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.

 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); 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); 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); 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); 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); 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); 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); 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);