InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 591-593
## Section 4.6.2.2. 1D incommensurately modulated structures |

A periodic deviation of atomic parameters from a reference structure (*basic structure*, BS) is called a *modulated structure* (MS). In the case of mutual incommensurability of the basic structure and the modulation period, the structure is called incommensurately modulated. Otherwise, it is called commensurately modulated. The modulated atomic parameters may be one or several of the following:

An incommensurately modulated structure can be described in a dual way by its *basic structure* and a *modulation function* . This allows the structure-factor formula to be calculated and a full symmetry characterization employing representation theory to be performed (de Wolff, 1984). A more general method is the *n*D description: it relates the *d*D aperiodic incommensurately modulated structure to a periodic structure in *n*D space. This simplifies the symmetry analysis and structure-factor calculation, and allows more powerful structure-determination techniques.

The *n*D embedding method is demonstrated in the following 1D example of a displacively modulated structure. A basic structure , with period *a* and , is modulated by a function , with the satellite vector , period , and *α* a rational or irrational number yielding a commensurately or incommensurately modulated structure (Fig. 4.6.2.1).

If the 1D IMS and its 1D modulation function are properly combined in a 2D parameter space , a 2D lattice-periodic structure results (Fig. 4.6.2.2). The actual atoms are generated by the intersection of the 1D physical (external, parallel) space with the continuous *hyperatoms*. The hyperatoms have the shape of the modulation function along the perpendicular (internal, complementary) space . They result from a convolution of the physical-space atoms with their modulation functions.

A basis (*D* basis) of the 2D hyperlattice is given by where *a* is the translation period of the BS and *c* is an arbitrary constant. The components of the basis vectors are given on a 2D orthogonal coordinate system (*V* basis). The components of the basis vector are simply the parallel-space period *a* of the BS and *α* times the perpendicular-space component of the basis vector . The vector is always parallel to the perpendicular space and its length is one period of the modulation function in arbitrary units (this is expressed by the arbitrary factor ). An atom at position **r** of the BS is displaced by an amount given by the modulation function , with . Hence, the perpendicular-space variable *t* has to adopt the value for the physical-space variable **r**. This can be achieved by assigning the slope *α* to the basis vector . The choice of the parameter *c* has no influence on the actual MS, *i.e.* the way in which the 2D structure is cut by the parallel space (Fig. 4.6.2.2*c*).

The basis of the lattice , reciprocal to Σ, can be obtained from the condition : with . The metric tensors for the reciprocal and direct 2D lattices for are The choice of an arbitrary number for *c* has no influence on the metrics of the physical-space components of the IMS in direct or reciprocal space.

The Fourier transform of the *hypercrystal* depicted in Fig. 4.6.2.2 gives the weighted reciprocal lattice shown in Fig. 4.6.2.3. The 1D diffraction pattern in physical space is obtained by a projection of the weighted 2D reciprocal lattice along as the Fourier transform of a section in direct space corresponds to a projection in reciprocal space and *vice versa*: Reciprocal-lattice points lying in physical space are referred to as *main reflections*, all others as *satellite reflections*. All Bragg reflections can be indexed with integer numbers in the 2D description . In the physical-space description, the diffraction vector can be written as , with for the satellite vector and the order of the satellite reflection. For a detailed discussion of the embedding and symmetry description of IMSs see, for example, Janssen *et al.* (2004).

A commensurately modulated structure with and , , and with , can be generated by shearing the 2D lattice Σ with a shear matrix : The subscript *D* (*V*) following the shear matrix indicates that it is acting on the *D* (*V*) basis. The shear matrix does not change the distances between the atoms in the basic structure. In reciprocal space, using the inverted and transposed shear matrix, one obtains

### References

Janssen, T., Janner, A., Looijenga-Vos, A. & de Wolff, P. M. (2004).*Incommensurate and commensurate modulated crystal structures.*In

*International Tables for Crystallography*, Vol. C, edited by E. Prince, ch. 9.8. Dordrecht: Kluwer Academic Publishers.

Wolff, P. M. de (1984).

*Dualistic interpretation of the symmetry of incommensurate structures.*

*Acta Cryst.*A

**40**, 34–42.