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Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of

Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of Materials”, B. Fultz and J. Howe, Springer- Verlag Berlin 2002. Chapter 8). Peak form for X-ray peaks: Gaussian Lorentizian Voigt, Psudo -Voigt:. Gaussian function. x 0. FWHM.

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Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of

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  1. Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. Chapter 8) Peak form for X-ray peaks: Gaussian Lorentizian Voigt, Psudo-Voigt:

  2. Gaussian function x0 FWHM

  3. Lorentzian function or Cauchy form x0 FWHM

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

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

  6. 2= FWHM

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

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

  9. 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)

  10. Size Broadening: Interference function Define deviation vector …

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

  12. Define Solution: x = 1.392 ~ 1.392 Define

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

  14. Strain broadening: Uniform strain  lattice constant change  Bragg peaks shift. Assume strain = d0 change to d0(1+ ). Diffraction condition: Peak shift In terms of  Larger shift for the diffraction peaks of higher order

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

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

  17. 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 gf(-x)g(x) * Integrate over x 4 f(x) 3 2 1 0 -2 -1 1 2 0 4 g(x) 3 2 1 0 -2 -1 1 2 0 Assume f and g are the functions on the right, the h() that we will get is 4 f(-x) 3 2 1 0 -2 -1 0 1 2

  18.  = -2  = -1 4 4 0 3 3 7/6 2 2 1 1 0 0 -2 2 -2 2 0 0  = 0  = 1 4 4 3 3 16/3 31/6 2 2 1 1 0 0 -2 2 -2 2 0 0 6 5  = 2 h() 4 4 3 0 3 2 2 1 1 0 0  -2 2 0 -2 0 2

  19. Convolution of Gaussians: Two functions f(): breadth Bf g(): breadth Bg  h() = f()*g(); breadth Bh http://www.tina-vision.net/docs/memos/2003-003.pdf

  20. Convolution of Lorentzians: Two Lorentzianfunctions: f(): breadth Bf g(): breadth Bg  h() = f()*g(); breadth Bh

  21. 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) Fourier transform the above three functions (DFT) l: [1/length], the range in k of the Fourier series is the interval –l/2 to l/2.

  22. 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

  23. 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

  24. Data from a perfect specimen Rachinger Correction (optional) f(k) Corrected data free of instrument broadening Stokes Correction G(n)= H(n)/F(n) F.T.-1 F.T. Data from the actual specimen Rachinger Correction (optional) h(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

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

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

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

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

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

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

  31. The convolution of two Gaussians Plot k2 vs G2 Slope = (k)2 (HWHM) G2

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

  33. The convolution of two Lorentzian Plot kvsG Slope = k (HWHM) G

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

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

  36. Nanocrystalline CeO2 Powder from P. Lamparter

  37. Nb film, WH plot from P. Lamparter

  38. from P. Lamparter

  39. anisotropy of shape or elastic constants, strains. and sizes  k2vsG2 or kvsG 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!

  40. 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

  41. Warren and AverbachMethod Fourier Methods with Multiple Orders size strain How to interpret A(L)?

  42. from P. Lamparter

  43. from P. Lamparter

  44. from P. Lamparter

  45. from P. Lamparter

  46. from P. Lamparter

  47. from P. Lamparter

  48. 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)

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