Is it really common to substitute "i" and "t" by a dot, i.e. instead of (hkil) and [uvtw] in crystallographic journals (hk.l) and [uv.w] are preferred? Is this the only advantage that h,k,i respectively u,v,t can be permuted to get equivalent planes and directions (at least for the highest symmetric crystal class 6/mmm)?
The answer is "yes" at both questions. Three indices carry all the necessary information (simply because space is 3-dim). However, the 4-indices notation is a convenient way to get equivalent planes and directions...
Thanks Michel, but if you use the dot instead of i and t you cannot recognize the equivalence of those directions and planes anymore, right?
Then from my point of view one can rather use Miller indexing since the definition of the reciprocal basis vectors is at least consistent for all crystal systems.
That's right Gert, but you can always calculate the value of the dot. For planes, we have: h+k+i = 0. For directions, it's more complex : [U,V,T,W] corresponds to [u,v,w], with W=w, u=U-T, v=V-T. Personally, I almost always prefer to use 3 indices...
I know some other equations for lattice directions which are finally equivalent: assuming [UVW] and [uvtw] - i.e. using capital letters for the Miller and small for the Miller-Bravais indices and Weber symbols - we have
[U V W ] = [ 2u+v 2v+u w]
or
[ u v t w ] = [ (2U-V)/3 (2V-U)/3 -(U+V)/3 W]
and u+v+t=0 .... as for (hkil) what is not surprizing since reciprocal and lattice vectors have the same direction.
Michel...in fact I agree with you, and this is actually also one part of the question: If there is no obvious advantage for the forth index (except of the recognition of equivalent planes and directions by their indices), and even substitute the additional index by a dot and has to recalculate it if necessary, why one uses them?
I myself also prefer the (hkl) and [uvw] indexing because of the consistence of the reciprocakl lattice description.
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