Wikipedia

Geometrical crystallography before X-rays

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 1891–4, physical crystallography and chemical crystallography.

Geometrical crystallography before X-rays covers the study of crystal form and the mathematical representation of crystal structure.[1] 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

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Image of stacked cannoballs
Cannonballs piled on a triangular (front) and rectangular (back) base, both face-centered cubic lattices.

The study of the geometrical properties of crystals began in the 16th century.[2] 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.[3] In 1550 Gerolamo Cardano made an early attempt to explain the shape of crystals as the result of a close packing of spheres.[4] In 1591 Thomas Harriot studied the close packing of cannonballs (spheres).[5] In 1597 Andreas Libavius recognized the geometrical characteristics of crystals and identified salts from their crystal shape.[6]

17th century

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Sphere packings in two dimensions
Drawing by Kepler (1611) of square (A) and hexagonal (B) packings[7]

In 1611 Johannes Kepler published Strena Seu de Nive Sexangula (A New Year's Gift of Hexagonal Snow)[8] which is considered the first treatise on geometrical[1] and atomistic[9] 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.[10] Kepler concluded that π/(3√2) = 0.74084 is the maximum possible density amongst any arrangement of spheres — this became known as the Kepler conjecture.[11] The conjecture was finally proved by Thomas Hales in 1998.[12]

In 1665 Robert Hooke attempted to explain crystal morphology based on the stacking of atoms.[13] In his work Micrographia[14] he reported on the regularity of quartz crystals observed with the recently invented microscope, and proposed that they are formed by spherules.[15]

Nicolas Steno rejected Paracelsus's proposed organic origin for crystals.[16] Steno first observed the law of constancy of interfacial angles when studying quartz crystals[17] (De solido intra solidum naturaliter contento, Florence, 1669),[18] 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.[19] Steno's work can be considered as the beginning of crystallography as an independent discipline.[20]

Diagram of cleavage planes in a crystal
Cleavage planes in a crystal of Iceland spar (Huygens, 1690) [21]

In 1678 Christiaan Huygens proposed a structural explanation of the double refraction of calcite based on ellipsoidal atoms.[22] Huygens discovered the polarization of light by Iceland spar, a transparent form of calcite, and published his results in his Traité de la Lumière.[23]

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)[24] and 1705 (De salibus dissertatio epistolaris physico-medico-mechanica)[25] 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.[26]

18th century

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In 1723 Moritz Anton Cappeller published Prodromus Crystallographiae, the first treatise on crystal shapes.[27] The introduction of the term crystallography is attributed to Cappeller.[28] In 1758 Roger Joseph Boscovich published[29] his atomic theory which stated that particles of matter were linked by attractive and repulsive forces[30] 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.[31]

Carl Linnaeus promoted a morphological, as opposed to a physical or chemical, approach to the study of crystals.[32] Linneaus published many accurate and detailed drawings of crystals, and identified the forms which were related by truncation.[33]

Hexagonal arrangement of spheres modelling the structure of niter (Lomonosov, 1749).[34]

In 1749 Mikhail Lomonosov postulated spherical atoms to study the structure of niter and rediscovered cubic close packing.[35] However, his work was not influential at the time.[36]

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.[37] 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.[38] In 1774 Abraham Gottlob Werner published his classification of minerals.[39] Werner's postulated seven primary forms, and showed that some geometrical forms could be derived from one another by truncation.[40]

With Jean-Baptiste L. Romé de l'Isle's Essai de cristallographie published in 1772[41] and Cristallographie published in 1783[42] the scientific approach to crystal structure began.[43] 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.[44] Romé de l'Isle noted that the angles are characteristic of a substance, thus generalizing the law of constancy of angles postulated by Steno.[45] 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.[46] However, Romé de l'Isle criticized René Just Haüy and Torbern Bergman for speculation on the internal structure of crystals without sufficient observational data.[47]

Drawing of calcite crystal assembled from small polyhedra
Calcite scalenohedron crystal constructed from small building blocks (molécules intégrantes) using the law of decrements of René Just Haüy.[48]

In 1781 René Just Haüy (often termed the "Father of crystallography")[49] 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).[50] This theory explained why all crystal planes are related by small rational numbers (the law of rational indices).[51] 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.[52] Haüy's molecular crystal structure theory assumed that molécules intégrantes were specific in shape and composition for every compound.[53] Haüy developed his mathematical theory of crystal structure over many years.[54] Haüy's theory turned out to be remarkably accurate, and gave crystallography a legitimate place among the sciences.[55]

Haüy's crystal structure theory was criticised as over-simplistic by William Hyde Wollaston in 1813[56] and by Henry James Brooke in 1819.[57] Haüy also tended to ignore experimental results that contradicted his structural theory, such as those achieved with the more accurate reflection goniometer[58] invented by Wollaston in 1809.[59] 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.[60] The discovery of the phenomena of isomorphism and polymorphism dealt a clear blow to Haüy's crystal structure theory.[61]

Atomism versus Dynamism

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Portrait of René Just Haüy
René Just Haüy
Portrait of Christian Samuel Weiss
Christian Samuel Weiss
Leaders of the Atomistic (French) and Dynamic (German) Schools of Crystallography
Portrait of Franz Ernst Neumann
Franz Ernst Neumann
Image of Gabriel Delafosse
Gabriel Delafosse
Researchers in the development of geometrical crystallography

Christian Samuel Weiss became familiar with Haüy's theory by translating the 4-volume Traité de mineralogie (1801).[62] Weiss added an appendix to volume 1 of the translation in which he first outlined his dynamical theory of crystals.[63] 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.[64] Weiss has been termed "the founder of geometric crystallography".[65]

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.[66] 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.[67] Weiss used crystallographic axes as the basis of his systematic classification of crystals.[68]

Weiss and his followers Moritz Ludwig Frankenheim and Johann F. C. Hessel studied the symmetry of crystals.[69] Up until 1800 the concept of symmetry did not have a very precise meaning,[70] however during the 19th century crystallography was progressively transformed into an empirical and mathematical science by the adoption of symmetry concepts.[71] "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."[72]

In 1829 Justus Günther Graßmann [de] published a study of the symmetries of the crystal systems using an algebra of linear combinations.[73] In 1832 Franz Ernst Neumann used symmetry considerations when studying double refraction in crystals.[74] 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.[75]

Gabriel Delafosse continued Haüy's work in France.[76] He was the first to use the terms lattice (réseau) and unit cell (maille).[77] 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.[78]

French scientists did not adopt the dynamic crystallographic theory, but they did attempted 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[79] crystals, which were not adequately accounted for by Haüy.[80] 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.[81]

Crystal systems

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Engraving of Christian Samuel Weiss
Christian Samuel Weiss
Lithograph of Friedrich Mohs
Friedrich Mohs
Lithograph of Carl Friedrich Naumann
Carl F. Naumann
Discoverers of the 7 crystal systems (1815–1830)

Christian Samuel Weiss introduced the concept of crystal systems in 1815.[82] 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.[83] 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.[84]

Friedrich Mohs established a classification system for minerals based solely on their external shape. Mohs distinguished four crystal systems rather than the seven identified by Weiss.[85] In 1822 Weiss and Mohs engaged in a priority dispute on who had first discovered the crystal systems.[86]

Crystal systems according to Mohs and Weiss
Mohs system Weiss system Modern system Required symmetries of the point group
Rhombohedral I. Three- and three-fold Trigonal 1 x 3-fold axis of rotation
II. Six-fold Hexagonal 1 x 6-fold axis of rotation
Pyramidal III. Four-fold Tetragonal 1 x 4-fold axis of rotation
Prismatic IV. Two- and two-fold Orthorhombic 3 x 2-fold axis of rotation or 1 x 2-fold plus 2 mirror planes
V. Two- and one-fold Monoclinic 1 x 2-fold axis of rotation or 1 mirror plane
VI. One- and one-fold Triclinic None
Tessular VII. Isotropic or spheroidal Cubic 4 x 3-fold axis of rotation

In 1824 Carl Friedrich Naumann confirmed Mohs' observation that the triclinic and monoclinic systems required inclined rather than orthogonal axes.[87] Naumann attempted a synthesis of the Weiss and Mohs systems by considering four different configuration of axes: orthogonal (three right angles), monoclinic (two right angles and one oblique one), diclinic (one right angle and two oblique ones), and triclinic (three oblique ones).[88] The diclinic system has not survived.[89]

Crystal classes

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Photograph of Moritz Ludwig Frankenheim
Moritz Ludwig Frankenheim
Drawing of Johann F. C. Hessel
Johann F. C. Hessel
Photograph of Axel Gadolin
Axel Gadolin
Discoverers of the 32 crystal classes (point groups) (1826–69)

In 1826 Moritz Ludwig Frankenheim published the first derivation of the 32 crystal classes,[90] but his work was forgotten for many decades.[91] In 1830, Johann Hessel[92] 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).[93] However, Hessel's work remained practically unknown for over 60 years and, in 1867, Axel Gadolin independently rediscovered his results.[94] Gadolin, who was unaware of the work of his predecessors,[95] found the crystal classes using stereographic projection to represent the symmetry elements of the 32 groups.[96] Gadolin's work had a clarity that attracted widespread attention, and caused Hessel's earlier work to be neglected.[97]

In 1884 Bernhard Minnigerode recognized the relationship with crystallography, and analyzed the 32 possible crystal classes in terms of group theory.[98] It was not recognized until later that it is precisely the mathematical group properties that make symmetry significant for crystals.[99]

Miller indices

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Photograph of William Hallowes Miller
William H. Miller
Drawing of Miller indices cubes
Miller indices
William H. Miller inventor of the Miller indices (1839)

The first to introduce indices to denote crystal planes was Christian Samuel Weiss.[100] 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.[101] In 1825 William Whewell, independently from Neumann, proposed essentially the same indices although he used the letters p, q and r.[102]

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).[103] 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.[104]

Bravais lattices

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Picture of Auguste Bravais
Auguste Bravais
Drawing of 14 Bravais lattices
Bravais lattices
Auguste Bravais and the 14 Bravais lattices (1848)

In 1835 Moritz Ludwig Frankenheim introduced the notion of lattice, independently of Ludwig August Seeber, and derived 15 lattice types;[105] these correspond to the 14 Bravais lattices, but Frankenheim double-counted one of the monoclinic lattices.[106]

In 1848 Auguste Bravais presented his work in deriving the 14 Bravais lattices.[107] The work was published in 1850,[108] and translated into English in 1949.[109] Bravais's work can be considered as drawing on a combination of the approaches of Haüy and Weiss.[110] 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.[111] 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".[112] 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.[113]

In 1851 Bravais showed that crystals preferentially cleaved parallel to lattice planes of high density.[114] 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.[115]

Space groups

[edit]
Drawing of a scalenohedron (saw-tooth crystal) constructed from spheres
Scalenohedron by Ludwig Seeber
Photograph of Leonhard Sohncke
Leonhard Sohncke
Photograph of Evgraf Fedorov
Evgraf Fedorov
Photograph of Arthur Schonflies
Arthur Moritz Schoenflies
Commemorative plaque for William Barlow
William Barlow plaque
Discoverers of the 230 space groups (1891–1894)

The identification of the 230 space groups has been extensively documented[116] and is now regarded as a major achievement of 19th century science.[117] 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.[118] 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.[119]

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.[120] Sohncke enumerated the space groups containing only the translations and rotations.[121] Sohncke credited previous researchers, especially Auguste Bravais and Camille Jordan.[122] He also rediscovered Seeber's 1824 paper on space lattices, and arranged a 1891 republication of Johann F. C. Hessel's 1830 work on the 32 crystal classes which had been previously overlooked.[123]

Rotoinversions and glide reflections were introduced by Evgraf Fedorov and Arthur Moritz Schoenflies to derive the 230 space groups. Fedorov[124] and Schoenflies[125] used different methods, but collaborated to reach the final list of space groups in 1891.[126] William Barlow also derived the 230 space groups in 1894 using a method based on patterns of oriented motifs.[127]

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.[128] An English synthesis of the work of Fedorov, Schoenflies and Barlow was made available by Harold Hilton in 1903.[129] 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.[1]

Progress towards the 230 space groups
Name Year Discovery
Christian Samuel Weiss 1815 7 crystal systems (using 2-, 3-, 4- and 6-fold rotation axes)
Ludwig August Seeber 1824 Concept of the space lattice
Moritz Ludwig Frankenheim 1835 15 lattice types (however one was double-counted)
Auguste Bravais 1848 14 space lattices (Bravais lattices) and 7 crystal systems
Leonhard Sohncke 1879 65 spatial arrangement of points (by adding rotation axes and screw axes to the 14 space lattices)
Evgraf Fedorov 1891 230 space groups (by adding rotoinversions and glide reflections)
Arthur Moritz Schoenflies 1891 230 space groups (using group theory)
William Barlow 1894 230 space groups (using patterns of oriented motifs)

By the beginning of the 20th century Paul Groth was able to define 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."[130]

Crystal structure prediction

[edit]
Sphere packing model of salt
Sphere packing by William Barlow, 1897 representing the structure of rock-salt (NaCl)[131]

Until the use of X-rays there was no way to determine the actual crystal structure of even the simplest substances such as salt (NaCl).[132] For example, in the 1880s, William Barlow proposed several crystal structures based on close-packing of spheres[133] some of which were validated later by X-ray crystallography; however the available data were too scarce in the 1880s to accept his models as conclusive.

In the period between the discovery of X-rays (1895) and X-ray diffraction (1912) Barlow and William Jackson Pope developed the principles of packing, and showed how to deduce the structures of some simple compounds.[134] William Johnson Sollas emphasised the importance of different atomic sizes in constructing simple crystals, and correctly concluded that the sodium and chlorine atoms in salt would be of different sizes.[135]

Research community

[edit]
Photograph of Paul Groth
Paul Groth
Title page of the first issue of the journal Zeitschrift für Krystallographie
Z. Kryst., 1, 1877
Founder of Zeitschrift für Krystallographie

Before the 20th century crystallography was not a well-established academic discipline. There were no academic positions specifically in crystallography. Workers in the field normally carried out their crystallographic research as an ancillary to other employment(s), or had independent means. The leading workers in the field of geometrical crystallography were employed as follows:

In the nineteenth century there were informal schools of geometrical crystallography researchers in France (Haüy, Delafosse, Bravais),[164] Germany (Weiss, Mohs, Frankenheim, Hessel, Seeber, Naumann, Neumann, Sohncke, Groth, Schoenflies)[165] and England (Wollaston, Whewell, Miller, Barlow).[166]

Until the founding of Zeitschrift für Krystallographie und Mineralogie by Paul Groth in 1877 there was no lead journal for the publication of crystallographic papers. The majority of crystallographic research was published in the journals of national scientific societies, or in mineralogical journals.[167] The inauguration of Groth's journal marked the emergence of crystallography as a mature science independent of geology.[168]

See also

[edit]

Citations

[edit]
  1. ^ a b c Senechal 1990b, p. 43.
  2. ^ Engel 1986, p. 10.
  3. ^ Agricola 1546; Agricola 1955, pp. 14, 119; Senechal 1990b, p. 43; Authier 2013, pp. 276–277; De Graef & McHenry 2012, p. 20.
  4. ^ Cardano 2013, p. 392; Authier 2013, p. 277.
  5. ^ Authier 2013, pp. 277–278.
  6. ^ Cuevas-Diarte & Reverter 2014, p. 3.
  7. ^ Kepler 1611, p. 9; Kepler 1966, p. 14.
  8. ^ Kepler 1611; Kepler 1966; Authier 2013, pp. 279–284.
  9. ^ Schneer 1960, p. 532.
  10. ^ Cuevas-Diarte & Reverter 2014, p. 3; Kepler 1611.
  11. ^ Authier 2013, pp. 284–286, 289–290.
  12. ^ Hales 2006; Szpiro 2003, pp. 201–214, 264–278.
  13. ^ Burke 1966, pp. 38–42.
  14. ^ Hooke 1665.
  15. ^ Cuevas-Diarte & Reverter 2014, p. 4; Authier 2013, pp. 292–296.
  16. ^ Oldroyd 1974, p. 153, "The accretion of crystalline matter provides the basis of the proposed explanations of crystal formation and an organic origin of mineral crystals is explicitly denied".
  17. ^ Senechal 1990a, p. 44; Authier 2013, pp. 299–305; Mascarenhas 2020.
  18. ^ Steno 1669; Steno 1916.
  19. ^ Ladd 2014.
  20. ^ Lalena 2006, p. 131.
  21. ^ Huyghens 1690, p. 94.
  22. ^ Moser & Robinson 2024; Burke 1966, pp. 41–42.
  23. ^ Cuevas-Diarte & Reverter 2014, p. 5; Authier 2013, pp. 305–306; Huyghens 1690.
  24. ^ Guglielmini 1705.
  25. ^ Guglielmini 1708.
  26. ^ Whewell 2011; Authier 2013, pp. 306–307; Senechal 1990b, p. 44.
  27. ^ Cappeller 1723.
  28. ^ Cuevas-Diarte & Reverter 2014, p. 5; Authier 2013, p. 272.
  29. ^ Boscovich 1922.
  30. ^ De Graef & McHenry 2012, p. 88.
  31. ^ Rowlinson 2002, pp. 49–51, 105–107; Senechal 1990b, p. 44.
  32. ^ Authier 2013, p. 308; Senechal 1990b, p. 44.
  33. ^ Linnaeus 1806.
  34. ^ Lomonosov 1959, p. 160.
  35. ^ Lomonosov 1959, pp. 149–181; Authier 2013, pp. 17–18; Vainshtein 2010, p. 12.
  36. ^ Senechal 1990b, p. 44; Lomonosov 1959, p. 149.
  37. ^ Bergman 1773; Bergman 1780.
  38. ^ Authier 2013, pp. 307–313.
  39. ^ Werner 1785; Werner 1962.
  40. ^ Burke 1966, p. 62; Senechal 1990b, p. 45.
  41. ^ Romé de L'Isle 1772; Mauskopf 1976, p. 10.
  42. ^ Romé de L'Isle 1783.
  43. ^ Burke 1966, pp. 62–77.
  44. ^ Carangeot 1783.
  45. ^ Metzger 1969; Phillips 1957.
  46. ^ Authier 2013, pp. 313–317.
  47. ^ Hooykaas 1981, p. 522.
  48. ^ Haüy 1801, pl. III, fig. 17.
  49. ^ Brock 1913; Adams 1918; Kraus 1918, p. 127; Fock 1895, p. 5.
  50. ^ Haüy 1782a; Haüy 1782b; Authier 2013, pp. 320–322.
  51. ^ Burke 1966, pp. 78–79.
  52. ^ Haüy 1784; Authier 2013, pp. 322–325.
  53. ^ Mauskopf 2012, p. 23.
  54. ^ Haüy 1801; Haüy 1815a; Haüy 1815b; Haüy 1822; Authier 2013, p. 328–333.
  55. ^ Cuevas-Diarte & Reverter 2014, pp. 6–7; Barlow & Miers 1901, p. 299, "It is not too much to say that nearly all the subsequent work on the subject has been but an expansion or modification of the work done by him."; Authier 2013, pp. 318–328.
  56. ^ Wollaston 1813.
  57. ^ Brooke 1819, p. 454.
  58. ^ Usselman 1986.
  59. ^ Wollaston 1809; Mohs 1823, p. 289; Burke 1966, p. 103; Authier 2013, p. 332.
  60. ^ Mitscherlich 1819; Tutton 1922, pp. 1221–1254; Melhado 1980.
  61. ^ Burke 1966, p. 132; Goodman 1969; Geiger 2016; Fock 1895, pp. 82–88.
  62. ^ Haüy 1801.
  63. ^ Weiss 1804; Authier 2013, p. 340.
  64. ^ Spencer 1910; Scholz 1996.
  65. ^ Hoffmann 2020, p. 42.
  66. ^ Eckert et. al. 1992, pp. 19–20; Authier 2013, pp. 337–340; Kaiser 1987, p. 83.
  67. ^ Weiss 1809; Senechal 1990b, p. 46; Authier 2013, pp. 342–343.
  68. ^ Burke 1966, pp. 148–166.
  69. ^ Authier 2013, pp. 337–340.
  70. ^ Hon & Goldstein 2008.
  71. ^ Scholz 1989a; Scholz 1989b; Scholz 1994; Katzir 2004.
  72. ^ Ewald 1962, p. 20–21.
  73. ^ Graßmann 1829; Scholz 1989b, pp. 119–121.
  74. ^ Neumann 1832.
  75. ^ Shinn 2013.
  76. ^ Delafosse 1843; Delafosse 1858; Delafosse 1860a; Delafosse 1860b.
  77. ^ Authier 2013, pp. 369–375.
  78. ^ Authier 2013, pp. 370–371.
  79. ^ Katzir 2004.
  80. ^ Senechal 1990b; Barlow & Miers 1901, p. 304, "... crystal structure consists in the similar repetition throughout space of identical units without regard to their shape or constitution".
  81. ^ Weiss 1814–1815.
  82. ^ Authier 2013, pp. 343–346.
  83. ^ Senechal 1990b, p. 46.
  84. ^ Mohs 1820, p. vii; Mohs 1821; Mohs 1822; Authier 2013, pp. 349–353.
  85. ^ Weiss 1822; Mohs 1823.
  86. ^ Naumann 1824.
  87. ^ Naumann 1826; Naumann 1830a; Naumann 1830b; Authier 2013, pp. 352–353.
  88. ^ Pertlik 2006.
  89. ^ Frankenheim 1826.
  90. ^ Burckhardt 1984; Burckhardt 1988, p. 34–39; Authier 2013, pp. 364–365.
  91. ^ Hessel 1830; Authier 2013, pp. 367–369.
  92. ^ Whitlock 1934; Wigner 1968, p. 796.
  93. ^ Barlow & Miers 1901, pp. 303, 309–310.
  94. ^ Burckhardt 1988, p. 59; Whitlock 1934.
  95. ^ Gadolin 1871; Authier 2013, p. 379.
  96. ^ Swartz 1909.
  97. ^ Minnigerode 1884; Minnigerode 1887.
  98. ^ Eckert et. al. 1992, p. 20.
  99. ^ Weiss 1816–1817; Authier 2013, pp. 346–347.
  100. ^ Neumann 1823; Authier 2013, pp. 353–355.
  101. ^ Whewell 1825; Authier 2013, pp. 355–357.
  102. ^ Miller 1839; Authier 2013, pp. 357–358.
  103. ^ McKie 1974; Nicol 1878, p. 676.
  104. ^ Frankenheim 1835.
  105. ^ Authier 2013, p. 366–367.
  106. ^ Authier 2013, pp. 375–378.
  107. ^ Bravais 1850.
  108. ^ Bravais 1949.
  109. ^ Boulliard et. al. 2022; Maitte 2001; Maitte 2013.
  110. ^ Senechal 1990b; Bonpunt 1999, p. 137.
  111. ^ Cromwell 1997, p. 322; Sohncke groups.
  112. ^ Jordan 1868, p. 168; Senechal 1990b, p. 48; Scholz 1994, p. 1272.
  113. ^ Bravais 1851; Bravais 1866; Ladd 2014, pp. 14–15 133–135.
  114. ^ Authier 2013, pp. 380–382; Senechal 1990b, p. 48.
  115. ^ Authier 2013, pp. 318–400; Burckhardt 1988, pp. 31–93; Scholz 1989c, pp. 17–153; Senechal 1990b, pp. 46–49.
  116. ^ Lalena 2006, p. 155; Authier 2013, p. 392.
  117. ^ Seeber 1824.
  118. ^ Authier 2013, pp. 14, 358–364.
  119. ^ Sohncke 1879; Sohncke 1888; Scholz 1989c, pp. 110–114, 121.
  120. ^ Senechal 1990a, p. 124; Authier 2013, pp. 382–386.
  121. ^ Kubbinga 2012, pp. 18–20; Scholz 1989c, pp. 111.
  122. ^ Sohncke 1891.
  123. ^ Fedorov 1891; Fedorov 1971.
  124. ^ Schönflies 1891.
  125. ^ Paufler & Filatov 2020; Authier 2013, pp. 392–400.
  126. ^ Barlow 1894; Paufler 2019.
  127. ^ Senechal 1990b, p. 48.
  128. ^ Hilton 1903; Authier 2013, p. 4.
  129. ^ Groth 1905, p. 506.
  130. ^ Barlow 1897, p. 547.
  131. ^ Authier 2013, p. 230.
  132. ^ Barlow 1883a; Barlow 1883b; Sohncke 1884; Barlow 1884; Kubbinga 2012, pp. 21–22; Mauskopf 2015.
  133. ^ Barlow 1897; Barlow & Pope 1906; Barlow & Pope 1907; Barlow & Pope 1910; Paufler 2019; Authier 2013, pp. 386–392.
  134. ^ Sollas 1898; Stillwell 1938, pp. 33–34.
  135. ^ Authier 2013, p. 309.
  136. ^ a b Authier 2013, p. 376.
  137. ^ Galiulin 2003; Shafranovskii & Belov 1962.
  138. ^ Correns 1972.
  139. ^ Authier 2013, p. 307.
  140. ^ Authier 2013, p. 281; Szpiro 2003, pp. 10–13, 16–18; MacTutor, Johannes Kepler.
  141. ^ Ewald 1962, p. 351; Freudenthal 1970; MacTutor, Arthur Moritz Schönflies.
  142. ^ Authier 2013, p. 358.
  143. ^ a b Authier 2013, p. 383; Burke 1970.
  144. ^ Authier 2013, p. 370; Taylor 1978.
  145. ^ Fischer 1972; H.A.M. 1928.
  146. ^ a b Authier 2013, p. 319; Hooykaas 1972; Kunz 1918; Boulliard et. al. 2022.
  147. ^ a b Authier 2013, p. 368; Burke 1972.
  148. ^ Authier 2013, p. 359; McKie 1974.
  149. ^ Authier 2013, p. 350.
  150. ^ Burke 1974; Gal 2007.
  151. ^ Authier 2013, p. 354; MacTutor, Franz Ernst Neumann.
  152. ^ Authier 2013, p. 338; Holser 1976.
  153. ^ Authier 2013, p. 357; MacTutor, William Whewell.
  154. ^ Mayer 1977.
  155. ^ a b Authier 2013, p. 301.
  156. ^ Authier 2013, p. 335; Goodman 1976; Usselman 2015.
  157. ^ Schuh 2007b, pp. 560–561.
  158. ^ Authier 2013, p. 293; MacTutor, Robert Hooke.
  159. ^ Birembaut 1971.
  160. ^ Authier 2013, p. 313; Hooykaas 1981.
  161. ^ Authier 2013, p. 387; Holser 1981.
  162. ^ Authier 2013, p. 34; MacTutor, Christiaan Huygens.
  163. ^ Mauskopf 1976; Kubbinga 2012; Lalena 2006.
  164. ^ Scholz 1989c; Kubbinga 2012; Lalena 2006.
  165. ^ Deas 1959; Authier 2013, pp. 333–337, 355–359, 386–400.
  166. ^ Schuh 2007a, pp. 371–373.
  167. ^ Kahr & McBride 1992, p. 12.

Works cited

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