diffuse scattering - equipes.lps.u-psud.fr€¦ · x-ray diffraction in crystals, imperfect...

$: DIFFUSE SCATTERING - equipes.lps.u-psud.fr€¦ · X-Ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies, A. Guinier, Dover publications. r A r B c A =c B =1/2, p=1/2$
DIFFUSE SCATTERING

Hercules 2008

Pascale Launois

Laboratoire de Physique des Solides (UMR CNRS 8502)Bât. 510

Université Paris Sud, 91 405 Orsay CEDEXFRANCE

[email protected]://www.lps.u-psud.fr/Utilisateurs/launois/

OUTLINE__________________________________________________________________________________________________________________

1. IntroductionGeneral expression for diffuse scatteringDiffuse scattering/propertiesA brief history

2. 1D chain- First kind disorder

Simulations of diffuse scattering/local orderAnalytical calculations

- Second kind disorder

3. Diffuse scattering analysisGeneral rulesTo go further

4. Examples- Barium IV: self-hosting incommensurate structure- Nanotubes@zeolite: structural characteristics of nanotubes from diffuse scattering analysis- DIPSΦ4(I3)0.74: a one-dimensional liquid

- DNA: ● B-DNA structure from diffuse scattering pattern ● coexistence of A- and B-form of DNA

5. Diffuse scattering in powder patterns

1. INTRODUCTION - General expression for diffuse scattering

● No long range order : second-kind disorder

Average structureORDER

Bragg peaks

2-body correlationsDISORDER

Diffuse scattering

● Average long range order : first-kind disorder

( ) ( ) ( ) ( ) mRsi

m nnnmnn

lkhlkh

nn esFFFGssF

VsI πδ 2

2

,,,,

2

*1⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−∝ ∑∑ +

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

( ) ∑∑ ++

+− ∝∝

m

Rsi

tnmnntmn

RRsimn

Rsin

mmnn eFFeFeFsI πππ 2

,

*

,

)(2*2

Variable omittedhere 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)

= ρ(r)2

∫∫

∫∫


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


Diffuse scattering ⇔ defects, disorder, local order, dynamics

Physical properties

LasersSemiconductor 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

Biologybiological 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 disorderAtoms 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 disorderAtoms 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 effectAtoms and vacancies : local order + size effects (interatomic distance d=>d+ε).

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


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 ε>0 and <0.

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


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


-Substitution disorder: intensity at low s.

Displacement disorder: no intensity around s=0.

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

For ε>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*.

sa*

ID ε=0ε>0


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

Analytical calculations

A atoms: red, B: purplecA=cB=1/2p= probability to have a ‘A(red)-B(purple)’ pair

a. Substitution disorder

4

2BA ff +

With αm=1-2pm, pm being the probability to have a pair AB at distance (ma);p0=0, p1=p.αm : Warren-Cowley coefficients

⎥⎦

⎤⎢⎣

⎡+

−∝ ∑

>

)2cos(214

0

2

smaff

Im

mBA

D πα

( ) ( ) ( ) ( ) smai

mnnnmnn

hnn esFFFahssF

asI π

δ222

*/1⋅⎟

⎠⎞⎜

⎝⎛ −+−∝ ∑∑ +

Demonstration

22 : BABA

AnfffffFFAn −

=+

−=>−=2

)(2

: BABABn

fffffFFBn −−=

+−=>−=

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

A(n)B(n+m)


( ) )**)((*2

mnmnnnnnnmnn FFFFsFFF +++ −−=−

( ) ⎥⎦⎤

⎢⎣⎡ −−−+−

−=−⇒ + mmmm

BAnnnmnn pppp

ffsFFF

21

21)1(

21)1(

21

4*

22

mp

n n+m

( ) mBA

nnnmnn

ffsFFF α

4*

22 −

=−⇒ +mm p21−=α

mmm −=≠= ααα ,0;10

( )

⎥⎦

⎤⎢⎣

⎡+

−=

−=⋅⎟

⎠⎞⎜

⎝⎛ −⇒

∑

∑∑

>

∞

−∞=+

)2cos(214

)2exp(4

*

0

2

222

smaff

smaiff

esFFF

mm

BA

mm

BAsmai

mnnnmnn

πα

παπ


mp

ppppprobprobprobprobp mmABmAABB

mABm )1()1( 11

1111−−

−− −+−=+=( ) mm

m p 121 αα =−=⇒

211

21

2

)2cos(211

4 απαα

+−−−

∝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

mm πα

παπα−

== ∑∑∞

=

∞

=

[ ] 1)2exp(1

1)2exp()2exp(11

1

1

−−−

=−= ∑∑∞

=

−

−∞= saisaismai

m

m

mm πα

παπα

n n+m

211

21

2

)2cos(211

4 απαα

+−−−

∝sa

ffI BA

D

p=1/2 : α1=0 : Laue formula

p>1/2 : α1<0 : positions of maxima = (2n+1) a*/2, n integer

P<1/2 : α1>0 : positions of maxima = n a*

A=B : no disorder : no diffuse scattering

4

2BA

D

ffI

−∝

Formula used in Diff1D


u -u

b. Displacement disorder

211

21

2

)2cos(211

4 απαα

+−−−

=sa

ffI BA

D

with andsuiAA eff π2→ sui

AB eff π2−→

( ) 211

2122

)2cos(2112sin

απααπ

+−−

=sa

sufI AD⇒

ID(s=0) =0

u=0 : no disorder : no diffuse scattering

Formula used in Diff1D



3. Size effectCombination of correlated displacement and substitution disorder

⎥⎦

⎤⎢⎣

⎡−+

−= ∑∑

>>

)2sin(22)2cos(214 00

2

smasmasmaff

Im

mm

mBA

D ππβπα

mst neighbors:

Mean-distance=ma ⇒

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

mAB

mBB

mBB

mAA

mAA madmadmad εεε +=+=+=

mAB

mBB

mAA

mABm

AB pp

2))(1( εεε +−−=

( ) ( ) AAAB

ABB

AB

B

fff

fff εαεαβ 111 11 +

−−+

−=

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

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

mAA πεπεπ +=+

( ) ( ) msdi

mnnnmnnD esFFFsI π22

* ⋅⎟⎠⎞⎜

⎝⎛ −∝ ∑ +

fB>fA

εBB>0 ⇒ β1>0 εAA<0

⇒ -β1sin(2πsa) >0 for s~<na, <0 for s~>na

sa*

ID εij=0Size effect

Formula used in Diff1D : m=1


BA

Second-kind disorderNo 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)=δ(z)+Σm>0(hm(z)+ h-m(z))⇒ Fourier transform: 1+2 Σm>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)2/2σ2) : H(s)~exp(-2π2s2σ2)exp(i2πsd)⇒ I(s)~sinh(4π2s2σ2/2)/ (cosh(4π2s2σ2/2)-cos(2πsd))

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 : width∝1/(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_____________________________________________________________________________________________________________________

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 3Tetragonal symmetryDiffuse 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


5.5 GPa 12.6 GPa

Phase Ibcc

Phase IIhcp

Phase IVTetragonal inc.

45 GPa

Phase Vhcp

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 disorderedand giving rise to strong diffuse scattering.The guest structures are incommensurate with the host.

An incommensurate host-guest structure in an element!


DiffuseScatteringzeolite

DS nanotubes

c*

(a) Standard source CuKα, LPS(b) ID1, ESRF

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


Small s, l=0: ID(s)∝fC2.(J0(2πsΦ/2))2

Nanotubes with Φ≈4Å. The smallest nanotubes. Vs superconductivity?Tang et al., Science 292, 2462 (2001)

4

2BA

D

ffI

−∝Laue formula :

Φ

s

s


Tetraphenyldithiapyranylidene iodine C34H24I2.28S2

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

4. EXAMPLES - DIPSΦ4(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 - DIPSΦ4(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 - DIPSΦ4(I3)0.75_____________________________________________________________________________________________________________________

3D ordered state below 182K

4. EXAMPLES - DIPSΦ4(I3)0.75_____________________________________________________________________________________________________________________

4. EXAMPLES - Structure of B-DNA_____________________________________________________________________________________________________________________

B form of DNA in fiber

The structure of DNAR. 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


Angle inversely related to helix radius

θ

P Spacingproportional to 1/P

pPitch = P

Spacing correspondsto 1/p

meridian


p=3.4Å

P=34Å

Extinction:t0=3P/8

P=34Å

p=3.4Å

t0

4. EXAMPLES - A- and B-form DNA coexist in a single crystal lattice_____________________________________________________________________________________________________________________

Crystal of A-DNA octamers

J. Doucet, J.-P. Benoit, W.B.T. Cruse, T. Prange and O. Kennard,

Nature 337, 190 (1989)


3.4Å: reinforcement of the molecular structure factor by the base stacking in B-DNA

DC : streaks at(n+1/2)2π/c

c=41.7Å

DC: streaks at2π/P

P=34Å(28-34Å)

What disorder for B-DNA octamersin the channels?


Diffuse peaks at (n+1/2)2π/c

● =one octamer, + =nothingor ● =2 octamers, + =nothing

Substitutional disorder between ● and +

+ + + + + +2c

B-DNA octamer length~2c/3

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

5. DIFFUSE SCATTERING in POWDER PATTERNS

C60 fullerenes

Single crystal

Powder

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

3.45 Å

5. DIFFUSE SCATTERING in POWDER PATTERNS

Random stacking of graphene layers

Powderaverage

Turbostratic carbon

● Diffraction peaks (00l)● Diffuse lines (hk)

0

2

4

6

8

10

0 5 Q (Å-1)

00 10 11 20 12 30l

?

The smallest scatteringangle

The scattering angle decreases

The scattering angle increasesagain

2θMin

I

2θ2θMin

Sawtooth peak :

● Symmetric (00l) peaks● Sawtooth shaped (hk) reflectionsPowder

average

● Diffraction peaks (00l)● Diffuse lines (hk)

0

2

4

6

8

10

0 5 Q (Å-1)

00 10 11 20 12 30l

0 2 4 6 8 10 12 14 16Q(Å-1)

(10)

(002

)

0 2 4 6 8 10 12 14 16Q(Å-1)

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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