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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
peak shape parameters sample displacement, sample transparency,
zero-shift errors
unit cell dimensions
preferred orientation, absorption, porosity,
extinction parameters
scale factor(s)
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Peak shape parameters
What determines peak shape?
Instrumental
source image
flat specimen
axial divergence
specimen transparency
receiving slit
monochromator(s)
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Peak shape parameters
What determines peak shape?
Spectral
inherent spectral width
most prominent effect - Kα1 Kα2 Kα3 Kα4 overlap
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Peak shape parameters
What determines peak shape?
Specimen
mosaicity
crystallite size
microstrain
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Peak shape parameters
Historical
Began with neutron diffraction
- peak shapes nearly Gaussian
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Peak shape parameters
1st basic peak parameter - FWHM
Caglioti formula: H = (U tan2 θ + V tan θ + W)1/2
i.e., FWHM varies with θ or 2θ
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Peak shape parameters
1st basic peak parameter - FWHM
Caglioti formula: H = (U tan2 θ + V tan θ + W)1/2
i.e., FWHM varies with θ or 2θ
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Peak shape parameters
X-ray case more complicated
not usually Gaussian
not usually Lorentzian
usually need to mix the two somehow
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Peak shape parameters
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Peak shape parameters
4 most common profile fitting fcns
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Peak shape parameters
4 most common profile fitting fcns
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Peak shape parameters
4 most common profile fitting fcns
Γ(z) = ∫ tz-1 et dt0
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Peak shape parameters
4 most common profile fitting fcns
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Peak shape parameters
But peaks are usually asymmetric - even after α2 stripping!
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Peak shape parameters
Finally - use NIST std 660 (LaB6) to determine broadening frominstrumental and spectral contributions & kept constant (U, V, W)
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Scale factors
Constant factor which scales calculated intensities of all reflectionsin pattern up to the observed intensities
Determined by least squares refinement
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Scale factors
Constant factor which scales calculated intensities of all reflectionsin pattern up to the observed intensities
Determined by least squares refinement
In GSAS, two types of scale factors:
histogram scale factor phase scale factor
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Scale factors
Constant factor which scales calculated intensities of all reflectionsin pattern up to the observed intensities
Determined by least squares refinement
In GSAS, two types of scale factors:
histogram scale factor phase scale factor
At any point in pattern, calc'd I is then obtained as above – thencompared with obs'd I
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Scale factor
When more than one phase, Sph calculated for each
w/o of each phase present automatically calc'd:
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Lattice parameters
In general
1/d2 = 1/V2 (S11h2 + S22k2 + S33l2 + 2S12hk + 2S13hl + 2S23kl)
S11 = b2c2 sin2 α
S22 = a2c2 sin2 β
S33 = a2b2 sin2 γ
S12 = abc2 (cos α cos β − cos γ)
S13 = ab2c (cos γ cos α − cos β)
S23 = a2bc (cos β cos γ − cos α)
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Lattice parameters
1/d2 = 1/V2 (S11h2 + S22k2 + S33l2 + 2S12hk + 2S13hl + 2S23kl)
GSAS:
