Crystallographic order

Crystallographic order

Long-range

Lattice periodicity

But what if structure is not perfectly periodic?

Non-crystallographic order

Long-range

Lattice periodicity


Bragg reflections disappear

Can't describe structure as a crystal

Can't determine all atom positions


Long-range

Lattice periodicity





Must determine character of local atomic environment

Determines properties in partially- & non-crystalline materials


Long-range

Lattice periodicity





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)


Q = (4π sin )/


S(Q) =I(Q)/<b>2


Q(S(Q) - 1)

PDF - g(r) crystalline Ni


Q = (4π sin )/


S(Q) =I(Q)/<b>2


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


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


radial distribution fcn:

R(r) = 4πr2o g(r)

CN = ∫r1

r2 R(r) dr


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)


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


PDF interpretation

a. directb. modeling

Direct:

a. peak position - ave. bond lengthsb. peak intensity - CNc. peak shape - probability

distribution


Bond lengths in silica


Peak intensities for carbons


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





R(r) = 4πr2o g(r)

CN = ∫r1

r2 R(r) dr

Modeling PDFs

Approach





R(r) = 4πr2o g(r)

CN = ∫r1

r2 R(r) dr

Iterate with origin on all other atoms

Modeling PDFs

Approach





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+

Instead, Cu atom site split


Peak intensities for carbons

Crystallographic order

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