I am currently looking at isotropic properties and let's say I want to determine my property in the [100] direction. Now, I basically want that my [100] direction be the same as the [100] direction in an experimental setup.

If I switch between the conventional unit cell and primitive unit cell, the direction changes. It changes with respect to the symmetry of an infinite crystal lattice (the primitive cell is a diagonal part of the conventional standard).

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How do I go about resolving this problem? There must be some standard to set the underlying coordinate systems, otherwise everyone would be getting confused by this.

  • $\begingroup$ What do you mean by "there [hkl] direction"? $\endgroup$ Aug 25, 2022 at 21:45

2 Answers 2


The Miller indices are defined with respect to your choice of unit cell. Usually, people use the same conventional unit cell, so they mean the same direction by [100]. If not, then they need to somehow sort it out to make sure that they talk about the same direction.

From http://xrayweb.chem.ou.edu/notes/symmetry.html :

When researchers discuss a particular material, they need to work from one standard or conventional description of the unit cell for that material. Thus crystallographers have chosen the following criteria for selecting unit cells. By convention the unit cell edges are chosen to be right-handed (a × b is the direction of c), to have the highest symmetry, and to have the smallest cell volume. If other symmetry considerations do not override, then the cell is chosen so that abc, and α, β, and γ all < 90° or all ≥ 90°.

In crystallography, it sometimes happens that similar crystals are processed with a different choice of the unit cell (different space group setting). Bringing them to the same reference setting is called re-indexing.


The conventional unit cell and primitive unit cell are just physical concepts used to explore the symmetry of the system. You do the calculation in such cells and then "expand" it to the whole system using symmetry operations.

Experimental techniques where you can growth single crystal will produce your material in a, let say, given crystal phase. For that phase, you can identify the conventional and primitive cells. Even if you use a method like Molecular Beam Epitaxy and use an oriented substrate, you will have a single crystal where you can define both type of cells.

The issue here is that if you run your calculations in a given direction, then the experimental measurements should be done in such a way that the property can be determine in the same direction of your calculations. Or if the experiment was already done, you can use its setup and run the calculation accordingly.


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