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) = (1/4 π Nr
2 ) ∑ ∑
 
 (r
r

) pair distribution function pair density function no. density of N atoms
Can get PDF from diffraction measurements

PDFs
Can get PDF from diffraction measurements
 (r) <––FT––> S(Q) (total scattering function) includes Bragg peaks, elastic & inelastic diffuse scattering

PDFs - examples
In (Ga
1-x
In x
)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) = 4 πr(  (r)  o
)

PDFs - examples
In (Ga
1-x
In x
)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 = (4 π sin  )/  total scattering structure function:
S(Q) =I(Q)/
< b
>
2 reduced structure function:
Q(S(Q) - 1)

PDFs - the total scattering method
Q = (4 π sin  )/  total scattering structure function:
S(Q) =I(Q)/
< b
>
2 reduced structure function:
Q(S(Q) - 1)

PDFs - the total scattering method
Q = (4 π 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 = (4 π sin  )/  total scattering structure
function:
S(Q) =I(Q)/
< b
>
2
reduced structure function:
Q(S(Q) - 1) pair distribution fcn:
 (r) =  o g(r) r ––> 0, g(r) ––> 0 r ––>
, g(r) ––> 1 reduced pair distribution fcn:
G(r) = 4 π r
2/ π ∫
 o
(g(r) - 1) =
Q(S(Q) - 1) sin (Qr) dQ large r - oscillates about 0 r ––> 0 , slope = -4 π r  o uncertainties const. w/ r

PDFs - the total scattering method reduced pair distribution fcn:
G(r) = 4 π r
2/ π
 o
(g(r) - 1) =
∫ Q(S(Q) - 1) sin (Qr) dQ large r - oscillates about 0 r ––> 0 , slope = -4 π r  o uncertainties const. w/ r for crystalline:
G(r) fairly const. w/ r for disordered:
G(r) falls off w/ r crystalline & exfoliated WS
2

PDFs - the total scattering method radial distribution fcn:
R(r) = 4 π r 2  o g(r)
CN = ∫ r1 r2
R(r) dr

PDFs - the total scattering method
More than 1 type of atom
If local structure around one type of atom well-defined:
 (r) =  o g(r) = (1/4 π Nr
2 ) ∑ ∑
 
 (r
r

) can define partial PDF g'

(r) = (1/4 π  o
Nr
2 ) ∑
 only 
∑
 only 
 (r
r

) g(r) = ∑ ∑ g'

(r)
 
S(Q) = ∑ ∑ S'

(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 noise
Can also get "differential PDFs" from XAFS data

PDFs - the total scattering method
PDF interpretation
a. direct b. modeling
Direct: a. peak position - ave. bond lengths b. peak intensity - CN c. 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 r n
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 r n
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) = 4 π r 2  o g(r)
CN = ∫ r1 r2
R(r) dr

Modeling PDFs
Approach
Develop model w/ set of N atoms at r n
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) = 4 π r 2  o 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 r n
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) = 4 π r 2  o g(r)
CN = ∫ r1 r2
R(r) dr
Iterate with origin on all other atoms
To account for different atomic species, multiply by b m b n
/
< 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 Ba
2
Cu
3
O
6+ 
XAFS - split oxygen site
Rietveld structure - no split

Modeling PDFs
Example - Y Ba
2
Cu
3
O
6+ 
XAFS - split oxygen site
Rietveld structure - no split

Modeling PDFs
Example - Y Ba
2
Cu
3
O
6+ 
Instead, Cu atom site split

PDFs - the total scattering method
Peak intensities for carbons