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Modulated structures belong to that kind of crystal structure in which the atoms suffer from certain compositional and/or positional fluctuation.
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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.
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.
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 ?
b* q a* Schematic diffraction pattern of an incommensurate modulated structure
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
Representation of one-dimensionally modulated incommensurate structures Lattice vectors in real- and reciprocal- space
situated at their average positions Modulated atoms Structure-factor formula
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
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