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Peak shape What determines peak shape? Instrumental source image flat specimen

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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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Presentation Transcript
slide1

Peak shape

What determines peak shape?

Instrumental

source image

flat specimen

axial divergence

specimen transparency

receiving slit

monochromator(s)

other optics

slide2

Peak shape

What determines peak shape?

Spectral

inherent spectral width

most prominent effect -

Ka1 Ka2 Ka3 Ka4 overlap

slide3

Peak shape

What determines peak shape?

Specimen

mosaicity

crystallite size

microstrain, macrostrain

specimen transparency

slide4

Peak shape

Basic peak parameter - FWHM

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

i.e., FWHM varies with , 2

slide5

Peak shape

Basic peak parameter - FWHM

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

i.e., FWHM varies with , 2

slide7

Peak shape

4 most common profile fitting fcns

slide8

(z) = ∫ tz-1 et dt

0

Peak shape

4 most common profile fitting fcns

slide9

Peak shape

4 most common profile fitting fcns

slide10

Peak shape

X-ray peaks usually asymmetric -

even after a2 stripping

slide11

Peak shape

Crystallite size - simple method

Scherrer eqn.

Bsize = (180/π) (K/ L cos )

Btot = Binstr + Bsize

2

2

2

slide12

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

slide13

Peak shape

Local strains also contribute to broadening

slide14

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 )

slide15

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

slide16

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)

slide17

Peak shape

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

slide18

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)

slide19

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)

slide20

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)

slide21

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

slide22

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

slide23

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

slide24

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

slide25

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

slide26

Peak shape

  • W-A showed
  • AL = ALS ALD (ALS indep of L; ALD dep on L)
  • ALD(h) = cos 2πL <L>h/a
slide27

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

slide28

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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