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Structural characterization
Part 2
Determining partial pair distribution functions
• X-ray absorption spectroscopy (XAS). • Atoms of different elements have absorption edges at
different energies. Structure from interference pattern of scattered electron waves from neighboring atoms.
• Neutron scattering using different isotopes • Different isotopes have different scattering lengths bi • Measurements on three samples of different isotopic
composition all three partial structure factors can be obtained
Extended X-ray Absorption Fine Structure
• EXAFS is an element specific technique • Probes the local structure atound each atom type • X-ray absorption spectrum is measured close to an X-ray absorption
edge of a particular element • Pre-edge region. • Absorption edge: Steep increase in X-ray absorption coefficient,
µ(E) • Post-edge region: Decreasing µ(E) with small oscillations • An X-ray photon is absorbed by an atom • A photoelectron is ejected and backscattered by neighboring atoms
X-ray absorption experiments • Performed at beam line of
a synchrotron • Transmission geometry
Source: Aksenov et al. 2006
Absorption process • Direct detection • Fluorescence detection • Auger electrons
Source: Rehr and Albers 2000 Source: Aksenov et al. 2006
Schematic picture of EXAFS
• Absorption of X-rays • Photoelectron ejected from
central atom • Electron waves scattered from
neighboring atoms interfere Source: Elliott
• Interference pattern extends 400-1000eV from the edge
Theoretical interpretation
Theory, continued • EXAFS equation can be
generalized to represent contributions from NR multiple scattering contributions of path length 2R.
• Electrons lose energy as they travel in the material – mean free path
• Limited range of tens of Å in EXAFS measurements
• A Fourier transform of χ(k) gives an effective reduced radial distribution function
• Peaks close to nearest and next-nearest neighbor distances
• After the first peak multiple scattering contributions are of increasing importance
• Should be taken into account in fitting to experimental data
Source: Ravel 2005
Analysis of EXAFS data • Obtain the EXAFS function
χ(k) and its Fourier transform χ(r) from fits to the experimental spectrum.
• Generate a simulated model structure consistent with the experimental data
• Reverse Monte-Carlo modelling and fitting to data
• Molecular dynamics simulations of amorphous structure for comparisons
• Model structures further analyzed
What do we learn?
• Partial pair distribution functions
• Interatomic spacings for nearest and next nearest neighbors, maybe further out
• Average coordination numbers • Coordination distributions • Bond angle distributions • Mean square deviations σ
(from Debye-Waller factor)
Reverse Monte-Carlo Modeling
• Choose interatomic potential • Minimize the energy • Minimize difference between experimental data and
simulation by varying the atomic configuration • Combine data from X-ray, neutron, EXAFS…. • Gives ”optimized” structural model that is consistent with
experiments • Not necessarily the ”true” structure • Shows important structural features of the material • Wide range of applications: Liquids, glasses, polymers,
crystals, magnetic materials
Example: Amorphous TiO2
Source: Carlos Triana
Neutron diffraction
2. Sample
2θ
3. Detector
Scan as a function of 2θ
fixed , ii kr
λ
ff kr
,λ
Isotope substitution • For a two component system the total structure factor S(Q) is made
up of 3 different partial structure factors Sij(Q) (~ scattering amplitude)
1 2 S11 S22 S12
( ) ( ) ( )1)(21)(1)()( 1221212222
2211
21
21 −+−+−= QSbbxxQSbxQSbxQS
Sij(Q) - partial structure factors ↓
gij(r) - partial pair distribution functions
Determining partial structure factors
• Different isotopes have different scattering lengths bi
• Measurements on three samples of different isotopic composition all three partial structure factors can be obtained
• Inversion to partial pair distribution functions or partial radial distribution functions
• Alternative: Combine ordinary and magnetic neutron scattering with X-ray scattering
Ex: Amorphous alloy Ni81B19
Source: Elliott: Physics of Amorphous materials
Gij(r)=4πrn0(g2,ij(r)-1)
Reverse Monte-Carlo Modeling
• Choose interatomic potential • Minimize the energy • Minimize difference between experimental data and
simulation by varying the atomic configuration • Combine data from X-ray, neutron, EXAFS…. • Gives ”optimized” structural model that is consistent with
experiments • Not necessarily the ”true” structure • Shows important structural features of the material • Wide range of applications: Liquids, glasses, polymers,
crystals, magnetic materials
-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
Q/Å-1
F N(Q
)
0 2 4 6 8 10 12 14 16
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
FX(Q
)
Q/Å-1
2 4 6 8 10
-2
0
2
FAg
(Q)
Q/Å-1 2 4 6 8
-4
-2
0
2
4
FI(Q
)
Q/Å-1
Neutron X-ray
Ag K EXAFS I LIII EXAFS
Ex: Glassy (AgI)x(AgPO3)1-x
(From R. McGreevy)
FSDP
(AgI)x(AgPO3)1-x
x=0
x=0.5
x=0
x=0.5
Ag+I P+O
AgI pushes apart phosphate chains -> FSDP
(From R. McGreevy)
Small angle scattering
• The study of structures on larger length scales
• Composites, particle aggregates • Porous materials
• X-rays, neutrons, light • SAXS, SANS, SALS
Scattering angle
• Crystalline materials – Bragg’s law: Scattering vector Q ~ d-1, where d is interplanar distance
• Q has dimension [m-1], hence large Q (large scattering angles) corresponds to small length scales
• At large Q we can resolve atomic distances • Small Q larger length scales • With small scattering angles (small Q) we can
study clustering on the nano-scale.
Small angle scattering
( ) ( ) ( ) ( )nm
i
V nm ddeV
QI nm rrrr rrQ −∫ ∫= ρρ1
( ) ( )∑ −=nm
inm
nmeffV
QI,
1 rrQ
At higher scattering angles/Q-values we get scattering from each atom.
At small angles/Q-values we have low resolution for individual atoms but see clusters of atoms in a volume V with scattering length density ρ(r).
Two-component material
• Consider as particles in a matrix • Define the scattering contrast by ρ(r)-ρ0 • Particle form factor
• Intensity per unit volume
• Structure factor S(Q) (assume isotropic particles)
( ) ( ) rr rQ deQf i
Vp
p
•∫ −= )( 0ρρ
)()()(2
QSQfVN
QI pp=
Spherical particles • Define
• Spheres of radius R – asymptotic expressions
• Radius of gyration often used for other shapes as well as for aggregates
• Radius of gyration Rg2=3R2/5 for spheres
• S is the surface area
2)()( QfQP p=
1)()(29)(2)(
1)5/1()()(
420
24
20
2220
2
>>−=−
=
<<−−=
− QRQRVQ
SQP
QRRQVQP
ρρρρπ
ρρ
Pair distribution function
• Number density np=Np/V • Relations between S(Q) and particle pair
distribution function analogous to those for atomic systems
• Isotropic materials
( )dQQrQrQSQnrg
drQrQrrgrnQ)S
p
p
)/))((sin1)((4)8(1)(
)/)((sin1)(41(213
2
22
−+=
−+=
∫∫
− ππ
π
Experimental techniques
• Limits: λ(nm) Q(Å-1) r (nm) • Light 400-600 5 10-5-3 10-3 200-10000 (SALS) • X-rays 0.1-0.4 10-2-15 0.05-50 (SAXS) • Neutrons 0.1-3 10-3-15 0.05-500 (SANS) • Complementary techniques
Small angle neutron scattering
Source: Per Zetterberg
Limiting expressions • Low Q: Guinier approx. • High Q: Porod approx.
• S is the total surface area • Influenced by particle
shape, size distributions: average of Rg
• Aggregation: correlation length ξ.
1/)(2)(
1)3/exp()()(42
0
2220
2
>>−=
<<−−=
g
gg
QRQSQP
QRRQVQP
ρρπ
ρρ
QR
Porod approx. compared to P(Q) for a sphere
Source: J. Teixeira in On Growth and Form
Fractal surfaces
• Smooth surface: S~r2
• Fractal surface: S~rDs
• Porod: P(Q)~(Qr)2/Q6 • Fractal surface:
• Slope between 3 and 4 • Proportionality constant is
a function of Ds.
sDQVQP −−∝ 620
2 /)()( ρρ
Lignite coal Ds=2.5
Bale and Schmidt, PRL 53 (1984) 586
Volume fractals • Pair distribution function g2(r)-1~rDf-3 • Structure factor
• S(Q)~1 at large Q and I(Q)=npP(Q) • Smaller Q: Fractal region
• Small Q: Guinier type law with correlation length ξ instead of Rg
( )∫ −+= drQrQrrgrnQ)S p )/)((sin1)(41( 22π
fff
f
DDD
D
QdyyyQQS
drQrQrrQ)S−−−
−
∫∫
~sin~)(
)/)((sin~(2
1
Gold colloidal aggregates • Model for g2(r)
• Slope between 1 and 3: Volume fractal Df~2
• SAXS exp. vs model
Source: P. Dimon et al, PRL 57 (1985) 598
Examples of porous materials
• Rocks, sandstones • Clays • Soils • Coals • Cement • Cellulose, cotton • Biomolecules, protein
aggregates • Food
• Some porous materials are built up of connected fractal aggregates
• Fractal surfaces are often present also in cases where the solid is non-fractal
• Examples of these two cases
Volume fractals: Silica aerogel • Extremely porous
continuous SiO2 solid network strucutre
• Combination of light and X-ray scattering data
• Df=2.1
• Smooth surfaces
Source: Schaefer et al, 1984
”Greige” Cotton • SAXS data • Guinier type cutoff
at low Q • Df=2.13 • Different kinds of
cotton have values in the range 2.1 to 2.7
• Aggregation of cellulose microcrystals
Q(nm-1)
Source: Lin et al, ACS Symp. Ser. 340 (1987) 233
Surface fractals: Sandstones • Sedimentary rocks • Structure and
properties interesting for oil industry
• ”Toy sandstones”: sand, crushed glass
• Example shows fractal surfaces in sandstones and shales.
Source: Po-zen Wong, Phys. Today 41 (1988)
Small angle neutron scattering (SANS)
Cement: A complex case • Calcium-silicate-hydrate
(CSH) aggregates • Volume fractal • D ~ 1.8 to 2.7 depending
on C/S and preparation
• Ordinary Portland cement during hydration
• Seems surface fractal
Source: Adenot et al. C.R. Acad. Sci. II, 317 (1993) 185. Source: Häussler et al. Phys. Scr. 50 (1994 )210.
Local porosity analysis
• Sintered glass beads • Diameter 250 µm
Source: R. Hilfer, Transport and relaxation phenomena in porous media
• Works for both fractal and non-fractal structures!
Example: Berea sandstone
• Local density function for different cell size L
• Local percolation probabilities for different L
Other techniques
• Nitrogen and water adsorption isotherms • Mercury porosimetry Pore size distributions
• X-ray microtomography for porous
structures