Diffraction by incommensurately        modulated phases                      Ë Î Ì Î Í Î ÑÎ Â -2 0 0 3                Моск...
At the end of this lecture you should be able to:          UNDERSTAND and INDEX         the reciprocal lattice of an    in...
b      aOne atom type A
[001]          b      a                 010One atom type A   100
[001]          b      a                 010One atom type A   100      a=3 Å      b=5 Å
[001]          b      a                   010                           1/5Å                  1/3ÅOne atom type A     100 ...
b          aAlternation A and B atoms
[001]              b          a                       010                            100Alternation A and B atoms
[001]              b          a                                     010                                          100Altern...
[001]              b                             q          a                                     010                     ...
All reflections      hklm         g  ha *  k b *  l c *  mq                                q  a* b*  c*         ...
mg G       b*        2                      [001]    1 q  b*    2                        0100                   q    01...
g  G  m.0.458b *q  0.458b *                           010                      q                     100
g  G  m.0.458b *q  0.458b *                     0001 0100                          010                            -    ...
Projections from 3+d reciprocal space &“simple” supercell in 3+d space  a2*    q                          a1*             ...
Projections from 3+d reciprocal space &   “simple” supercell in 3+d spacea2*=e2+q           a2*     e2           q        ...
Basis vectors Basis vectors of the reciprocal lattice       a1 *  a *       a2 *  b *       a3 *  c *       a4 *  e4 ...
q=0.5b*             q=0.458b*             0100               0100      1000              1000q=0.33b*            q=0.25b* ...
[001]              0100               010             q      0001              0002       100       1000            1 1  g...
0001 0100     010       -  q 0101            hk0m: no conditions1001000
Superspace groups: position and phase                (r,t)           ( Rr + v, t + ){R|v} is an element of the space gro...
Incommensurately modulated        materials
Reële ruimte   Reciproke ruimte R*a                       000 b*                        a*                        200
Reële ruimte                 Reciproke ruimte R*                   3aa                        a* q=                       ...
Reële ruimte               Reciproke ruimte R*a                   3a                         a*                   (3+Δ)a q...
Exercise   Index the following experimental  diffraction patterns and derive thepossible superspace group(s) from the     ...
First, index the commensurate parent structure LaSrCuO4, this will help you with indexing the next, incommensurate one.Giv...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.                                         001  The reflection in ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.                                         001  The reflection in ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.         010          020         110                        002...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.         010          020          110                        00...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.         010         020         110                        002 ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.                        002                2.61 Å   002         ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.                        002                  2.61 Å   002       ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.         100         200         020                        002 ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.         100         200         020                        002 ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.          110          220          ½½0                         ...
Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.          110          220          ½½0                         ...
002           110002               110           200      200                        020
002           110002               110           200      200                        020
Determine the reflection condition for h0l.                                         110                        002       0...
Determine the reflection condition for h0l.                                         110                        002       0...
Determine the reflection condition for hhl.                           002             110       002                       ...
Determine the reflection condition for hhl.                            002              110       002                     ...
Determine the reflection condition for hk0.                           002                   110       002                 ...
Determine the reflection condition for hk0.                           002                   110       002                 ...
Solved reflection conditions                                          110                      002                     - 0...
Determine space group• International Tables: I---• Most symmetrical I4/mmm
Incommensurate vs. basic cell    LaSrCuO3.52    Hadermann et al., Journal of Materials    Chemistry, 17, 22, 2007, 2344-2350
Identify andindex thesubcellreflections.
Identify andindex thesubcellreflections.
Propose amodulation vector.
q=αa*α<0.5
q=αa*α>0.5
[010]        002q=αa*                  200α<0.5
Index the satellite indicated in green.001                                -0001100
Index the satellite indicated in green.0001
Index the next satellite indicated in green.0002   -2001   -2002
-   -2002
Which of the indicated vectors corresponds to themodulation vector chosen on the previous slides?
Which of the indicated vectors corresponds to themodulation vector chosen on the previous slides?
Is the proposed   vector still valid?               yes               no
Is the proposed   vector still valid?              no
You need  q1        q1=αa*+βb*  q2        q2= -αa*+βb*  q1 ànd q2            α=β<0.25
You need              q1=αa*+βb*              q2= -αa*+βb*  q1 ànd q2              α=β<0.25
Index the reflection indicated in red.                             20011                                  -               ...
Index the reflection indicated in red.                                -                             20011
Index the others patterns consistently        with this new choice.      [010]               [110]         [001]          ...
Index the reflection indicated in yellow.                                                -                                ...
Index the reflection indicated in yellow.                                               -                                 ...
Index the reflection indicated in yellow.                                               -                                 ...
Index the reflection indicated in yellow.                                               -                                 ...
Index the reflection indicated in yellow.                                               -                                 ...
Index the reflection indicated in yellow.                                               -                                 ...
Derive the reflection conditions for hklmn.                                               -                               ...
Derive the reflection conditions for hklmn.                                               -                               ...
Derive the reflection conditions for hhlm0.                                                -                              ...
Derive the reflection conditions for hklm.                                               -                                ...
Derive the reflection conditions for hklm.                                                 -                              ...
Determine the superspace group.                                                     -                                     ...
You know cell parameters,  modulation vector and   superspace group.
At the end of this lecture you should be able to:          UNDERSTAND and INDEX         the reciprocal lattice of an    in...
Aperiodic crystal workshop 2013: TEM
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Aperiodic crystal workshop 2013: TEM

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This lecture was given at the 2nd International School on Aperiodic Crystals in Bayreuth, Germany. It is an updated version of the lecture given on the 1st school, that can be found between my lectures as "TEM for incommensurately modulated materials".

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Aperiodic crystal workshop 2013: TEM

  1. 1. Diffraction by incommensurately modulated phases Ë Î Ì Î Í Î ÑÎ Â -2 0 0 3 Московский Государственный Университет им. М.В. Ломоносова Факультет Наук о Материалах КАФЕДРА НЕОРГАНИЧЕСКОЙ ХИМИИ Лаборатория Направленного Неорганического Синтеза Исаева А.А. Í Î ÂÛ Å Í ÈÇØ ÈÅ Ñ ÅØ ÀÍ Í Û Å ÁËÎ ×Í Û Å Ì Õ ÀËÜÊÎ ÃÅÍ ÈÄÛ Í ÈÊÅËß Научные руководители д.х.н., проф. Б.А. Поповкин к.х.н., асс. ФНМ А.И. Баранов Joke Hadermann
  2. 2. At the end of this lecture you should be able to: UNDERSTAND and INDEX the reciprocal lattice of an incommensurately modulated material.
  3. 3. b aOne atom type A
  4. 4. [001] b a 010One atom type A 100
  5. 5. [001] b a 010One atom type A 100 a=3 Å b=5 Å
  6. 6. [001] b a 010 1/5Å 1/3ÅOne atom type A 100 a=3 Å b=5 Å
  7. 7. b aAlternation A and B atoms
  8. 8. [001] b a 010 100Alternation A and B atoms
  9. 9. [001] b a 010 100Alternation A and B atoms m Reflections at g  G  b* 2
  10. 10. [001] b q a 010 100Alternation A and B atoms m Reflections at g  G  b* 2 g  ha *  k b *  l c *  mq 1 q  b* 2
  11. 11. All reflections hklm g  ha *  k b *  l c *  mq q  a* b*  c* g  G  mqBasic structurereflections g  G  0q  G hkl0
  12. 12. mg G b* 2 [001] 1 q  b* 2 0100 q 010 0001 1001 100 1000
  13. 13. g  G  m.0.458b *q  0.458b * 010 q 100
  14. 14. g  G  m.0.458b *q  0.458b * 0001 0100 010 - q 0101 100 1000
  15. 15. Projections from 3+d reciprocal space &“simple” supercell in 3+d space a2* q a1* (Example in 1+1 reciprocal space)
  16. 16. Projections from 3+d reciprocal space & “simple” supercell in 3+d spacea2*=e2+q a2* e2 q a1* (Example in 1+1 reciprocal space)
  17. 17. Basis vectors Basis vectors of the reciprocal lattice a1 *  a * a2 *  b * a3 *  c * a4 *  e4  qq   a *   b*   c*
  18. 18. q=0.5b* q=0.458b* 0100 0100 1000 1000q=0.33b* q=0.25b* 0100 0100 1000 1000
  19. 19. [001] 0100 010 q 0001 0002 100 1000 1 1 g  G  m[ , ,0] 3 3 1 1q  a *  b *  0c * 3 3
  20. 20. 0001 0100 010 - q 0101 hk0m: no conditions1001000
  21. 21. Superspace groups: position and phase (r,t) ( Rr + v, t + ){R|v} is an element of the space group of the basic structure is a phase shift and  is ±1 Example Pnma(01/2)s00 Space group of the components of q symmetry-operators basic structure for the phase
  22. 22. Incommensurately modulated materials
  23. 23. Reële ruimte Reciproke ruimte R*a 000 b* a* 200
  24. 24. Reële ruimte Reciproke ruimte R* 3aa a* q= 1/3a*
  25. 25. Reële ruimte Reciproke ruimte R*a 3a a* (3+Δ)a q=1/3a* q=(1/3-δ)a*
  26. 26. Exercise Index the following experimental diffraction patterns and derive thepossible superspace group(s) from the reflection conditions. The example is incommensurately modulated LaSrCuO4-x.
  27. 27. First, index the commensurate parent structure LaSrCuO4, this will help you with indexing the next, incommensurate one.Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 1/6.50 Å 1/2.61 Å1/6.50 Å 1/1.85 Å 1/2.61 Å 1/1.85 Å (Simulated ED patterns)
  28. 28. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 001 The reflection in the red box is: 002 200 1/6.50 Å 1/2.61 Å 1/6.50 Å 1/1.85 Å 1/2.61 Å 1/1.85 Å
  29. 29. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 001 The reflection in the red box is: 002 200 1/6.50 Å 1/2.61 2.61 Å Å 1/6.50 Å 1/1.85 Å 1/2.61 2.61 Å Å 1/1.85 1.85 Å Å
  30. 30. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 010 020 110 002 1/2.61 Å 002 1/1.85 Å 1/2.61 Å 1/1.85 Å
  31. 31. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 010 020 110 002 1/2.61 Å 002 1/1.85 Å 1/2.61 Å 1/1.85 Å
  32. 32. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 010 020 110 002 2.61 Å 002 020 2.61 Å 1.85 Å
  33. 33. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 002 2.61 Å 002 020 2.61 Å 1.85 Å
  34. 34. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 002 2.61 Å 002 020 2.61 Å 1/1.85 Å 1/1.85 Å 020
  35. 35. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 100 200 020 002 2.61 Å 002 020 2.61 Å 1/1.85 Å 020
  36. 36. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 100 200 020 002 2.61 Å 002 020 2.61 Å 1.85 Å 020
  37. 37. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 110 220 ½½0 002 2.61 Å 002 020 2.61 Å 200 020
  38. 38. Given data:cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å. 110 220 ½½0 002 2.61 Å 002 020 2.61 Å 200 020
  39. 39. 002 110002 110 200 200 020
  40. 40. 002 110002 110 200 200 020
  41. 41. Determine the reflection condition for h0l. 110 002 002 110 200 200h0l: h+l=2n h = 2n l = 2n
  42. 42. Determine the reflection condition for h0l. 110 002 002 110 200 200h0l: h+l=2n
  43. 43. Determine the reflection condition for hhl. 002 110 002 110 200 200h0l: h+l=2n hhl: h+l=2n h = 2n l = 2n
  44. 44. Determine the reflection condition for hhl. 002 110 002 110 200 200h0l: h+l=2n hhl: l = 2n
  45. 45. Determine the reflection condition for hk0. 002 110 002 110 200 200h0l: h+l=2n hhl: l = 2n hk0: h+k=2n h = 2n k = 2n
  46. 46. Determine the reflection condition for hk0. 002 110 002 110 200 200h0l: h+l=2n hhl: l = 2n hk0: h+k=2n
  47. 47. Solved reflection conditions 110 002 - 002 110 110 200 200h0l:h+l=2n hhl:l=2n hk0:h+k=2nAlso (from rest of the zones) hkl: h+k+l=2n.
  48. 48. Determine space group• International Tables: I---• Most symmetrical I4/mmm
  49. 49. Incommensurate vs. basic cell LaSrCuO3.52 Hadermann et al., Journal of Materials Chemistry, 17, 22, 2007, 2344-2350
  50. 50. Identify andindex thesubcellreflections.
  51. 51. Identify andindex thesubcellreflections.
  52. 52. Propose amodulation vector.
  53. 53. q=αa*α<0.5
  54. 54. q=αa*α>0.5
  55. 55. [010] 002q=αa* 200α<0.5
  56. 56. Index the satellite indicated in green.001 -0001100
  57. 57. Index the satellite indicated in green.0001
  58. 58. Index the next satellite indicated in green.0002 -2001 -2002
  59. 59. - -2002
  60. 60. Which of the indicated vectors corresponds to themodulation vector chosen on the previous slides?
  61. 61. Which of the indicated vectors corresponds to themodulation vector chosen on the previous slides?
  62. 62. Is the proposed vector still valid? yes no
  63. 63. Is the proposed vector still valid? no
  64. 64. You need q1 q1=αa*+βb* q2 q2= -αa*+βb* q1 ànd q2 α=β<0.25
  65. 65. You need q1=αa*+βb* q2= -αa*+βb* q1 ànd q2 α=β<0.25
  66. 66. Index the reflection indicated in red. 20011 - 20011 - 20011
  67. 67. Index the reflection indicated in red. - 20011
  68. 68. Index the others patterns consistently with this new choice. [010] [110] [001] 020 200002 002 - 110 200002 [010] 200
  69. 69. Index the reflection indicated in yellow. - 20011 00010 00001 20011 - 20011 - 20011
  70. 70. Index the reflection indicated in yellow. - 20011 00010 00001 - 20011
  71. 71. Index the reflection indicated in yellow. - 20011 00010 00001 10110 10111 - 10111
  72. 72. Index the reflection indicated in yellow. - 20011 00010 00001 - 10111
  73. 73. Index the reflection indicated in yellow. - 20011 00010 00001 00002 - 00002 - 00001
  74. 74. Index the reflection indicated in yellow. - 20011 00010 00001 - 00002
  75. 75. Derive the reflection conditions for hklmn. - 20011 00010 00001 hkl:h+k+l=2n hhl:l=2n hklmn: h+k+l+m+n=2i h+k+l=2i m+n=2i
  76. 76. Derive the reflection conditions for hklmn. - 20011 00010 00001 hkl:h+k+l=2n hhl:l=2n hklmn: h+k+l=2i
  77. 77. Derive the reflection conditions for hhlm0. - 20011 00010 00001 hkl:h+k+l=2n hhl:l=2n hhlm0: l=2i h=2i l,m=2i
  78. 78. Derive the reflection conditions for hklm. - 20011 00010 00001 hkl:h+k+l=2n hhl:l=2n hhlm0: but hhlm0:m=2i l,m=2i is sufficient
  79. 79. Derive the reflection conditions for hklm. - 20011 00010 00001 hkl:h+k+l=2n hhl:l=2n hklmn:h+k+l=2i hhlm0:m=2i (-hhl0n: n=2i)
  80. 80. Determine the superspace group. - 20011 00010 00001 I4/mmm(α α0, -αα0)00mg http://stokes.byu.edu/iso/ssg.php Stokes et al., Acta Cryst. A67, 45-55 (2011)
  81. 81. You know cell parameters, modulation vector and superspace group.
  82. 82. At the end of this lecture you should be able to: UNDERSTAND and INDEX the reciprocal lattice of an incommensurately modulated material. Recap? http://www.slideshare.net/johader

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