Law of rational indices

The International Union of Crystallography (IUCr) gives the following definition: "The law of rational indices states that the intercepts, OP, OQ, OR, of the natural faces of a crystal form with the unit-cell axes a, b, c are inversely proportional to prime integers, h, k, l. They are called the Miller indices of the face.

René Just Haüy showed in 1784[10] that the known interfacial angles could be accounted for if a crystal were made up of minute building blocks (molécules intégrantes), such as cubes, parallelepipeds, or rhombohedra.

The 'rise-to-run' ratio of the stepped faces of the crystal was a simple rational number p/q, where p and q are small multiples of units of length (generally different and not more than 6).

[12]: 333 In 1830, Johann Hessel[13] proved that, as a consequence of the law of rational indices, morphological forms can combine to give exactly 32 kinds of crystal symmetry in Euclidean space, since only two-, three-, four-, and six-fold rotation axes can occur.

[14][15]: 796  However, Hessel's work remained practically unknown for over 60 years and, in 1867, Axel Gadolin independently rediscovered his results.

Parallel equidistant planes divide the axis OA into an integral number of equal intercepts, Oa 1 , a 1 a 2 , etc., and the same holds for OB and OC. If OA is divided into h parts, OB into k parts, and OC into l parts, the planes (a 1 b 1 c 1 , a 2 b 2 c 2 , etc.) are identified by the Miller indices ( hkl ). The diagram shows plane (243). [ 1 ]
Miller indices of a plane ( hkl ) and a direction [ hkl ]. The intercepts on the axes are at a/ h , b/ k and c/ l .
Calcite scalenohedron crystal constructed from small building blocks ( molécules intégrantes ) using the method of René Just Haüy (1801) in his Traité de Minéralogie . [ 5 ]
Rhombic dodecahedron assembled from progressively smaller cubic building blocks. Garnet has this crystal habit with {110} crystal faces. [ 21 ]