Three integers name every plane in a crystal. Take where a plane crosses the three axes, invert each intercept, clear to the smallest whole numbers — and (h k l) drops out: a label so exact that Bragg's law diffracts from it and the lattice stacks by it. Down the center, data flows: the intercepts go in, the engine inverts, the indices and interplanar spacing come out. The blue team builds and defends it; the red team tries to break it.
source Miller, A Treatise on Crystallography (1839) — archive.org/details/ATreatiseOnCrystallography. Year fixed, link marked AMBER (scanned facsimile, no canonical DOI). Rendered, not quoted.
The index is not the intercept; it is its reciprocal. Four moves and it is mechanical:
1 read the intercepts on the a, b, c axes (in units of the cell edge). 2 take the reciprocal of each — an infinite intercept (parallel to that axis) becomes 0. 3 clear the fractions to the smallest common integers. 4 that ordered triple is (h k l).
Live trace of the current plane:
| axis | intercept | reciprocal | index |
|---|
Miller (1839) gave the crystal a coordinate grammar: every face and cleavage plane became a triple of integers instead of a fraction of an intercept. That triple is the shared token of the next spheres.
the-braggs-law diffracts off the very d-spacings this window computes — nλ = 2d·sinθ, with d = a/√(h²+k²+l²) feeding straight in. the-bravais-lattice supplies the stack of atoms these planes cut. Each sphere is the next one's premise.
The blue team's live check: re-run the reciprocal-and-clear procedure on the known plane (1/2, 1, ∞) and re-verify d(110)=a/√2. Expect (2 1 0). If red skips the reciprocal, this badge is where it shows.
A plane meets the three crystal axes at fractional distances along the cell edges: on a, on b, on c. A plane parallel to an axis never meets it — its intercept is ∞.
| intercept | meaning | reciprocal |
|---|---|---|
| 1 | crosses at the cell edge | 1 |
| 1/2 | crosses at half the edge | 2 |
| 2 | crosses at twice the edge | 1/2 |
| ∞ | parallel — never crosses | 0 |
These three numbers — and the cell edge a — are the whole input. The reciprocal column is what the panel below clears to integers.
Change any control — the indices and d-spacing are computed on the spot from reciprocal-and-clear, never looked up.
What the machine produces, proven: the integer triple (h k l) and, for a cubic cell, the interplanar spacing d = a/√(h²+k²+l²) — so d(100)=a, d(110)=a/√2, d(111)=a/√3 exactly, and higher-index planes sit closer together. The current plane is above; these laws are the output.
The blue team's witness (left) re-verifies these live; the red team (right) tries to skip the reciprocal.
For hexagonal crystals the 3-index form even hides the symmetry, so crystallographers switch to 4-index Miller–Bravais (h k i l) with i = −(h+k). The reciprocal-and-clear rule is exact — but it is a labeling, and the geometry it labels is only as simple as the cell.
"(h k l) are the axis intercepts." Cut. They are the reciprocals of the intercepts, cleared to integers — using the intercepts directly is exactly the tamper in window 6.
"Miller invented crystal indices." Cut, corrected. Reciprocal-intercept notation traces to Whewell and Grassmann/Frankenheim; Miller systematized it in the 1839 Treatise — which is why the name stuck.
"Higher indices mean bigger planes." Kept, corrected. Larger h²+k²+l² means more closely spaced planes — smaller d, weaker higher-angle diffraction.
The red team's move: use the intercepts directly as the indices — skip the reciprocal. Now (1/2, 1, ∞) is mis-indexed and the cubic d-spacing comes out wrong. The blue team's witness (window 7) is watching.
Skip the inversion and (1/2, 1, ∞) no longer clears to (2 1 0), so d = a/√(h²+k²+l²) is computed on the wrong triple — the witness recomputes, disagrees with the known plane, and turns red. Nothing is faked; the attack is real and it is caught.