Upload
others
View
10
Download
0
Embed Size (px)
Citation preview
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