Last update 16/ 8/ 2011
in English/ in Portuguese
Here one of 29 examples of two dimensional objects periodically repeated
in space according to the rectangular plane group p2mm
can be observed every time this page is uploaded. The rectangular cell
with the mirror planes
are displayed in red with the origin in its upper left corner. Binary
axes are located at the intersections of reflection
planes but the two flod axis symbol is not shown. After a click on
a gray rectangle above, the figure will guide the reader in order to localize
the corresponding location given by coordinates of the respective equivalent
positions according to the multiplicity, on column "M." in the table
between this text and the figure, or the range of the asymmetric unit.
The coordinates of the mouse pointer on the figure can be observed
in the status bar.
The periodic structure of objects studied in plane group p2mm will have at least one object in the location of symmetry equal to one of the point groups: 1, m, and 2mm. Any object of the periodic structure has its respective point symmetry. If the object is located over point group m or over pont group 2mm its position is classified as special. Otherwise if the object is located only over point symmetry 1 its position is classified as general.
If a disk centered in any general position is present in the randomly selected example in this page it will always have multiplicity equal 4, which is the highest for p2mm group and its point symmetry, "P.s" will be equal "1", as can be observed in the table. If this is the case then each general equivalent position is obtained by the action of the symmetry operation in the same column, for example "-x, -y" coordinates obtained by the rotation of the coordinates "x, y" about the binary axis perpendicular to this screen located at coordinates "0, 0" in the figure. Next general equivalent position is obtained by the reflection of coordinate x from position "x, y" by the third symmetry operator named mirror plane oriented perpendicular to the screen at the origin and any y coordinate. The fourth general equivalent position is obtained after the reflection of coordinate y from the starting point at "x, y" by a mirror plane containing the origin and perpendicular to the previous plane and perpendicular to this screen.
If a disk in this randomly selected example is centered in a special position point symmetry, "P.s" equal to "m", announced in the table close to the screen coordinates "x=70, y=420" it will always have multiplicity equal 2 and be operated by a mirror plane that reflects point with coordinates "0, y" to the equivalent position "0, -y" in the figure.
If a disk in this randomly selected example is centered in a special position with point symmetry "m", announced in the table close to the screen coordinates " x=70, y=470" it will always have multiplicity equal 2 and be operated by a mirror plane that reflects point with coordinates "x, 0" to "-x, 0" in the figure.
Finally, if a disk in this randomly selected example is centered in a special position with point symmetry "2mm", announced close to the bottom of the table, it will always have multiplicity equal 1 and located on the intersection of mirror planes, on a binary axis.
If appropriate, objects at symmetry equivalent positions related by a mirror plane will be enantiomorfic.
Exercise
Find the numerical coordinates of the symmetry elements to operate as many disk center coordinates given in the above table as necessary in order to have at least the coordinates of all the disks centers in the given rectangular cell and the coordinates of all the disks centers in a next rectangular cell.
Reference
International Tables for Crystallography (2005). Vol A, edited by T. Hahn, Dordrecht: Springer.
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