1. 1 Refinement parameters What are the parameters to be determined? atom positional parameters atom thermal motion parameters atom site occupancy parameters background function parameters sample displacement, sample transparency, zero-shift errors peak shape parameters unit cell dimensions preferred orientation, absorption, porosity, extinction parameters scale factor(s)

2. 2 Refinement parameters What are the parameters to be determined? atom positional parameters atom thermal motion parameters atom site occupancy parameters background function parameters sample displacement, sample transparency, zero-shift errors peak shape parameters unit cell dimensions preferred orientation, absorption, porosity, extinction parameters scale factor(s)

3. 3 Atom positional parameters I hkl ~ |F hkl | 2 F hkl =  ƒ j e 2πi (hx j + ky j + lz j ) need x j, y j, z j for all atoms in unit cell – except for symmetry-related atom positions certain "special position" coordinates

4. 4 Atom positional parameters Ex. – R3m If atoms in 36i, need x,y,z

5. 5 Atom positional parameters Ex. – R3m If atoms in 36i, need x,y,z If atoms in 18g, need x

6. 6 Atom positional parameters Ex. – R3m If atoms in 36i, need x,y,z If atoms in 18g, need x If atoms in 3a, no parameters

7. 7 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/I o (T) = exp(–16π 2  2 (sin 2  / 2 )

8. 8 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/I o (T) = exp(–16π 2  2 (sin 2  / 2 ) mean square amplitude of atomic vibration for isotropic motion

9. 9 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/I o (T) = exp(–16π 2  2 (sin 2  / 2 ) Usually considered part of atomic scattering factor ƒ j = ƒ oj exp(-8π 2  j 2 (sin 2  / 2 ) = ƒ oj exp(-B j (sin 2  / 2 ) B is "temperature factor"

10. 10 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/I o (T) = exp(–16π 2  2 (sin 2  / 2 ) Usually considered part of atomic scattering factor ƒ j = ƒ oj exp(-8π 2  j 2 (sin 2  / 2 ) = ƒ oj exp(-B j (sin 2  / 2 ) B is "temperature factor" Generally, B approx. 0.5 - 1.0 Å 2, larger for many organic materials, & never negative

11. 11 Atomic displacement parameters When motion is anisotropic:

12. 12 Atomic displacement parameters When motion is anisotropic: (sin 2  / 2 = 1/4 d* 2 d* = ha* + kb* + lc*

13. 13 Atomic displacement parameters When motion is anisotropic: B ij = 8π 2 U ij

14. 14 Atomic displacement parameters Need very high quality data for anisotropic parameters detn B ii are lengths of thermal ellipsoid semi-major and semi-minor axes All Bs describe orientation of ellipsoids wrt lattice vectors

15. 15 Atomic displacement parameters Depending on site symmetry, some  s may be equivalent & some = 0 Ex. - NaNO 3 R3c but can use hexagonal cell (2nd setting)

16. 16 Atomic displacement parameters for  relationships use tables in Pryor and Willis - Thermal Vibrations in Crystallography, pp 104-110 Na, N O

17. 17 Atomic displacement parameters From structure refinement: Na, N

18. 18 Atomic displacement parameters From structure refinement: Na, N

19. 19 Atomic displacement parameters From structure refinement: O (tilted 49° wrt c axis)

20. 20 Atomic displacement parameters Need very high quality data for anisotropic parameters detn B ii are lengths of thermal ellipsoid semi-major and semi-minor axes All Bs describe orientation of ellipsoids wrt lattice vectors Need: B ii > 0 B ii B jj > B ij 2 B 11 B 22 B 33 + B 12 2 B 13 2 B 23 2 > B 11 B 23 2 + B 22 B 13 2 + B 33 B 12 2

21. 21 Site occupancy ƒ j = g j ƒ oj g = 1 - fully occupied g = 0 - unoccupied

22. 22 Site occupancy ƒ j = g j ƒ oj g = 1 - fully occupied g = 0 - unoccupied Two cases: vacancies – must correspond to stoichiometry substitutions –  g i = 1 (including vacancies) & must correspond to stoichiometry

23. 23 Refinement parameters What are the parameters to be determined? atom positional parameters atom thermal motion parameters atom site occupancy parameters background function parameters sample displacement, sample transparency, zero-shift errors peak shape parameters unit cell dimensions preferred orientation, absorption, porosity, extinction parameters scale factor(s)

24. 24 Refinement parameters What contributes to background? general instrumental scattering air scattering fluorescence incoherent scattering TDS – thermal diffuse scattering amorphous material – internal or external

25. 25 Refinement parameters Common background function - polynomial b i =  B m (2  i ) m determine Bs to get backgrd intensity b i at i th point m=0 N

26. 26 Refinement parameters Common background function - polynomial b i =  B m (2  i ) m determine Bs to get backgrd intensity b i at i th point Many other functions b i = B 1 +  B m cos(2  m-1 ) Amorphous contribution b i = B 1 + B 2 Q i +  (B 2m+1 sin(Q i B 2m+2 ))/ Q i B 2m+2 Q i = 2π/d i m=0 N N m=2 m=1 N-2