Electron diffraction of commensurately and incommensurately modulated materials

1. Electrondiffraction of commensurately and incommensuratelymodulatedmaterials Joke Hadermann www.slideshare.net/johader/

2. Modulation =

3. Incommensurate/commensurate

4. Basic cell, one plane b a Oneatom type A

5. Basic cell EDP [001] b 010 a 100 Oneatom type A

6. basic cell, SF [001] b 010 a 100 Oneatom type A

7. double cell model b a Alternation A and B atoms

8. double cell, EDP= [001] b 010 a 100 Alternation A and B atoms

9. double cell, g vectors= [001] b 010 a 100 Alternation A and B atoms Reflections at

10. [001] double cell start choice 010 100

11. [001] double cell, supercell 010 100

12. [001] double cell, q-vector 010 100

13. [001] double cell, supercell indices b a 010 b’ 010 100 100 a’

14. [001] double cell, q-vector indices b a 010 100 q

15. [001] double cell, satellites weaker b’ 010 100 a’

16. [001] double cell, SF b’ 010 100 a’

17. [001] double cell, odd vs. even b’ 010 100 a’ If k=2n+1 If k=2n

18. general modulation along main If the periodicity of the modulation in direct space is nb: Extra ref.: Canusesupercell:

19. overview 2b Extra reflections [001] b’ 010 a’ 010 100

20. overview 3b Extra ref.: [001] b’ 010 a’ 010 100

21. overview 4b Extra ref.: [001] b’ 010 a’ 010 100

22. Modulationnótalongmainaxis of basicstructure b b a a

23. 3 x d110 Modulation nót along main axis of basic structure (110) b b a a

24. 3 x d110 clear Modulation nót along main axis of basic structure (110) b a

25. 3x d110 ED, g Modulation nót along main axis of basic structure (110) [001] b 010 a 110 100

26. 110, indexed in basic [001] 010 1/3 1/3 0 2/3 2/3 0 110 100

27. 110, indexed in 3a x 3b [001] 010 030 11 0 22 0 110 100 330 300

28. 110, indexed in correct supercell [001] - 120 010 010 100 110 100

29. 110, indexed in correct supercell, complete [001] - 120 010 010 110 100 200 110 100 - 300 210

30. 110, P matrix reciprocal relation [001] b’* b* a’* a*

31. 110, P matrix rec to direct [001] b’* b* a’* a*

32. 110, P to direct cell b’ b a a’

33. advantage b’ b a a’

34. general supercell ,,=p/n Càntakesupercell e.g. n x basiccell parameter

35. the trouble with 0.458 ,,=p/n Càntakesupercell e.g. n x basiccell parameter 0.458=229/500 ! Approximations: 5/9=0.444, 4/11=0.455, 6/13=0.462,… Different cells, spacegroups, inadequate forrefinements,…

36. The q-vectorapproach All reflections hklm Basicstructurereflections hkl0

37. double, ED, g [001] double ED, g b 010 a 100

38. [001] double in q b 010 a 100

39. double indexed with q double indexed with q [001] 0100 q 010 0001 1001 100 1000

40. 0.458: q indicated 010 q 100

41. 0.458 indexed with q 0100 0001 010 - q 0101 100 1000

42. all four with q 0100 0100 1000 1000 0100 0100 1000 1000

43. [001] 110, with q 0100 010 q 0001 0002 100 1000

44. advantages of the q-vector method Advantages of the q-vector method: - subcellremainsthe same - alsoapplicable to incommensuratemodulations

45. Incommensuratelymodulatedmaterials Loss of translationsymmetry

46. LaCaCuGa(O,F)5 Example of a compositionalmodulation LaCaCuGa(O,F)5: amountF variessinusoidally Hadermann et al., Int.J.In.Mat.2, 2000, 493

47. Bi2201 Example of a displacivemodulation Bi-2201 Picture from Hadermann et al., JSSC 156, 2001, 445

48. Reciprocal space: reflections only Projectionsfrom 3+d reciprocalspace & “simple” supercell in 3+d space q (Example in 1+1 reciprocalspace)

49. e2 + q =a2* Projectionsfrom 3+d reciprocalspace & “simple” supercell in 3+d space a2*=e2+q a2* e2 q a1* (Example in 1+1 reciprocalspace)

50. reciprocal unit cell a1* x a2* Projectionsfrom 3+d reciprocalspace & “simple” supercell in 3+d space a2*=e2+q a2* e2 q a1* (Example in 1+1 reciprocalspace)