1. What’s a modulated structure ?
Muti-dimensional direct methods of
solving modulated structures
Incommesurate modulation
in Bi-based supercondutors
from electron crystallography

2. What’s a Modulated Structure ?
T = 0 (mod t) or MOD (T, t) = 0
Commensurate modulation Þ
superstructures
T ¹ 0 (mod t) or MOD (T, t) ¹ 0
Incommensurate modulation Þ
incommensurate structures T

3. Schematic diffraction pattern of an incommensurate modulated structureb* q

4. Conclusion In the reciprocal space:
The diffraction pattern of an incommen-surate modulated crystal is the projection of a 4- or higher-dimensional weighted lattice
In the direct space:
An incommensurate modulated structure is the “hypersection” of a 4- or higher-dimensional periodic structure cut with the 3-dimensional physical space

5. Representation of one-dimensionally modulated incommensurate structures
Lattice vectors in real- and reciprocal- space

6. Structure-factor formula
Modulated
atoms
situated at their
average positions

7. Modified Sayre Equations in multi-dimensional space

8. incommensurate modulated structures
Strategy of solving
incommensurate modulated structures
i) Derive phases of main reflections
using
ii) Derive phases of satellite reflections
using
iii) Calculate the multi-dimensional Fourier map
iv) Cut the resulting Fourier map with the
3-D ‘hyperplane’ (3-D physical space)
v) Parameters of the modulation functions are
measured directly on the multi-dimensional
Fourier map

9. Electron Crystallographic Study of Bi-based Superconductors using
Multi-dimensional Direct Methods

10. Why Electrons ? 1. Electrons are better for studying
minute and imperfect crystalline samples
2. Electron microscopes are
the only instrument that can produce
simultaneously EM’s and ED’s for the same
crystalline sample at atomic resolution
3. Electrons are better for revealing
light atoms in the presence of heavy atoms

11. Scattering of X-rays and Electrons
by Different Elements Bi Sr Ca Cu
X-rays
Relative scattering power
Sinq /l ~ 0.31
Electrons O O

12. Bi-based Superconductors
Bi2Sr2Can-1CunO2n+4+x
n = n = n = 3
Bi Bi Bi-2223
Bismuth
bi-layer
Bi-O
Bi-O
Bi-O
Sr-O
Cu-O
Ca-O c
Perovskite
layer
Sr-O
Cu-O
Ca-O
Sr-O
Cu-O
Bi-O
Bismuth
bi-layer
Bi-O
Bi-O

13. Electron diffraction analysis of the Bi-2223 superconductor
Space group:
P [Bbmb]
a = 5.49,
b = 5.41,
c = 37.1Å;
q = 0.117b*
*The average
structure is
known*

14. Bi-2223 [100] projected potential
Space group: P [Bbmb]
a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b*
RsymM = 0.12 (Nref. =42)
RsymS = 0.13 (Nref. = 70)
Rm = 0.16
Rs = 0.17

15. Bi-2223 4-dimensional metal atoms a3 a4 cut at a2 = 0 and projected
down the a1 axis
Space group: P [Bbmb]
a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b*
a1 = a, a2 = b d,
a3 = c, a4 = d

16. Image Processing of Bi-2212
EM image from
Dr. S. Horiuchi
Space group: N [Bbmb]
a = 5.42, b = 5.44, c = 30.5Å;
q = 0.22b* + c* FT
FT-1
Phase
extension

17. Image Processing of Bi-2212 (continued)8 4 1 Bi Sr Cu Ca
Original image
Enhanced image c
Oxygen
in Cu-O layer b

18. O atoms on the Cu-O layer
Electron diffraction analysis of Bi-2201
Space group: P[B 2/b] -1]; a = 5.41, b = 5.43, c = 24.6Å,
b = 90o; q = 0.217b* c*
O atoms on the Cu-O layer
RT = 0.32 Rm = 0.29 RS1 = 0.29 RS2 = 0.36 RS3 = 0.52
Bi-O
Sr-O
Cu-O b c
O (extra)

19. Experimental B and M
Bi-2201
Influence of
thermal motion (B) and Modulation (M) to the dynamical diffraction
B set to zero
B,M set to zero
M set to zero

20. Bi-2201 The effect of sample thickness ~300Å ~200Å ~5Å ~100Å
Oxygen in
Cu-O layer
Bi-O
Sr-O
Cu-O
Extra oxygen