Trigonal (3m) Space Groups

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

To the right of each space group is a listing of coordinate points. These are thecoordinates to which a general point (x,y,z) is transformed by the space group. Origins(called "equivalent points" in the International Tables), areadditional points around which the points are transformed. For example, (0,0,0) refers toa corner of the unit cell, (1/2,1/2,1/2) to the center. All space groups have origin(0,0,0). For a space group with an additional origin (1/2,1/2,1/2), point (x,y,z) is alsotransformed to (1/2+x,1/2+y,1/2+z) and so on.

3-fold and 6-fold coordinates are tabulated with respect to axes intersecting at 60 degrees. In this oblique coordinate system, coordinates tend to be simple

156-p3m1.gif (3284 bytes)

156     P3m1

(+x,+y,+z); (-y,+x-y,+z); (+y-x,-x,+z);
(-y,-x,+z); (+x,+x-y,+z); (+y-x,+y,+z);

157-p31m.gif (2756 bytes)

157     P31m

(+x,+y,+z); (-y,+x-y,+z); (+y-x,-x,+z);
(+y,+x,+z); (-x,+y-x,+z); (+x-y,-y,+z);

158-p3c1.gif (3001 bytes)

158     P3c1

(+x,+y,+z); (-y,+x-y,+z); (+y-x,-x,+z);
(-y,-x,1/2+z); (+x,+x-y,1/2+z); (+y-x,+y,1/2+z);

159-p31c.gif (2492 bytes)

159     P31m

(+x,+y,+z); (-y,+x-y,+z); (+y-x,-x,+z);
(+y,+x,1/2+z); (-x,+y-x,1/2+z); (+x-y,-y,1/2+z);

Rhombohedra are cubes distorted along one of the three fold axes. Coordinates for rhombohedral symmetries can be described in terms of the rhombohedral cell edges or with respect to the threefold symmetry axis system. Coordinates here are given with respect to the threefold symmetry axis system.

160-r3m.gif (5082 bytes) 160      R3m

Origins (0,0,0); (1/3, 2/3, 2/3); (2/3, 1/3, 1/3);

(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z);
(-y, -x, +z); (+x, +x-y, +z); (+y-x, +y, +z);

161-r3c.gif (4732 bytes) 161 R3c

Origins (0,0,0); (1/3, 2/3, 2/3); (2/3, 1/3, 1/3);

(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z);
(-y, -x, 1/2+z); (+x, +x-y, 1/2+z);
(+y-x, +y, 1/2+z);

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Created 13 October 1999, Last Update 11 June 2020