From hexagonal to orthorhombic: AlN
We describe the orthogonalization of a hexagonal atomic structure for TEM and STEM image simulations. Orthogonalzation is required for structure models where one of the cell angles is not 90 degrees. The material in this example is aluminum nitride (AlN) and we want to prepare a structure model for imaging along the  zone-axis of the crystal by using the program CellMuncher.
Quick guide to structure model orthogonalization
The steps required to orthogonalize a structure model for STEM and TEM image simulations are listed here in short. More detailed explanations are given below.
Initial structure file of hexagonal AlN
The example is based on an initial structure data file as given by the text block below.
Please copy the text block and save it as file
the initial structure file!
Alternative you may create the structure file with the program BuildCell by the command
The structure data is taken from H. Schulz & K.H. Thiemann, Solid State Communications 23 (1977) 815-818.
In  projection it displays as shown by the image below.
Calculation of the orthogonal cell dimensions
By orthogonalizing a structure model, we try to find a (usually larger) periodic unit of the structure, where all cell axes are oriented with 90 degree angles against each other. In the present example of AlN, where the angle between a- and b-axis is 120 degrees, the orthogonalization is a mere 2-dimensional problem, as illustrated below, because the angles alpha between the x- and z-axis, and beta between the y- and z-axis are both 90 degree already.
In this case, an orthogonal unit cell is found with the new basis vectors a' = a and b' = 2b - a.
In general, the solution is not alway as simple, in particular, when the angle gamma is not 120 deg, or when the original lattice constants are not equal, i.e. a ≠ b. In such general cases, the solution requires a linear combination with very large multiples of the original lattice vectors to span a perfectly periodic new unit. There are also cases, where even no exact solution exists. It may be possible to finde approximate solutions.
The following sequence of operations with the program CellMuncher will transform the hexagonal AlN structure model into a model with an orthorhombic unit cell.
-f AlN-00.cel -o AlN-01.cel --swap-axes=yz
-f AlN-01.cel -o AlN-02.cel --orthogonalize-plane=xz,2
to cut an orthogonal block from the initial structure (see the CellMuncher documentation). The option
is the most direct way of creating an orthogonal super-cell of any given structure. However, it requires a few pre-calculation to select a proper block, depending very much on the details of the input structure.
-f AlN-02.cel -o AlN-03.cel --swap-axes=yz
-f AlN-03.cel -o AlN-04.cel --periodic=x --periodic=y --periodic=z
-f AlN-04.cel -o AlN_001.cel --remove-close-atoms=0.2
denotes the minimum allowed interatomic distance in the structure. A value of 0.2 Å is usually sufficient.
The structure obtained after the transformtion to an orthorhombic cell is shown below in three different projections.
The structure definition file
AlN_001.cel obtained after the above transformations
to an orthorhombic cell is listed.
Summary of commands:
Following is the complete sequence of all commands used above. You may copy this sequence and execute it by one call from a batch file or shell script.
Translation from CEL to CIF for visualization
Use the command
CellMuncher -f dummy-file-name.cel -o dummy-file-name.cif -w CIF
Note that the Biso values in the CIF output of CellMuncher is on a wrong scale. Correct these values before using the CIF file in simulations.