Recently, I submitted a paper in a crystallographic journal where I used Miller-Bravais indices (hkil) for lattice planes, and Weber (Schweizer) symbols [uvtw] for lattice directions. One reviewer complained the indexing with the following words: "The fourth symbol for a direction has no meaning (please check this in a good crystallography book). The general accepted rule for indexing directions is to write [hk.l]." There are a few things which puzzle me. a) the meaning of four indices are discussable for directions in lattice as well as in reciprocal lattice. Maybe they are more used for lattice planes since symmetry-equivalent planes are easier to recognize by permutation (like for cubic crystals), but this is also correct for lattice directions. b) the use of [hk.l] instead of [uv.w], and c) the application of [uv.w] in general. As I know, [uv.w] is used instead of [uvtw] in order to express that t is a redundant information, but still is valid: t=-(u+v). The reviewer however claimed: "Everything is o.k. for planes, where h+k+i = 0 or i = -(h+k). A bit more complicated is the application for directions. You can´t use in a one by one mapping the formula i = -(h+k)." I believe he is wrong. I found a very convincing and comprehensive description in McHenry, DeGraef: Structure of Materials.
Here a short comparison of the indexing of identical directions described in crystal as well as reciprocal lattice....
Hello, I am by no means an expert in crystallography; however, Marx, Gross: Festkörperphysik seems to agree with you.
Of course, there's redundancy when we write (hkil) or [uvtw]. But these symbols elegantly respect the symmetry of the hexagonal crystal. It's always seemed silly to me that some authors like to replace one of the indices by a dot. That looks ugly and serves no useful purpose.
@ E. Lord: Maybe I am wrong, but from my point of view this advantage only matches the hexagonal but not the trigonal indexing (OK, only partially). The major problem from my point of view is that the reciprocal lattice has another definition. Easier of course but again different to the general definition.
In fact it is the same imperfect "suggestion" as for cubic indexing where people automatically conclude (perhaps even for all other crystal systems) that any permutation of indices reflects all equivalent planes or direction. In fact, this is ONLY for the highest symmetry correct...and this symmetry does not reflect the majority of crystalline phases. Therefore, I in contrast to you, Eric, I do not like this simplification per se. It is definitely convenient :-), I totally agree...
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