Geometrical crystallography before X-rays describes how geometrical crystallography developed as a science up to the discovery of X-rays by Wilhelm Conrad Röntgen in 1895. In the period before X-rays, crystallography can be divided into three broad areas: geometric crystallography culminating in the discovery of the 230 space groups in 1891–1894, physical crystallography and chemical crystallography.
Geometrical crystallography before X-rays covers the study of crystal form and the mathematical representation of crystal structure. It includes the atomism and dynamism theories of crystal structure, the invention of the Miller indices, and the discovery of the 7 crystal systems, the 32 crystal classes, the 14 Bravais lattices, and the 230 space groups.
16th century
The study of the geometrical properties of crystals began in the 16th century. In 1546 Georgius Agricola published a study of mineralogy in which morphology, or geometrical shape, was one of the characteristics used to classify minerals such as quartz. In 1550 Gerolamo Cardano made an early attempt to explain the shape of crystals as the result of a close packing of spheres. In 1591 Thomas Harriot studied the close packing of cannonballs (spheres). In 1597 Andreas Libavius recognized the geometrical characteristics of crystals and identified salts from their crystal shape.
17th century
In 1611 Johannes Kepler published Strena Seu de Nive Sexangula (A New Year's Gift of Hexagonal Snow) which is considered the first treatise on geometrical and atomistic crystallography. Kepler studied the packing of spheres, in order to explain the hexagonal symmetry of snow crystals. Kepler demonstrated that in a compact packing each sphere has six neighbours in the same plane, three in the plane above, and three in the plane below, for a total of twelve touching spheres. Kepler concluded that /(3) = 0.74084 is the maximum possible density amongst any arrangement of spheres — this became known as the Kepler conjecture. The conjecture was finally proved by Thomas Hales in 1998.
In 1665 Robert Hooke attempted to explain crystal morphology based on the stacking of atoms. In his work Micrographia he reported on the regularity of quartz crystals observed with the recently invented microscope, and proposed that they are formed by spherules.
Nicolas Steno rejected Paracelsus's proposed organic origin for crystals. Steno first observed the law of constancy of interfacial angles when studying quartz crystals (De solido intra solidum naturaliter contento, Florence, 1669), and noted that, although the crystals of a substance differed in appearance from one to another, the angles between corresponding faces were always the same. Steno's work can be considered as the beginning of crystallography as an independent discipline.
In 1678 Christiaan Huygens proposed a structural explanation of the double refraction of calcite based on ellipsoidal atoms. Huygens discovered the polarization of light by Iceland spar, a transparent form of calcite, and published his results in his Traité de la Lumière.
A geometrical theory of crystal structure based on polyhedra was proposed by Domenico Guglielmini. Guglielmini's publications of 1688 (Riflessioni filosofiche dedotte dalle figure de Sali) and 1705 (De salibus dissertatio epistolaris physico-medico-mechanica) concluded that basic forms (cube, rhombohedron, hexagonal prism, and octahedron) of various salt crystals are characteristic of each substance, are identical in form, indivisible, and have faces with identical inclinations to each other.
18th century
In 1723 Moritz Anton Cappeller published Prodromus Crystallographiae, the first treatise on crystal shapes. The introduction of the term crystallography is attributed to Cappeller. In 1758 Roger Joseph Boscovich published his atomic theory which stated that particles of matter were linked by attractive and repulsive forces and that the solid so formed was compressible rather than rigid; this would become relevant in the 19th century when Haüy theorised that crystals were constructed from identical units stacked up without spaces.
Carl Linnaeus promoted a morphological, as opposed to a physical or chemical, approach to the study of crystals. Linneaus published many accurate and detailed drawings of crystals, and identified the forms which were related by truncation.
In 1749 Mikhail Lomonosov postulated spherical atoms to study the structure of niter and rediscovered cubic close packing. However, his work was not influential at the time.
In 1773 Torbern Bergman, a leader in the field of chemical analysis, described the crystal forms of calcite and stated that all the forms could be built up from the cleavage rhombohedron. Bergman, building on the previous work of Linnaeus, developed a classification of minerals based on chemical characteristics, with subclasses organized by their external shapes, and defined seven primary crystal forms. In 1774 Abraham Gottlob Werner published his classification of minerals. Werner's postulated seven primary forms, and showed that some geometrical forms could be derived from one another by truncation.
With Jean-Baptiste L. Romé de l'Isle's Essai de cristallographie published in 1772 and Cristallographie published in 1783 the scientific approach to crystal structure began. Romé de l'Isle described over 500 crystal forms and accurately measured the interfacial angles of a great variety of crystals, using the goniometer designed by his student Arnould Carangeot. Romé de l'Isle noted that the angles are characteristic of a substance, thus generalizing the law of constancy of angles postulated by Steno. Romé de l'Isle considered that the shape of a crystal is a consequence of the packing of elemental particles, and defined six primitive forms. However, Romé de l'Isle criticized René Just Haüy and Torbern Bergman for speculation on the internal structure of crystals without sufficient observational data.
In 1781 René Just Haüy (often termed the "Father of crystallography") discovered that crystals always cleave along crystallographic planes. Based on this observation, and the fact that the inter-facial angles in each crystal species always have the same value, Haüy concluded that crystals must be periodic and composed of regularly arranged layers of tiny polyhedra (molécules intégrantes). This theory explained why all crystal planes are related by small rational numbers (the law of rational indices). In 1784 René-Just Haüy published Essai d'une théorie sur la structure des cristaux, appliquée à plusieurs genres de substances cristallisées in which he stated his law of decrements (décroissement): a crystal is composed of molecules arranged periodically in three dimensions without leaving any gaps. Haüy's molecular crystal structure theory assumed that molécules intégrantes were specific in shape and composition for every compound. Haüy developed his mathematical theory of crystal structure over many years. Haüy's theory turned out to be remarkably accurate, and gave crystallography a legitimate place among the sciences.
Haüy's crystal structure theory was criticised as over-simplistic by William Hyde Wollaston in 1813 and by Henry James Brooke in 1819. Haüy also tended to ignore experimental results that contradicted his structural theory, such as those achieved with the more accurate reflection goniometer invented by Wollaston in 1809. In 1819 Eilhard Mitscherlich discovered the law of isomorphism which states that compounds which contain the same number of atoms, and have similar structures, tend to exhibit similar crystal forms. The discovery of the phenomena of isomorphism and polymorphism dealt a clear blow to Haüy's crystal structure theory.
Atomism versus Dynamism
Christian Samuel Weiss became familiar with Haüy's theory by translating the 4-volume Traité de mineralogie (1801). Weiss added an appendix to volume 1 of the translation in which he first outlined his dynamical theory of crystals. In contrast to Haüy, Weiss took a purely geometric approach to external crystal morphology, completely disregarding any attempt at modelling the internal structure of crystals. Weiss has been termed "the founder of geometric crystallography".
Weiss rejected Haüy's static "atomistic" theory of crystals instead using a "dynamic" approach that was typical of the German natural philosophers of the early 19th century. Weiss understood the external forms of crystals as a consequence of internal attractions and repulsions, and that generative forces were expressed in definite directions which could be observed as one or more axes of rotation. Weiss used crystallographic axes as the basis of his systematic classification of crystals.
Weiss and his followers Moritz Ludwig Frankenheim and Johann F. C. Hessel studied the symmetry of crystals. Up until 1800 the concept of symmetry had a variety of meanings, however during the 19th century crystallography was progressively transformed into an empirical and mathematical science by the adoption of symmetry concepts. "In the first half of the 19th century the paramount symmetry problem was that of point symmetry: to enumerate all possible combinations of symmetry elements which pass through a common point, the origin, and therefore leave this point single. The crystallographic symmetry elements were observed to be exclusively 2, 3, 4 and 6-fold axes, mirror planes, and centres of inversion."
In 1829 published a study of the symmetries of the crystal systems using an algebra of linear combinations. In 1832 Franz Ernst Neumann used symmetry considerations when studying double refraction in crystals. By the second half of the 19th century the study of crystals was focused more on their geometry and mathematical analysis than their physical properties.
Gabriel Delafosse continued Haüy's work in France. He was the first to use the terms lattice (réseau) and unit cell (maille). He stated that the orientation of the axes in a substance is constant, which implies symmetry of translation (a defining feature of a lattice), and that the external symmetry of a crystal reflects its inner symmetry, namely the symmetry of the constituent atoms and their arrangement. In other words, the law of symmetry applies to both the inside and the outside of a crystal.
French scientists did not adopt the dynamic crystallographic theory, but they did attempt to learn from it. Delafosse built on Haüy's crystallographic approach by stating that the structure and physical properties of crystals should exhibit the same symmetry. Delafosse aimed to resolve the apparent counter-examples to Haüy's law of symmetry by explaining that the symmetry of the physical phenomena revealed the inner structure of crystals. This structure is sometimes more complex than the external morphology. Crystals, in these cases, are of lower symmetry than the lattice. This substructure explained the behaviour of hemihedral crystals, which were not adequately accounted for by Haüy. Delafosse argued that Haüy's molécules intégrante did not necessarily have a physical reality, but rather that its polyhedral form should be regarded instead as the space surrounding a lattice point.
Crystal systems
Christian Samuel Weiss introduced the concept of crystal systems in 1815. Weiss defined seven crystal systems: five based on three orthogonal axes (cubic, tetragonal, orthorhombic, monoclinic and triclinic), and two (trigonal and hexagonal) based on three axes in a plane at 60° to each other and a fourth axis orthogonal to the plane. The number and type of the crystal systems of Weiss correspond to the modern systems apart from the triclinic and monoclinic cases which have non-orthogonal axes.