Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin Chapter 8) Peak form for X-ray peaks: Gaussian Lorentizian Voigt, Psudo-Voigt:

Gaussian function x0x0 FWHM

Lorentzian function or Cauchy form FWHM x0x0

Voigt: convolution of a Lorentzian and a Gaussian Complex error function FWHM most universal; more complex to fit.

Lorentzian function or Cauchy form FWHM Gaussian function FWHM pseudo-Voigt:  : Cauchy content, fraction of Cauchy form.

2  = FWHM

Lineshapes: disturbed by the presence of K  1 and K  2. Decouple them if necessary: Rachinger Correction for K  1 and K  2 separation: Assume: (1) K  1 and K  2 identical lines profiles (not necessarily symmetrical); (2) I p of K  2 = ½ I p of K  1.

… … General form Example: Separated by 3 unit I i : experimental intensity at point i I i (  1 ): part of I i due to due to K  1

Diffraction Line Broadening and Convolution Sources of Broadening: (1) small sizes of crystalline (2) distributions of strains within individual crystallites, or difference in strains between crystallites (3) The diffractometer (instrumental broadening)

Size Broadening: Interference function Define deviation vector …

I Half width half maximum (HWHM): particular usually small  Solve graphically

Define ~ Solution: x = Define

FWHM In X-ray, 2  is usually used, define B in radians If the  is used instead of 2 , K should be divided by 2. Scherrer equation, K is Scherrer constant

Strain broadening: Uniform strain  lattice constant change  Bragg peaks shift. Assume strain =   d 0 change to d 0 (1+  ). Diffraction condition: In terms of  Peak shift Larger shift for the diffraction peaks of higher order

Distribution of strains  diffraction peaks broadening Strain distribution  relate to kk is the HWHM of the diffraction G along

Instrument broadening: Main Sources: Combining all these broadening by the convolution procedure  asymmetric instrument function convolution

The Convolution Procedure: instrument function f(x) and the specimen function g(x) the observed diffraction profile, h(  ). The convolution steps are * Flip f(x)  f(-x) * Shift f(-x) with respect to g(x) by  f(-x)  f(  -x) * Multiply f and g f(  -x)g(x) * Integrate over x f(x)f(x) g(x)g(x) Assume f and g are the functions on the right, the h(  ) that we will get is f(-x) 0

 =  =  =  =  =  0 31/616/3 0 7/6 0 h()h() 5 6

Convolution of Gaussians: Two functions f(  ): breadth B f g(  ): breadth B g  h(  ) = f(  )*g(  ); breadth B h

Convolution of Lorentzians: Two Lorentzian functions: f(  ): breadth B f g(  ): breadth B g  h(  ) = f(  )*g(  ); breadth B h

Fourier Transform and Deconvolutions: Remove the blurring, caused by the instrument function: deconvolution (Stokes correction). Instrument broadening function: f(k) (*k is function of  ) True specimen diffraction profile: g(k) Measured by the diffractometer: h(K) l: [1/length], the range in k of the Fourier series is the interval –l/2 to l/2. Fourier transform the above three functions (DFT)

The function f and g vanished outside of the k range  Integration from -  to  is replaced by –l/2 to l/2 Orthogonality condition vanishes by symmetry

Convolution in k-space is equivalent to a multiplication in real space (with variable n/l). The converse is also true. Important result of the convolution theorem! Deconvolution: {G(n)} is obtained from

Data from a perfect specimen Data from the actual specimen Rachinger Correction (optional) Rachinger Correction (optional) f(k)f(k) Stokes Correction G(n)= H(n)/F(n) h(k)h(k) F.T. Corrected data free of instrument broadening F.T. -1 g(k)g(k) “Perfect” specimen: chemical composition, shape, density similar to the actual specimen (  specimen roughness and transparency broadening are similar) * E.g.: For polycrystalline alloy, the specimen is usually obtained by annealing

f(k), g(k), and h(k): asymmetric  F.T. complex coeff.

real part g(k) is real and can be reconstructed as

Simultaneous Strain and Size Broadening: True sample diffraction profile: strain broadening and size broadening effect Take advantage of the following facts: Crystalline size broadening is independent of G Strain broadening depends linearly on G Usually, know one to get the other Both unknown

Williamson-Hall Method Easiest way! Requires an assumption of the shape of the peaks: Kinematical crystal shape factor intensity Gaussian function characteristic of the strain broadening convolution

Assume a Gaussian strain distribution (quick falloff for strain larger than the yield strain)  (  )

Approximate the size broadening part with a Gaussian function Good only when strain broadening >> size broadening (see page 9) characteristic width

The convolution of two Gaussians Plot  k 2 vs G 2 (k)2(k)2 G2G2 Slope = (HWHM)

Approximate the size broadening and strain broadening : Lorentzian functions Size: Strain:

The convolution of two Lorentzian Plot  k vs G kk G Slope = (HWHM)

The following pages are from: am.mpg.de/nanoschool2004/lectures- I/Lamparter.pdf

Ball-milled Mo from P. Lamparter  (FWHM) G L 2

Nanocrystalline CeO 2 Powderfrom P. Lamparter

Nb film, WH plot from P. Lamparter

anisotropy of shape or elastic constants, strains. and sizes   k 2 vs G 2 or  k vs G not linear Using a series of diffraction e.g. (200), (400) {(600) overlap with (442), can not be used}  provide a characteristic size and characteristic mean-square strain for each crystallographic direction!

E  k fit better than  k in this case  elastic anisotropic is the main reason for the deviation of  k to G. Ball-milled bcc Fe-20%Cu

Warren and Averbach Method Fourier Methods with Multiple Orders sizestrain How to interpret A(L)?

from P. Lamparter

Williamson-Hall Method Easy to be done Only width of peaks needed Warren-Averbach Method More mathematics Precise peak shapes needed Distributions of size and microstrain Relation to other properties(dislocations)