r

sc m nm n

I f e f e


( )*2 mni d rm n

m nf f e

π ⋅= ∑∑

0* 2sins sd θ

λ λ−

= =0s λ

s λ

*d

2θmnr

** cosmn mnd d rr φ= ⋅

φ

Two possible approaches - #2 real space2. Average over all possible orientations of rmn in space

54


mnr

0s λ

s λ

*d

( )*2 mni d rD m n

m n

I f f eπ ⋅

∝ ∑∑ ( )*

*

sin 22

mnm n

m n mn

d rf f

d rπ

π= ∑∑

Two possible approaches - #2 real space

2. Average over all possible cosφ values:rmn is allowed to take all possibleorientations in space


* cosmmn nd rd r φ= ⋅

φ

Debye formula



55


Debye formula for one (fcc) unit cell

*

*

sin 22

mnD m n

m n mn

d rI f f

d rπ

π= ∑∑

( ) ( ) ( ) ( )72sin 2 48sin 3 2 24sin 2 8sin 330sin142 3 2 2 3D

ka ka ka kakaIkaka ka ka ka

= + + + + +

0mnr =

a

2a a 3 2a 2a 3a

14 72 30 48 24 8

SCATTERING FROM A NANOCRYSTAL POWDER

56


0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

11

12

00

22

0

31

12

22

40

0

33

14

20

42

2

33

35

11

44

0

Inte

nsity

(a.u

.)

2θ (degrees)

Scattering from (random oriented) Cu unit cells . Mo Kα (0.07093 nm)

( ) ( ) ( ) ( )72sin 2 48sin 3 2 24sin 2 8sin 330sin142 3 2 2 3D

ka ka ka kakaIkaka ka ka ka

= + + + + +

Cu bars: multiplicity




57


0 10 20 30 40 50 60 700

20

40

60

80

100

120

1401

11

20

0

22

0

31

12

22

40

0

33

14

20

42

2

33

3,5

11

44

0

Inte

nsity

(a.

u.)

2θ (degrees)

Scattering from (random oriented) Cu unit cells . Mo Kα (0.07093 nm)2 2

*dk uDI LP e−

⋅ ⋅

Cu bars: ICSD #46699


58


Scattering from bcc-Fe cubic crystals . Cu Kα (0.15406 nm)


0 20 40 60 80 100 120 140 1600

20k

40k

60k

80k

100k

120k

(222)

(311)

(220)

(211)

(200)

Inte

nsity

2θ (degrees)

(110)

0 20 40 60 80 100 120 140 1600

500

1000

1500

2000

(222)

(311)

(220)

(211)

(200)

Inte

nsity

2θ (degrees)

(110)

5x5x5 15x15x15

3x3x31x1x1

0 20 40 60 80 100 120 140 1600

100

200

300

400

(222)

(311)

(220)

(211)

(200)

Inte

nsity

2θ (degrees)

(110)

0 20 40 60 80 100 120 140 1600

5

10

15

20

(222)

(311)

(220)

(211)

(200)

Inte

nsity

2θ (degrees)

(110)



59




0 20 40 60 80 100 120 140 1600

20k

40k

60k

80k

100k

120k

(222)

(311)

(220)

(211)

(200)

Inte

nsity

2θ (degrees)

(110)15x15x15

7471 atoms

60




0 2 4 6 8 10 12 140

100k

200k

300k

400k

500k

Inte

nsity

2θ (degrees)

15x15x157471 atoms



61




0 2 4 6 8 10 12 14100

1k

10k

100k

1M

10M

100MIn

tens

ity

2θ (degrees)

15x15x157471 atoms

I(0) = 55.815.841 = 74712

62



Twinned spherical domains (fcc, 28897 Au atoms, nominal size 9.8 nm)

0 20 40 60 80 100 120 140 160105

106

107

108

109

1010

1011

1012

Inte

nsity

2θ (degrees)

K. Beyerlein, Univ. of Trento



63


The Debye formula approach can be extended to a distribution of nanocrystals, also including a radial, homogeneous strain

POWDER DIFFRACTION LINE PROFILE ANALYSIS

C - cubooctahedron

I - icosahedron

D - decahedron

A. Cervellino et al., J. Appl. Cryst. 36 (2003) 1148

64



graphene(sp2 carbon single-layer)

L. Gelisio, Univ. of Trento



65



Carbon nanotubes

L. Gelisio, Univ. of Trento

66



20 40 60 80 100 120 140 1600

1x106

2x106

3x106

4x106

Inte

nsity

(a.

u.)

2θ (degrees)

cluster of nanocrystalline grains: 7.6 million atomsatomistic simulations – molecular dynamics

A. Leonardi , Univ. of Trento



67


QUESTIONS ??

68


Two possible approaches

1. Reciprocal space approach(Laue – Wilson)

2. Direct space (or Real space) approach(Debye)




69


CRYSTAL, NANOCRYSTAL AND POWDER: SIZE EFFECT

LARGE crystal

100

110

020120220

NANOcrystal

5 nm

NANOcrystal

POWDER 10 nm

70


-2 -1 0 1 2

-2

-1

0

1

2


2 λ

Ewald sphere

0s λ

s λ2θ

d ∗

Powder Diffractionsphere



71


-2 -1 0 1 2

-2

-1

0

1

2

2 λ


72


Powder made of simple-shape crystallites(one param., convex solids: sphere, cube, tetrahedron, octahedron,…)

( )* Kd

Dββ =

DIFFRACTION FROM A POLYCRYSTALLINE MATERIAL

Kβ , Scherrer constant, changes with crystallite shapes and (hkl)

Paul Scherrer (1890–1969)

Scherrer formula (1918)



73


Powder made of simple-shape crystallites(one param., convex solids: sphere, cube, tetrahedron, octahedron,…)


Kβ , Scherrer constant, changes with crystallite shapes and (hkl). For a sphere:

( )* Kd

Dββ =

Paul Scherrer (1890–1969)

Scherrer formula (1918)

43

Kβ =

74


TRADITIONAL LPA: INTEGRAL BREADTH METHODS

( )* Kd

Dββ =( )2

cosK

Dβ λ

β θθ

=

Scherrer formula2θ space recipr. space

Profile information can be represented by the Integral Breadth β, (peak area / peak maximum). Assuming domain size effects only:

20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

Inte

nsity

(co

unts

)

2θ (degrees)

20 30 40 50 600

1000

2000

3000

4000

5000

6000

7000

Inte

nsity

(co

unts

)

2θ (degrees)



75


CAVEAT #1 – what is the true result of the Scherrer formula ?


The integral breadth still provides a valid mean size value, but:

( )* 3

4V

K Md KD M

βββ = =

< >where M3, M4 are 3rd and 4th moments of g(D) ( )( )i

iM D g D dD= ∫

M1à meanM2 - M1

2 à variance

For a DISTRIBUTION of crystallitesg(D)

D

76


( )*2 2 iLd LcI F e π ε∝

b

ac

a

L=n·a

undistorted

b

ac

a

L=n·a

dL

distorted

Quite complex: unit cells at distance L=na can be displaced and rotated.Neglecting rotation and considering an average strain dL Lε =

MICROSTRAIN EFFECT IN POWDER DIFFRACTION

CAVEAT #2 – what is the effect of lattice distortions ?



77


Considering both domain size and lattice distortion (microstrain) effects

( ) ( )* 2 cosdβ β θ θ λ = ⋅

INTEGRAL BREADTH METHODS

( ) 1 / 222 2 tancosV

KD

β λβ θ ε θ

θ≈ +

< >

(as a first order approximation):

78


( )* *2V

Kd e d

Dββ = + ⋅

< >

Considering both domain size and lattice distortion (microstrain) effects

( ) ( )* 2 cosdβ β θ θ λ = ⋅

in a β(d*) vs. d* plot, intercept and slope of linear regression are related, respectively, to <D>V and e


0 2 4 6 8 10 120,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

β(d*

) (n

m-1)

d * (nm -1) [ =2sinθ/λ ]

Williamson-Hall (WH) plot



79


Peak profiles invariably overlap in powder patterns. This can make it difficult to extract profile information directly from observed data


20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

Inte

nsity

(co

unts

)

2θ (degrees)

20 40 60 80 100 120 140

100

1000

10000

111

200

220

311

222

400

331

420

422

511/

333

440

531

600/

442

620

533

622

Inte

nsity

(co

unts

)

2θ (degrees)

80


… a step forward



81


20 40 60 80 100 120 1400

1000200030004000500060007000 Sol-gel cerium oxide powder, 1h @ 400°C

Inte

nsity

(co

unts

)

2θ (degrees)

Microstructural Parameters

DiffractionPattern

WPPMPhysical Model

WPPM is based on a direct modelling of the experimental pattern, based on physical models of the microstructure and lattice defects:

WHOLE POWDER PATTERN MODELLING

20 40 60 80 100 120 140

01000200030004000500060007000

Nanocrystalline cerium oxidefrom xerogel, 1h @ 400°C

Inte

nsity

(co

unts

)

2θ (degrees)M. Leoni, R. Di Maggio, S. Polizzi & P. Scardi, J. Am. Ceram. Soc. 87 (2004) 1133.

P.Scardi & M. Leoni, Acta Cryst. A 57 (2001) 604. P.Scardi & M. Leoni, Acta Cryst. A 58 (2002) 190

82


0 2 4 6 8 10 120

5

10

15

20

25

30

35

40 TEM WPPM

Freq

uenc

y

Grain diameter (nm)

5 nm

*hkld

*hk ld

ABC

AB

AB

C

*hkld

*hk ld

(…)


20 40 60 80 100 120 1400

1000200030004000500060007000 Sol-gel cerium oxide powder, 1h @ 400°C

Inte

nsity

(co

unts

)

2θ (degrees)20 40 60 80 100 120 140

01000200030004000500060007000

Nanocrystalline cerium oxidefrom xerogel, 1h @ 400°C

Inte

nsity

(co

unts

)

2θ (degrees)M. Leoni, R. Di Maggio, S. Polizzi & P. Scardi, J. Am. Ceram. Soc. 87 (2004) 1133.

P.Scardi & M. Leoni, Acta Cryst. A 57 (2001) 604. P.Scardi & M. Leoni, Acta Cryst. A 58 (2002) 190



83


Microstructural Parameters

DiffractionPattern

WPPMPhysical Model

WPPM is based on a direct modelling of the experimental pattern, based on physical models of the microstructure and lattice defects:


How does it work ??

84


DIFFRACTION LINE PROFILE: CONVOLUTION OF EFFECTS

So far we consider that different effects affecting the line profile simply ‘add’, i.e., the peak width is the sum of different components.

According to the Williamson-Hall formula,

( )* *2ββ = + ⋅< >V

Kd e d

D

‘size’ ‘strain’

Actually, this is not the general case …



85



A diffraction peak is a convolution ( ) of profile components produced by different sources: instrumental profile (IP), domain size (S), microstrain (D), faulting (F), anti-phase domain boundaries (APB), stoichiometry fluctuations (C), grain surface relaxation (GSR), etc.

⊗

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ...IP S D F APB C GRSI s I s I s I s I s I s I s I s= ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

h = g ⊗ f

What is the difference between convolution and sum of effects ??

86



What is the difference between convolution and sum of effects ??

2θ 2θ2θ

⊗

g profile, slit (box) function; f profile, bell-shape function (e.g. gaussian)

( ) ( ) ( )= ⊗IP SI s I s I sExample: let’s just consider instrument (IP) and domain size (S):

( ) ( )( )= −∫ IP SI s I t I s t dt



87



h fg= ⊗

2θ

2θ

2θ

2θ

2θ

2θ

2θ

2θ

2θ

2θ

88



A diffraction peak is a convolution ( ) of profile components produced by different sources: instrumental profile (IP), domain size (S), microstrain (D), faulting (F), anti-phase domain boundaries (APB), stoichiometry fluctuations (C), grain surface relaxation (GSR), etc.

⊗


h = g ⊗ f

the Fourier Transform of I(s) is the product of the FTs of the single profile components



89


the Fourier Transform of I(s) is the product of the FTs of the single profile components

( ) ( ) 2e hkliL sLI s C d Lπ ⋅∞

− ∞

∝ ⋅ ∫


The diffraction profile results from a convolution of effects:

WPPM : HOW DOES IT WORK ??

P. Scardi & M. Leoni J. Appl. Cryst. 39 (2006) 24 - P. Scardi & M. Leoni, Acta Cryst. A58 (2002) 190

( ) ...IP S D F F APBi pV hkl hklhkl hkl hkl

i

C A T A A A iB A= = ⋅ ⋅ ⋅ + ⋅ ⋅ ∏instr. profile lattice defects / straindomain

size/shape

90


( ) ( ) ( ) ( )2 2 21 exp ln2 exp 2IPpV s sT L k L k Lπ σ π σ= − ⋅ − ⋅ + − ⋅ Instrumental profile

Ai(L) EXPRESSIONS (ANALYTICAL OR NUMERICAL FORM)

Domain size effect: µ, σ

( ) 23,3

0 ,3

ln (3 )( )

22

cl nS c n

nn l

L K n MA L H Erfc L

M

µ σ

σ−

=

⋅ − − − = ⋅ ⋅

∑

Dislocation (strain) effect: ρ, Re,(Chkl)

( )22 * 2 *

1( ) exp2

Dhklhkl ehklA L b C d L f L Rπ ρ = − ⋅

( )

( )

*22

1 22 2

12( ) 1 3 2 3

( ) 3 6 12 12

hkl

o

oLo

oFhkl

F ohkl L

o

LLdhA L

LLB LL L

σα β α

σ β β α β α

⋅= − − +

= − ⋅ ⋅ ⋅ − − − +

Anti-Phase Domains: γ

( )( ) 2 2 2

2( ) expAPB

hklhkl

h k LA L

d h k lγ − + ⋅

= −+ +

ABC

AB

ABC

*hkld

*hk ldFaulting: α (def.), β (twin)

*hkld

*hk ld

0 2 4 6 8 10 120

5

10

15

20

25

30

35

40 TEM WPPM

Fre

quen

cy

Grai n d iame te r (nm)5 nm

( )2 2 2 2 2 2

22 2 2hkl

h k k l l hC A B A B Hh k l

+ += + ⋅ = + ⋅

+ +



91


WPPM APPLICATIONS

TWO EXAMPLES


92


10 20 30 40 50 60 70 80 90 100

0

10000

20000

30000

40000

50000

60000

70000

80000

10 15 20 25 3 0 35

10000

20000

30000

40000

50000

60000

70000

80000

Inte

nsity

(co

unts

)

2θ (degrees)

Inte

nsity

(co

unts

)

2θ (degrees)

Xerogel obtained by vacuum-drying: broad diffraction lines of nanocrystalline fcc phaseESRF ID31 - glass capillary, λ=0.6325 nm

WPPM APPLICATIONS: NANOCRYSTALLINE CERIA

P. Scardi, M. Leoni, ECS Transactions, 3 (2006) 125



93


WPPM APPLICATIONS: NANOCRYSTALLINE CERIAXerogel obtained by vacuum-drying: broad diffraction lines of nanocrystalline fcc phase

10 20 30 40 50 60 70 80 90 100

0

10000

20000

30000

40000

50000

60000

70000

80000

Inte

nsity

(co

unts

)

2θ (degrees)

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0<D>=2.25(10) nm

p(D

) (a

.u.)

D (nm)

ESRF ID31 - glass capillary, λ=0.6325 nm

P. Scardi, M. Leoni, ECS Transactions, 3 (2006) 125

94


NANOCRYSTALLINE Fe-1.5%Mo POWDER Planetary ball milling - production of nanocrystalline Fe-1.5%Mo

Ωω



95


10 20 30 40 50 60 70 80 90 100

0

50

100

150

200

250

300

350

400

450

Inte

nsity

(x10

3 cou

nts)

2θ (degrees)

10 20 30 40 50 60 70 80 90 100

10

100

Inte

nsity

(x10

3 cou

nts)

2θ (degrees)

2 hours

Ball milled Fe1.5Mo (Fritsch P4) – data collected at ESRF – ID31 λ=0.0632 nm

NANOCRYSTALLINE Fe-1.5%Mo POWDER

M. D’Incau, M. Leoni & P. Scardi, J. Mat. Research, 22 (2007) 1744

96


10 20 30 40 50 60 70 80 90 100

0

50

100

150

200

250

Inte

nsity

(x10

3 cou

nts)

2θ (degrees)

(b)

10 20 30 40 50 60 70 80 90 100

10

100

Inte

nsity

(x10

3 cou

nts)

96 hours






97


10 20 30 40 50 60 70 80 90 100

0

50

100

150

200

250

In

tens

ity (x

103 c

ount

s)

2θ (degrees)

(b)



0 20 40 60 80 100 120 1400,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

Ball milling time (h)

Dis

loca

tion

dens

ity, ρ

(x1

016 m

-2)

0

20

40

60

80

100

120

140

160M

ean domain size, D

(nm)


98


0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 00 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0 0 h 2 h 16 h 32 h 64 h 128 h

Dom

ain

size

dis

tribu

tion,

g(D

)

D (n m )

0 20 40 60 80 100 120 1400.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ball milling time (h)

Dis

loca

tion

dens

ity, ρ

(x1

016 m

-2)

0

20

40

60

80

100

120

140

160

Mean dom

ain size, D (nm

)

Ball milled Fe1.5Mo (Fritsch P4) – data collected at ESRF – ID31 λ=0.0632 nmIn addition to mean values, WPPM provides the size distribution





99


Diffraction Analysis of Materials MicrostructureE.J. Mittemeijer & P. Scardi, editors.Berlin: Springer-Verlag, 2004.

[email protected]

Powder Diffraction: Theory and PracticeR.E. Dinnebier & S.J.L. Billinge, editors.Cambridge: RSC Publishing, 2008. Cap. XIII, p.376

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