Modified Sayre equations in multidimensional space

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 hi is the ith component of the vector h, bi forms the set of basic vectors defining a multidimensional reciprocal unit cell. The right-hand side of (1) can be split into three parts, i.e.

,     (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:

1. derive the phases of main reflections using Equation (4);
2. derive the phases of satellite reflections using Equation (6) or (7);
3. calculate a multidimensional Fourier map using the observed structure factor magnitudes and the phases from i) and ii);
4. 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;
5. parameters of the modulation functions are measured directly on the multidimensional Fourier map resulting from iii).
References
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. Acta Cryst. A49, 704-708.
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, Acta Cryst. A43, 820-824.
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. Acta Cryst. A52, 640-644.
Sayre, D. (1952). The squaring method: a new method for phase determination, Acta Cryst. 5, 60-65.
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. Acta Cryst. A50, 511-515.