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Methodische Aspekte der Strukturcharakterisierung nanokristalliner Materie
Reinhard B. Neder
Institut für MineralogieUniversität Würzburg
Nanoparticles
● Extended clusters ~1 nm to 300 nm
● Properties intermediate between molecule and crystal
● Optical properties depend on size
Open topics for nanoparticles
� TEM: preparation; core/shell; statistics� XRD: Debye-Scherrer invalid: size + disorder� Exciton: depends on bulk theory� small angle: available edges
Zanchet et al. J. Phys. Chem B (2000),104,11013
Au particlesscale = 5nm
determination of size
Open topics for nanoparticles
16 Å
32 Å
Dhkl = 0.94 λ
β sin Θ
Open topics for nanoparticles
� TEM: preparation; core/shell; statistics� XRD: Debye-Scherrer invalid: size + disorder� Exciton: depends on bulk theory� small angle: available edges
determination of size
Open topics for Nanoparticles
● structure
small crystal <=> glass <=> unique structure
core = rim ? surface relaxation ?
homogeneous structure ⇔ heterogeneous structure
Open topics for Nanoparticles
● structure� TEM: lattice planes ==> well ordered� XRD: too few, broad peaks� EXAFS: local order only
Zanchet et al. J. Phys. Chem B (2000),104,11013
thiol passivated Au nanoparticles
Open topics for Nanoparticles
● structure� TEM: lattice planes ==> well ordered� XRD: too few, broad peaks� EXAFS: local order only
Wu et al. J. Phys. Chem B (2000), 106, 4569
CeO2
Open topics for Nanoparticles
● structure� TEM: lattice planes ==> well ordered� XRD: too few, broad peaks� EXAFS: local order only
Wu et al. J. Phys. Chem B (2000), 106, 4569
CeO2
Open topics for Nanoparticles
● structure� TEM: lattice planes ==> well ordered� XRD: too few, broad peaks ==> no direct structure� EXAFS: local order only
Characterisation techniquesTEM
Powder diffraction
Pair distribution function PDF
Absorption spectroscopy
Small angle scattering
standard / anomalous
real space refinement
complementary to PDF,chemically selective
chemically sensitive viaanomalous small angle
CdS-Glutathione
contradictory size information 15 to 30 Å
Cd
S
Bond length Cd-S- within core- to Glutathione molecule
Structure
Data Collection
BW5, HASYLAB
λ=0.088 ÅE=140 keVT=15 Ksealed capillaryQmax = 30 Å-1
Experimental Data
Only three broad maxima
Inset corresonds toexperiment with Cu-Kα
Normalized Structure Factor
Experimental PDF
narrow first maximum at 2.525 Å
broad, asymmetric second maximum at 4.11 Å
Experimental PDF
longest distance
broad maxima
Experimental PDF
weak maxima at ~1.5 Å
Analysis of first Maximum
R = 2.525 Å; σ = 0.063 ÅN = 3.4
ONE Cd-S distance
Cd-Sinorganic
Cd-Sorganic
Analysis of second Maximum
R = 4.13 Å; σ = 0.15 Å N = 5.8 R = 3.85 Å; σ = 0.12 Å N = 2.7
TWO Cd-Cd distances=> two Cd-S-Cd angles
Cd-Sorganic -Cd = 100 ° Cd-Sinorganic -Cd = 109 °
Summary of direct Interpretation
Chem. analysis Cd1 S0.5 Glutathione0.5
RAMAN spectroscopy No H-S modes ==> Glutathione bound to Cd
PDF 1. peak Cd-S = 2.525 Å; σ = 0.063 Å
2. peak Cd-Sinorganic-S = 109(5)°Cd-Sorganic -S = 100(4)°
PDF longer distances highly disordered
PDF longest distances Diameter ~ 18 Å
ZnO Nanoparticles
012
110010
002011
013
Rietveld refinement
R wp 18 %Wurtzite
size 9.5 nmFWHM 60 = 1.0
anisotropicline widths
ZnO Nanoparticles
012110
Rietveld refinement
R wp 7 %Wurtzite
size 3.2 nmFWHM 60 = 3.0
deviations at 012 and 110
textureanisotropic shapestacking faults
ZnO Nanoparticles
012
110103
Single line fit
hkl FWHM Size
012 3.75 2.42
110 2.72 3.45
103 2.68 3.60
textureanisotropic shapestacking faults
ZnO Nanoparticles Fitting by Debye
Sum over all atom pairsno restrictions on sample structure
Debye formula :
< | F(h) |2 > = Σ j fj2 + Σ i Σ j,j ≠ i fi fj sin ( 2π h rij) / (2π h rij)
open to finite particle with any shapedefects like stacking faults etc.
ZnO Nanoparticles Fitting by Debye
< | F(h) |2 > = Σ j fj2 + Σ i Σ j,j ≠ i fi fj sin ( 2π h rij) / (2π h rij)
= N cJΣ J fJ2 + 2 Σ I Σ J fi fj Σ i Σ j,j > i sin ( 2π h rij) / (2π h rij)
Debye formula :
ZnO Nanoparticles Fitting by Debye< | F(h) |2 > = N cJΣ J fJ
2 + 2 Σ I Σ J fi fj Σ i Σ j,j > i sin ( 2π h rij) / (2π h rij)
for all atom ifor all atoms j > i
compile distance rij into histogram for type IJcompile relative fraction of atoms type I
for all atom pairs IJfor all h
multiply histogram by sin ( 2π h rij) / (2π h rij) (from lookup table) multiply by 2*fi fj
for all atom type Ifor all h
add fi2 * relative amount
creating ZnO Nanoparticles
Calculate powder pattern
Repeat and average
create a large single Wurtzite layer A/B
Stack along c (with faults)
Cut to proper size
{110} and {001}
Repeat with new set of parameter
using a Differential Evolutionary Scheme
Differential Evolution
P1
P2
= trial (d,d)donor
trial (d,p)
trial (p,d)
donor base
parent
choose parent
difference vector
choose difference vector
difference vector * factor
add to donor base to get donorcross-over between parent and donorcompute cost function, keep better of parent/trial
100 * arctan ( | x – 100.23 |
0.05) + noise
Sample for Differential evolution
80.3 * arctan ( | x – 48.188 |
0.87)
Sample for Differential evolution
99.93 * arctan ( | x – 100.23 |
0.049)
Sample for Differential evolution
creating ZnO Nanoparticles
Calculate powder pattern
Repeat and average
create a large single Wurtzite layer A/B
Stack along c (with faults)
Cut to proper size
{110} and {001}
Repeat with new set of parameter
using a Differential Evolutionary Scheme
ZnO Nanoparticles Fitting by DebyeDebye formula
ZnO Wurtzite Structure
acoveral Usize in a-b planesize along cz(oxygen)Stacking probability
R = 8.8 %
ZnO Nanoparticles
Debye formula
Rietveld Rietveld Debyea 3.269 3.256c 5.250 5.224z(O) 0.3876 0.3861B 1.1 1.5
Rietveld Debyesize 3.2 3.6 / 3.8prob --- 0.14
Aspects of PDF Simulation
Finite particle size
Structure more cystal like?more glass like ?complete ?
Proper densityProper coordination distributionProper distance distributions
Best strategy trial and errorRMCevolutionary algorithm
PDF of Nano versus Bulk
Bulk: Number of interatomic pairs increases with r2
4 π ρ0 r
DISCUS uses r !
Nei
ghbo
urs
PDF of Nano versus Bulk
Nanoparticle: A longest vector exists
DISCUS uses r !
4 π ρ0 r tanh(shape(r-diameter)) 4 π ρ0 r
Diameter
Nei
ghbo
urs
Crystal of nanoparticles
Periodic boundary conditions lead to PDF maximawell beyond the particle diameter.
Diameter
Crystal of nanoparticles
Periodic boundary conditions with random orientationdestroy PDF maxima beyond the particle diameter.
Individual nanoparticle
PDF with modified background function
Incomplete structure
Nanoparticle with core and stabilizing molecules
Vectors within core defined by model structure
free molecules
ill defined vectors
scale factor
volume ratio
ZnO Pair Distribution Function
sharp maxima
few stacking faults
Size ~ 9.5 nm
laboratory data
ZnO Pair Distribution Function
laboratory data
simulation based on periodic structure
ZnO Pair Distribution Function
sharp maxima
diameter ~ 5.5 nm
single line fit5.0 nm
dia-
met
er
laboratory data
Rietveld 3.7 nm
ZnO Pair Distribution Function
a 3.256 3.264c 5.238 5.250z(O) 0.3817 0.3836size 38 63
Rietveld PDF
prismatic crystalsno stacking faultsacz(O)Bsize
laboratory data
Size dependent propertiesa and c increase with decreasing particle size Δ vol = 0.7%
smaller particles are less anisotropic in shape
Conclusions
Rietveld/Scherrer equation underestimates particle size
modified Debye formula allows computation of powder pattern for complex nanoparticles
modified PDF calculation allows calculation for finite objects and partially defined objects Differential Evolution is a powerful fitting technique
Acknowledgements
V.I. Korsunskiy
C. Barglik-ChoryG. Mueller
C. KumpfF. NiederdraenkP. Luczak
German Science Foundation SFB410 II-VI Semiconductors
S. DembskiC. GrafC. Ruehl