diffuse scattering• crystals - barium iv: self-hosting incommensurate structure -...

DIFFUSE SCATTERING

Hercules 2012

Pascale Launois

Laboratoire de Physique des Solides (UMR CNRS 8502)

Bât. 510

Université Paris Sud, 91 405 Orsay CEDEX

FRANCE

pascale.launois@u-psud.fr

OUTLINE __________________________________________________________________________________________________________________

1. Introduction

General expression for diffuse scattering

Diffuse scattering/properties

A brief history

2. 1D chain - First kind disorder

Simulations of diffuse scattering/local order

Analytical calculations

- Second kind disorder

3. Diffuse scattering analysis

General rules

To go further

Note: static or dynamic origin…

4. Examples • Crystals

- Barium IV: self-hosting incommensurate structure

- Nanotubes@zeolite: structural characteristics of nanotubes from diffuse scattering analysis

- DIPSF4(I3)0.74: a one-dimensional liquid

- Graphene

• Fibers

- DNA: structure from diffuse scattering pattern

• Powders

- C60 fullerenes

- Turbostratic graphite

- Peapods C60@single walled carbon nanotube

1. INTRODUCTION - General expression for diffuse scattering

● No long range order : second-kind disorder

Average structure

ORDER

Bragg peaks

2-body correlations

DISORDER

Diffuse scattering

● Average long range order : first-kind disorder

( ( ( ( mRsi

mn

nnmnn

lkh

lkhn

n esFFFGssFV

sI 22

,,

,,

2

*1

Fn = FT(electronic density within a unit cell ‘n’)

(

m

Rsi

tnmnn

tmn

RRsi

mn

Rsi

nmmnn eFFeFeFsI 2

,

*

,

)(2*2

Variable omitted

here after

P. Launois, S. Ravy and R. Moret, Phys. Rev. B 52 (1995) 5414

s

I(s)

x104

Diffuse scattering :

not negligible with respect to

Bragg peaks

Conservation law :

(Diffuse + Bragg) (s)

= r(r)2

1. INTRODUCTION - General expression for diffuse scattering

From Welberry and Butler, J. Appl. Cryst. 27 (1994) 205

(a) (b)

Same site occupancy (50%)

Same two-site correlations (0)

(a): completely disordered;

(b): large three-site correlations

Same scattering patterns!

Scattering experiments:

probe 2-body correlations

Ambiguities can exist

=> Use chemical-physical constraints

1. INTRODUCTION - General expression for diffuse scattering

Diffuse scattering defects, disorder, local order, dynamics

Physical properties

Lasers

Semiconductor properties (doping)

Hardness of alloys (Guinier-Preston zones)

Plastic deformation of crystals (defect motions)

Ionic conductivity (migration of charged lattice defects)

Geosciences Related to growth conditions

Biology

biological activity of proteins: intramolecular activity...

1. INTRODUCTION - Diffuse scattering vs properties

A few dates ...

•1918 : calculation of diffuse scattering for a disordered binary alloy (von Laue)

•1928 : calculation of scattering due to thermal motions (Waller)

•1936 : first experimental results for stacking faults (Sykes and Jones)

•1939 : measurement of scattering due to thermal motion

Diffuse scattering measurements and analysis : no routine methods

Many recent progresses :

- power of the sources (synchrotron...)

- improvements of detectors

- computer simulations

1. INTRODUCTION - A brief history

2. 1D CHAIN LATTICE –1st kind of disorder-simulations _____________________________________________________________________________________________________________________

Program ‘Diff1D’

to simulate diffuse scattering for a 1D chain :

Substitution disorder

Atoms and vacancies on an infinite 1D periodic lattice (concentrations=1/2).

The local order parameter p is the probability of ‘atom-vacancy’ nearest-neighbors pairs.

Displacement disorder

Atoms on an infinite 1D periodic lattice,

with displacements (+u) or (-u) (same concentration of atoms shifted right and left).

The local order parameter p is the probability of ‘(+u)/(-u)’ nearest-neighbors pairs.

Size effect

Atoms and vacancies : local order + size effects (interatomic distance d=>de.

DIFF1D (program : D. Petermann, P. Launois, LPS)

2. 1D CHAIN LATTICE –1st kind of disorder-simulations _____________________________________________________________________________________________________________________

QUESTIONS :

- What parameter p would create a random vacancy distribution?

- Describe and explain the effect of correlations (diffuse peak positions and widths).

- What is the main difference between substitution and displacement disorder for the

diffuse scattering intensity?

- Size effect : compare diffuse scattering patterns with substitutional disorder and with

displacement disorder correlated with substitutional disorder. Discuss the cases e>0 and

<0.

- Simulate the diffuse scattering above a second order phase transition, when lowering

temperature.

2. 1D CHAIN LATTICE –1st kind of disorder-simulations _____________________________________________________________________________________________________________________

ANSWERS :

- Random : p=1/2

- p>1/2 : positions of maxima = (2n+1) a*/2, n integer (ABAB: doubling of the period)

p<1/2 : positions of maxima = n a* (small domains: AAAA, BBBBB)

Width (correlation length)-1

Substitution, p=0.6 Substitution, p=0.75

Displacements, p=0.75

Substitution disorder, p=1/2

2. 1D CHAIN LATTICE –1st kind of disorder-simulations _____________________________________________________________________________________________________________________

-Substitution disorder: intensity at low s.

Displacement disorder: no intensity around s=0.

-Size effect : intensity asymmetry around s=na* (n integer).

For e>0: atom-atom distance larger than vacancy-vacancy distance,

atom scattering factor > that of a vacancy (0) => positive contribution just before na*,

negative contribution just after na*.

s a*

ID e=0

e>0

2. 1D CHAIN LATTICE –1st kind of disorder-simulations _____________________________________________________________________________________________________________________

2. 1D CHAIN LATTICE –1st kind of disorder-calculations _____________________________________________________________________________________________________________________

Analytical calculations

A atoms: red, B: purple

cA=cB=1/2

p= probability to have a ‘A(red)-B(purple)’ pair

a. Substitution disorder

4

2

BA ff

With am=1-2pm, pm being the probability

to have a pair AB at distance (ma);

p0=0, p1=p.

am : Warren-Cowley coefficients

)2cos(214

0

2

smaff

Im

m

BA

D a

( ( ( ( smai

mnnnmnn

hnn esFFFahssF

asI

222

*/1

Demonstration

22 : BABA

An

fffffFFAn

2

)(

2 : BABA

Bn

fffffFFBn

A(n)A(n+m) B(n)A(n+m) B(n)B(n+m)

A(n)B(n+m)

2. 1D CHAIN LATTICE –1st kind of disorder-calculations _____________________________________________________________________________________________________________________

( )**)((*2

mnmnnnnnnmnn FFFFsFFF

(

mmmm

BA

nnnmnn ppppff

sFFF2

1

2

1)1(

2

1)1(

2

1

4*

2

2

mp

n n+m

( m

BA

nnnmnn

ffsFFF a

4*

22

mm p21a

mmm aaa ,0;10

(

)2cos(214

)2exp(4

*

0

2

2

22

smaff

smaiff

esFFF

m

m

BA

m

m

BAsmai

mnnnmnn

a

a

2. 1D CHAIN LATTICE –1st kind of disorder-calculations _____________________________________________________________________________________________________________________

mp

ppppprobprobprobprobp mmAB

m

AABB

m

ABm )1()1( 11

1111

( mm

m p 121 aa

2

11

2

1

2

)2cos(21

1

4 aa

a

sa

ffI

BA

D

X-Ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies, A. Guinier, Dover publications.

Interactions between nearest neighbors only :

)2exp(1

1)2exp()2exp(

10

1

0 saisaismai

m

m

m

ma

aa

1)2exp(1

1)2exp()2exp(

11

1

1

saisaismai

m

m

m

ma

aa

n n+m

2

11

2

1

2

)2cos(21

1

4 aa

a

sa

ffI

BA

D

p=1/2 : a1=0 : Laue formula

p>1/2 : a1<0 : positions of maxima = (2n+1) a*/2, n integer

P<1/2 : a1>0 : positions of maxima = n a*

A=B : no disorder : no diffuse scattering

4

2

BA

D

ffI

Formula used in Diff1D

2. 1D CHAIN LATTICE –1st kind of disorder-calculations _____________________________________________________________________________________________________________________

u -u

b. Displacement disorder

2

11

2

1

2

)2cos(21

1

4 aa

a

sa

ffI

BA

D

with and sui

AA eff 2 sui

AB eff 2

( 2

11

2

122

)2cos(21

12sin

aa

a

sasufI AD

ID(s=0) =0

u=0 : no disorder : no diffuse scattering

Formula used in Diff1D

2. 1D CHAIN LATTICE –1st kind of disorder-calculations _____________________________________________________________________________________________________________________

2. 1D CHAIN LATTICE –1st kind of disorder-calculations _____________________________________________________________________________________________________________________

3. Size effect

Combination of correlated displacement and substitution disorder

)2sin(22)2cos(214 00

2

smasmasmaff

Im

m

m

m

BA

D a

mst neighbors:

Mean-distance=ma

)1(),1(),1( mAB

mAB

mBB

mBB

mAA

mAA madmadmad eee

mAB

mBB

mAA

mABm

AB pp

2))(1( ee

e

( ( AA

AB

ABB

AB

B

ff

f

ff

feaea 111 11

Warren, X-Ray Diffraction, Addison-Wesley pub.

)...2exp()21())1(2exp( smaismaismai mAA

mAA ee

( ( msdi

mnnnmnnD esFFFsI

22

*

fB>fA

eBB>0 1>0

eAA<0

-1sin(2sa) >0 for s~<na, <0 for s~>na

s a*

ID eij=0

Size effect

Formula used in Diff1D : m=1

2. 1D CHAIN LATTICE –1st kind of disorder-calculations _____________________________________________________________________________________________________________________

B A

Second-kind disorder

No long range order

X-Ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies, A. Guinier, Dover publications.

rA rB

cA=cB=1/2, p=1/2

a=(rA+rB)/2

(in a* unit)

Cumulative fluctuations in distance Peak widths increase with s

2. 1D CHAIN LATTICE –2nd kind of disorder _____________________________________________________________________________________________________________________

1D liquid:

* distribution function h1(z) of the first neighbors of an arbitrary reference unit

* hm(z)= h1(z)* h1(z) * …* h1(z)

Total distribution function: S(z(zSm0(hm(z)+ h-m(z))

Fourier transform: 1+2 Sm>0(Hm(s)+ H*m(s))=1+2 Re(H(s)/(1-H(s)))

where H(s)=FT(h1)

Gaussian liquid: h1(z)~exp(-(z-d)2/2s2) : H(s)~exp(-22s2s2)exp(i2sd)

I(s)~sinh(42s2s2/2)/ (cosh(42s2s2/2)-cos(2sd))

See e.g. E. Rosshirt, F. Frey, H. Boysen and H. Jagodzinski, Acta Cryst. B 41, 66 (1985)

2. 1D CHAIN LATTICE –2nd kind of disorder-calculations _____________________________________________________________________________________________________________________

3. DIFFUSE SCATTERING ANALYSIS _____________________________________________________________________________________________________________________

A few properties of the diffuse scattering

• WIDTH

- 1st kind disorder : width1/(correlation length)

modulations larger than Brillouin Zone : no correlations;

diffuse planes : 1D ordered structures with no correlations;

diffuse lines : disorder between ordered planes

- Width increases with s: 2nd kind of disorder

• POSITION

local ordering in direct space (AAAA, ABABAB...)

• INTENSITY

Scattering at small s : contrast in electronic density (substitution disorder...)

No scattering close to s=0 : displacements involved

Extinctions => displacement directions... I~f(s.u) : s u I=0

To go further and understand/evaluate microscopic interactions ...

calculations.

-Analytical.

Modulation wave analysis (for narrow diffuse peaks),

Correlation coefficients analysis (for broad peaks).

-Numerical.

Model of interactions (with chemical and physical constraints)

Direct space from Monte-Carlo or molecular dynamics simulations;

Or use of mean field theories ...

Scattering pattern.

3. DIFFUSE SCATTERING ANALYSIS _____________________________________________________________________________________________________________________

3. DIFFUSE SCATTERING ANALYSIS _____________________________________________________________________________________________________________________

Diffuse scattering origin can be static or dynamic

4. EXAMPLES - Barium IV _____________________________________________________________________________________________________________________

From R.J. Nelmes, D.R. Allan, M.I. McMahon and S.A. Belmonte, Phys. Rev. Lett. 83 (1999) 4081

l: -3 -2 -1 0 1 2 3 Tetragonal symmetry

Diffuse planes l/3.4Å

Synchrotron radiation Source,

Daresbury Lab.

and laboratory source

Diffuse planes in reciprocal space in direct space ?

No diffuse scattering in the plane l=0 in direct space ?

Diffuse planes in reciprocal space

Disorder between chains in direct space

No diffuse scattering in the plane l=0 displacements along the chain axes

)()( usfsID

No contribution for s u

4. EXAMPLES - Barium IV _____________________________________________________________________________________________________________________

5.5 GPa 12.6 GPa

Phase I

bcc

Phase II

hcp

Phase IV

Tetragonal inc.

45 GPa

Phase V

hcp

Tetragonal ‘host’, with ‘guest’ chains in

channels along the c axis of the host. These

chains form 2 different structures, one well

crystallized and the other highly disordered

and giving rise to strong diffuse scattering.

The guest structures are incommensurate

with the host.

An incommensurate host-guest structure in

an element!

4. EXAMPLES - Barium IV _____________________________________________________________________________________________________________________

Diffuse

Scattering

zeolite

DS nanotubes

c*

From P. Launois, R. Moret, D. Le Bolloc’h, P.A. Albouy, Z.K. Tang, G. Li and J.Chen,

Solid State Comm. 116 (2000) 99; ESRF Highlights 2000, p. 67

Diffuse scattering in the plane l=0 close to the origin => ?

4. EXAMPLES - Nanotubes inside zeolite _____________________________________________________________________________________________________________________

Zeolite AlPO4-5 (AFI) Nanotubes inside the zeolite channels

Aluminium

or phosphor

Oxygen

Intensity at small s Contrast in electronic density :

partial occupation of the channel - different between channels

Schematic drawing of the nanotubes electronic density

projected in the z=0 plane

4. EXAMPLES - Nanotubes inside zeolite _____________________________________________________________________________________________________________________

Small s, l=0: ID(s)fC2.(J0(2sF/2))2

Nanotubes with F4Å. The smallest nanotubes. Vs superconductivity?

Tang et al., Science 292, 2462 (2001)

4

2

BA

D

ffI

Laue formula :

F

s

s

4. EXAMPLES - Nanotubes inside zeolite _____________________________________________________________________________________________________________________

Tetraphenyldithiapyranylidene iodine C34H24I2.28S2

From P.A. Albouy, J.P. Pouget and H. Strzelecka, Phys. Rev. B 35 (1987) 173

4. EXAMPLES - DIPSF4(I3)0.75

_____________________________________________________________________________________________________________________

LURE (synchrotron)

c axis= horizontal

Diffuse scattering in planes l/(9.79Å)

l values

- Extinctions?

- Intensity distribution between the different diffuse planes?

- Widths of the diffuse planes?

4. EXAMPLES - DIPSF4(I3)0.75

_____________________________________________________________________________________________________________________

No intensity for l=0 displacement disorder along the chain axes

Widths of the diffuse planes increase with l

second-type disorder

One-dimensional liquid at Room Temperature

The most intense planes : l=3,4, 6 and 7 : due to the molecular

structure factor of the triiodide anions.

4. EXAMPLES - DIPSF4(I3)0.75

_____________________________________________________________________________________________________________________

3D ordered state below 182K

4. EXAMPLES - DIPSF4(I3)0.75

_____________________________________________________________________________________________________________________

4. EXAMPLES - Graphene

Electron

scattering

J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth & S. Roth, Nature 446, 60 (2007)

J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, D. Obergfell, S. Roth, C. Girit, A. Zettl,

Solid State Comm. 143 (2007) 101-109

J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth & S. Roth, Nature

446, 60 (2007)

4. EXAMPLES - Structure of B-DNA

_____________________________________________________________________________________________________________________

B form of DNA in fiber

The structure of DNA

R. Franklin + J.D. Watson and F.H.C. Crick (1953)

I~<FB-DNA FB-DNA*

>

Gives the double helix shape

&

indicates its method of replication

4. EXAMPLES - Structure of B-DNA

_____________________________________________________________________________________________________________________

Angle inversely

related to helix radius

q

P Spacing

proportional

to 1/P

p

Pitch = P

Spacing

corresponds

to 1/p

meridian

4. EXAMPLES - Structure of B-DNA

_____________________________________________________________________________________________________________________

p=3.4Å

P=34Å

Extinction:

t0=3P/8

P=34Å

p=3.4Å

t0

P. Launois, S. Ravy and R. Moret, Phys. Rev. B 52 (1995) 5414

4. EXAMPLES – C60 powder

Fullerenes C60

Single crystal

Powder

W.A. Kamitakahara et al., Mat. Res. Soc. Symp. Proc. 270 (1992)167

3.45 Å

4. EXAMPLES – Turbostratic carbon

Random stacking of graphene layers

Turbostratic carbon

● Diffraction peaks (00l)

● Diffuse lines (hk)

0

2

4

6

8

10

0 5 Q (Å-1)

00 10 11 20 12 30 l

Powder

average

?

The smallest scattering

angle

The scattering angle

decreases

The scattering angle increases

again

2θMin

I

2θ 2θMin

Sawtooth peak :

● Symmetric (00l) peaks

● Sawtooth shaped (hk) reflections Powder

average

● Diffraction peaks (00l)

● Diffuse lines (hk)

0

2

4

6

8

10

0 5 Q (Å-1)

00 10 11 20 12 30 l

0 2 4 6 8 10 12 14 16

Q(Å-1)

(10

)

(00

2)

0 2 4 6 8 10 12 14 16

Q(Å-1)

4. EXAMPLES – Peapods

L

K. Hirahara et al.,

Phys. Rev. B (2001)

Qz=0

2/L

4/L

2/L

4/L

(Qz)

Powder average

→ sawtooth peak

0,60 0,65 0,70 0,75 0,80

Inte

nsity

Q(Å-1)

1D crystal

after powder average

2/L

0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80

Inte

nsity

Q(Å-1)

L=9.8A,sigma=0.1A

L=9.8A,sigma=0.5A

L=9.8A,sigma=1A

• 1D liquid

• Quantification of the increase of liquid fluctuations with increasing T

C. Bousige, S. Rols, P. Launois and collaborators (submitted)

For other examples, see e.g. the review articles:

- Interpretation of Diffuse X-ray Scattering via Models of Disorder,

T.R. Welberry and B.D. Butler, J. Appl. Cryst. 27 (1994) 205

- Diffuse X-ray Scattering from Disordered Crystals,

T.R. Welberry and B.D. Butler, Chem. Rev. 95 (1995) 2369

-Diffuse scattering in protein crystallography,

J.-P. Benoit and J. Doucetn Quaterly Reviews of Biophysics 28 (1995) 131

-Diffuse scattering from disordered crystals (minerals),

F. Frey, Eur. J. Mineral. 9 (1997) 693

- Special issue of Z. Cryst. on ‘Diffuse scattering’

Issue 12 (2005) Vol. 220

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