Hexagonal (6*, 6/m and 622) Space Groups

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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To the right of each space group is a listing of coordinate points. These are the coordinates to which a general point (x,y,z) is transformed by the space group. Origins (called "equivalent points" in the International Tables), are additional points around which the points are transformed. For example, (0,0,0) refers to a 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 also transformed 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

174-p6x.gif (2090 bytes) 174 P6* =3/m
(+x, +y, +z);(-y, +x-y, +z);(+y-x, -x, +z);
(+x, +y, -z);(-y, +x-y, -z);(+y-x, -x, -z);
175-p6_m.gif (3117 bytes)
175 P6/m
(+x, +y, +z);(-y, +x-y, +z);(+y-x, -x, +z);
(-x, -y, -z);(+y, +y-x, -z);(+x-y, +x, -z);
(-x, -y, +z);(+y, +y-x, +z);(+x-y, +x, +z);
(+x, +y, -z);(-y, +x-y, -z);(+y-x, -x, -z);
176-p6km.gif (2989 bytes)
176 P6.3/m
(+x, +y, +z);(-y, +x-y, +z);(+y-x, -x, +z);
(-x, -y, -z);(+y, +y-x, -z);(+x-y, +x, -z);
(-x, -y, 1/2+z);(+y, +y-x, 1/2+z);(+x-y, +x, 1/2+z);
(+x, +y, 1/2-z);(-y, +x-y, 1/2-z);(+y-x, -x, 1/2-z);
 
177-p622.gif (5046 bytes) 177 P622
(+x, +y, +z);(-y, +x-y, +z);(+y-x, -x, +z);
(-x, -y, -z);(+y, +y-x, -z);(+x-y, +x, -z);
(+y, +x, -z);(-x, +y-x, -z);(+x-y, -y, -z);
(-y, -x, -z);(+x, +x-y, -z);(+y-x, +y, -z);
 
178-p6i2.gif (4596 bytes) 178 P6.122
(+x, +y, +z);(-y, +x-y, 1/3+z);(+y-x, -x, 2/3+z);
(-x, -y, 1/2+z);(+y, +y-x, 5/6+z);(+x-y, +x, 1/6+z);
(+y, +x, 1/3-z);(-x, +y-x, 2/3-z);(+x-y, -y, -z);
(-y, -x, 5/6-z);(+x, +x-y, 1/6-z);(+y-x, +y, 1/2-z);
 
179-p6p2.gif (4606 bytes) 179 P6.522
(+x, +y, +z);(-y, +x-y, 2/3+z);(+y-x, -x, 1/3+z);
(-x, -y, 1/2+z);(+y, +y-x, 1/6+z);(+x-y, +x, 5/6+z);
(+y, +x, 2/3-z);(-x, +y-x, 1/3-z);(+x-y, -y, -z);
(-y, -x, 1/6-z);(+x, +x-y, 5/6-z);(+y-x, +y, 1/2-z);
 
180-p6j2.gif (4688 bytes) 180 P6.222
(+x, +y, +z);(-y, +x-y, 2/3+z);(+y-x, -x, 1/3+z);
(-x, -y, +z);(+y, +y-x, 2/3+z);(+x-y, +x, 1/3+z);
(+y, +x, 2/3-z);(-x, +y-x, 1/3-z);(+x-y, -y, -z);
(-y, -x, 2/3-z);(+x, +x-y, 1/3-z);(+y-x, +y, -z);
181-p6o2.gif (4723 bytes) 181 P6.422
(+x, +y, +z);(-y, +x-y, 1/3+z);(+y-x, -x, 2/3+z);
(-x, -y, +z);(+y, +y-x, 1/3+z);(+x-y, +x, 2/3+z);
(+y, +x, 1/3-z);(-x, +y-x, 2/3-z);(+x-y, -y, -z);
(-y, -x, 1/3-z);(+x, +x-y, 2/3-z);(+y-x, +y, -z);
182-p6k2.gif (4819 bytes) 182 P6.322
(+x, +y, +z);(-y, +x-y, +z);(+y-x, -x, +z);
(+y, +x, -z);(-x, +y-x, -z);(+x-y, -y, -z);
(-x, -y, 1/2+z);(+y, +y-x, 1/2+z);(+x-y, +x, 1/2+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 15 December 2011

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