Presentation is loading. Please wait.

Presentation is loading. Please wait.

Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. Chapter.

Similar presentations


Presentation on theme: "Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. Chapter."— Presentation transcript:

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 x0x0 FWHM

3 Lorentzian function or Cauchy form FWHM x0x0

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

5 Lorentzian function or Cauchy form FWHM Gaussian function FWHM pseudo-Voigt:  : 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: 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.

8 … … 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

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 ~ 1.392 Solution: x = 1.392 Define

13 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

14 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

15 Distribution of strains  diffraction peaks broadening Strain distribution  relate to k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 g f(  -x)g(x) * Integrate over x 0 12-2 0 1 2 3 4 0 12-2 0 1 2 3 4 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 12-2 0 1 2 3 4 f(-x) 0

18 0 1 2 3 4 0 2-2  = -1 0 1 2 3 4 0 2-2  = 1 0 1 2 3 4 0 2-2  = -2 0 1 2 3 4 0 2-2  = 0 0 1 2 3 4 0 2-2  = 2 0 1 2 3 4 2-2  0 31/616/3 0 7/6 0 h()h() 5 6

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

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

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) 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)

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 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

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

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

27 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

28 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

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 Good only when strain broadening >> size broadening (see page 9) characteristic width

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

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

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

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

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

36 Nanocrystalline CeO 2 Powderfrom P. Lamparter

37 Nb film, WH plot from P. Lamparter

38

39 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!

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

42 from P. Lamparter

43

44

45

46

47

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)


Download ppt "Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. Chapter."

Similar presentations


Ads by Google