1 refinement parameters what are the parameters to be determined? atom positional parameters atom...
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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)
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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)
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Atom positional parameters
Ihkl ~ |Fhkl|2 Fhkl = 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
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Atom positional parameters
Ex. – R3m
If atoms in 36i, need x,y,z
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Atom positional parameters
Ex. – R3m
If atoms in 36i, need x,y,z
If atoms in 18g, need x
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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
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Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π22 (sin2/2)
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Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π22 (sin2/2) mean square amplitude of atomic vibration for isotropic motion
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Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π22 (sin2/2)
Usually considered part of atomic scattering factor
ƒj = ƒoj exp(-8π2j2 (sin2/2) = ƒoj exp(-Bj (sin2/2)
B is "temperature factor"
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Atomic displacement parameters
Debye-Waller factor (see R. W. James, Optical Principles of
the Diffraction of X-rays)
I(T)/Io(T) = exp(–16π22 (sin2/2)
Usually considered part of atomic scattering factor
ƒj = ƒoj exp(-8π2j2 (sin2/2) = ƒoj exp(-Bj (sin2/2)
B is "temperature factor"
Generally, B approx. 0.5 - 1.0 Å2, larger for many organic
materials, & never negative
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Atomic displacement parameters
When motion is anisotropic:
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Atomic displacement parameters
When motion is anisotropic:
(sin2/2 = 1/4 d*2 d* = ha* + kb* + lc*
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Atomic displacement parameters
When motion is anisotropic:
Bij = 8π2 Uij
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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
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Atomic displacement parameters
Depending on site symmetry, some s may be
equivalent & some = 0
Ex. - NaNO3
R3c but can use hexagonal cell (2nd setting)
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Atomic displacement parameters
for relationships use tables in Pryor and Willis - Thermal
Vibrations in Crystallography, pp 104-110
Na, N O
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Atomic displacement parameters
From structure refinement:
Na, N
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Atomic displacement parameters
From structure refinement:
Na, N
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Atomic displacement parameters
From structure refinement:
O
(tilted 49° wrt c axis)
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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 B13
2 B232 > B11 B23
2 + B22B132 + B33 B12
2
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Site occupancy
ƒj = gj ƒoj
g = 1 - fully occupied
g = 0 - unoccupied
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Site occupancy
ƒj = gj ƒoj
g = 1 - fully occupied
g = 0 - unoccupied
Two cases:
vacancies – must correspond to stoichiometry
substitutions – gi = 1 (including vacancies) &
must correspond to stoichiometry
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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)
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Refinement parameters
What contributes to background?
general instrumental scattering
air scattering
fluorescence
incoherent scattering
TDS – thermal diffuse scattering
amorphous material – internal or external
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Refinement parameters
Common background function - polynomial
bi = Bm (2i)m
determine Bs to get backgrd intensity bi at ith point
m=0
N
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Refinement parameters
Common background function - polynomial
bi = Bm (2i)m
determine Bs to get backgrd intensity bi at ith point
Many other functions
bi = B1 + Bm cos(2m-1)
Amorphous contribution
bi = B1 + B2 Qi + (B2m+1 sin(QiB2m+2))/ QiB2m+2
Qi = 2π/di
m=0
N
N
m=2
m=1
N-2