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[prcantos]

Hi. I wonder if the Miller-Bravais indices notation is really useful in Crystallography. In other words, what is the real utility of the extra index i in (hkil) in hexagonal and trigonal systems? The i-index is redundant because h+k+i=0, so you can always leave it out. Is there any heavy reason to keep it?

[Pete Richards]

Thank you Pablo for raising an important question.

You are right that the third index i is redundant in the hexagonal/trigonal system referred to a hexagonal unit cell with a=b, beta=120°. Most recent authors that I am aware of (me included) are dropping this i index, but being careful to say that they are doing so, and why. For hexagonal minerals, this is pretty trivial. But for trigonal/rhombohedral minerals, there is another rhombohedral frame of reference that uses three equal axes separated by a rhombohedral angle alpha normally not equal to 90°. In the case of calcite, the three axes would be parallel to the edges of the cleavage rhombohedron, and the angle alpha would be the angle between each of these axes. This system is also written using {hkl} notation, but the numbers would be different from those with respect to the Miller-Bravais hexagonal cell setting.

The use of the rhombohedral cell setting is diminishing through time. I think (on minimal experience) that the strong-hold of this setting was England. But there is enough use of this frame of reference in the standard reference works that it is important to be clear which frame of reference one is using!

[Bob Carnein]

I suppose that, for anyone who uses the Miller indices every day, shortening the index by one digit makes sense (though how important it is may be debatable). Having two systems coexistent at the same time is confusing to students. In terms of visualization, I think the "old" system for hexagonal minerals is clearer, but there certainly are good arguments to abandon it. On the other hand, what, after all, is "really necessary" anyway?

[Pete Richards]

Having two coexisting systems of notation is indeed confusing. Bad enough within the hexagonal cell framework; doubly confusing with the rhombohedral framework. It is always critical to state clearly the frame of reference!

(Technical note: There are many examples of minerals that, over time, have gone from being referred to one lattice to later being referred to another one. A very common case is in the tetragonal system, where the question is whether the primary prism on a crystal is {100} or {110}. Given this choice (and a bit more information...), the morphological unit cell and the indices of all the common forms follow automatically. But if you/they change the definition of a crystal by switching between those two prisms, the whole morphological system also changes, and must be adjusted accordingly. Typically, this does not happen. Consequently, we see in many famous texts crystal drawings that cannot be reproduced by using the forms listed in the text and the contemporary unit cell, or cleavages that are not where they should be. This is the most chaotic part of current-day morphological crystallography!)

I agree that there is a certain comfort in the four-index Miller-Bravais notation, especially if you are used to it. Although I have largely eliminated the third index in my writings, I must admit that I have to think twice when working with a hexagonal/trigonal mineral and looking at only three indices.

With face-specific indices, it gets worse. (1 1 -2 3), (-2 1 1 3), and (1 -2 1 3) are part of the top of the same trigonal form (hexagonal dipyramid in the case of calcite). The cyclic transposition of the first three numbers helps us to understand that the set of three faces are equivalent, only the order of the indices is different. But if we have (1 1 3), (-2 1 3), and (1 -2 3), that relationship is obscured.

One of the main arguments for abandoning the four-index system is a purely practical one: with it you will be guaranteed that at least one index (usually i if we're talking about forms, not specific faces) will be negative. Negative index values are written with a bar over the number, not a minus sign in front. This is a royal pain with contemporary word processors and drawing programs, and must be as much of a pain for those doing the layout for journal or collector's magazine articles as it is for the authors of those manuscripts. I could not do it with the indices above.

This solution, such as it is, of dropping the third index only works to eliminate superior bars for the hexagonal/trigonal system, and one must be clear about usage to avoid confusion. But with, for example, the triclinic bar-1 symmetry class of the feldspars, there will inevitably be forms for which at least one of the indices is negative. You're out of luck in this case.

One solution is to study only minerals with higher symmetry!

[prcantos]

Ok, thank you very much.

I supposed that the main feature of the four-indices system was the clarity.

[Pete Richards]

Sorry to be so long in replying - other activities intervened!

Below is a rhombohedron; it happens to be the cleavage rhombohedron of calcite. The top drawing shows the hexagonal axis set, with a unique (three-fold) vertical axis and three equivalent horizontal axes 120° apart from each other.

The bottom drawing shows the same rhombohedron with the rhombohedral axis set, composed of three equal axes all separated by an angle alpha. In this case the rhombohedron is congruent with the unit cell, so the axes are parallel to the edges of the rhombohedron. One possible set of axes would be the three upper edges, another possible set would be the three bottom edges. One of these sets would be the negatives of the other set, though really the axes should be though of as extending in both directions from the origin, which I placed at the center of the crystal.

[Pete Richards]

[prcantos]

Ok, thank you, Peter. What an eye for 3D graphics! You are right with the angles in the rhombohedron.