By definition, as you have already stated, h, k, l should be relatively prime. So (002) should not be allowed. It should be reduced to (001) by dividing all indices by 2..

Also, by definition, Miller indices of a plane specifies only its orientation, but not its position. Thus the the plane passing through the centre of the unit cell also have the same orientation and so the same Miler indices.

In diffraction, we do use indices like (002). But here, to be precise, they are not indices of planes, but indices of diffraction peak (or Bragg reflection). The common multiplier in these cases indicate the order of the diffraction. Thus (002) diffraction is the 2nd. order diffraction from the (001) planes. To avoid this ambiguity, some authors drop the use of parenthesis for the indices of diffraction peaks. So they will call the the diffraction just 002 and not (002).

But really if you wish to distinguish between two parallel planes, one passing through the corners and one passing through the centres, you may redefine your Miller Indices and insist that (hkl) represents both orientation and position. But in that case, you also have to specify that h, k, l need not be relatively prime and one should not multiply or divide by any common integer. Equivalently, you also loose the freedom to choose your origin freely. So your Miller indices will now refer to a fixed origin. In that case (002) will be the mid plane and the (001) will be the plane passing through the centre. But these will be single planes and not family of parallel planes. So they cannot be associated with any interplanar spacings.

By definition, as you have already stated, h, k, l should be relatively prime. So (002) should not be allowed. It should be reduced to (001) by dividing all indices by 2..

Also, by definition, Miller indices of a plane specifies only its orientation, but not its position. Thus the the plane passing through the centre of the unit cell also have the same orientation and so the same Miler indices.

In diffraction, we do use indices like (002). But here, to be precise, they are not indices of planes, but indices of diffraction peak (or Bragg reflection). The common multiplier in these cases indicate the order of the diffraction. Thus (002) diffraction is the 2nd. order diffraction from the (001) planes. To avoid this ambiguity, some authors drop the use of parenthesis for the indices of diffraction peaks. So they will call the the diffraction just 002 and not (002).

But really if you wish to distinguish between two parallel planes, one passing through the corners and one passing through the centres, you may redefine your Miller Indices and insist that (hkl) represents both orientation and position. But in that case, you also have to specify that h, k, l need not be relatively prime and one should not multiply or divide by any common integer. Equivalently, you also loose the freedom to choose your origin freely. So your Miller indices will now refer to a fixed origin. In that case (002) will be the mid plane and the (001) will be the plane passing through the centre. But these will be single planes and not family of parallel planes. So they cannot be associated with any interplanar spacings.

As far as I know, hkl are Laue indices and "need" to be used for interferences, i.e. in XRD or TEM interferences should be called hkl and not (hkl) (as sometimes misleadingly used). Or is this not true?

Is nowhere discussed what should be done in case of centered unit cells?

I think we are in agreement that hkl, and not (hkl), should be used for indexing reflections (Bragg) or interferences (Laue). The (hkl) should be reserved for Miller indices of planes.

Since Miller indices (hkl) , as defined conventionally, include any or all planes parallel to a plane with intercepts a/h, b/k and c/l, there is no possibility of distinguishing different planes of the same orientation.

However, if one wants to distinguish them, in some cases, like in close-packed crystals, stacking sequence conventions such as ABAB for HCP and ABCABC for CCP is used. The ABAB case of HCP is close to what you want for your FCC lattice. Here Both A and B are the same (0001) plane but A passes through the corners of the unit cell whereas B passes through its centre.

May be in the case of (001) FCC you can say that the stacking sequence is ABAB with A passing through the corners and B passing through the centres.

Well, there are a few things which do not match. a) hcp and ccp are crystal STRUCTURES, and we are talking about crystal LATTICES. Moreover, hcp is described by a primitive lattice, i.e. one lattice point represents two atoms (where non of them is at 0,0,0 as you explained. One is at 1/3, 2/3, 1/3 and the second at 2/3,1/3,2/3.) Since both fcc and bcc are structures, I don't think it is a good example (or I do not see the benefit, sorry).

Anyway, I totally agree with you that hkl are the description of reciprocal lattice points which perfectly match to the interference orders, whereas (hkl) stand for a lattice plane describing all parallel lattice planes (of the same normal direction). BTW: orientation is a very "sensitive" term used for crystal orientation description. I would prefer alignment.

To me it still sounds surprising that this topic should never have been discussed before? I mean, for an F-lattice d(100) describes the lattice parameter ...but also the d-value...although there is another lattice plane in between :-). Is this only a terminological problem...?

I think it is not only a trivial terminological problem, but can be genuinely confusing at an introductory level. To cite a personal example - one is taught that slip occurs easily on widely-spaced planes (a consequence of the 'a' term in Peierls stress). Applying the d= a/sqrt(h^2+k^2+l^2) formula for interplanar spacing of cubic systems, it appears that (001) planes are spaced further apart than (111). Yet, the slip plane in FCC is (111). Some people try to explain it off on basis of planar density (only the partial truth), but once the half-distance planes are considered, one readily sees that (111) planes are further spaced.

Of course, it is true that d= a/sqrt(h^2+k^2+l^2) holds for the simple cubic lattice, not for centred lattices. But this distinction is rarely made, and anyway, unless the (002) type definition is used, it seems difficult to arrive at an equivalent formula for FCC/BCC.

Eagerly looking forward to hearing opinions on this, especially on any mistake I might be making.

After thinking about this problem (and the comment of Aashranth) the solution for the additional but not considered lattice planes was quite simple. If lattice centerings are used (A,B,C,I,F), always then an additional lattice plane exist when the corresponding reflex is absent. For example: for an F-lattice, only if h+k+l (h,k,l prime to each other!) no additional lattice plane exists (with the half lattice plane distance 1/2 d(hkl) ). This means, that for (111) the same lattice plane distance as for the P-lattice exist whereas for the I-lattice the half results. For (123) the (real) lattice plane distance in an F-lattice is half that big as for the P-lattice, whereas for the I-lattice they are identical (h+k+l=2n).

BTW: For R either the same lattice plane distance results (if -h+k+l=3n) as for P, or the "real" lattice plane is 3-times shorter.

However, the question still remains: Is this already discussed somewhere, how the lattice plane distances are concerned if the lattice is not primitive? Do we have to use that one described by the primitive lattice (and all formulae in textbooks) or do we have to use the "real" distance between lattice planes defined by existing lattice points?

or in other words: if an interference is systematically absent (caused by the centered lattice) the real lattice plane distance is halved or divided by three. The R lattice is then of course described by hexagonal basis vectors and not by the rhombohedral.

I've just found a contribution from Nespolo:The ash heap of crystallography: restoring forgotten basic knowledge. J. Appl. Cryst. 2015.

He points out that the assumption of coprime h,k,l is only reasonable for primitive lattices. For centered lattices lattice planes (200) are possible as long as they run through lattice points, i.e. he allows in maximum (hkl) with a factor of two for all (!) centered lattices, and a factor of 3 for R only, i.e. for R (001), (002) and (003) are possible, whereas for e.g. I and F (100) and (200) are definable. In contrast, 100 (the interference) is absent for F, whereas 400 exists (in contradiction to (400) since there are no lattice points) .... This is an alternative concept to the one described above. I would propose to follow the indexing discussed by Nespolo since it has in common that lattice planes are then existing if lattice points define them.