It has been proved by Hao, Liu and Fan (1987) that the Sayre equation (1952) can easily be extended into multidimensional space. We have

, (1)

here **h **is a multidimensional reciprocal vector
defined as

, (2)

where *h _{i}* is the

, (3)

Where subscript *m* stands for main reflections while
subscript
*s* stands for satellites. Since the intensities of satellites
are on average much weaker than those of main reflections, the last summation
on the right-hand side of (3) is negligible in comparison with the second,
while the last two summations on the right-hand side of (3) are negligible
in comparison with the first. Letting *F*(**h**) on the left-hand
side of (3) represents only the structure factor of main reflections we
have to first approximation

, (4)

On the other hand, if *F*(**h**) on the left-hand
side of (3) corresponds only to satellites, it follows that

. (5)

For ordinary incommensurate modulated structures the first summation on the right-hand side of (5) has vanished. Because any three-dimensional reciprocal lattice vector corresponding to a main reflection will have zero components in the extra dimensions so that the sum of two such lattice vectors could never give rise to a lattice vector corresponding to a satellite. We then have

. (6)

For composite structures (Fan *et al*.
1993; Sha *et al.*, 1994; Mo *et
al**.*, 1996) on the other hand, since the average structure
itself is a 4- or higher-dimensional periodic structure, the first summation
on the right-hand side of (5) does not vanish. We have instead of (6) the
following equation:

. (7)

Equation (4) indicates that the phases of main reflections can be derived by a conventional direct method neglecting the satellites. Equation (6) or (7) can be used to extend phases from the main reflections to the satellites respectively for ordinary incommensurate modulated structures or composite structures. This provides a way to determine the modulation functions objectively. The procedure will be in the following stages:

- derive the phases of main reflections using Equation (4);
- derive the phases of satellite reflections using Equation (6) or (7);
- calculate a multidimensional Fourier map using the observed structure factor magnitudes and the phases from i) and ii);
- cut the resulting Fourier map with a 3-dimensional ‘hyperplane’ to obtain an ‘image’ of the incommensurate modulated structure in the 3-dimensional physical space;
- parameters of the modulation functions are measured directly on the multidimensional Fourier map resulting from iii).

Fan, H.F., van Smaalen, S., Lam, E.J.W. & Beurskens, P.T. (1993). Direct methods for incommensurate intergrowth compounds I. Determination of the modulation.

Hao, Q., Liu, Y. W. and Fan, H. F. (1987). Direct methods in superspace I. Preliminary theory and test on the determination of incommensurate modulated structures,

Mo, Y.D., Fu, Z.Q., Fan, H.F., van Smaalen, S. Lam, E.J.W. & Beurskens, P.T. (1996). Direct methods for incommensurate intergrowth compounds III. Solving the average structure in multidimensional space.

Sayre, D. (1952). The squaring method: a new method for phase determination,

Sha, B.D., Fan, H.F., van Smaalen, S., Lam, E.J.W. & Beurskens, P.T. (1994). Direct methods for incommensurate intergrowth compounds II. Determination of the modulation using only main reflections.