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. 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. Atom positional parameters
Ihkl ~ |Fhkl|2 Fhkl = S ƒj e2πi (hxj + kyj + lzj)
need xj, yj, zj for all atoms in unit cell –
except for
symmetry-related atom positions
certain "special position" coordinates

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

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

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. Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π2m2 (sin2q)/l2)

8. Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π2m2 (sin2q)/l2)
mean square amplitude of atomic vibration for isotropic motion

9. Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π2m2 (sin2q)/l2)
Usually considered part of atomic scattering factor
ƒj = ƒoj exp(-8π2mj2 (sin2q)/l2) = ƒoj exp(-Bj (sin2q)/l2)
B is "temperature factor"

10. Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π2m2 (sin2q)/l2)
Usually considered part of atomic scattering factor
ƒj = ƒoj exp(-8π2mj2 (sin2q)/l2) = ƒoj exp(-Bj (sin2q)/l2)
B is "temperature factor"
Generally, B approx Å2, larger for many organic
materials, & never negative

11. Atomic displacement parameters
When motion is anisotropic:

12. Atomic displacement parameters
When motion is anisotropic:
(sin2q)/l2 = 1/4 d*2 d* = ha* + kb* + lc*

13. Atomic displacement parameters
When motion is anisotropic:
Bij = 8π2 Uij

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

15. Atomic displacement parameters
Depending on site symmetry, some bs may be
equivalent & some = 0
Ex. - NaNO3
R3c but can use hexagonal cell (2nd setting)

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

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

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

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

20. Atomic displacement parameters
Need very high quality data for anisotropic parameters detn
Bii are lengths of thermal ellipsoid semi-major and semi-minor axes
All Bs describe orientation of ellipsoids wrt lattice vectors
Need:
Bii > 0
Bii Bjj > Bij2
B11 B22 B33 + B122 B132 B232 > B11 B232 + B22B132 + B33 B122

21. Site occupancy
ƒj = gj ƒoj
g = 1 - fully occupied
g = 0 - unoccupied

22. Site occupancy ƒj = gj ƒoj g = 1 - fully occupied g = 0 - unoccupied
Two cases:
vacancies – must correspond to stoichiometry
substitutions – S gi = 1 (including vacancies) &
must correspond to stoichiometry

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

25. Refinement parameters
Common background function - polynomial
bi = S Bm (2qi)m
determine Bs to get backgrd intensity bi at ith point N
m=0

26. Refinement parameters
Common background function - polynomial
bi = S Bm (2qi)m
determine Bs to get backgrd intensity bi at ith point
Many other functions
bi = B1 + S Bm cos(2qm-1)
Amorphous contribution
bi = B1 + B2 Qi + S (B2m+1 sin(QiB2m+2))/ QiB2m+2
Qi = 2π/di N
m=0 N
m=2
N-2
m=1