Composite structures can be considered as coherent combinations of two or more

1. Modulated structures belong to that kind of crystal structure in which the atoms suffer from certain compositional and/or positional fluctuation. If the period of fluctuation is commensurate with that of the three-dimensional unit cell then a superstructure results, otherwise an incommensurate modulated structure is obtained.

2. Composite structures can be considered as coherent combinations of two or more modulated structures. Each of the structures is characterized by a unit cell and a set of modulation wave vectors. Composite structures differ from ordinary incommensurate modulated structures in that they do not have a three-dimensional periodic basic structure.

3. T T T T T t T = 0 (mod t) or MOD (T, t) = 0 Commensurate modulation Þ superstructures T¹ 0 (mod t) or MOD (T, t) ¹ 0 Incommensurate modulation Þ incommensurate structures What’s a Modulated Structure ?

4. b* q a* Schematic diffraction pattern of an incommensurate modulated structure

5. 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

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

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

8. The Sayre equation in multi-dimensinal space For phasing main reflections to solve the averaged structure For phasing satellites of composite structures For phasing satellites of ordinary incommensurate modulated structures Modified Sayre Equations in multi-dimensional space

9. using using Strategy of solving incommensurate modulated structures i) Derive phases of main reflections ii) Derive phases of satellite reflections 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