Last Update 31/ 3/ 2013
in English/ in Portuguese
This page may be useful to help a beginner in the study of crystallography to find the coordinates of a rectangular unit cell c2mm origin in a field of disks organized in a periodic pattern. The statusbar must be installed and working on the browser in order to read the mouse coordinates. The valid solution found with this resource will be according to the International Tables for Crystallography, with the intersection of two perpendicular mirror planes and the binary axis on the origin. This will bring the hunting to end.
Suggested procedure
A click on the mouse button will show a circle centered on the intersection
of two perpendicular straight lines in red with its respective coordinates
in the statusbar.
If the red circle covers the 4 black points at the center of 4 black
disks the origin of the c2mm unit cell is exactly on the center
of the red circle, with the valid coordinates x and y for
the origin readable on the status bar and the two perpendicular straight
lines are now two perpendicular mirror planes of group c2mm.
A screenshot can be pasted into a graphical editor and the figure can be enlarged to the comfortable size in order to observe if all of the 4 black points are covered by the red circle. On the other hand, if the user keeps note of the coordinates of the center of the red circle, any necessary correction can be calculated after a visual analysis. For example: if the center of the red circle has a valid x and the y coordinate has two units more than the valid value, in this case two of the 4 mentioned black points will be two coordinate units far away from the upper part of the red circle and out of it. In this case the lower part of the red circle will be two units of coordinate far away from the other two black points as well and both points will be inside the red circle. This applet can be repeated at will.
A screenshot can be pasted into a graphical editor and the figure can be enlarged to the comfortable size in order to observe if the 4 black points are really covered by the red circle. On the other hand, if the user keeps note of the coordinates of the center of the red circle, any necessary correction can be calculated after a visual analysis. For example: if the center of the red circle has a valid x and the y coordinate has two units more than the valid value, two of the 4 mentioned black points will be two coordinate units far away from the upper part of the red circle and out of it. In this case the lower part of the red circle will be two units of coordinate far away from the other two black points as well and both points will be inside the red circle. This applet can be repeated at will.
In future applications of this method to find the coordinates of unit cells of groups p2mm, p2mg, p2gg and c2mm in other periodical patterns, draw one or more concentric circles on a transparent plastic and glide it on the periodical pattern figure until the valid coordinates of the origin are found, according to the requirements of each symmetry group. The number of intercepted objects by the red circle must be even and all the couple of objects inside the borderlines of the periodic pattern must be related by the actual binary axis.
Exercises
1) Write the coordinates of three couple of disk centers related by the binary axis located on the origin.
2) Write the coordinates of three couple of disk centers related by the mirror plane on the origin parallel to x.
3) Write the coordinates of three couple of disk centers related by the mirror plane on the origin parallel to y.
4) Use this method to find the origin of a rectangular unit cell of symmetry groups p2mm, p2mg, p2gg and c2mm to be selected from page plane symmetry groups.
Reference
International Tables for Crystallography (2005). Vol. A, edited by T. Hahn, Dordrecht: Springer.
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