1. Peak shape
What determines peak shape?
Instrumental
source image
flat specimen
axial divergence
specimen transparency
receiving slit
monochromator(s)
other optics

2. Peak shape
What determines peak shape?
Spectral
inherent spectral width
most prominent effect -
Ka1 Ka2 Ka3 Ka4 overlap

3. Peak shape
What determines peak shape?
Specimen
mosaicity
crystallite size
microstrain, macrostrain
specimen transparency

4. Peak shape
Basic peak parameter - FWHM
Caglioti formula: H = (U tan2  + V tan  + W)1/2
i.e., FWHM varies with , 2

5. Peak shape
Basic peak parameter - FWHM
Caglioti formula: H = (U tan2  + V tan  + W)1/2 (not Lorentzian)
i.e., FWHM varies with , 2

6. Peak shape

7. Peak shape
4 most common profile fitting fcns

8. 4 most common profile fitting fcns
Peak shape
4 most common profile fitting fcns
(z) = ∫ tz-1 et dt

9. Peak shape
4 most common profile fitting fcns

10. Peak shape
X-ray peaks usually asymmetric -
even after a2 stripping

11. Crystallite size - simple method
Peak shape
Crystallite size - simple method
Scherrer eqn.
Bsize = (180/π) (K/ L cos )
Btot = Binstr + Bsize 2 2 2

12. Peak shape
Crystallite size - simple method
Scherrer eqn.
Bsize = (180/π) (K/ L cos )
104Å Bsize = (180/π) (1.54/ 104 cos 45°) = ° 2
103Å Bsize = 0.125° 2
102Å Bsize = 1.25° 2
10Å Bsize = 12.5° 2

13. Peak shape
Local strains also contribute to broadening

14. 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 )

15. 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 )
Lorentzian
(Bobs − Binst) = Bsize + Bstrain
Gaussian 2

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

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

18. 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)

19. Local strains also contribute to broadening The Warren-Averbach method
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 y

20. Local strains also contribute to broadening The Warren-Averbach method
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 y

21. 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

22. Local strains also contribute to broadening The Warren-Averbach method
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

23. Local strains also contribute to broadening The Warren-Averbach method
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

24. Local strains also contribute to broadening The Warren-Averbach method
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

25. Local strains also contribute to broadening The Warren-Averbach method
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

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

27. AL = ALS ALD (ALS indep of L; ALD dep on L)
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
ln An
n=2
n=3 l2

28. 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.