Modulated Structures CIF Dictionary version 1.0

_cell_subsystem_matrix_W_

Names

Category: cell_subsystem

Data type: numb

Must appear in a looped list containing '_cell_subsystem_code'
Enumeration default: 0

Definition

    	
   In the case of composites, for each subsystem the matrix W as
   defined in van Smaalen, S. (1991) Phys. Rev. B43, 11330 (see also
   van Smaalen [(1995). Cryst. Rev., 4, 79-202]. Its dimension must
   match (_cell_modulation_dimension + 3)*
   (_cell_modulation_dimension + 3).
   Intergrowth compounds are composed of several periodic
   substructures in which the reciprocal lattices of two different
   subsystems are incommensurate in at least one direction. The
   indexation of the whole diffraction diagram with integer indices
   requires more than three reciprocal basic vectors. However the
   distinction between main reflections and satellites is not so
   obvious as in normal incommensurate structures. Indeed true
   satellites are normally difficult to locate for composites and
   the modulation wave vectors are reciprocal vectors of the
   other subsystem(s) referred to the reciprocal basis on one
   of them. The choice of the enlarged reciprocal basis
   {a*, b*, c*, q~1~,..., q~d~} is completely arbitrary but
   the reciprocal basis of each subsystem is always known through
   the W matrices. These matrices [(3+d)x(3+d)-dimensional], one for
   each subsystem, can be blocked as follows:

                     (Z^\n^~3~    Z^\n^~d~)
              W^\n^= (                    )
                     (V^\n^~3~    V^\n^~d~)

   the dimension of each block being (3x3), (3xd), (dx3) and (dxd)
   for Z^\n^~3~, Z^\n^~d~, V^\n^~3~ and V^\n^~d~ respectively. For
   example, Z^\n^ expresses the reciprocal basis of each subsystem
   in terms of the basis {a*, b*, c*, q~1~ ,..., q~d~}.
   W^\n^ also gives the irrational components of the modulation wave
   vectors of each subsystem in its own three-dimensional reciprocal
   basis {a~\n~*, b~\n~*, c~\n~*} and the superspace group of
   a given subsystem from the unique superspace group of the
   composite.

   The structure of these materials is always described by a set of
   incommensurate structures, one for each subsystem. The atomic
   coordinates, modulation parameters and wave vectors used for
   describing the modulation(s) are always referred to the (direct
   or reciprocal) basis of each particular subsystem. Although
   expressing the structural results in the chosen common basis is
   possible (using the matrices W) it is less confusing to use
   this alternative description. Atom coordinates are only put in a
   common basis when interatomic distances are calculated.
   Usually the triplet of reciprocal vectors {a*, b* and c*}
   span the lattice of main reflections of one of the subystems and
   therefore its W matrix is the unit matrix.

   For composites described in a single data block using
   *_subsystem_code pointers, the cell parameters, the superspace
   group and the measured modulation wave vectors (see
   CELL_WAVE_VECTOR below) correspond to the reciprocal basis
   described in _cell_reciprocal_basis_description and coincide
   with the reciprocal basis of the specific subsystem (if any)
   whose W matrix is the unit matrix. The cell parameters and the
   symmetry of the remaining subsystems can be derived using the
   appropriate W matrices. In any case (single or multiblock CIF)
   the values assigned to the items describing the atomic parameters
   (including the wave vectors used to describe the modulations)
   are always the same and are referred to the basis of each
   particular subsystem. Such a basis will be explicitly given in a
   multiblock CIF or should be calculated (with the appropriate W
   matrix) in the case of a single block description of the
   composite.