Crystallographic order
description
Transcript of Crystallographic order
Crystallographic order
Long-range
Lattice periodicity
But what if structure is not perfectly periodic?
Non-crystallographic order
Long-range
Lattice periodicity
But what if structure is not perfectly periodic?
Bragg reflections disappear
Can't describe structure as a crystal
Can't determine all atom positions
Non-crystallographic order
Long-range
Lattice periodicity
But what if structure is not perfectly periodic?
Bragg reflections disappear
Can't describe structure as a crystal
Can't determine all atom positions
Must determine character of local atomic environment
Determines properties in partially- & non-crystalline materials
Non-crystallographic order
Long-range
Lattice periodicity
But what if structure is not perfectly periodic?
Bragg reflections disappear
Can't describe structure as a crystal
Can't determine all atom positions
Must determine character of local atomic environment
Determines properties in partially- & non-crystalline materials
New, unfamiliar view - the PDF
PDFs
Relative atomic positions (positional correlations) described by distances {r }
Then, distance distribution is
(r) = o g(r) = (1/4πNr2) ∑ ∑ (r - r)
pair density function
no. density of N atoms
pair distribution function
Can get PDF from diffraction measurements
PDFs
(r) <––FT––> S(Q) (total scattering function)
includes Bragg peaks, elastic & inelastic diffuse scattering
Can get PDF from diffraction measurements
PDFs - examples
In (Ga1-xInx)As, lattice constant varies w/ x
Implies (Ga, In)-As bond length varies w/ x…..?? Actually, only relative nos. of Ga-As & In-As bonds change…..bond lengths constant
Here's the evidence
G(r) = 4πr((r) - o)
PDFs - examples
In (Ga1-xInx)As, lattice constant varies w/ x
Implies (Ga, In)-As bond length varies w/ x…..?? Actually, only relative nos. of Ga-As & In-As bonds change…..bond lengths constant
Details: note
localized strain effects
PDFs - examples
PDFs - the total scattering method
Q = (4π sin )/
total scattering structure function:
S(Q) =I(Q)/<b>2
reduced structure function:
Q(S(Q) - 1)
PDFs - the total scattering method
Q = (4π sin )/
total scattering structure function:
S(Q) =I(Q)/<b>2
reduced structure function:
Q(S(Q) - 1)
PDFs - the total scattering method
Q = (4π sin )/
total scattering structure function:
S(Q) =I(Q)/<b>2
reduced structure function:
Q(S(Q) - 1)
PDF - g(r) crystalline Ni
PDFs - the total scattering method
Q = (4π sin )/
total scattering structure function:
S(Q) =I(Q)/<b>2
reduced structure function:
Q(S(Q) - 1) large r - oscillates about 0
r ––> 0, slope = -4πro
uncertainties const. w/ r
pair distribution fcn:
(r) = o g(r)
r ––> 0, g(r) ––> 0r ––> ∞, g(r) ––> 1reduced pair distribution fcn:
G(r) = 4πro(g(r) - 1) =2/π∫Q(S(Q) - 1) sin (Qr) dQ
PDFs - the total scattering method
crystalline & exfoliated WS2
large r - oscillates about 0
r ––> 0, slope = -4πro
uncertainties const. w/ r
for crystalline:G(r) fairly const.
w/ rfor disordered:
G(r) falls off w/ r
reduced pair distribution fcn:
G(r) = 4πro(g(r) - 1) =2/π∫Q(S(Q) - 1) sin (Qr) dQ
PDFs - the total scattering method
radial distribution fcn:
R(r) = 4πr2o g(r)
CN = ∫r1
r2 R(r) dr
PDFs - the total scattering method
g(r) = ∑ ∑ g'(r)
S(Q) = ∑ ∑ S'(r)
More than 1 type of atom
If local structure around one type of atom well-defined:
(r) = o g(r) = (1/4πNr2) ∑ ∑ (r - r)
can define partial PDF
g'(r) = (1/4π o Nr2) ∑only ∑only (r - r)
PDFs - the total scattering method
g(r) = ∑ ∑ g'(r)
S(Q) = ∑ ∑ S'(r)
To get g'(r) s, need sets of independent, high quality diffraction patterns
patterns similar - differences sometimes
lost in noiseCan also get "differential PDFs" from XAFS data
PDFs - the total scattering method
PDF interpretation
a. directb. modeling
Direct:
a. peak position - ave. bond lengthsb. peak intensity - CNc. peak shape - probability
distribution
PDFs - the total scattering method
Bond lengths in silica
PDFs - the total scattering method
Bond lengths in silica
PDFs - the total scattering method
Peak intensities for carbons
PDFs - the total scattering method
Peak widths in InAs & Ni
InAs
Ni
Modeling PDFs
Approach
Develop model w/ set of N atoms at rn
Put origin on random atom
Find distance to every other atom
Add unit value to R(r) for each atom at that distance
Modeling PDFs
Approach
Develop model w/ set of N atoms at rn
Put origin on random atom
Find distance to every other atom
Add unit value to R(r) for each atom at that distance
R(r) = 4πr2o g(r)
CN = ∫r1
r2 R(r) dr
Modeling PDFs
Approach
Develop model w/ set of N atoms at rn
Put origin on random atom
Find distance to every other atom
Add unit value to R(r) for each atom at that distance
R(r) = 4πr2o g(r)
CN = ∫r1
r2 R(r) dr
Iterate with origin on all other atoms
Modeling PDFs
Approach
Develop model w/ set of N atoms at rn
Put origin on random atom
Find distance to every other atom
Add unit value to R(r) for each atom at that distance
R(r) = 4πr2o g(r)
CN = ∫r1
r2 R(r) dr
Iterate with origin on all other atoms
To account for different atomic species, multiply by bmbn/<b>2
Modeling PDFs
How good is model?
Compare w/ PDF calc'd from scattering data (real space)
Or, can calc scattering data (Fourier space)
Model frequently has adjustable parameters
Use Rietveld refinement procedure and watch residuals
Modeling PDFs
Example - Y Ba2 Cu3 O6+
XAFS - split oxygen site
Rietveld structure - no split
Modeling PDFs
Example - Y Ba2 Cu3 O6+
XAFS - split oxygen site
Rietveld structure - no split
Modeling PDFs
Example - Y Ba2 Cu3 O6+
Instead, Cu atom site split
PDFs - the total scattering method
Peak intensities for carbons