Diffraction from nanocrystalline...

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X-ray Diffraction

[email protected]

Special thanks to: Luca Gelisio, Alberto Leonardi, Luca Rebuffi, Cristy L. Azanza Ricardo,Mirco D’Incau, Andrea Troian, Emmanuel Garnier, Mahmoud Abdellatief

• Basic aspects of x-ray crystallography and powder diffraction

• Diffraction from nanocrystalline materials

mailto:[email protected]

P. Scardi – Diffraction from nanocrystalline materialsICTP School - Trieste, 04.04.2016 62

FROM SINGLE CRYSTAL TO POWDER DIFFRACTION

( )( )

24sc

PD

I s dI s

sπ

Ω∝ ∫ ( )2F I s=

perfect (infinite) crystalD

( ) ( )2* mni S rsc m n

m nI s f f e π ⋅∝ ∑∑

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

1. Traditional reciprocal space approach : sum & average

P. Scardi – Diffraction from nanocrystalline materialsICTP School - Trieste, 04.04.2016

Debye formula (Direct Space)

L real nanocrystals are complex objects

DIFFRACTION FROM NANOCRYSTALLINE MATERIALS

CdS-CdSe OCTAPODS

non-crystallographic (e.g. multiply twinned) nanoparticles, 2D and highly disordered layer systems:Ø translational symmetry: not verifiedØ large strain / misfit – complex local atomic arrangement



( )( )2 2

24

mni s r

m nPD

f e dI s

s

π

π

⋅ Ω=

∑∑∫

2. Direct (real) space approach : average & sum

rmnrmn



( )( )2 2

24

mni s r

m nPD

f e dI s

s

π

π

⋅ Ω=

∑∑∫

( ) ( )2 sin 22

mnPD

m n mn

srI s f

srπ

π= ∑∑

( ) ( )2 2 cos 22

0

sin 21 2 sin4 2

mn mni s r isr mnmn

mn mn

sre e r d

r sr

ππ π φ π

π φ φπ π

⋅ = =∫

Debye Scattering Equation (DSE)

rmn

2. Direct (real) space approach : average & sum

rmnrmn


( )2 sin 2( )

2mn

PDm n mn

srI s f

srπ

π= ∑∑

DSE APPLICATION TO NON-CRYSTALLOGRAPHIC NPsDebye Scattering Equation (DSE)


( )2 sin 2( )

2mn

PDm n mn

srI s f

srπ

π= ∑∑

DSE APPLICATION TO GRAPHENE AND RELATED MATERIALSDebye Scattering Equation (DSE)

L. Gelisio et al., J. Appl. Cryst. 43 (2014) 647


( )2 sin 2( )

2mn

PDm n mn

srI s f

srπ

π= ∑∑

DSE APPLICATION TO GRAPHENE AND RELATED MATERIALSDebye Scattering Equation (DSE)

Carbon nanotubes

L. Gelisio, PhD Thesis, Univ. of Trento, 2014


0 1 2 3 4 50

2000400060008000

10000120001400016000

0.00.1 4.4 4.6 4.8 5.0 5.20

100

200

300600700800

B n

rn (nm)B mn

rmn (nm)

(a)

69

( ) ( )2 2sin 2 sin 2( )

2 2mn mn

PD mnm n mnmn mn

sr srI s f f B

sr srπ π

π π= ≡∑∑ ∑

DSE CALCULATION BY ATOMIC DISTANCE HISTOGRAMDebye Scattering Equation (DSE)

Atomic distance histogram (Bmn) for a cubic crystal with 8x8x8 sc unit cells (a) and corresponding powder pattern according to IPD(s), with f=1, unit cell parameter, a0=0.361 nm (b). P. Scardi & L. Gelisio, “Diffraction from nanocrystalline materials”, Chapter XVIII in Synchrotron Radiation, ed. S. Mobilio et al. Springer 2015.

0 20 40 60 80 100 120 140 1600123

45678

100

110

111

200

210

211

220

221

300/

310

311 22

232

032

1

400

410

322

330/

411

331

420

421

0 2 4 6 8 10 12 14 16 18 200.01

0.1

1

10

100

1000

Inte

nsity

2θ (degrees)

Inte

nsity

2θ (degrees)

(b)

In the coming months, look for a special issue of Acta Crystallographyca A, edited by Billinge, Cervellino, Neder & ScardiTotal Scattering methods – the 100 Years of the Debye Scattering Equation (DSE2015 conference, Cavalese (I) June 2015)


PAIR DISTRIBUTION FUNCTION (PDF)

Zernike & Prins (1927): for amorphous specimens, volume V, N atoms, theradial distribution function (RDF) is:

( ) ( ) ( )2 20 2

0

4 4 8 1 2I s

r r r r s Sin sr dsNf

π ρ π ρ π π∞

≅ + −

∫( )RDF r =

intensity in absolute units:

( ) 2

2

I s N ff

−

( )a d cc

− − → =

2f( ) ( )i M Compton


PDF AND SYNCHROTRON RADIATION

SR is mandatory to improve resolution!

à S. J. L. Billinge, Z. Kristallogr. 219 (2004) 117

1950

1999


PAIR DISTRIBUTION FUNCTION (PDF)

( ) ( )0 0

21 1 2s S s Sin sr dsr

πρ

∞

= + − ∫

( ) ( ) 04G r r rπ ρ ρ= −

( ) ( )24RDF r r rπ ρ=

reduced radial distribution function

radial distribution function

( ) ( ) 0g r rρ ρ= pair distribution function - PDF

( ) ( )2

I sS s

Nf=

à S. J. L. Billinge, Z. Kristallogr. 219 (2004) 117


PDF AND SYNCHROTRON RADIATION

SR is mandatory to improve resolution!

à Courtesy of R. Neder


PDF OF NANOPARTICLE SYSTEMS


Effect of finite size and shape of the nanoparticle


PDF OF NANOPARTICLE SYSTEMS


Indication of stacking faults


PDF ANALYSIS OF NANOPARTICLE SYSTEMS


Au nanoparticle + ligand

P. Scardi – Diffraction from nanocrystalline materialsICTP School - Trieste, 04.04.2016 77à Courtesy of R. Neder

PDF ANALYSIS OF NANOPARTICLE SYSTEMSAu nanoparticle + ligand

Bottom-up modelling DISCUSDIFFEV


TOTAL SCATTERING TECHNIQUES

K. Page et al., J.Appl.Cryst. 44 (2011) 327

PDF approach Debye Scattering Equation

P. Scardi et al., Phys. Rev. B91 (2015) 155414


TOTAL SCATTERING TECHNIQUES

K. Page et al., J.Appl.Cryst. 44 (2011) 327

PDF approach Debye Scattering Equation

P. Scardi & L. Gelisio, Nat. Sci. Reports 6, 22221 (2016)



( )( )

24sc

PD

I s dI s

sπ

Ω∝ ∫ ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ...IP S D F APB C GRSF I s I s I s I s I s I s I s= ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

1. Traditional reciprocal space approach : sum & average

( ) ( )2 sin 22

mnPD

m n mn

srI s f

srπ

π= ∑∑

Direct (real) space approach: average & sum Debye Scattering Equation (DSE)

2. Total Scattering methods

( ) ( ) ( ) ( )20 0 0

11 12

rg r Q S Q Sin Qr dQ

rρρ π ρ

∞

= = + − ∫

Pair Distribution Function (PDF)

( ) ( ) ( )20

11 4 22 V

I s N f r r Sin sr drs

π ρ ρ ππ

= + −

∫



relaxed(energy minimization)

geometrical

L. Gelisio, K.R. Beyerlein & P. Scardi, Thin Solid Films (2012). In press.

Debye Scattering Equation

( )2 sin 2( )

2mn

PDm n mn

srI s f

srπ

π= ∑∑

à toward an integration between atomistic modelling and diffraction analysis: real structure of nanoparticle systems

Current research / future trends


à toward an integration between atomistic modelling and diffraction analysis: plastically deformed nanocrystalline systems; grain boundary, line and planar defects

10 20 30 40 50 60 70 80 90 1000,0

0,5

1,0

1,5

2,0

2,5

Inte

nsity

(a.

u.)

2θ (degrees)

10 20 30 40 50 60 70 80 90 100

0.01

0.1

1

Inte

nsity

(a.

u.)

2θ (degrees)

DIFFRACTION FROM NANOCRYSTALLINE MATERIALSCurrent research / future trends

20 30 40 50 60 70 80 90 100 110 1200

5000

10000

15000

20000

20 30 40 50 60 70 80 90 100 110 1201

10

100

1000

10000

Inte

nsity

Q = 4πsinθ/λ (nm-1)20 30 40 50 60 70 80 90 100 110 120

-10000

1000

Res

idua

l

Inte

nsity

Q = 4πsinθ/λ (nm-1)

A. Leonardi & P. Scardi, Met. Mat.Trans A (2015). In press


GENERAL REFERENCES

B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading, MA, 1969.A. Guinier, X-ray Diffraction, Freeman & Co, S. Francisco, 1963.A.J.C. Wilson, X-ray Optics, 2nd ed., Methuen & Co, London, 1962.H.P. Klug & L.E. Alexander, X-ray Diffraction procedures, Wiley, New

York, 1974.B.D. Cullity, Elements of X-ray Diffraction, Addison-Wesley, Reading

Ma, 1978.

Powder Diffraction: Theory and PracticeR.E. Dinnebier & S.J.L. Billinge, editors.

Cambridge: Royal Society of Chemistry, 2008.

P. Scardi, Chapter 13 on Line Profile Analysis:


REFERENCES - [email protected]

Extending the Reach of Powder Diffraction Modelling by User Defined MacrosP. Scardi & R. E. Dinnebier, editorsA special issue of Materials Science Forum, 2010.

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

Synchrotron Radiation. Basics, Methods and Applications

S. Mobilio, F. Boscherini, C. Meneghini, editors.Springer-Verlag, 2015



REFERENCES - [email protected]

State-of-the-art Line Profile Analysis based on Whole Powder Pattern Modelling

The Debye Scattering Equation for studying static and dynamic disorder in nanocrystals

doi: 10.1038/srep20712 (2016)

doi: 10.1038/srep22221 (2016)


X-ray [email protected]

per aspera ad astra


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