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# Modulated materials with electron diffraction

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This lecture was given at the International School of Crystallography in Erice 2011, on the topic of Electron Crystallography. It explains the very basics of how to index commensurately and incommensurately modulated materials. It was meant for a 40 minute lecture.

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### Modulated materials with electron diffraction

2. 2. Modulation =<br />
3. 3. Incommensurate/commensurate<br />
4. 4. Basic cell, one plane<br />b<br />a<br />Oneatom type A<br />
5. 5. Basic cell EDP<br />[001]<br />b<br />010<br />a<br />100<br />Oneatom type A<br />
6. 6. basic cell, SF<br />[001]<br />b<br />010<br />a<br />100<br />Oneatom type A<br />
7. 7. double cell model<br />b<br />a<br />Alternation A and B atoms<br />
8. 8. double cell, EDP=<br />[001]<br />b<br />010<br />a<br />100<br />Alternation A and B atoms<br />
9. 9. double cell, g vectors=<br />[001]<br />b<br />010<br />a<br />100<br />Alternation A and B atoms<br />Reflections at<br />
10. 10. double cell start choice<br />[001]<br />010<br />100<br />
11. 11. double cell, supercell<br />[001]<br />010<br />100<br />
12. 12. double cell, q-vector<br />[001]<br />010<br />100<br />
13. 13. double cell, supercell indices<br />[001]<br />b<br />a<br />010<br />b’<br />010<br />100<br />100<br />a’<br />
14. 14. double cell, q-vector indices<br />[001]<br />b<br />a<br />010<br />100<br />q<br />
15. 15. double cell, satellites weaker<br />[001]<br />b’<br />010<br />100<br />a’<br />
16. 16. double cell, SF<br />[001]<br />b’<br />010<br />100<br />a’<br />
17. 17. double cell, odd vs. even<br />[001]<br />b’<br />010<br />100<br />a’<br />If k=2n+1<br />If k=2n<br />
18. 18. general modulation along main<br />If the periodicity of the modulation in direct space is nb:<br />Extra ref.:<br />Canusesupercell:<br />
19. 19. overview 2b<br />Extra reflections<br />[001]<br />b’<br />010<br />a’<br />010<br />100<br />
20. 20. overview 3b<br />Extra ref.:<br />[001]<br />b’<br />010<br />a’<br />010<br />100<br />
21. 21. overview 4b<br />Extra ref.:<br />[001]<br />b’<br />010<br />a’<br />010<br />100<br />
22. 22. Modulationnótalongmainaxis of basicstructure<br />b<br />b<br />a<br />a<br />
23. 23. 3 x d110<br />Modulation nót along main axis of basic structure<br />(110)<br />b<br />b<br />a<br />a<br />
24. 24. 3 x d110 clear<br />Modulation nót along main axis of basic structure<br />(110)<br />b<br />a<br />
25. 25. 3x d110 ED, g<br />Modulation nót along main axis of basic structure<br />(110)<br />[001]<br />b<br />010<br />a<br />110<br />100<br />
26. 26. 110, indexed in basic<br />[001]<br />010<br />1/3 1/3 0<br />2/3 2/3 0<br />110<br />100<br />
27. 27. 110, indexed in 3a x 3b<br />[001]<br />010<br />030<br />11 0<br />22 0<br />110<br />100<br />330<br />300<br />
28. 28. 110, indexed in correct supercell<br />[001]<br />-<br />120<br />010<br />010<br />100<br />110<br />100<br />
29. 29. 110, indexed in correct supercell, complete<br />[001]<br />-<br />120<br />010<br />010<br />110<br />100<br />200<br />110<br />100<br />-<br />300<br />210<br />
30. 30. 110, P matrix reciprocal relation<br />[001]<br />b’*<br />b*<br />a’*<br />a*<br />
31. 31. 110, P matrix rec to direct<br />[001]<br />b’*<br />b*<br />a’*<br />a*<br />
32. 32. 110, P to direct cell<br />b’<br />b<br />a<br />a’<br />
33. 33. advantage<br />b’<br />b<br />a<br />a’<br />
34. 34. general supercell<br />,,=p/n<br />Càntakesupercell<br />e.g. n x basiccell parameter<br />
35. 35. the trouble with 0.458<br />,,=p/n<br />Càntakesupercell<br />e.g. n x basiccell parameter<br />0.458=229/500 !<br />Approximations: <br />5/9=0.444, 4/11=0.455, 6/13=0.462,…<br />Different cells, spacegroups, inadequate forrefinements,…<br />
36. 36. The q-vectorapproach<br />All<br />reflections<br />hklm<br />Basicstructurereflections<br />hkl0<br />
37. 37. double, ED, g<br />double ED, g<br />[001]<br />b<br />010<br />a<br />100<br />
38. 38. double in q<br />[001]<br />b<br />010<br />a<br />100<br />
39. 39. double indexed with q<br />double indexed with q<br />[001]<br />0100<br />q<br />010<br />0001<br />1001<br />100<br />1000<br />
40. 40. 0.458: q indicated<br />010<br />q<br />100<br />
41. 41. 0.458 indexed with q<br />0100<br />0001<br />010<br />-<br />q<br />0101<br />100<br />1000<br />
42. 42. all four with q<br />0100<br />0100<br />1000<br />1000<br />0100<br />0100<br />1000<br />1000<br />
43. 43. 110, with q<br />[001]<br />0100<br />010<br />q<br />0001<br />0002<br />100<br />1000<br />
44. 44. advantages of the q-vector method<br />Advantages of the q-vector method:<br />- subcellremainsthe same<br /> - alsoapplicable to incommensuratemodulations<br />
45. 45. Incommensuratelymodulatedmaterials<br />Loss of translationsymmetry<br />
46. 46. LaCaCuGa(O,F)5<br />Example of a compositionalmodulation<br />LaCaCuGa(O,F)5: amountF variessinusoidally<br />Hadermann et al., Int.J.In.Mat.2, 2000, 493<br />
47. 47. Bi2201<br />Example of a displacivemodulation<br />Bi-2201<br />Picture from Hadermann et al., JSSC 156, 2001, 445<br />
48. 48. Reciprocal space: reflections only<br />Projectionsfrom 3+d reciprocalspace & “simple” supercell in 3+d space<br />q<br />(Example in 1+1 reciprocalspace)<br />
49. 49. e2 + q =a2*<br />Projectionsfrom 3+d reciprocalspace & “simple” supercell in 3+d space<br />a2*=e2+q<br />a2*<br />e2<br />q<br />a1*<br />(Example in 1+1 reciprocalspace)<br />
50. 50. reciprocal unit cell a1* x a2*<br />Projectionsfrom 3+d reciprocalspace & “simple” supercell in 3+d space<br />a2*=e2+q<br />a2*<br />e2<br />q<br />a1*<br />(Example in 1+1 reciprocalspace)<br />
51. 51. Basis vectors of the reciprocal lattice<br />
52. 52. 3+1 D direct lattice: the modulation<br />Example: q= γc*<br />(Displacivemodulationalong c)<br />c<br />
53. 53. vector e4=a4<br />Example: q= γc*<br />(Displacivemodulationalong c)<br />c<br />c<br />u<br />c<br />1<br />z<br />t<br />1<br />0<br />x4<br />e4=a4<br />
54. 54. vector a3 = c-gamma e4<br />Example: q= γc*<br />(Displacivemodulationalong c)<br />c<br />γ<br />c<br />a3<br />a3<br />u<br />c<br />1<br />z<br />x3<br />a3 = c - γe4<br />x3 = 0<br />t<br />1<br />0<br />x4<br />e4=a4<br />
55. 55. unit cell<br />Example: q= γc*<br />(Displacivemodulationalong c)<br />c<br />γ<br />c<br />a3<br />a3<br />u<br />c<br />1<br />z<br />x3<br />a3 = c - γe4<br />x3 = 0<br />t<br />1<br />0<br />x4<br />e4=a4<br />
56. 56. projection back to first unit cell<br />Example: q= γc*<br />(Displacivemodulationalong c)<br />c<br />γ<br />c<br />c<br />a3<br />a3<br />u<br />c<br />1<br />1<br />z<br />x3<br />a3 = c - γe4<br />x3 = 0<br />t<br />1<br />0<br />0<br />x4<br />e4=a4<br />
57. 57. modulation function<br />Example: q= γc*<br />(Displacivemodulationalong c)<br />c<br />Modulationfunction u<br />γ<br />z = z0 + u(x4)<br />c<br />c<br />a3<br />a3<br />u<br />c<br />1<br />z<br />x3<br />a3 = c - γe4<br />x3 = 0<br />t<br />1<br />0<br />0<br />x4<br />e4=a4<br />
58. 58. restores the periodicity<br />Example: q= γc*<br />(Displacivemodulationalong c)<br />c<br />Modulationfunction u<br />γ<br />z = z0 + u(x4)<br />c<br />c<br />a3<br />a3<br />u<br />c<br />1<br />z<br />x3<br />a3 = c - γe4<br />x3 = 0<br />t<br />1<br />0<br />0<br />x4<br />e4=a4<br />In 3+1D: again unit cell, translationsymmetry<br />
59. 59. Basis vectors<br />Basis vectors in reciprocal space<br />Basis vectors in direct space<br />
60. 60. Superspace groups<br />Superspacegroups: position and phase<br />(r,t) ( Rr + v, t + ) <br />{R|v} is an element of the space group of the basic structure<br /> is a phase shift and is ±1<br />Example<br />Pnma(01/2)s00<br />Spacegroup of the basicstructure<br />components of q<br />symmetry-operatorsfor the phase<br />
61. 61. Separate the basicreflections (m=0) from the satellites (m≠0)<br />
62. 62. Separate the basicreflections (m=0) from the satellites (m≠0)<br />-shouldform a regular 3D lattice<br />-highestsymmetrywithlower volume <br />
63. 63. satellites change positions<br />Separate the basic reflections (m=0) from the satellites (m≠0)<br />Hint fromchanges vs. composition, temperature,…<br />
64. 64. Select the modulation vector<br />Possibly multiple solutions<br />
65. 65. two possible modulation vectors<br />2000<br />2000<br />2400<br />2200<br />2200<br />0003<br />0103<br />-<br />2002<br />-<br />2403<br />0002<br />0002<br />0001<br />0101<br />q<br />q<br />x<br />0200<br />0200<br />hklm: h+k=2n, k+l=2n, h+l=2n<br />Fmmm(10)<br />HKLm: H+K+m=2n, K+L+m=2n, <br />L+H=2n<br />Xmmm(00)<br />
66. 66. Conditionsfor the basiccell and modulation vector<br />2000<br />2400<br />2200<br />0003<br />-<br />2403<br />0002<br />(qr,qi) in correspondencewithchosencrystal system & centeringbasiccell<br />0001<br />q<br />0200<br />
67. 67. Possibleirrationalcomponents in the different crystalsystems<br />Example of derivation: seelecturenotes.<br />
68. 68. Compatibility of rationalcomponentswithcentering types<br />Example of derivation: seelecturenotes.<br />
69. 69. comparison bulk with ED<br />
70. 70. Summary<br />Commensuratemodulations:<br />supercell<br />q-vector<br />Incommensuratemodulations<br /> (Commensurateapproximation)<br />q-vector<br />q-vector -> (3+1)D Superspace<br />