Are there an infinite combination of (h,k,l) Miller Indices, or would there be some sort of limit to the possible combinations?
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1I think there can't be an infinite combination of Miller indices for a given crystallographic system, given the finite distance between the lattice points. – Kksen Jul 30 '21 at 03:22
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In this case, is there a way for me to calculate how many possible combinations there are? Thanks for the help! – Zeno Chen Jul 30 '21 at 03:39
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I would add that it is the combination of the finite distance between lattice points and the finite number of atoms in a real crystal that limit the possible combinations. Though, practically speaking, it is rare to deal with very large indices. – gigo318 Jul 30 '21 at 03:40
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@ZenoChen Is this purely a hypothetical question, or do you have a problem you are trying to solve? You will need to define the volume, density (i.e. lattice constant) for the material you are interested in. – gigo318 Jul 30 '21 at 03:48
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Yes, I am working on an experiment involving Bragg Diffraction, and I need to determine all the possible Miller Indices in order to calculate the intensity of my photon beam. I think the volume is 1mm^3 and the density is 3.6g/cm^3. – Zeno Chen Jul 30 '21 at 03:57
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I see. If you are calculating something like a structure factor then my point is just a technicality and not really needed for practical calculations. @innating's answer gives the possible acceptable values of h,k,l given the lattice coordinates x, y, z. If this isn't sufficient to address your problem, I would suggest reposing your question with regards to the Bragg diffraction problem you are trying to solve. – gigo318 Jul 30 '21 at 04:15
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I see. I made a mistake and gigo318 is correct, the finite size of the crystal is the real constraint that limits the number of the combination to a finite number. – Kksen Jul 30 '21 at 04:35
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@KamKahSen in practice, finite-size effects of the crystal never enter in the calculations. – GiorgioP-DoomsdayClockIsAt-90 Jul 30 '21 at 08:18
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Let us start from the definition of Miller indices:
the set of all the Miller indices is in one-to-one correspondence with the set of all the possible planes passing through the points of Bravais Lattice (BL) points. The BL is a theoretical concept useful for describing the translational symmetry of a crystal. A BL is infinite, contains infinite points and infinite sets of parallel planes passing through them. Therefore, there is no upper limit to the set of the possible Miller indices.
This fact can be seen easily by noting that each triple ($hkl$) denotes the family of planes orthogonal to the direction $h {\bf b}_1+k {\bf b}_2+l {\bf b}_3$, where is a basis of the Reciprocal Lattice. Linearly independent triples denote different families. Being $(hkl) \in \mathbb Z^3$, there are infinite independent triples, therefore, infinite independent directions in the reciprocal lattice and finally infinite Miller indices. There is no contradiction with the existence of a minimum distance between the sites of the BL since the sites in two nearest neighbor planes have not to be one in front of the other.
In conclusion, it is not possible to list all the Miller indices. What can be done is listing those corresponding to all the planes with an interplanar distance larger than some threshold.
A final remark on finite crystals. Of course, every real crystal has to be finite, and this trivial fact would induce a natural cut-off of the values of Miller indices. However, for macroscopic crystals, a direct connection between the crystal size and the upper limit of the Miller indices is not usually done.
