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I’ve learnt that in crystallography, there are many crystallographic directions that are “equivalent.” For example, in a cubic crystal system, the [011] and [110] and [101] directions are “crystallographically equivalent.” See the image below:

enter image description here

I cannot begin to fathom how they are equivalent. These are vectors after all, and from what I can tell with my rudimentary knowledge of maths, these vectors are not equivalent in direction. Yet all materials science textbooks and powerpoints insist that, for cubic systems, any permutation of positive and negative indices will yield a same equivalent direction. That is, the [011], [110], and [101] vectors are… the same vector?

I share the same confusion with planes, even if they aren’t vectors. For cubic systems, just as for directions, any arrangement of positive and negative indices produces an equivalent plane. For example, (100) and (010) are the same plane, when to me they clearly look different.

So what exactly is “crystallographic equivalence”?

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    $\begingroup$ It's all about symmetry. Cubic system is not just $a=b=c,\,\alpha=\beta=\gamma=90^\circ$; it comes with certain symmetry which makes many things equivalent. $\endgroup$ Sep 22, 2016 at 15:18
  • $\begingroup$ They are the same chemically - that is all that is important in this case. $\endgroup$ Sep 22, 2016 at 15:22
  • $\begingroup$ Perhaps it is easier to understand if you assume that the structure is not cubic but extend the lengths so that $a \ne b \ne c$ , (keeping right angles) and plot out some atom positions in 2D then make vectors and planes . Next make the lengths equal and see what happens. You should find that (100) (010) etc are interchangeable. $\endgroup$
    – porphyrin
    Sep 22, 2016 at 15:40
  • $\begingroup$ Ivan is right it is all about symmetry. Think of it this way. You make the cube as a ball and stick model as shown. In your mind you number the balls according to their position. You turn your back and I reorder the cube. Now which ball was your ball 1? You can't tell because all the balls are the same. Now if you make the cube using one red ball and 7 black balls there is no symmetry and you cvould always tell what ball was which. $\endgroup$
    – MaxW
    Sep 22, 2016 at 17:29
  • $\begingroup$ More specifically, following up on @IvanNeretin - it is about the allowed symmetry operations on the crystal structure. These are operations that take the crystal structure back to itself. In your cubic example, you can rotate 90 degrees about the z-axis, taking x -> y and y->-x, but the structure does not change. Thus, [100] and [010] are equivalent (and, similarly, so is [001]). $\endgroup$
    – Jon Custer
    Sep 22, 2016 at 17:54

2 Answers 2

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Directions & sites are said to be crystallographically equivalent, because there are symmetry operators that relate them. So

  • learn about simpler symmetry operations (proper axes, mirror planes, centres of inversion) or advanced ones (like screw axes, glide planes, etc.)

  • learn how to recognize them in every-day's life. It may be helpful for crystallography, and in other fields too: IR / Raman spectroscopy, for example, not only for understanding (descriptive), but for forcasting (predictive). Tutorials like this represent an entry.

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    $\begingroup$ All of this in chemistry is an application of an area of mathematics called group theory. $\endgroup$
    – MaxW
    Sep 22, 2016 at 20:07
  • $\begingroup$ @MaxW I completely agree. Character tables to recognize the IR active vibrations, too. $\endgroup$
    – Buttonwood
    Sep 22, 2016 at 20:13
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    $\begingroup$ you could have a look at molecule-viewer.com to try out finding symmetry elements in each of the commonly met symmetry species in molecules. $\endgroup$
    – porphyrin
    Sep 23, 2023 at 8:32
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    $\begingroup$ @Buttonwood, I prefer to use molecule-viewer.com because it allows you to workout the point group by adding sym elements one by one etc, and there are examples of all point groups found in practice (except diatomics). I found that the automatic estimation is ok if the structure is also v good but sometimes the experimental structure is poor and so automatic estimation is poor but the point group obvious by eye. However, using this automatic method is a good starting point. $\endgroup$
    – porphyrin
    Sep 29, 2023 at 6:56
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    $\begingroup$ @porphyrin If it is about crystallographic model data (as in the question, then more space group symmetry than point group symmetry of an isolated molecule), Platon's Addsym can be a valuable assistant. It however requires a model at hand (.cif file), the hkl reflections (well, for a couple of years the .cif obtained with ShelX include them) and works even better with the structure factor .fcf file sometimes still available. Without reading .hkl/.fcf, visualizer Jmol lacks some information of the experimentally measured electron density map $\endgroup$
    – Buttonwood
    Sep 29, 2023 at 23:52
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Crystallographic equivalence arises because we can change our coordinate systems. For example, $[1\,1\,1]$ direction and $[\bar{1} \,\bar{1}\,1]$ are equivalent in the cubic class because we can change the coordinate system and end up getting the value of $[1\,1\,1]$ for the same vector:

equivalent crystallographic directions

$$ [1\,1\,1] = \color{red}{[\bar{1}\,\bar{1}\,1]} = \overrightarrow{\mathrm{CE}} \\ \color{red}{[1\,1\,1]} = [\bar{1}\,\bar{1}\,1] = \overrightarrow{\mathrm{AG}} $$

There are multiple ways to represent the exact same directions. There is a set of red axis and a set of black axis and we can clearly see that $[1\,1\,1]$ in red is $[\bar{1}\,\bar{1}\,1]$ in black although they represent the same direction $\overrightarrow{\mathrm{AG}}.$

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