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DIFFUSE SCATTERING - Université Paris-Saclay · X-Ray Diffraction in Crystals, Imperfect Crystals...
Transcript of DIFFUSE SCATTERING - Université Paris-Saclay · X-Ray Diffraction in Crystals, Imperfect Crystals...
DIFFUSE SCATTERING
Hercules 2013
Pascale Launois
Laboratoire de Physique des Solides (UMR CNRS 8502)
Bât. 510
Université Paris Sud, 91 405 Orsay CEDEX
FRANCE
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
1938
Guinier-Preston zones
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
For other examples, see e.g. the review articles:
- Diffuse X-ray Scatterin g from Disordered Crystals,
T.R. Welberry and B.D. Butler, Chem. Rev. 95 (1995) 2369
-The interpretation and analysis of diffuse scattering using Monte Carlo
simulation methods
T.R. Welberry and D.J. Goosens, Acta Crys. A 64 (2008) 23
-Diffuse scattering in protein crystallography,
J.-P. Benoit and J. Doucet in 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
Free access: http://www.extenza-eps.com/OLD/loi/zkri