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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 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 - K a1 K a2 K a3 K a4 overlap Peak shape What determines peak shape? Spectral inherent spectral width most prominent effect - K a1 K a2 K a3 K a4 overlap

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Peak shape What determines peak shape? Specimen mosaicity crystallite size microstrain, macrostrain specimen transparency 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 tan 2 + V tan + W) 1/2 i.e., FWHM varies with , 2 Peak shape Basic peak parameter - FWHM Caglioti formula: H = (U tan 2 + 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 tan 2 + V tan + W) 1/2 (not Lorentzian) i.e., FWHM varies with , 2 Peak shape Basic peak parameter - FWHM Caglioti formula: H = (U tan 2 + V tan + W) 1/2 (not Lorentzian) i.e., FWHM varies with , 2

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Peak shape

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4 most common profile fitting fcns Peak shape 4 most common profile fitting fcns

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Peak shape 4 most common profile fitting fcns Peak shape 4 most common profile fitting fcns (z) = ∫ t z-1 e t dt 0

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Peak shape 4 most common profile fitting fcns Peak shape 4 most common profile fitting fcns

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Peak shape X-ray peaks usually asymmetric - even after a 2 stripping Peak shape X-ray peaks usually asymmetric - even after a 2 stripping

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Peak shape Crystallite size - simple method Scherrer eqn. B size = (180/ π) ( K / L cos ) B tot = B instr + B size Peak shape Crystallite size - simple method Scherrer eqn. B size = (180/ π) ( K / L cos ) B tot = B instr + B size 2 2 2 2 2 2

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Peak shape Crystallite size - simple method Scherrer eqn. B size = (180/ π) ( K / L cos ) Peak shape Crystallite size - simple method Scherrer eqn. B size = (180/ π) ( K / L cos ) 10 4 ÅB size = (180/ π) ( 1.54 / 10 4 cos 45°) = 0.0125° 2 10 3 Å B size = 0.125° 2 10 2 Å B size = 1.25° 2 10Å B size = 12.5° 2 10 4 ÅB size = (180/ π) ( 1.54 / 10 4 cos 45°) = 0.0125° 2 10 3 Å B size = 0.125° 2 10 2 Å B size = 1.25° 2 10Å B size = 12.5° 2

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Peak shape Local strains also contribute to broadening 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 - B strain = (4 tan ) size broadening - B size = ( K / L cos ) Peak shape Local strains also contribute to broadening Williamson & Hall method (1953) Stokes & Wilson (1944): strain broadening - B strain = (4 tan ) size broadening - B size = ( K / L cos )

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Peak shape Local strains also contribute to broadening Williamson & Hall method (1953) Stokes & Wilson (1944): strain broadening - B strain = (4 tan ) size broadening - B size = ( K / L cos ) Lorentzian ( B obs − B inst ) = B size + B strain Gaussian ( B obs − B inst ) = B size + B strain Peak shape Local strains also contribute to broadening Williamson & Hall method (1953) Stokes & Wilson (1944): strain broadening - B strain = (4 tan ) size broadening - B size = ( K / L cos ) Lorentzian ( B obs − B inst ) = B size + B strain Gaussian ( B obs − B inst ) = B size + B strain 2 2 2 2

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Peak shape strain broadening - B strain = (4 tan ) size broadening - B size = ( K / L cos ) Lorentzian ( B obs − B inst ) = B size + B strain (B obs − B inst ) = (K / L cos ) + 4 (tan θ ) (B obs − B inst ) cos = (K / L) + 4 (sin θ ) Peak shape strain broadening - B strain = (4 tan ) size broadening - B size = ( K / L cos ) Lorentzian ( B obs − B inst ) = B size + B strain (B obs − B inst ) = (K / L cos ) + 4 (tan θ ) (B obs − B inst ) cos = (K / L) + 4 (sin θ )

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Peak shape (B obs − B inst ) cos = (K / L) + 4 (sin θ ) Peak shape (B obs − B inst ) cos = (K / L) + 4 (sin θ )

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Peak shape (B obs − B inst ) cos = (K / L) + 4 (sin θ ) For best results, use integral breadth for peak width (width of rectangle with same area and height as peak) Peak shape (B obs − B inst ) cos = (K / L) + 4 (sin θ ) 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) 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) f(y) g(z) z x yy

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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) 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) f(y) g(z) z x yy

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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 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 ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 h 3 = (2 a 3 sin )/ Peak shape Local strains also contribute to broadening The Warren-Averbach method Warren found: Power in peak ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 h 3 = (2 a 3 sin )/ n

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Peak shape Local strains also contribute to broadening The Warren-Averbach method Warren found: Power in peak ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 h 3 = (2 a 3 sin )/ sine terms small - neglect Peak shape Local strains also contribute to broadening The Warren-Averbach method Warren found: Power in peak ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 h 3 = (2 a 3 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 ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 h 3 = (2 a 3 sin )/ n = m'- m; Z n - distortion betwn m' and m cells N n = no. n pairs/column of cells Peak shape Local strains also contribute to broadening The Warren-Averbach method Warren found: Power in peak ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 h 3 = (2 a 3 sin )/ n = m'- m; Z n - distortion betwn m' and m cells N n = 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 ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 W-A: A L = A L S A L D (A L S indep of L; A L D dep on L) L = na Peak shape Local strains also contribute to broadening The Warren-Averbach method Warren found: Power in peak ~ ∑ {A n cos 2 π nh 3 + B n sin 2 π nh 3 } A n = N n /N 3 W-A: A L = A L S A L D (A L S indep of L; A L D dep on L) L = na n

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Peak shape W-A showed A L = A L S A L D (A L S indep of L; A L D dep on L) A L D (h) = cos 2 π L h/a W-A showed A L = A L S A L D (A L S indep of L; A L D dep on L) A L D (h) = cos 2 π L h/a

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Peak shape W-A showed A L = A L S A L D (A L S indep of L; A L D dep on L) A L D (h) = cos 2 π L h/a Procedure: ln A n (l) = ln A L S -2 π 2 l 2 W-A showed A L = A L S A L D (A L S indep of L; A L D dep on L) A L D (h) = cos 2 π L h/a Procedure: ln A n (l) = ln A L S -2 π 2 l 2 n=0 n=1 n=2 n=3 l2l2 ln A n

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Advantages vs. the Williamson-Hall Method ﾊ･ 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 ﾊ･ 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.

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