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)

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)

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

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

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

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

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)

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

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"

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. 0.5 - 1.0 Å2, larger for many organic materials, & never negative

Atomic displacement parameters When motion is anisotropic:

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

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

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

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

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

Atomic displacement parameters From structure refinement: Na, N

Atomic displacement parameters From structure refinement: Na, N

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

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

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

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

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)

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

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

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