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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? Spectral inherent spectral width most prominent effect -

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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?

Spectral

inherent spectral width

most prominent effect -

Ka1 Ka2 Ka3 Ka4 overlap


Peak shape

What determines peak shape?

Specimen

mosaicity

crystallite size

microstrain, macrostrain

specimen transparency


Peak shape

Basic peak parameter - FWHM

Caglioti formula: H = (U tan2  + V tan  + W)1/2

i.e., FWHM varies with , 2


Peak shape

Basic peak parameter - FWHM

Caglioti formula: H = (U tan2  + V tan  + W)1/2 (not Lorentzian)

i.e., FWHM varies with , 2



Peak shape

4 most common profile fitting fcns


(z) = ∫ tz-1 et dt

0

Peak shape

4 most common profile fitting fcns


Peak shape

4 most common profile fitting fcns


Peak shape

X-ray peaks usually asymmetric -

even after a2 stripping


Peak shape

Crystallite size - simple method

Scherrer eqn.

Bsize = (180/π) (K/ L cos )

Btot = Binstr + Bsize

2

2

2


Peak shape

Crystallite size - simple method

Scherrer eqn.

Bsize = (180/π) (K/ L cos )

104Å Bsize = (180/π) (1.54/ 104 cos 45°) = 0.0125° 2

103Å Bsize = 0.125° 2

102Å Bsize = 1.25° 2

10Å Bsize = 12.5° 2


Peak shape

Local strains also contribute to broadening


Peak shape

Local strains also contribute to broadening

Williamson & Hall method (1953)

Stokes & Wilson (1944):

strain broadening - Bstrain = <>(4 tan )

size broadening - Bsize = (K/ L cos )


2

2

2

2

Peak shape

Local strains also contribute to broadening

Williamson & Hall method (1953)

Stokes & Wilson (1944):

strain broadening - Bstrain = <>(4 tan )

size broadening - Bsize = (K/ L cos )

Lorentzian

(Bobs − Binst) = Bsize + Bstrain

Gaussian

(Bobs − Binst) = Bsize + Bstrain


Peak shape

strain broadening - Bstrain = <>(4 tan )

size broadening - Bsize = (K/ L cos )

Lorentzian

(Bobs − Binst) = Bsize + Bstrain

(Bobs − Binst) = (K / L cos ) + 4 <ε>(tan θ)

(Bobs − Binst) cos  = (K / L) + 4 <ε>(sin θ)


Peak shape

(Bobs − Binst) cos  = (K / L) + 4 <ε>(sin θ)


Peak shape

(Bobs − Binst) cos  = (K / L) + 4 <ε>(sin θ)

For best results, use integral breadth for peak width (width of rectangle with same area and height as peak)


x

y

z

y

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)


x

y

z

y

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)


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

Warren found:

Power in peak ~ ∑ {An cos 2πnh3 + Bn sin 2πnh3}

An = Nn/N3 <cos 2πlZn>

h3 = (2 a3 sin )/

n


Peak shape

Local strains also contribute to broadening

The Warren-Averbach method

Warren found:

Power in peak ~ ∑ {An cos 2πnh3 + Bn sin 2πnh3}

An = Nn/N3 <cos 2πlZn>

h3 = (2 a3 sin )/

sine terms small - neglect

n


Peak shape

Local strains also contribute to broadening

The Warren-Averbach method

Warren found:

Power in peak ~ ∑ {An cos 2πnh3 + Bn sin 2πnh3}

An = Nn/N3 <cos 2πlZn>

h3 = (2 a3 sin )/

n = m'- m; Zn - distortion betwn m' and m cells

Nn = no. n pairs/column of cells

n


  • Peak shape

  • Local strains also contribute to broadening

  • The Warren-Averbach method

  • Warren found:

  • Power in peak ~ ∑ {An cos 2πnh3 + Bn sin 2πnh3}

  • An = Nn/N3 <cos 2πlZn>

    • W-A:

    • AL = ALS ALD (ALS indep of L; ALD dep on L)

  • L = na

n


Peak shape

  • W-A showed

  • AL = ALS ALD (ALS indep of L; ALD dep on L)

  • ALD(h) = cos 2πL <L>h/a


Peak shape

  • W-A showed

  • AL = ALS ALD (ALS indep of L; ALD dep on L)

  • ALD(h) = cos 2πL <L>h/a

  • Procedure:

  • ln An(l) = ln ALS -2π2 l2<Zn2>

n=0

n=1

n=2

ln An

n=3

l2


  • 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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