Diffrazione da Materiali Policristallini

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Crystallite Size Analysis –Nanomaterials

This tutorial was created from a presentation by Professor Paolo Scardi and Dr. Mateo Leone fromthe University of Trento, Italy. The presentation was given at an ICDD workshop held during the2008 EPDIC-11 Meeting in Warsaw, Poland. Professor Paolo Scardi is shown above on the right.

The tutorial includes the theory and examples of the particle size algorithm and display features that

are embedded in PDF-4+! It also demonstrates how this simulation can be used for the study ofnanomaterials.

The ICDD is grateful to both Paolo and Mateo as well as the University of Trento for allowing theICDD to use their data for this tutorial.

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Note

This presentation can be directly viewedusing your browser.

It can also be saved, and viewed, withMicrosoft® PowerPoint®. The authors have

made additional comments in the notessection of this presentation, which can

be viewed within PowerPoint®, but is notvisible using the browser.

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ICDD PDF-4+ 2008ICDD PDF-4+ 2008EPDIC Workshop 3EPDIC Workshop 3

Featuring:Featuring:ICDD PDF-4+/DDView+ 2008:ICDD PDF-4+/DDView+ 2008:

Applications to NanomaterialsApplications to Nanomaterials

Prof. Paolo Scardi (University of Trento), ICDD Director-at-Large

Dr. Matteo Leoni (University of Trento), ICDD Regional Chair (Europe)

September 19, 2008

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PDF Card

• Contains Diffraction Data of Material

• Multiple d-Spacing Sets– Fixed Slit Intensity– Variable Slit Intensity– Integrated Intensity– New: Footnotes for

d-Spacings (*)• Options

– 2D/3D Structure– Bond Distances/Angles– Electron Patterns– New: PD3 Pattern– Diffraction Pattern

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Diffraction Pattern

• Simulated digitized pattern

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• pseudo-Voigt (pV)• Modified Thompson-

Cox-Hastings pV• Gaussian• Lorentzian• Particle Size

Profile Settings

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• Particle Size (Gamma distribution of diameters of spherical coherent-scattering domains)

Particle Size

Diffraction Pattern

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? Given the variety of available profile functions, why bother with a new one???

Answer: because the size distribution matters!!!

Profile Settings

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PIONEERS IN POWDER DIFFRACTION: PAUL SCHERRER

2 2

2 B

I d

I

The Scherrer formula [Gottinger Nachrichten 2 (1918) 98]

ln 22cos

h

h – full width at half maximum – effective domain size – wavelength – Bragg angle

Cerium oxide powder from xerogel, 1 h @400°C

Paul Scherrer (1890–1969)

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EFFECTIVE SIZE AND GRAIN SIZE

What is the meaning of L, the ‘effective size’ of the Scherrer formula?

2cosL

L ≠ D

5 nm

D

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SCHERRER FORMULA AND SIZE DISTRIBUTION

4

3

1V

MLK M

In most cases, crystalline domains have a distribution of sizes (and shapes).

Scherrer constant a shape factor, generally function of hkl (4/3 for spheres)

<D> = M1 meanM2 - M1

2 variance

2cosVL

( )iiM D g D dD

Distribution ‘moments’

Scherrer formula is still valid

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4

3

1V

ML L

K MD

EFFECTS OF A SIZE DISTRIBUTION

Example: lognormal distributions of spheres, g(D) (mean , variance )

5 nm

D

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<L>V

13.33D

Lognormal distribution of spheres: 2 2exp ln ) 2 2p D D D

2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size


P. Scardi, Size-Strain V (Garmisch (D) Sept. 2007). Z. Kristallogr. 2008. In press

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13.33D

13.23D <L>V

2 2exp ln ) 2 2p D D D Lognormal distribution of spheres:

2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size



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13.33D

11.82D

<L>V


2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size



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13.33D

8.25D <L>V


2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size



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13.33D

4.53D <L>V


2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size



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13.33D

1.95D <L>V


2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size



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<L>V


2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size



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2 2D exp

23 4 7 2VL exp

mean diameter

‘Scherrer’ size



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EFFECT OF A BIMODAL SIZE DISTRIBUTION

If the size distribution is multimodal, a single (“mean size”) number is of little use, and possibly misleading!!

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? Given the variety of available profile functions, why bother with a new one???

Answer: because the size distribution matters!!!

Profile Settings

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“Particle size” option in DDView+

s s=2sin/ Gamma distribution : mean variance

: mean variance

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M. Leoni & P. Scardi, “Nanocrystalline domain size distributions from powder diffraction data”, J. Appl. Cryst. 37 (2004) 629

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PDF 04-001-2097 cerium oxide

“Particle size” option in DDView+PDF 04-001-2097 cerium oxide

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TEM: 4.5 nmXRD/WPPM:4.4nm

PDF 04-001-2097 cerium oxide

M. Leoni & P. Scardi, “Nanocrystalline domain size distributions from powder diffraction data”, J. Appl. Cryst. 37 (2004) 629

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Validating the procedure


20 30 40 50 60 70 80 90 1000

2000

4000

6000

8000

10000

12000

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Inte

nsity

(a.

u.)

2theta (deg)

Diffraction pattern of Au generated by Debye equation

0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0

Freq

uenc

y (a

.u.)

Spherical domain size (nm)

Lognormal distribution of spherical domains (m=1.2, s=0.15) <D>=3.36 nm

K. Beyerlein, A. Cervellino, M. Leoni, R.L. Snyder & P. Scardi. EPDIC-11 (Warsaw (PL) Sept. 2008)

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0 1 2 3 4 5 6 7 8 90.0

0.2

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1.0

Freq

uenc

y (a

.u.)


Lognormal (m=1.2, s=0.15) <D>=3.36 nm DDView+ Gamma (m=3.36, s=50) <D>=3.36 nm

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0 1 2 3 4 5 6 7 8 90.0

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1.0

Freq

uenc

y (a

.u.)




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0 1 2 3 4 5 6 7 8 90.0

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1.0

Freq

uenc

y (a

.u.)




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0 1 2 3 4 5 6 7 8 90.0

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1.0

1.2

Freq

uenc

y (a

.u.)




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0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

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Freq

uenc

y (a

.u.)




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0 1 2 3 4 5 6 7 8 90.0

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uenc

y (a

.u.)



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• DDView+ “Particle size” is NOT a profile fitting!• NO other sources of line broadening (e.g., instrumental profile, dislocations, etc.) are considered!

Use this feature only for estimating domain size, especially in nano materials where the line width/shape is dominated by the size effects.

• Gamma distribution is flexible and handy, but in some cases it MIGHT NOT work! Domains might not be spherical!

In all other cases, use a Line Profile Analysis software, e.g., PM2K, based on the WPPM algorithm ([email protected])

CAVEAT!CAVEAT!

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International Centre for Diffraction Data

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Phone: 610.325.9814

Fax: 610.325.9823

Thank you for viewing our tutorial. Additional tutorials are available at the ICDD web site (

www.icdd.com).

http://www.icdd.com/

Diffrazione da Materiali Policristallini

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