Tetragonal (4/m 2/m 2/m) 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.

139i4_mm.gif (4858 bytes)

139); i4/mmm

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

(+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)

140i4_mc.gif (3837 bytes)

140); i4/mcm

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

(+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)

141i4i_a.gif (5325 bytes)

141); i41/amd

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

(+x,+y,+z); (-x,-y,+z);
(+x, 1/2+y,1/4-z); (-x, 1/2-y, 1/4-z)
(-x,+y,+z); (+x,-y,+z);
(-x, 1/2+y,1/4-z); ( ); (+x, 1/2-y, 1/4-z)
(+y,+x,-z); (-y,-x,-z);
(+y, 1/2+x,1/4+z); (-y, 1/2-x, 1/4+z)
(-y,+x,-z); (+y,-x,-z);
(-y, 1/2+x,1/4+z); (+y, 1/2-x, 1/4+z)

142i4i_a.gif (5371 bytes)

142); i41/acd

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

(+x,+y,+z); (+x,-y,1/2+z);
(+x, 1/2+y,1/4-z); (+x, 1/2-y, 3/4-z)
(-x,-y,+z); (-x,+y,1/2+z);
(-x, 1/2-y,1/4-z); ( ); (-x, 1/2+y, 3/4-z)
(-y,+x,-z); (+y,+x,1/2-z);
(-y, 1/2+x,1/4+z); (+y, 1/2+x, 3/4+z)
(+y,-x,-z); (-y,-x,1/2-z);
(+y, 1/2-x,1/4+z); (-y, 1/2-x, 3/4+z)

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Created 13 October 1999, Last Update 11 June 2020