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| title | chunk | source | category | tags | date_saved | instance |
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| Geometrical crystallography before X-rays | 3/4 | https://en.wikipedia.org/wiki/Geometrical_crystallography_before_X-rays | reference | science, encyclopedia | 2026-05-05T16:17:27.087973+00:00 | kb-cron |
In 1826 Moritz Ludwig Frankenheim published the first derivation of the 32 crystal classes, but his work was forgotten for many decades. In 1830, Johann Hessel 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 (the crystallographic restriction). However, Hessel's work remained practically unknown for over 60 years and, in 1867, Axel Gadolin independently rediscovered his results. Gadolin, who was unaware of the work of his predecessors, found the crystal classes using stereographic projection to represent the symmetry elements of the 32 groups. Gadolin's work had a clarity that attracted widespread attention, and caused Hessel's earlier work to be neglected. In 1884 Bernhard Minnigerode recognized the relationship with crystallography, and analyzed the 32 possible crystal classes in terms of group theory. It was not recognized until later that it is precisely the mathematical group properties that make symmetry significant for crystals.
== Miller indices ==
The first to introduce indices to denote crystal planes was Christian Samuel Weiss. In the Weiss system a face is denoted by its three intercepts, ma, nb, pc, with three orthogonal axes, where a, b and c are unit lengths along these axes (in modern notation (1/m 1/n 1/p)). In 1823 Franz Ernst Neumann suggested that the inverse of the Weiss indices (m n p) were simpler and easier to use. In 1825 William Whewell, independently from Neumann, proposed essentially the same indices although he used the letters p, q and r. William Hallowes Miller, a student of Whewell and subsequently his successor in the Chair of Mineralogy at Cambridge University, introduced the Miller indices in his book A Treatise on Crystallography (1839). The Miller indices are essentially the same as those of Neumann and Whewell but Miller used the letters h, k and l (h k l). Miller's indices were accepted by his contemporaries because of their algebraic convenience, and it is his notation that is currently used in crystallography.
== Bravais lattices ==
In 1835 Moritz Ludwig Frankenheim introduced the notion of lattice, independently of Ludwig August Seeber, and derived 15 lattice types; these correspond to the 14 Bravais lattices, but Frankenheim double-counted one of the monoclinic lattices. In 1848 Auguste Bravais presented his work in deriving the 14 Bravais lattices. The work was published in 1850, and translated into English in 1949. Bravais's work can be considered as drawing on a combination of the approaches of Haüy and Weiss. Bravais constructed his mathematical lattices as finite sets of points in space, thus avoiding the need for the packing of spheres or polyhedra to represent physical atoms or molecules. He defined axes, planes and centres of inversion as symmetry elements, and identified all of their possible combinations. Bravais assumed that every atom or molecule in the lattice had the same orientation; in 1879 Leonhard Sohncke removed this restriction to derive his "Sohncke groups". Camille Jordan acknowledged Bravais' work on the combination of symmetry elements in his group theory paper Mémoire sur les groupes des mouvements published in 1868–9. In 1851 Bravais showed that crystals preferentially cleaved parallel to lattice planes of high density. This is sometimes referred to as Bravais's law or the law of reticular density and is an equivalent statement to the law of rational indices.
== Space groups ==
The identification of the 230 space groups has been extensively documented and is now regarded as a major achievement of 19th century science. The space groups became important in the 20th century after the discovery of X-ray diffraction and the founding of the field of X-ray crystallography. Ludwig August Seeber first put forward the concept of the space lattice in 1824. He proposed that crystals were assembled from minute particles represented by spheres rather than stacked parallelepipeds without any gaps as Haüy had theorised (compare the scalenohedron diagrams of Haüy and Seeber). Seeber attempted to reconcile the atomistic and dynamic approaches by the regular arrangement of particles with attractive and repulsive forces between them; the gaps between the particles allow for expansion or contraction in response to external physical forces. In 1879 Leonhard Sohncke combined the 14 Bravais lattices with the rotation axes and the screw axes to arrive at his 65 spatial arrangements of points in which chiral crystal structures form. Sohncke enumerated the space groups containing only the translations and rotations. Sohncke credited previous researchers, especially Auguste Bravais and Camille Jordan. He also rediscovered Seeber's 1824 paper on space lattices, and arranged an 1891 republication of Johann F. C. Hessel's 1830 work on the 32 crystal classes which had been previously overlooked. Rotoinversions and glide reflections were introduced by Evgraf Fedorov and Arthur Moritz Schoenflies to derive the 230 space groups. Fedorov and Schoenflies used different methods, but collaborated to reach the final list of space groups in 1891. William Barlow also derived the 230 space groups in 1894 using a method based on patterns of oriented motifs. Schoenflies work was more influential than Fedorov's because he published his work in German rather than Russian, and Schoenflies' notation was more convenient and became widely adopted. An English synthesis of the work of Fedorov, Schoenflies and Barlow was made available by Harold Hilton in 1903. Fedorov went on to derive the 17 plane groups in 1891 and to study space-filling polyhedra. The discovery of the space groups was not universally recognized as an important scientific breakthrough at the time, but after the invention of X-ray crystallography their physical significance was fully appreciated.
By the beginning of the 20th century Paul Groth defined the geometric structure of a crystal as follows: "A crystal—considered as indefinitely extended—consists of n interpenetrating regular-point systems; each of which is formed from similar atoms; each of these point systems is built up from a number of interpenetrating space lattices, each of the latter being formed from similar atoms occupying parallel positions."
== Crystal structure prediction ==