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".

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. At the end of this lecture you should be able to:
UNDERSTAND and INDEX
the reciprocal lattice of an
incommensurately modulated material.

3. b
a
One atom type A

4. [001]
b
a 010
One atom type A 100

5. [001]
b
a 010
One atom type A 100
a=3 Å
b=5 Å

6. [001]
b
a 010
1/5Å
1/3Å
One atom type A 100
a=3 Å
b=5 Å

7. b
a
Alternation A and B atoms

8. [001]
b
a 010
100
Alternation A and B atoms

9. [001]
b
a 010
100
Alternation A and B atoms
m
Reflections at g  G  b*
2

10. [001]
b q
a 010
100
Alternation A and B atoms
m
Reflections at g  G  b*
2
g  ha *  k b *  l c *  mq
1
q  b*
2

11. All reflections
hklm g  ha *  k b *  l c *  mq
q  a* b*  c*
g  G  mq
Basic structure
reflections g  G  0q  G
hkl0

12. m
g G b*
2 [001]
1
q  b*
2
0100
q 010
0001
1001
100
1000

13. g  G  m.0.458b *
q  0.458b *
010
q
100

14. g  G  m.0.458b *
q  0.458b *
0001 0100
010
-
q 0101
100
1000

15. Projections from 3+d reciprocal space &
“simple” supercell in 3+d space
a2*
q a1*
(Example in 1+1
reciprocal space)

16. Projections from 3+d reciprocal space &
“simple” supercell in 3+d space
a2*=e2+q
a2*
e2
q
a1*
(Example in 1+1
reciprocal space)

17. Basis vectors
Basis vectors of the reciprocal lattice
a1 *  a *
a2 *  b *
a3 *  c *
a4 *  e4  q
q   a *   b*   c*

18. q=0.5b* q=0.458b*
0100 0100
1000 1000
q=0.33b* q=0.25b*
0100 0100
1000 1000

19. [001]
0100
010
q
0001
0002
100
1000
1 1
g  G  m[ , ,0]
3 3
1 1
q  a *  b *  0c *
3 3

20. 0001 0100
010
-
q 0101
hk0m: no conditions
100
1000

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. Incommensurately modulated
materials

23. Reële ruimte Reciproke ruimte R*
a
000 b*
a*
200

24. Reële ruimte Reciproke ruimte R*
3a
a
a* q=
1/3a*

25. Reële ruimte Reciproke ruimte R*
a
3a
a*
(3+Δ)a q=1/3a*
q=(1/3-δ)a*

26. Exercise
Index the following experimental
diffraction patterns and derive the
possible superspace group(s) from the
reflection conditions.
The example is incommensurately
modulated LaSrCuO4-x.

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 Å
1/6.50 Å
1/1.85 Å 1/2.61 Å
1/1.85 Å
(Simulated ED patterns)

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 002 110
002 110 200
200
020

40. 002 110
002 110 200
200
020

41. Determine the reflection condition for h0l.
110
002
002 110 200
200
h0l: h+l=2n
h = 2n
l = 2n

42. Determine the reflection condition for h0l.
110
002
002 110 200
200
h0l: h+l=2n

43. Determine the reflection condition for hhl.
002 110
002 110 200
200
h0l: h+l=2n hhl: h+l=2n
h = 2n
l = 2n

44. Determine the reflection condition for hhl.
002 110
002 110 200
200
h0l: h+l=2n hhl:
l = 2n

45. Determine the reflection condition for hk0.
002 110
002 110 200
200
h0l: h+l=2n hhl: l = 2n hk0: h+k=2n
h = 2n
k = 2n

46. Determine the reflection condition for hk0.
002 110
002 110 200
200
h0l: h+l=2n hhl: l = 2n hk0: h+k=2n

47. Solved reflection conditions
110
002 -
002 110 110 200
200
h0l:h+l=2n hhl:l=2n hk0:h+k=2n
Also (from rest of the zones) hkl: h+k+l=2n.

48. Determine space group
• International Tables: I---
• Most symmetrical I4/mmm

49. Incommensurate vs. basic cell
LaSrCuO3.52
Hadermann et al., Journal of Materials
Chemistry, 17, 22, 2007, 2344-2350

50. Identify and
index the
subcell
reflections.

51. Identify and
index the
subcell
reflections.

52. Propose a
modulation vector.

53. q=αa*
α<0.5

54. q=αa*
α>0.5

56. [010]
002
q=αa*
200
α<0.5

57. Index the satellite indicated in green.
001
-
0001
100

58. Index the satellite indicated in green.
0001

59. Index the next satellite indicated in green.
0002
-
2001
-
2002

60. -
-
2002

61. Which of the indicated vectors corresponds to the
modulation vector chosen on the previous slides?

62. Which of the indicated vectors corresponds to the
modulation vector chosen on the previous slides?

63. Is the proposed
vector still valid?
yes
no

64. Is the proposed
vector still valid?
no

65. You need
q1 q1=αa*+βb*
q2 q2= -αa*+βb*
q1 ànd q2
α=β<0.25

66. You need
q1=αa*+βb*
q2= -αa*+βb*
q1 ànd q2
α=β<0.25

67. Index the reflection indicated in red.
20011
-
20011
-
20011

68. Index the reflection indicated in red.
-
20011

69. Index the others patterns consistently
with this new choice.
[010] [110] [001]
020 200
002 002
-
110
200
002 [010]
200

70. Index the reflection indicated in yellow.
-
20011
00010
00001
20011
-
20011
-
20011

71. Index the reflection indicated in yellow.
-
20011
00010
00001
-
20011

72. Index the reflection indicated in yellow.
-
20011
00010
00001
10110
10111
-
10111

73. Index the reflection indicated in yellow.
-
20011
00010
00001
-
10111

74. Index the reflection indicated in yellow.
-
20011
00010
00001
00002
-
00002
-
00001

75. Index the reflection indicated in yellow.
-
20011
00010
00001
-
00002

76. 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

77. Derive the reflection conditions for hklmn.
-
20011
00010
00001
hkl:h+k+l=2n
hhl:l=2n
hklmn:
h+k+l=2i

78. 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

79. 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

80. 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)

81. 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)

82. You know cell parameters,
modulation vector and
superspace group.

83. 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