NetAutomorphism

This tool allows to analyse the automorphisms of a net (based on a quotient graph as for example created by Topolan). It shows for a quotient graph

All automorphisms.
For each automorphisms the corresponding rotational matrix, the type of rotarion/ reflection, the intrinsic translation, and the orbits of edges and nodes.

Last but not least, for automorphisms that correspond to a translation (in particular those corresponding to a smaller lattice), a reduced quotient graph is calculated. For more details see Georg Thimm and Björn Winkler. Net Topologies, Space Groups, and Crystal Phases Zeitschrift für Kristallographie, 221(12), 749-758, September 2006

The following image shows a screen shot of the main display showing a part of the automorphisms of a 2 dimensional quotient graph (2- and 3-dimensional graphs are acceptable). Automorphisms that lead to a reduction of the quotient graph are highlighted in orange. The columns in the table show

the rotational part of the Seitz symbol.
The type of the symmetry element and its orientation (if applicable, for rotation a vector defining the direction of the axis, for reflections a vector perpendicular to the plane)
The intrinsic vector(s) for the symmetry ("-" indicates that this is not caclulated; initiate the calculation using the "Settings" menue).
The orbits of the edges.
The orbits of the nodes.

The following image shows a window displaying a reduced graph. The tabulators allow to switch between the reduced QG, the orbits that define the reduction, and the coordination sequences for the graph.

The graphical user interface (updated 28 April 2008) of NetAutomorphism uses JLinAlg (bundeled with the GUI) and the eclipse modelling framework (available with eclipse).

Requirements:

Java
The class path to include the jar files of crys.jar, org.eclipse.emf.common (or similar) and org.eclipse.emf.ecore jar files.

The GUI can be run using the command

java -classpath crys.jar:org.eclipse.emf.ecore_2.4.1.v200808251517.jar:org.eclipse.emf.common_2.4.0.v200808251517.jar crystal.util.gui.NetAuto

The menus of the the main and allow to

Load a QG and calculate all automorphisms or only translational automorphisms (the later is computationally less expensive).
Save a reduced QG
Save either the entire table with the automorphisms or a reduced QG in LaTeX format.
Set whether to display all automorphisms or only those inducing a reduction of the QG.
Set whether to display all intrinsic translations.
Set whether to display node and edge maps
Interrupt the calculation of the automorphisms.

Net Isomorphism

This tool allows to compare two QG for isomorphism (under the constraint that their base graphs are isomorphic). The GUI can be run using the command

java NetIso

and has the same software requrements as the net automorphism tool.

JMOL modification: Export structures as Quotient graphs

The jar file and the sources of a modified version of Jmol (see also here). Can be used to create quotient graphs for the perusal with the crystal net tool. Do do so

start Jmol,
load a a CIF file,
create the structure (from the menus: "Reload {444 666 1}" or similar),
Export as QG...

Creating constraints and embeddings from automorphisms

Once a QG is loaded and automorphisms are calculated, the mouse can be used to select the automorphisms by which the embedding of the QG should be constrained. Then, the menu "Constraints" gives acces to either solving the constraints and saving them to a file in LaTeX or plain text format (compatible with maxima), or to create a CIF file from it. Before the CIF file can be saved, it may be necessary to fix the free variables in the constraining equations by completing the table of fractional coordinates.

Examples

Examples for QGs: halite, cuprite, rutile, square net with 3x3 nodes per unit cell, square net with 2x3 nodes per unit cell, diamond, coesite, calcite, calcite with for a centered unit cell, strontium feldspar, , 2-dimenional triangle net with 2x2 nodes per unit cell, magnesite for a centered cell, and magnesite.