c2mm UNIT CELL ORIGIN HUNTER

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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Table of subjects.
Presentation
Chemistry Analytical Chromatography
Elemental organic analysis
Volumetric analysis, simulation
Crystallography 3 fold screw axis
4 fold inversion axis on tetrahedron
5 fold rotation axis absent in crystallography
c2mm
c2mm unit cell origin hunter
Binary axis and reflection plane in stereographic projection
Bravais lattices
Conic sections under symmetry operators
Converting from spherical coordinates to stereographic projection
Crystal lattice and unit cell
Determination of unit cell
Elements of symmetry in action - animation
Elements of symmetry in action - cube game
Elements of symmetry in action - dodecahedron game
Elements of symmetry in action - icosahedron game
Elements of symmetry in action - octahedron game
Elements of symmetry in action - tetrahedron game
Ewald sphere and crystal measurements
Extinctions
Five classes in the cubic system
Five classes in the rhombohedral system
From tetrahedron to prism
Gnomonic projection
Improper symmetry axis
Miller indices
Miller indices - animation
Miller indices - cube game
Miller indices - octahedron game
Miller indices - rhombic dodecahedron game
Miller indices - tetrahedron game
Mirror plane
Mirror planes and Miller indices game - tetrahedron
Orientations of the cube
p2mm
Plane symmetry groups
Question on point group
Rotation axis in octahedron and Werner compounds
Rotation axis on tetrahedron and organic molecules
Rotation of objects about an arbitrary axis
Rotation of the parallel and stereographic projections of the cube
Rotation of the stereographic and parallel projection of the cube III
Seven faces in stereographic projection
Seven classes in the hexagonal system
Seven classes in the tetragonal system
Six elements of symmetry in seven orientations
Spherical projection of the octahedron
Stereographic projection
Stereographic projection of six polyhedra in different orientations
Straight line equations and symmetry elements
Symmetry, 2 fold axis
Symmetry, 2, 3 and 6 fold axis in benzene
Symmetry, 3 fold axis in the cube
Symmetry, 4 fold axis in the cube
Symmetry, 4 fold axis in the unit cell of gold
Symmetry elements and Miller indices game
Symmetry elements and Miller indices game - octahedron
Symmetry in art and in crystallography
Three classes in the monoclinic system
Three classes in the orthorhombic system
Twin crystals
Two classes in the triclinic system
Unit cell in hexagonal net
General Butane conformations
Density
Electrochemical cell
Ethane conformations
Oxidation and reduction
Resources of chemical-ICT: water, health and symmetry
Solid and liquid gold