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
∫∫
∫∫
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
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)
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 ε>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 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-simulations_____________________________________________________________________________________________________________________
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. 1D CHAIN LATTICE –1st kind of disorder-calculations_____________________________________________________________________________________________________________________
( ) )**)((*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
2. 1D CHAIN LATTICE –1st kind of disorder-calculations_____________________________________________________________________________________________________________________
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
2. 1D CHAIN LATTICE –1st kind of disorder-calculations_____________________________________________________________________________________________________________________
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
2. 1D CHAIN LATTICE –1st kind of disorder-calculations_____________________________________________________________________________________________________________________
2. 1D CHAIN LATTICE –1st kind of disorder-calculations_____________________________________________________________________________________________________________________
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
2. 1D CHAIN LATTICE –1st kind of disorder-calculations_____________________________________________________________________________________________________________________
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
4. EXAMPLES - Barium IV_____________________________________________________________________________________________________________________
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!
4. EXAMPLES - Barium IV_____________________________________________________________________________________________________________________
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
4. EXAMPLES - Nanotubes inside zeolite_____________________________________________________________________________________________________________________
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
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 - 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
4. EXAMPLES - Structure of B-DNA_____________________________________________________________________________________________________________________
Angle inversely related to helix radius
θ
P Spacingproportional to 1/P
pPitch = P
Spacing correspondsto 1/p
meridian
4. EXAMPLES - Structure of B-DNA_____________________________________________________________________________________________________________________
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)
4. EXAMPLES - A- and B-form DNA coexist in a single crystal lattice_____________________________________________________________________________________________________________________
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?
4. EXAMPLES - A- and B-form DNA coexist in a single crystal lattice_____________________________________________________________________________________________________________________
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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