Tetragonal (P42/mmm) 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.

167-r3xc.gif (10398 bytes) 131    P42/mcm

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

167-r3xc.gif (10398 bytes) 132     P42/mmc

(+x, +y, +z); (-x, -y, +z); (-x, +y, 1/2+z); (+x, -y, 1/2+z);
(+x, +y, -z); (-x, -y, -z); (-x, +y, 1/2-z); (+x, -y, 1/2-z);
(+y, +x, +z); (-y, -x, +z); (-y, +x, 1/2+z); (+y, -x, 1/2+z);
(+y, +x, -z); (-y, -x, -z); (-y, +x, 1/2-z); (+y, -x, 1/2-z);
167-r3xc.gif (10398 bytes) 133     P42/nbc

(+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);
163-p3xc.gif (4642 bytes) 134     P42/nnm

(+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);
164-p3xm.gif (4792 bytes) 135     P42/mbc

(+x, +y, +z); (-y, +x, 1/2+z);
(1/2+x, 1/2-y, +z); (1/2+y, 1/2+x, 1/2+z);
(+x, +y, -z); (-y, +x, 1/2-z);
(1/2+x, 1/2-y, -z); (1/2+y, 1/2+x, 1/2-z);
(-x, -y, +z); (+y, -x, 1/2+z);
(1/2-x, 1/2+y, +z); (1/2-y, 1/2-x, 1/2+z);
(-x, -y, -z); (+y, -x, 1/2-z);
(1/2-x, 1/2+y, -z); (1/2-y, 1/2-x, 1/2-z);
165-p3xc.gif (4492 bytes) 136     P42/mnm

(+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) 137     P42/nmc

(+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) 138     P42/ncm

(+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 15 December 2011

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