# Refinement parameters What are the parameters to be

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) 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 Atom positional parameters Ihkl ~ |Fhkl|2 Fhkl =

j e2i (hxi (hxj + kyj + lzj) need xj, yj, zj for all atoms in unit cell except for symmetry-related atom positions certain "special position" coordinates 3 Atom positional parameters Ex. R3m If atoms in 36i, need x,y,z

4 Atom positional parameters Ex. R3m If atoms in 36i, need x,y,z If atoms in 18g, need x 5 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 6 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/Io(T) = exp(16i (hx22 (sin2/2)

7 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/Io(T) = exp(16i (hx22 (sin2/2) mean square amplitude of atomic vibration for

isotropic motion 8 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/Io(T) = exp(16i (hx22 (sin2/2) Usually considered part of atomic scattering factor j = oj exp(-8i (hx2j2 (sin2/2) = oj exp(-Bj (sin2/2)

B is "temperature factor" 9 Atomic displacement parameters Debye-Waller factor (see R. W. James, Optical Principles of the Diffraction of X-rays) I(T)/Io(T) = exp(16i (hx22 (sin2/2) Usually considered part of atomic scattering factor j = oj exp(-8i (hx2j2 (sin2/2) = oj exp(-Bj (sin2/2) B is "temperature factor"

Generally, B approx. 0.5 - 1.0 2, larger for many organic materials, & never negative 10 Atomic displacement parameters When motion is anisotropic: 11 Atomic displacement parameters When motion is anisotropic:

(sin2/2 = 1/4 d*2 d* = ha* + kb* + lc* 12 Atomic displacement parameters When motion is anisotropic: Bij = 8i (hx2 Uij

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

Atomic displacement parameters Depending on site symmetry, some s may be equivalent & some = 0 Ex. - NaNO3 R3c but can use hexagonal cell (2nd setting) 15 Atomic displacement parameters for relationships use tables in Pryor and Willis Thermal Vibrations in Crystallography, pp 104-110

Na, N O 16 Atomic displacement parameters From structure refinement: Na, N 17

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

(tilted 49 wrt c axis) 19 Atomic displacement parameters Need very high quality data for anisotropic parameters detn Bii are lengths of thermal ellipsoid semi-major and semiminor 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 20 Site occupancy j = gj oj g = 1 - fully occupied g = 0 - unoccupied 21

Site occupancy j = gj oj g = 1 - fully occupied g = 0 - unoccupied Two cases: vacancies must correspond to stoichiometry substitutions &

gi = 1 (including vacancies) 22 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) 23 Refinement parameters What contributes to background? general instrumental scattering

air scattering fluorescence incoherent scattering TDS thermal diffuse scattering amorphous material internal or external 24 Refinement parameters Common background function - polynomial N

bi = B m=0 m (2i)m determine Bs to get backgrd intensity bi at ith point

25 Refinement parameters Common background function - polynomial N bi = B m=0

m (2i)m determine Bs to get backgrd intensity bi at ith point Many other functions N bi = B1 +m=2 Bm cos(2m-1)

Amorphous contribution N-2 bi = B1 + B2 Qi m=1 + (B2m+1 sin(QiB2m+2))/ QiB2m+2 Qi = 2i (hx/di 26

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