Index

Modulated structures dictionary (msCIF) version 3.2.1

_cell_subsystem.matrix_W

Definition:

   

      In the case of composites, for each subsystem the matrix W as
      defined in van Smaalen (1991);  [see also van Smaalen (1995) or 
      van Smaalen (2012)].
      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
      indexing 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 as
      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 of 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. Atomic coordinates are only
      referred to a common basis when interatomic distances are
      calculated. Usually, the reciprocal vectors {a*, b* and c*\}\
      span the lattice of main reflections of one of the subsystems 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.

      References: Smaalen, S. van (1991). 
              Phys. Rev. B, 43, 11330-11341.
              Symmetry of composite crystals
              Smaalen, S. van (1995). 
              Crystallogr. Rev. 4, 79-202.
              Incommensurate crystal structures
              Smaalen, S. van(2012). 
              Incommensurate Crystallography. Oxford University Press.

Type: Real

Values appear in Matrix context.

Dimension: [11,11]

Evaluation method:

     With  m  as  cell_subsystem

      cell_subsystem.matrix_W = [

    [ m.matrix_W_1_1, m.matrix_W_1_2, m.matrix_W_1_3, m.matrix_W_1_4, m.matrix_W_1_5, m.matrix_W_1_6, 
      m.matrix_W_1_7, m.matrix_W_1_8, m.matrix_W_1_9, m.matrix_W_1_10, m.matrix_W_1_11],
    [ m.matrix_W_2_1, m.matrix_W_2_2, m.matrix_W_2_3, m.matrix_W_2_4, m.matrix_W_2_5, m.matrix_W_2_6, 
      m.matrix_W_2_7, m.matrix_W_2_8, m.matrix_W_2_9, m.matrix_W_2_10, m.matrix_W_2_11],
    [ m.matrix_W_3_1, m.matrix_W_3_2, m.matrix_W_3_3, m.matrix_W_3_4, m.matrix_W_3_5, m.matrix_W_3_6, 
      m.matrix_W_3_7, m.matrix_W_3_8, m.matrix_W_3_9, m.matrix_W_3_10, m.matrix_W_3_11],
    [ m.matrix_W_4_1, m.matrix_W_4_2, m.matrix_W_4_3, m.matrix_W_4_4, m.matrix_W_4_5, m.matrix_W_4_6, 
                           m.matrix_W_4_7, m.matrix_W_4_8, m.matrix_W_4_9, m.matrix_W_4_10, m.matrix_W_4_11],
    [ m.matrix_W_5_1, m.matrix_W_5_2, m.matrix_W_5_3, m.matrix_W_5_4, m.matrix_W_5_5, m.matrix_W_5_6, 
                           m.matrix_W_5_7, m.matrix_W_5_8, m.matrix_W_5_9, m.matrix_W_5_10, m.matrix_W_5_11],
    [ m.matrix_W_6_1, m.matrix_W_6_2, m.matrix_W_6_3, m.matrix_W_6_4, m.matrix_W_6_5, m.matrix_W_6_6, 
                           m.matrix_W_6_7, m.matrix_W_6_8, m.matrix_W_6_9, m.matrix_W_6_10, m.matrix_W_6_11],
    [ m.matrix_W_7_1, m.matrix_W_7_2, m.matrix_W_7_3, m.matrix_W_7_4, m.matrix_W_7_5, m.matrix_W_7_6, 
                           m.matrix_W_7_7, m.matrix_W_7_8, m.matrix_W_7_9, m.matrix_W_7_10, m.matrix_W_7_11],
    [ m.matrix_W_8_1, m.matrix_W_8_2, m.matrix_W_8_3, m.matrix_W_8_4, m.matrix_W_8_5, m.matrix_W_8_6, 
                           m.matrix_W_8_7, m.matrix_W_8_8, m.matrix_W_8_9, m.matrix_W_8_10, m.matrix_W_8_11],
    [ m.matrix_W_9_1, m.matrix_W_9_2, m.matrix_W_9_3, m.matrix_W_9_4, m.matrix_W_9_5, m.matrix_W_9_6, 
                           m.matrix_W_9_7, m.matrix_W_9_8, m.matrix_W_9_9, m.matrix_W_9_10, m.matrix_W_9_11],
    [ m.matrix_W_10_1, m.matrix_W_10_2, m.matrix_W_10_3, m.matrix_W_10_4, m.matrix_W_10_5, m.matrix_W_10_6, 
                              m.matrix_W_10_7, m.matrix_W_10_8, m.matrix_W_10_9, m.matrix_W_10_10, m.matrix_W_10_11],
    [ m.matrix_W_11_1, m.matrix_W_11_2, m.matrix_W_11_3, m.matrix_W_11_4, m.matrix_W_11_5, m.matrix_W_11_6, 
                              m.matrix_W_11_7, m.matrix_W_11_8, m.matrix_W_11_9, m.matrix_W_11_10, m.matrix_W_11_11]  ]

Category:
cell_subsystem