Basic Aspects of x-ray crystallography and powder diffractionindico.ictp.it/.../212/material/slides/0.pdf · • High energy X-rays: to extend the accessible region of reciprocal

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Basic Aspects of x-ray crystallography and powder diffraction

- Diffraction from nanocrystalline materials

[email protected]

Special thanks to: Luca Rebuffi & Binayak Mukherjee

P. Scardi – Diffraction from nanocrystalline materials ICTP School - Trieste, 10.05.2018 2

PRESENTATION OUTLINE

PART I May 10, 9:00 – 10:30 • Powder diffraction: basic elements • Nanocrystalline & severely deformed materials

PART II May 10, 17:00 – 18:30 • Computer lab:

hands-on session with TOPAS


1912 - THE DISCOVERY OF X-RAY DIFFRACTION

copper sulfate (triclinic) random orientation

zinc blende (cubic)


1916 - FROM SINGLE-CRYSTAL TO POWDER DIFFRACTION

1916 DEBYE-SCHERRER CAMERA

s

so

s-so


2 sinhkl BMP P d nN

P

X-ray beam

Diffracted X-ray beam

X-ray 1

X-ray 2

(hkl) planes

ATOMIC PLANES AND DIFFRACTION CONDITION

hkl

B B

Interference of X-rays scattered by atomic planes

BRAGG LAW


in Bragg condition:

all waves in phase

irrespective of depth from the surface.

B

not in Bragg condition:

phase relations change with depth

at a certain depth, a reflected wave is in antiphase with the surface: the

two waves cancel each- other

antiphase


BRAGG LAW

B


DIFFRACTION PATTERN FROM A POLYCRYSTALLINE

B

in antiphase: cancel each-other

not cancelled

B

D

2B 21 22

1

2

D2

L

limit of cancellation


BRAGG LAW


SCHERRER EQUATION

12

cos BL L

D

Peak width is inversely proportional to the crystalline domain thickness

L

2B

D2

L

2B

D2

L

2B

D2


MgO powder


X-RAY DIFFRACTION (XRD) FROM SMALL CRYSTALS

J.T. Randall, The diffraction of X-rays and electrons by amorphous, solids, liquids, and gases, 1934


XRD FROM SMALL CRYSTALS: SCHERRER EQUATION

... ... ... The theory provides for the half-value width h of the maximum defined in the known manner, which occurs at the angle to the incident X-ray beam, the value:

/L is the ratio of the wavelength of the monochromatic X-

rays used to the edge of the crystal presumed to be cube-

shaped

(2)

h


Determination of the size and internal structure of

colloid particles by means of x-rays

Presented by P. Debye in the meeting of

by

L

cry

sta

l siz

e

peak width


PRACTICE: SCHERRER EQUATION

42 43 44 45 46 47 48 49 50 51 52

0

1000

2000

3000

48.34°46.64°

Inte

nsity

(co

un

ts)

2 (degrees)

2B=47.49°

150150

3250

1700

FWHM=1.7°

?L

L : effective crystalline domain size (in nm) h : full width at half maximum (FWHM in radians) = 0.15406 nm (X-ray wavelength)

( )ln 2

2cos 2B

h

L

(220) peak of nanocrystalline CeO2


LABORATORY vs SYNCHROTRON RADIATION XRD

30 40 50 60 70 80 90 100 110 120 130 140 150

10

100

1000

Inte

nsity (c

ounts

)

2 (degrees)

Fig. 1 in P. Scardi & L. Gelisio, Chapter XVIII,

“Diffraction from nanocrystalline materials”

Powder diffraction data from a ball milled Fe1.5%Mo powder collected on a traditional laboratory instrument (Rigaku PMG-VH, Bragg-

Brentano geometry) with CuK radiation (=0.1540598 nm). On the right: schematic of reciprocal space with extension of the limiting

sphere (radius 2/).

Powder diffractometer geometry

10 20 30 40 50 60 70 80 90 10010

100

1000

Inte

nsity (c

ounts

x 1

00)

2 (degrees)

Powder diffraction data from a ball milled Fe1.5%Mo powder collected (b) on ID31 (now ID22) at ESRF, Grenoble (F) (=0.0632 nm).

On the right: schematic of reciprocal space with extension of the limiting sphere (radius 2/).

1916 Debye-Scherrer geometry

(the return !)


LABORATORY vs SYNCHROTRON RADIATION XRD

10 20 30 40 50 60 70 80 90 10010

100

1000

Inte

nsity (c

ounts

x 1

00)

2 (degrees)

Powder diffraction data from a ball milled Fe1.5%Mo powder collected (b) on ID31 (now ID22) at ESRF, Grenoble (F) (=0.0632 nm).

On the right: schematic of reciprocal space with extension of the limiting sphere (radius 2/).

1916 Debye-Scherrer geometry

(the return !)

• High intensity (brilliancy): better counting statistics / shorter data collection time ( fast kinetics, in situ/in operando studies)

• Highly collimated beam: narrow instrumental profile for high resolution / accuracy (in the measurement of peak position, intensity, width, and shape)

• High energy X-rays: to extend the accessible region of reciprocal space (collect more peaks, more information, proper evaluation of asymptotic trend of intensity in PDF analysis, etc.)

• Tuning X-ray energy: e.g., for haldling absoprtion problems, or to exploit absorption thresholds (anomalous scattering)


DIFFRACTION FROM NANOCRYSTALLINE POWDER

( ) ( ) ( )2 2*m ni s r i s r

sc m n

m n

I s f e f e

( )2* mni s r

m n

m n

f f e

sx [Å-1]

sy [

Å-1

]

( )

( )2*

24

mni s r

m n

m nPD

f f e d

I ss

s=Q/2=2sin / 2 sind s d d

S

mrnr

mnr



( ) ( ) ( )2 2*m ni s r i s r

uc m n

m n

I s f e f e

n n n nr u a v b w c

* * *s ha kb lc

( )2

2

1

n n n

Ni u h v k w l

n

n

f e

2

F

(un,vn,wn)

(0,0,0)

(½,1,½)

(0,1,1)

(1,1,1)

(1,1,0)

(0,1,0)

(1,½,½)

(0,½,½)

(½,0,½)

(½,½,1)

(½,½,0)

(1,0,0)

(1,0,1)

(0,0,1)

D Na a

Traditional "reciprocal space" approach: factorize unit cell intensity

F, the structure factor

Intensity from one unit cell, Iuc |F|2



( )2

,F s D

line profile function

DFactorize structural contribution: the line profile function

sx [Å-1]

sy [

Å-1

]

S

mrnr

mnr

( )

( )2*

24

mni s r

m n

m nPD

f f e d

I ss



( ) ( )2

,PD sphereI s F s D

100 ÅD

( ) ( )2

0cos 2

D

F A L sL dL

0 20 40 60 80 100 120 1400.0

0.2

0.4

0.6

0.8

1.0

A(L)=V(L)/V0

L (Å)

Fourier Transform of peak profile: the Common Volume Function

V(L)

( )3

3

3 11

2 2

L LA L

D D

[hkl]

sx [Å-1]

sy [

Å-1

]

s

P. Scardi – Diffraction from nanocrystalline materials ICTP School - Trieste, 10.05.2018

4 5 6 7 8 9 100

200

400

600

800

1000

Inte

nsity [

a.u

.]

s [Å-1]

4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6

1

10

100

1000

Inte

nsity [a

.u.]

s [Å-1]

18


( ) ( )2

,PD sphereI s F s D

100 ÅD

( ) ( )2

0cos 2

D

F A L sL dL

0 20 40 60 80 100 120 1400.0

0.2

0.4

0.6

0.8

1.0

A(L)=V(L)/V0

L (Å)


V(L)

( )3

3

3 11

2 2

L LA L

D D

[hkl]


10 20 30 40 50 60 70 80 90 100

0

10k

20k

30k

40k

50k

60k

70k

80k

Inte

nsity

(co

unts

)

2 (degrees)

19


( )( )

( )2 2ln 2

1/2

1

2

Dg D e

D

6 8 10 12 14 16 18 20 22 24 26 28 30 32

10k

20k

30k

40k

50k

60k

70k

80k

Inte

nsity (c

ounts

)

2 (degrees)

Xerogel cerium oxide powder

5 nm

ESRF – ID31 (now ID22) =0.0632 nm

H0=½, H1=-¾, H2=0, H3=¼ 0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0<D>=2.25(10) nmg(D)

D (nm)

Scardi, P. (2008). Powder Diffraction: Theory and Practice, edited by R. E. Dinnebier & S. J. L. Billinge, ch. 13, pp. 376–413. Cambridge: Royal Society of Chemistry

( )

( )( )

232

0

ln 3erfc exp 2 6

22

ln,sphere

n

n

n

A L

L n nH n L


4 5 6 7 8 9 100

200

400

600

800

1000

Inte

nsity [

a.u

.]

s [Å-1]

20

( ) ( )max2

0cos 2

L

F A L sL dL


( ) ( )2

,PD cubeI s F s D

80.6 ÅD

4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6

1

10

100

1000

Inte

nsity [a

.u.]

s [Å-1]


0 20 40 60 80 100 120 140 1600.0

0.2

0.4

0.6

0.8

1.0

V(L)/V0

L (Å)

[111]

(111) (111)

(111)

V(L)

( )2 3

111 2 3

31

3 3

L L LA L

D D D


4 5 6 7 8 9 100

200

400

600

800

1000

Inte

nsity [

a.u

.]

s [Å-1]

21


80.6 ÅD

4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6

1

10

100

1000

Inte

nsity [a

.u.]

s [Å-1]


0 20 40 60 80 100 120 140 1600.0

0.2

0.4

0.6

0.8

1.0

V(L)/V0

L (Å)

[111]

(111) (111)

(111)

[200]

( )200 1L

A LD

(200)

(200)

(200)

( ) ( )2

,PD cubeI s F s D ( ) ( )max2

0cos 2

L

F A L sL dL

( )2 3

111 2 3

31

3 3

L L LA L

D D D


DIFFRACTION FROM NANOCRYSTALLINE POWDER Any shape A.Leonardi et al., J.Appl.Cryst. 45 (2012) 1162

Wulff polyhedra

24 26 28 30 32 34 36

2 (degrees)

CO0 CO25 CO100

(111)

(200)



Pd nanocrystals

P. Scardi et al.

Phys.Rev.B 91(2015)155414


DIFFRACTION FROM NANOCRYSTALLINE POWDER … what I did not consider (so far…)

Courtesy of A. Young & F. Tsung

Boston College,

: microstructure !

most metals:

a0

a0+Da CeO2

Surface relaxation

Perez-Demydenko & Scardi, Phil. Mag. 97 (2017) 2317


DIFFRACTION FROM NANOCRYSTALLINE POWDER … what I did not consider (so far…) : microstructure !

Severe Plastic Deformation

(Zhu et al. J. Mater. Res. 18 (2003) 1908)


DIFFRACTION FROM NANOCRYSTALLINE POWDER microstructure perturbation of “perfect crystal structure”

Any shape A.Leonardi et al., J.Appl.Cryst. 45 (2012) 1162

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

(s

)

S = s - shkl

(Å)

( ) ( ) ( )max2

0cos 2

LSI s F A L sL dL

( ) ( )max * 2

0

LS isL

LI s A L FF e dL

( )max2 2

0( ) ...

LS D F F isLI s F A A A iB e dL

domain size inhomog.strain faulting



Pd nanocrystals

( )max2 2

0

LS D IP isLI s F A A T e dL

domain size

inhomog. strain

instrum. profile

Inhomogeneous displacement ( strain, e = DL/L ): ( )

2 2 22 <ΔLhklsDA L e

P. Scardi et al. Phys.Rev.B 91(2015)155414


0 20 40 60 80 100 120 140 160 180 200 220 240 2600.00

0.05

0.10

0.15

0.20

0.25

0.30

0 20 40 60 80 100 120 140 160 180 200 220 240 260

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 20 40 60 80 100 120 140 160 180 200 220 240 2600.00

0.05

0.10

0.15

0.20

0.25

0.30

[hhh]

[hh0]

D

L2 h

hh

1/2 (Å

)

DL

2 hh

0

1/2 (Å

)

DL

2 h00

1/2 (Å

)

L (Å)

28


( )max2 2

0

LS D IP isLI s F A A T e dL

domain size

inhomog. strain

instrum. profile

Pd nanocrystals

P. Scardi et al. Phys.Rev.B 91(2015)155414

MD trunc. cube

Inhomogeneous displacement ( strain, e = DL/L ):

Radial

110 cross-section

Planar

Molecular Dynamics (MD) atomic displacement maps

MD perfect cube

MD sphere

Warren plot (root mean square displacement vs L)

XRD

( )2 2 22 <ΔLhklsDA L e


WHOLE POWDER PATTERN MODELLING – WPPM

( )max2 2

0( ) ...

LS D F F IP isLI s F A A A iB T e dL

domain size inhomog.strain faulting instrum. profile

( ) ( ) ( )( ) ( ) ( ) ...IP S D F APBI s I s I s I s I s I s

Diffraction profile as a convolution of (independent) effects:


High-energy grinding (ball-milling) of an iron alloy powder: AstaloyTM Fe1.5Mo

Rebuffi et al., Sci. Reports 6 (2016) 20712

SEM TEM

HREM





( ) ( ) ( ) ( )2 2S D IP isL

hkl hklI s F A L A L T L e dL

XRD data : MCX beamline, Italian synchrotron ELETTRA


Size effect

Inhom. Displ. effect


0 2 4 6 8 10 12 14 160.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(222)

(110)

DL

2>

1/2

L (nm)

(200)

33


XRD data : MCX beamline, Italian synchrotron ELETTRA



Re= 4.3 nm

r = 4.5x1016 m-2

Warren’s plot

2 dislocations

per crystalline

domain !

( )

22 2 *

4

hkl

hkl e

C bL L f L R

r

D



DIFFRACTION FROM NANOCRYSTALLINE POWDER Understanding XRD line profile analysis result by Molecular Dynamics simulations

<D> = 9.3 nm

s.d. = 5.9 nm

atomistic model

by space tessellation

Molecular Dynamics (MD)

simulation



DIFFRACTION FROM NANOCRYSTALLINE POWDER Understanding XRD line profile analysis result by Molecular Dynamics simulations


DIFFRACTION FROM NANOCRYSTALLINE POWDER Temperature Diffuse Scattering – TDS



Debye

Density of States

Temperature Diffuse Scattering – TDS in small crystals

P. Scardi – Diffraction from nanocrystalline materials ICTP School - Trieste, 10.05.2018 38 P. Scardi et al. J. Appl. Cryst. 50 (2017) 508

DIFFRACTION FROM NANOCRYSTALLINE POWDER TDS in ball-milled FeMo nanocrystals: static and dynamic contributions

Argonne 11bm, 30keV – 100, 200, 300K

10 20 30 40 50

0

5000

10000

15000

20000

25000

30000

35000

Inte

nsity

(co

unts

)

2 (degrees)

10 20 30 40 50100

1000

10000

Inte

nsity

(co

unts

)

2 (degrees)

10 20 30 40 50

0

5000

10000

15000

20000

25000

Inte

nsity

(co

unts

)

2 (degrees)

10 20 30 40 50

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Inte

nsity (

counts

)

2 (degrees)

Expt. data

capillary + fcc phase

TDS

10 20 30 40 50

0

5000

10000

15000

20000

25000

Inte

nsity

(co

unts

)

2 (degrees)

10 20 30 40 50

0

50

100

150

Inte

nsity (

counts

)

2 (degrees)

300K

200K

100K

100K 200K 300K

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

T (K)B

ISO (Å

2)

2.858

2.860

2.862

2.864

2.866

2.868

2.870

2.872

a0 (Å

)

TDS

bulk Fe

20% increase in BS

ball milled Fe


DIFFRACTION FROM NANOCRYSTALLINE POWDER EXAFS & XRD : Mean Square Displacement (MSRD, MSD) and DCF

6900 7200 7500 7800 8100 8400 8700-0.6

-0.4

-0.2

0.0

0.2

0.4

(E

)

Energy (eV)

Ni edge

4 6 8 10 12 14 16

0.0

0.5

1.0

1.5

2.0

2.5

Magnitude

K (Å-1)

Fe foil

b.m. FeMo_RT

b.m. FeMo_78K0 50 100 150 200 250 300

0,000

0,002

0,004

0,006

0,008

0,010

0,012

MS

RD

, (2

x<

u2 s>

) (Å

2)

T (K)

0,000

0,002

0,004

0,006

0,008

0,010

0,012

Fe foil

(bulk)

22 qu D

+20%

+20% 2

||u D

2 2

|| 2 qu MSRD u DCF D D

EXAFS Debye-Waller parm. < 𝛥𝑢||2 > : 1° coordination shell

XRD Debye-Waller parm. < 𝛥𝑢𝑞2 >=B/82 : average over whole crystal

P. Scardi et al. J. Appl. Cryst. 50 (2017) 508

20% increase in BS



P. Scardi et al. J. Appl. Cryst. 50 (2017) 508

What is the main reason for the 20% increase in BS (static disorder) ?

Molecular Dynamics simulation of a cluster of 50 Fe grains (size distribution from XRD/WPPM)

MSD - Mean Square Displacement < 𝛥𝑢𝑞2 > (B=82< 𝛥𝑢𝑞

2 >)

results for average-size grain (G5)

no defects

g.b. only

1 edge disloc.

and g.b.

vacancies

and g.b.

DB = BG5 – Bbulk 22% 22%+ ~2% 22%+ ~0% (<<1%)


FROM SINGLE CRYSTAL TO POWDER DIFFRACTION

( )( )

24

sc

PD

I s dI s

s

( )

2F I s

perfect (infinite) crystal

D

( ) ( )2* mni S r

sc m n

m n

I s f f e

( ) ( ) ( ) ( ) ( ) ( ) ( ) ...IP S D F APB C GSRI s I s I s I s I s I s I s

Traditional reciprocal space approach : sum & average


PRESENTATION OUTLINE

PART II May 10, 17:00 – 18:30 • Computer lab:

hands-on session with TOPAS


Debye formula

(Direct Space)

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:

translational symmetry: not verified

large strain / misfit – complex local atomic arrangement



Direct (real) space approach : average & sum

( )

( )2*

24

mni s r

m n

m nPD

f f e d

I ss



( )

( )2 2

24

mni s r

m nPD

f e d

I ss

( )( )2 sin 2

2

mn

PD

m n mn

srI s f

sr

( ) ( )2 2 cos 2

2

0

sin 212 sin

4 2mn mn

i s r isr mn

mn

mn mn

sre e r d

r sr

Debye Scattering Equation (DSE)

rmn

Direct (real) space approach : average & sum

rmn rmn


( )2 sin 2( )

2

mn

PD

m n mn

srI s f

sr

DSE APPLICATION TO NON-CRYSTALLOGRAPHIC NPs



( )2 sin 2( )

2

mn

PD

m n mn

srI s f

sr

DSE APPLICATION TO GRAPHENE AND RELATED MATERIALS


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


( )2 sin 2( )

2

mn

PD

m n mn

srI s f

sr

DSE APPLICATION TO GRAPHENE AND RELATED MATERIALS


Carbon nanotubes


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 Practice

R.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 Macros

P. Scardi & R. E. Dinnebier, editors

A special issue of Materials Science Forum, 2010.

Diffraction Analysis of Materials Microstructure

E.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

Documents

Basic Aspects of x-ray crystallography and powder diffractionindico.ictp.it/.../212/material/slides/0.pdf · • High energy X-rays: to extend the accessible region of reciprocal