Pm3m, Pn3n

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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For the isometric space groups, we encounter a visualization problem. There is no longer a single principal symmetry direction we can look along. The diagonal 3-fold symmetry axes rotate the motif into planes perpendicular to the plane of the diagram so they are seen edgewise. So for isometric space groups,  three modes of visualization are employed. First is an oblique drawing of the cubic unit cell with the R motif on a smaller cube. Second is an oblique drawing with stereograms replacing the small cube. The stereograms are drawn in standard crystallographic style without any attempt to represent the projections in perspective. Finally there is a view of the unit cell and stereograms viewed perpendicular to a face. For cases like screw axes where only one octant of the cube or stereogram might be present, only that octant is portrayed.

221    Pm3m

Hexoctahedral point groups in a simple cubic lattice

(+x, +y, +z); (+z, +x, +y); (+y, +z, +x); (+x, +z, +y); (+y, +x, +z); (+z, +y, +x)
(+x, -y, -z); (+z, -x, -y); (+y, -z, -x); (+x, -z, -y); (+y, -x, -z); (+z, -y, -x)
(-x, +y, -z); (-z, +x, -y); (-y, +z, -x); (-x, +z, -y); (-y, +x, -z); (-z, +y, -x)
(-x, -y, +z); (-z, -x, +y); (-y, -z, +x); (-x, -z, +y); (-y, -x, +z); (-z, -y, +x)
(-x, -y, -z); (-z, -x, -y); (-y, -z, -x); (-x, -z, -y); (-y, -x, -z); (-z, -y, -x)
(-x, +y, +z); (-z, +x, +y); (-y, +z, +x); (-x, +z, +y); (-y, +x, +z); (-z, +y, +x)
(+x, -y, +z); (+z, -x, +y); (+y, -z, +x); (+x, -z, +y); (+y, -x, +z); (+z, -y, +x)
(+x, +y, -z); (+z, +x, -y); (+y, +z, -x); (+x, +z, -y); (+y, +x, -z); (+z, +y, -x)

222    Pn3n

432 point groups at the corners of a cube with the mirror image 432 group  in the center. Not an I lattice because the corner and center points are different.

(+x, +y, +z); (+z, +x, +y); (+y, +z, +x); (-x, -z, -y); (-y, -x, -z); (-z, -y, -x)
(+x, -y, -z); (+z, -x, -y); (+y, -z, -x); (-x, +z, +y); (-y, +x, +z); (-z, +y, +x)
(-x, +y, -z); (-z, +x, -y); (-y, +z, -x); (+x, -z, +y); (+y, -x, +z); (+z, -y, +x)
(-x, -y, +z); (-z, -x, +y); (-y, -z, +x); (+x, +z, -y); (+y, +x, -z); (+z, +y, -x)

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

 


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Created 23 July 2001, Last Update 15 December 2011

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