Peak shape

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Peak shape What determines peak
shape? Instrumental source image flat
specimen axial divergence specimen
transparency receiving slit monochromator(s)
other optics
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Peak shape What determines peak
shape? Spectral inherent spectral width most
prominent effect - Ka1 Ka2 Ka3 Ka4 overlap
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Peak shape What determines peak
shape? Specimen mosaicity crystallite
size microstrain, macrostrain specimen
transparency
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Peak shape Basic peak parameter - FWHM Caglioti
formula H (U tan2 ? V tan ?
W)1/2 i.e., FWHM varies with ?, 2?
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Peak shape Basic peak parameter - FWHM Caglioti
formula H (U tan2 ? V tan ? W)1/2 (not
Lorentzian) i.e., FWHM varies with ?, 2?
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Peak shape
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Peak shape 4 most common profile fitting fcns
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Peak shape 4 most common profile fitting fcns
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Peak shape 4 most common profile fitting fcns
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Peak shape X-ray peaks usually asymmetric -
even after a2 stripping
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Peak shape Crystallite size - simple
method Scherrer eqn. Bsize (180/p) (K??/ L
cos ?) Btot Binstr Bsize
2
2
2
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Peak shape Crystallite size - simple
method Scherrer eqn. Bsize (180/p) (K??/ L
cos ?)
104Å Bsize (180/p) (1.54?/ 104 cos 45)
0.0125 2? 103Å Bsize 0.125 2? 102Å Bsize
1.25 2? 10Å Bsize 12.5 2?
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Peak shape Local strains also contribute to
broadening
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Peak shape Local strains also contribute to
broadening Williamson Hall method
(1953) Stokes Wilson (1944) strain
broadening - Bstrain lt?gt?(4 tan ?) size
broadening - Bsize (K??/ L cos ?)
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Peak shape Local strains also contribute to
broadening Williamson Hall method
(1953) Stokes Wilson (1944) strain
broadening - Bstrain lt?gt?(4 tan ?) size
broadening - Bsize (K??/ L cos ?) Lorentzian
(Bobs - Binst) Bsize Bstrain Gaussian
(Bobs - Binst) Bsize Bstrain
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Peak shape strain broadening - Bstrain lt?gt?(4
tan ?) size broadening - Bsize (K??/ L cos
?) Lorentzian (Bobs - Binst) Bsize
Bstrain (Bobs - Binst) (K? / L cos ?) 4
ltegt(tan ?) (Bobs - Binst) cos ? (K? / L) 4
ltegt(sin ?)
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Peak shape (Bobs - Binst) cos ? (K? / L)
4 ltegt(sin ?)
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Peak shape (Bobs - Binst) cos ? (K? / L)
4 ltegt(sin ?) For best results, use integral
breadth for peak width (width of rectangle with
same area and height as peak)
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Peak shape Local strains also contribute to
broadening The Warren-Averbach method (see
Warren X-ray Diffraction, Chap 13) Begins with
Stokes deconvolution (removes instrumental
broadening)
h(x) (1/A) ? g(x) f(x-z) dz (y x-z)
h(x)
g(z)
f(y)
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Peak shape Local strains also contribute to
broadening The Warren-Averbach method h(x)
g(z) represented by Fourier series Then F(t)
H(t)/G(t)
h(x) (1/A) ? g(x) f(x-z) dz (y x-z)
h(x)
g(z)
f(y)
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Peak shape Local strains also contribute to
broadening The Warren-Averbach method h(x)
g(z) represented by Fourier series Then F(t)
H(t)/G(t) F(t) is set of sine cosine
coefficients
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Peak shape Local strains also contribute to
broadening The Warren-Averbach method Warren
found Power in peak ? An cos 2pnh3 Bn sin
2pnh3 An Nn/N3 ltcos 2plZngt h3 (2 a3
sin ?)/?
n
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Peak shape Local strains also contribute to
broadening The Warren-Averbach method Warren
found Power in peak ? An cos 2pnh3 Bn sin
2pnh3 An Nn/N3 ltcos 2plZngt h3 (2 a3
sin ?)/? sine terms small - neglect
n
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Peak shape Local strains also contribute to
broadening The Warren-Averbach method Warren
found Power in peak ? An cos 2pnh3 Bn sin
2pnh3 An Nn/N3 ltcos 2plZngt h3 (2 a3
sin ?)/? n m'- m Zn - distortion betwn m'
and m cells Nn no. n pairs/column of cells
n
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Peak shape Local strains also contribute to
broadening The Warren-Averbach method Warren
found Power in peak ? An cos 2pnh3 Bn sin
2pnh3 An Nn/N3 ltcos 2plZngt W-A AL
ALS ALD (ALS indep of L ALD dep on L) L
na
n
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Peak shape
W-A showed AL ALS ALD (ALS indep of L ALD
dep on L) ALD(h) cos 2pL lt?Lgth/a
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Peak shape
W-A showed AL ALS ALD (ALS indep of L ALD
dep on L) ALD(h) cos 2pL lt?Lgth/a Procedure
ln An(l) ln ALS -2p2 l2ltZn2gt
n0
n1
n2
ln An
n3
l2
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Advantages vs. the Williamson-Hall Method
???Produces crystallite size distribution.?More
accurately separates the instrumental and sample
broadening effects.?Gives a length average size
rather than a volume average size.Disadvantages
vs. the Williamson-Hall Method ???More prone to
error when peak overlap is significant (in other
words it is much more difficult to determine the
entire peak shape accurately, than it is to
determine the integral breadth or
FWHM).?Typically only a few peaks in the pattern
are analyzed.