Strain Anisotropy in X-ray Line Profile Analysis

1. DislocationModel of StrainAnisotropy Tamás Ungár Diffraction Laboratory, Department of Matrials Physics Eötvös University Budapest, Hungary Riso-DTU Masters-PhD-Postdoc course X-ray Line Profile Analysis Riso, September-November, 2009

2. Note: Today ALL(!) evaluations are possibly done by convolutional whole profile or whole pattern fitting but the qualiative features are better seen in Williamson-Hall or Warren-Averbach plots

3. Strain . . . . . . . . . . . . . . . . . . . . . . . . O Size Twinning or faulting Schematic picture of line broadening

4. What is strain-anisotropy?

5. Ball-milled WC Gillies, D.C. & Lewis, D. Powder Metallurgy, 11 (1968) 400. anisotropy: Breadths in a Williamson-Hall plot are anisotropic in terms of hkl indices

6. Ball-milled WC Gillies, D.C. & Lewis, D. Powder Metallurgy, 11 (1968) 400. strain-anisotropy: global increase of breadths

7. 300  1 100 Evaluation of strain by ignoring strain anisotropy Ball-milled WC Gillies, D.C. & Lewis, D. Powder Metallurgy, 11 (1968) 400.

8. Brief history of strain-anisotropy

9. 4 consecutive papers of A.J.C. Wilson and coworkers: 1) A.R Stokes & A.J.C. Wilson: The diffraction of X rays by distorted crystal aggregates - I Proc. Phys. Soc. 56 (1944) 174-181 2) A.J.C. Wilson: The diffraction of X-rays by distorted-crystal aggregates. II. Diffraction by bent lamellae Acta Cryst. 2 (1949) 220-222 3) J.N. Eastabrook & A.J.C. Wilson: The Diffraction of X-Rays by Distorted-Crystal Aggregates III Remarks on the Interpretation of the Fourier Coefficients Proc. Phys. Soc. B 65 (1952) 67-75 4) A.J.C. Wilson: The diffraction of X-rays by distorted-crystal aggregates. IV Diffraction by a crystal with an axial screw dislocation Acta Cryst. 5 (1952) 318-322

10. A.R Stokes & A.J.C. Wilson: The diffraction of X rays by distorted crystal aggregates - I Proc. Phys. Soc. 56 (1944) 174-181 . . . It is found that the "apparent strain" is given by η . . . where 2 = A + BH and Strain is hkl dependent

11. Unfortunately: . . . This equation is verified within the rather large experimental error for metal filings and wire. Details of the experimental work will be published elsewhere. . . . 2 = A + BH “Details of the experimental work” were never published

12. The next appearance of strain anisotropy: structure refinement by the Rietveld method using neutron diffraction data Caglioti G, Paoletti A, Ricci FP. Nucl. Instrum. 3 (1958) 223 introduce the term: strain anisotropy strain anisotropy is disturbing in Rietveld structure refinement

13. First suggestion to make use of strain anisotropy for dislocation analysis: P. Klimanek & R. Kuzel: X-ray Diffraction Line Broadening Due to Dislocations in Non-Cubic Materials. I. General Considerations and the Case of Elastic Isotropy Applied to Hexagonal Crystals J. Appl. Cryst. (1988). 21, 59-66 R. Kuzel & P. Klimanek: X-ray Diffraction Line Broadening Due to Dislocations in Non-Cubic Materials. II. The Case of Elastic Anisotropy Applied to Hexagonal Crystals J. Appl. Cryst. (1988). 21,363-368 R. Kuzel & P. Klimanek: X-ray Diffraction Line Broadening Due to Dislocations in Non-Cubic Crystalline Materials. III. Experimental Results for Plastically Deformed Zirconium J. Appl. Cryst. (1989). 22, 299-307

14. Orientation factor Slip systems from X-ray line broadening in hexagonal crystals: R. Kužel jr., P. Klimanek, J. Appl. Cryst., 22 (1989) 299-307. Pioneering wok, however, NOT easy to implement

15. Strain-anisotropy is a general feature in line broadening

16. Strain anisotropy Copper deformed by Equal Channel Angular Pressing (ECAP) Williamson-Hall plot

17. Strain anisotropy Warren-Averbach plot of inert-gas condensed copper

18. Warren would have suggested to evaluate this plot as:

19. Structural investigations of submicrocrystalline metals obtained by high-pressure torsion deformation R. Kužel, Z. Matej, V. Cherkaska, J. Pešicka, J. Cıžek, I. Procházka, R.K. Islamgaliev, Journal of Alloys and Compounds, 2005 Williamson-Hall plot

20. Structural investigations of submicrocrystalline metals obtained by high-pressure torsion deformation R. Kužel, Z. Matej, V. Cherkaska, J. Pešicka, J. Cıžek, I. Procházka, R.K. Islamgaliev, Journal of Alloys and Compounds, 2005 Williamson-Hall plot SPD: SeverePlasticDeformation

21. Soft magnetism in mechanically alloyed nanocrystalline materials T. D. Shen, R. B. Schwarz, and J. D. Thompson, PHYSICAL REVIEW B 72, 14431 (2005) Fe80Cu20(at. %) Strain anisotropy

22. Revealing the powdering methods of black makeup in Ancient Egypt by fitting microstructure based Fourier coefficients to the whole x-ray diffraction profiles of galena (PbS) T. Ungár, P. Martinetto, G. Ribárik, E. Dooryhée, Ph. Walter, M. Anne, J.Appl.Phys. 91 (2002) 2455 Warren-Averbach plot

23. Simple model based on dislocations

24. b g b g T T gb  0 gb = 0 dislocation is visible dislocation is invisible strong contrastweak contrast strong line broadening weak line broadening

25. Analogous with transmission electron microscopy (TEM) K.Shiramine, Y.Horisaki, D.Suzuki, S.Itoh,Y.Ebiko, S.Muto, Y. Nakata, N.Yokoyama, Threading dislocations in InAsquantum dot structure, Journal of Crystal Growth, 205 (1999) 461-466 g = 004 g = 222 g = 222

26. 22g2L2 <> ln An = ln- Fundamental equation of line broadening Warren & Averbach (1952): size Fourier coefficients mean square strain

27. for dislocations [Krivoglaz, Wilkens]: ln(Re/L) Cis the contrast factor of dislocations C = C (g,b,l,cij)

28. Copper deformed by Equal Channel Angular Pressing (ECAP) modified Williamson-Hall plot

29. Inert-gas condensed copper modified Warren-Averbach method

30. Average crystallite size in the inert-gas condensed copper specimen classical Warren-Averbach analysis 200 – 400 reflections <Lo> 7 nm modified Warren-Averbach analysis all reflections <Lo> 18 nm TEM size: 18 nm T.Ungár, S.Ott, P.G.Sanders, A.Borbély, J.R.Weertman, Acta Materialia, 10, 3693-3699 (1998)

31. Soft magnetism in mechanically alloyed nanocrystalline materials T. D. Shen, R. B. Schwarz, and J. D. Thompson, PHYSICAL REVIEW B 72, 14431 (2005) Fe80Cu20(at. %)

32. Ball-milled galena (PbS) T. Ungár, P. Martinetto, G. Ribárik, E. Dooryhée, Ph. Walter, M. Anne, J.Appl.Phys. 91 (2002) 2455 modified Warren-Averbach method

33.  1014 m-2 TEM Dislocations structure in ball-milled PbS (galena) P. Martinetto, J. Castaing, and P. Walter, P. Penhoud and P. Veyssiere, J. Mater. Res., Vol. 17, No. 7, Jul 2002

34. nanocrystalline SiC sintered at 1800 oC by 5.5 GPa J. Gubicza, S. Nauyoks, L. Balogh, J. Labar, T.W. Zerda, T. Ungár, J. Mater. Res. 22, 1314-1321 (2007) modified Williamson-Hall plot Williamson-Hall plot Twinning Twins + dislocations 100 nm

35. When the concept of average dislocation contrast factors: C may not work:  stronglytextured materials  single crystals Each reflection needs an individual contrast factor

36. Diffraction on a single grain in an MgSiO3 perovskite P. Cordier, T. Ungár, L. Zsoldos, G. Tichy, Dislocations creep in MgSiO3 perovskite at conditions of the Earth's uppermost lower mantle, Nature, 428 (2004) 837-840.

37. Depth P, T 3 GPa 1100°C 100 km Pyroxenes (Mg, Fe)SiO3 (Ca, Mg, Fe)2Si2O6 Olivine (Mg, Fe)2SiO4 Garnets (Mg, Fe, Ca)3 Al2Si3O12 Upper mantle 13 GPa 1400°C 410 km Wadsleyite (Mg, Fe)2SiO4 Garnets (Mg, Fe, Ca)3(Al, Si)2Si3O12 Transition zone 520 km Ringwoodite (Mg, Fe)2SiO4 23 GPa 1600°C 670 km Magnesiowustite (Mg, Fe)O Lower mantle Perovskite (Mg, Fe, Al)(Si, Al)SiO3-x CaSiO3 135 GPa 3500°C 2900 km Schematic composition of Earth's mantle

38. 1 mm

40. Strain anisotropy

41. Strain anisotropy Williamson-Hall plot

42. Measured dislocation contrast factors: Cm

43. Modified Williamson-Hall plot

44. 9 - 12 5 - 8 13 - 20 1 - 4 Conceivable Burgers vectors: 1 - 20

45. Measuredandcalculated dislocation contrast factors: Cm, Ccalc

46. The only Burgers vectors that survived the search-match by comparing the measured and calculated dislocation contrast factors [010] [100]