Superspace groups

Normally, solids are crystalline, i.e. they have a three-dimensional periodic order with three-dimensional translational symmetry. However, this is not always so. Aperiodic crystals do have a long-distance order, but no three-dimensional translational symmetry. In a formal (mathematical) way, they can be treated with lattices having translational symmetry in four- or five-dimensional space , the so-called superspace their symmetry corresponds to a four- or five-dimensional superspace group. The additional dimensions are not dimensions in real space, but have to be taken in a similar way to the fourth dimension in space-time. In space-time the position of an object is specified by its spatial coordinates x, y, z the coordinate of the fourth dimension is the time at which the object is located at the site x, y, z. 

P. M. de Wolff, W. van Aalst, The Four-dimensional Space Group of y-Na2Co3. Acta Crystallogr. A 1972, 28, Sill A. Janner, T. Janssen, Superspace Groups. Physica A 1979, 99,47-76. 

All superspace groups for one dimensionally modulated structures (756 superspace... 

Superspace groups for ID and 2D Modulated Structures http //quasi.nims. go.jp/yamamoto/spgr.html... 

The structure periodicity of composite incommensurate inclusion compounds generally requires four basis reciprocal lattice vectors for lattice description and a fourdimensional superspace group to characterize its symine-try. The use of the superspace approach in stmeture determination of ehaimel inclusion compounds is scarce but was applied recently in the study of urea inclusion compounds. " For a commensurate inclusion compound. 

It is important to emphasize the benefits of understanding the symmetry properties (and structural properties) of the composite inclusion compound in a superspace group, rather than restricting the structural description of such materials at the level of the separate basic structures. Knowledge of the structural properties of the separate basic structures contains no information on the modulations that each subsystem experiences in the real inclusion... 

A more recent study of LiFeBOg by Janssen et al. [33] refined it in the fourdimensional superspace group C2/c(a0y)00, with a = 1/2 and y = 0. The Fe and Li sites are not split in their modulated superstructure, and Li ions are at long-range-ordered tetrahedral sites. This modulation of Li sites can slightly... 

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