Tetragonal (p4/mmm) 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.

162-p3xm.gif (5009 bytes) 123    p4/mmm

(+x, +y, +z); (-x ,-y, +z); (+x, +y, -z); (-x, -y, -z);
(-x, +y, +z); (+x, -y, +z); (-x, +y, -z); (+x, -y, -z);
(-y, +x, +z); (+y, -x, +z); (-y, +x, -z); (+y, -x, -z);
(+y, +x, +z); (-y, -x, +z); (+y, +x, -z); (-y, -x, -z);
163-p3xc.gif (4642 bytes) 124    p4/mcc

(+x, +y, +z); (-x,-y,+z); (+x, +y, -z); (-x, -y, -z);
(-y, +x, +z); (+y,-x,+z); (-y, +x, -z); (+y, -x, -z);
(-x, +y, 1/2+z); (+x,-y, 1/2+z); (-x, +y, 1/2-z); (+x, -y, 1/2-z)
(+y, +x, 1/2+z); (-y,-x, 1/2+z); (-y, -x, 1/2-z); (+y, +x, 1/2-z)
164-p3xm.gif (4792 bytes) 125    p4/nbm

(+x, +y, +z); (-x, -y, +z);
(1/2+x, 1/2+y, -z); (1/2-x, 1/2-y, -z);
(-x, +y, -z); (+x, -y, -z);
(1/2+x, 1/2-y, +z); (1/2-x, 1/2+y, +z);
(-y, +x, +z); (+y, -x, +z);
(1/2-y, 1/2+x, -z); (1/2+y, 1/2-x, -z);
(+y, +x, -z); (-y, -x, -z);
(1/2+y, 1/2+x, +z); (1/2-y, 1/2-x, +z);
165-p3xc.gif (4492 bytes) 126    p4/ncc
(+x, +y, +z); (-x, -y, +z);
(1/2+x, 1/2+y, 1/2-z); (1/2-x, 1/2-y, 1/2-z);
(-x, +y, -z); (+x, -y, -z);
(1/2-x, 1/2+y, 1/2+z); (1/2+x, 1/2-y, 1/2+z);
(-y, +x, +z); (+y, -x, +z);
(1/2-y, 1/2+x, 1/2-z); (1/2+y, 1/2-x, 1/2-z);
(+y, +x, -z); (-y, -x, -z);
(1/2+y, 1/2+x, 1/2+z); (1/2-y, 1/2-x, 1/2+z);
166-r3xm.gif (10626 bytes) 127    p4/mbm

(+x, +y, +z); (-x, -y, +z);
(1/2+x, 1/2-y, -z); (1/2-x, 1/2+y, +z);
(+x, +y, -z); (-x, +y, -z);
(1/2+x, 1/2-y, -z); (1/2-x, 1/2+y, -z);
(-y, +x, +z); (+y, -x, +z);
(1/2+y, 1/2+x, +z); (1/2-y, 1/2-x, +z);
(-y, +x, -z); (+y, -x, -z);
(1/2+y, 1/2+x, -z); (1/2-y, 1/2-x, -z);
167-r3xc.gif (10398 bytes) 128    p4/mnc Fix figure

(+x, +y, +z); (-x, -y, +z);
(1/2+x, 1/2-y, 1/2-z); (1/2-x, 1/2+y, 1/2+z);
(+x, +y, -z); (-x, -y, -z);
(1/2+x, 1/2-y, 1/2-z); (1/2-x, 1/2+y, 1/2-z);
(-y, +x, +z); (+y, -x, +z);
(1/2+y, 1/2+x, 1/2+z); (1/2-y, 1/2-x, 1/2+z);
(-y, +x, -z); (+y, -x, -z);
(1/2+y, 1/2+x, 1/2-z); (1/2-y, 1/2-x, 1/2-z);
167-r3xc.gif (10398 bytes) 129    p4/nmm

(+x, +y, +z); (-x, -y, +z);
(1/2+x, 1/2+y, -z); (1/2-x, 1/2-y, -z);
(-x, +y, +z); (+x, -y, +z);
(1/2-x, 1/2+y, -z); (1/2+x, 1/2-y, -z);
(-y, +x, -z); (+y, -x, -z);
(1/2-y, 1/2+x, +z); (1/2+y, 1/2-x, +z);
(+y, +x, -z); (-y, -x, -z);
(1/2+y, 1/2+x, +z); (1/2-y, 1/2-x, +z);
167-r3xc.gif (10398 bytes) 130    p4/ncc

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

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Created 11 December 2001, Last Update 11 June 2020