Bilbao Crystallographic Server SYMMODES Help

The software package SYMMODES (Symmetry Modes) provides the necessary tools for a group-theoretical analysis of a structural phase transition characterized by a symmetry change G-H, with G > H. The package combines modules of the programs SUBGROUPGRAPH, WYCKSPLIT and the program SOPD (Subgroup Graph Order-Parameter Displacements). The latter has been developed by H. Stokes and D. Hatch (Brigham Young University, Utah, USA) and is a dedicated segment of the package ISOTROPY.

Input Data

• Step 1: Identification of group-subgroup pair.
  • Option 1:
    • Enter the sequential numbers of the supergroup G and subgroup H as given in International Tables for Crystallography volA(referred to as ITA) or choose them from the table of space groups. Please note the default choices of the space-group settings which are used in this program.
    • Specify the index of H in G.
    • Choose a subgroup H_j from a list of subgroups H_j of G of the specified index.
  • Option 2:
    • Enter the ITA numbers of the supergroup(G) and subgroup(H) or choose them from the table of space groups. Please note the default choices of the space-group settings which are used in this program.
    • Define the transformation between the bases of G and H consisting of a rotational (3x3) matrix and an origin shift (3x1) column.

• Step 2: Specify the high symmetry structure by choosing the Wyckoff positions of the occupied atomic orbits.

Examples and screenshots

Step 1:Identification of the group-subgroup pair.

Example: (option 1)

Supergroup Number	62
Subgroup Number	33
Index	2

This is the look and feel of the first form in the option 1. The user must specify the supergroup(G) and subgroup(H) numbers and the index [i].

Given the space groups G, H and their index, the program returns a list of all possible subgroups H_j (of the space group type H) with this index. Each subgroup is specified by the transformation matrix relating its basis to that of G. Different transformation matrices leading to the same subgroup H_j are listed under 'Identical'. The subgroups H_j are distributed into classes of conjugate subgroups. The user is expected to check from the list the relevant subgroup H_j.

Example: (option 2)

Supergroup Number	62
Subgroup Number	33
Transformation Matrix	1 0 0 0 0 0 1 0 0 -1 0 0

This is the look and feel of the first form in option 2. The user must specify the supergroup(G) and subgroup(H) ITA numbers and the transformation matrix that relates the bases of G and H.

Step 2: Wyckoff Positions of the high-symmetry structure

Given the group-subgroup pair G > H, the program returns a list of the Wyckoff positions of G. The user is expected those Wyckoff positions which correspond to the occupied orbits of G.

Example:

G = Pnma and occupied Wyckoff positions 8d and 4c.

Output Data

The results of the symmetry analysis can be summarized as follows:

1. Given the space-group types of G and H, and their index(in the option 1), the program constructs the lattice of maximal subgroups relating G and H and provides:
   • All possible subgroups H_j of the type of H with their distribution into classes of conjugate subgroups.
   • The corresponding transformation matrices relating the (conventional) bases of G to each of the maximal subgroups H_j.
   • The particular group-subgroup graph of chains of maximal subgroups for the group-subgroup pair G > H_j.

   Example: To see the results for the symmetry break Pnma > Pna2₁ with index 2, click here.

2. For a given symmetry break G > H_j and a crystal structure specified by the Wyckoff positions of the occupied atomic orbits, the user is supplied with:
   • The number and the patterns of the displacive modes: For each k vector relevant to the symmetry break G to H the program lists all displacive (primary and secondary) symmetry modes. The modes characterized by the same irreducible representation, order parameter direction and isotropy subgroup are listed in a table form for each of the selected Wyckoff positions. The labels of the irreducible representations follow the notation of Miller and Love (1967) [*]. The isotropy subgroup is specified by ITA number, the Hermann-Mauguin and Schoënflies symbols and the transformation matrix that relates its basis to that of G. The different columns under 'Displacements' correspond to different symmetry modes. The polarization vectors for all atoms in the unit cell of G are represented by triplets of numbers indicating the relative displacive amplitudes. The symmetry mode patterns are referred to the coordination system of G.

     Example: Symmetry modes for the group-subgroup pair Pnma > Pna2₁, index 2 and Wyckoff positions 8d, 4c.

     K vector: GM = (0,0,0)

     Irrep	GM1+
     Order Parameter	(a)
     Isotropy Subgroup H	62 Pnma D2h-16
     Transformation Matrix	[ 1 0 0 ] [ 0] [ 0 1 0 ] [ 0] [ 0 0 1 ] [ 0]

     WP 8d	Displacements
     (x,y,z)	(1,0,0)	(0,1,0)	(0,0,1)
     (x+1/2,-y+1/2,-z+1/2)	(1,0,0)	(0,-1,0)	(0,0,-1)
     (-x,y+1/2,-z)	(-1,0,0)	(0,1,0)	(0,0,-1)
     (-x+1/2,-y,z+1/2)	(-1,0,0)	(0,-1,0)	(0,0,1)
     (-x,-y,-z)	(-1,0,0)	(0,-1,0)	(0,0,-1)
     (-x+1/2,y+1/2,z+1/2)	(-1,0,0)	(0,1,0)	(0,0,1)
     (x,-y+1/2,z)	(1,0,0)	(0,-1,0)	(0,0,1)
     (x+1/2,y,-z+1/2)	(1,0,0)	(0,1,0)	(0,0,-1)

     WP 4c	Displacements
     (x,1/4,z)	(1,0,0)	(0,0,1)
     (x+1/2,1/4,-z+1/2)	(1,0,0)	(0,0,-1)
     (-x,3/4,-z)	(-1,0,0)	(0,0,-1)
     (-x+1/2,-1/4,z+1/2)	(-1,0,0)	(0,0,1)

     Irrep	GM4-
     Order Parameter	(a)
     Isotropy Subgroup H	33 Pna2_1 C2v-9
     Transformation Matrix	[ 1 0 0 ] [ 0] [ 0 0 1 ] [ 0] [ 0 -1 0 ] [ 0]

     WP 8d	Displacements
     (x,y,z)	(1,0,0)	(0,1,0)	(0,0,1)
     (x+1/2,-y+1/2,-z+1/2)	(-1,0,0)	(0,1,0)	(0,0,1)
     (-x,y+1/2,-z)	(-1,0,0)	(0,1,0)	(0,0,-1)
     (-x+1/2,-y,z+1/2)	(1,0,0)	(0,1,0)	(0,0,-1)
     (-x,-y,-z)	(1,0,0)	(0,1,0)	(0,0,1)
     (-x+1/2,y+1/2,z+1/2)	(-1,0,0)	(0,1,0)	(0,0,1)
     (x,-y+1/2,z)	(-1,0,0)	(0,1,0)	(0,0,-1)
     (x+1/2,y,-z+1/2)	(1,0,0)	(0,1,0)	(0,0,-1)

     WP 4c	Displacements
     (x,1/4,z)	(0,1,0)
     (x+1/2,1/4,-z+1/2)	(0,1,0)
     (-x,3/4,-z)	(0,1,0)
     (-x+1/2,-1/4,z+1/2)	(0,1,0)

   • The splitting of the Wyckoff positions during the symmetry break G > H_j.

     For each one of the selected Wyckoff positions of G, the program returns

     • The representatives of the Wyckoff positions in the basis of the group and in the basis of the subgroup.
     • The subgroup Wyckoff positions (into which the group Wyckoff positions have been split to) specified by 'Wyckoff Multiplicity' and 'Wyckoff Letter'.
     • The representatives of the subgroup Wyckoff positions.

   Example: Splitting scheme of the Wyckoff position 8d for the symmetry break Pnma > Pna2₁.

   Representative			Subgroup Wyckoff position
   No	group basis	subgroup basis	name[n]	representative
   1	(x, y, z )	(x, y, z )	4a1	(x1, y1, z1 )
   2	(-x, y+1/2, -z )	(-x, -y, z+1/2 )		(-x1, -y1, z1+1/2 )
   3	(x+1/2, y, -z-1/2 )	(x+1/2, -y+1/2, z )		(x1+1/2, -y1+1/2, z1 )
   4	(-x+1/2, y+1/2, z-1/2 )	(-x+1/2, y+1/2, z+1/2 )		(-x1+1/2, y1+1/2, z1+1/2 )
   5	(-x+1/2, -y, z-1/2 )	(-x+1/2, y+1/2, -z )	4a2	(x2, y2, z2 )
   6	(x+1/2, -y+1/2, -z-1/2 )	(x+1/2, -y+1/2, -z+1/2 )		(-x2, -y2, z2+1/2 )
   7	(-x, -y, -z )	(-x, -y, -z )		(x2+1/2, -y2+1/2, z2 )
   8	(x, -y+1/2, z )	(x, y, -z+1/2 )		(-x2+1/2, y2+1/2, z2+1/2 )