Crystallographic order

1. Crystallographic order Long-range Lattice periodicity But what if structure is not perfectly periodic?

2. 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

3. 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

4. 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

5. PDFs   pair distribution function pair density function no. density of N atoms Relative atomic positions (positional correlations) described by distances {r} Then, distance distribution is (r) = o g(r) = (1/4πNr2) ∑ ∑ (r - r) Can get PDF from diffraction measurements

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

7. PDFs - examples Here's the evidence G(r) = 4πr((r) - o) 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

8. 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

9. PDFs - examples

10. 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)

11. 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)

12. 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

13. PDFs - the total scattering method pair distribution fcn: (r) = o g(r) r ––> 0, g(r) ––> 0 r ––> ∞, g(r) ––> 1 reduced pair distribution fcn: G(r) = 4πro(g(r) - 1) = 2/π∫Q(S(Q) - 1) sin (Qr) dQ 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 = -4πro uncertainties const. w/ r

14. PDFs - the total scattering method reduced pair distribution fcn: G(r) = 4πro(g(r) - 1) = 2/π∫Q(S(Q) - 1) sin (Qr) dQ crystalline & exfoliated WS2 large r - oscillates about 0 r ––> 0, slope = -4πro uncertainties const. w/ r for crystalline: G(r) fairly const. w/ r for disordered: G(r) falls off w/ r

15. PDFs - the total scattering method radial distribution fcn: R(r) = 4πr2o g(r) CN = ∫r1r2 R(r) dr

16. 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) = (1/4πNr2) ∑ ∑ (r - r) can define partial PDF g'(r) = (1/4π oNr2) ∑only  ∑only (r - r) • g(r) = ∑ ∑ g'(r) • S(Q) = ∑ ∑ S'(r)

17. PDFs - the total scattering method     • g(r) = ∑ ∑ g'(r) • S(Q) = ∑ ∑ S'(r) • To getg'(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

18. 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

19. PDFs - the total scattering method • Bond lengths in silica

20. PDFs - the total scattering method • Bond lengths in silica

21. PDFs - the total scattering method • Peak intensities for carbons

22. PDFs - the total scattering method • Peak widths in InAs & Ni InAs Ni

23. 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

24. 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) = 4πr2o g(r) • CN = ∫r1r2 R(r) dr

25. 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) = 4πr2o g(r) • CN = ∫r1r2 R(r) dr • Iterate with origin on all other atoms

26. 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) = 4πr2o g(r) • CN = ∫r1r2 R(r) dr • Iterate with origin on all other atoms • To account for different atomic species, multiply by bmbn/<b>2

27. 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

28. Modeling PDFs • Example - Y Ba2 Cu3 O6+ • XAFS - split oxygen site • Rietveld structure - no split

29. Modeling PDFs • Example - Y Ba2 Cu3 O6+ • XAFS - split oxygen site • Rietveld structure - no split

30. Modeling PDFs • Example - Y Ba2 Cu3 O6+ • Instead, Cu atom site split

31. PDFs - the total scattering method • Peak intensities for carbons