Syngonia (from the Greek. Σύν "according to, together, near" + γωνία "angle";letters “Similarity”) - classification of crystallographic symmetry groups , crystals and crystal lattices depending on the coordinate system ( coordinate frame ); symmetry groups with a single coordinate system are combined into one syngony. Crystals belonging to the same system have similar angles and edges of unit cells .
The crystal system is a classification of crystals and crystallographic groups based on a set of symmetry elements that describe a crystal and belong to a crystallographic group.
Lattice system - classification of crystal lattices depending on their symmetry .
In the literature there is a confusion of all three concepts: syngonies [1] , the crystal system [2], and the lattice system [3] - which are often used as synonyms .
In the Russian-language literature, the term "grid system" is not used yet. Usually the authors mix this concept with the crystal system. In the book "Fundamentals of Crystallography" [4], the authors use the term "Grid syngony" (" By the symmetry of nodes, spatial gratings can be divided into seven categories, called syngony gratings "). In the same authors, syngonies are called systems (“ The most established classification of groups is their division into six systems based on the symmetry of the facet complexes ”).
Content
Syngony
Historically, the first classification of crystals was the division into syngonies, depending on the crystallographic coordinate system. The axes of symmetry of the crystal were chosen for the coordinate axes, and in their absence, the crystal edges. In the light of modern knowledge about the structure of crystals, the transmissions of the crystal lattice correspond to such directions, and the Brava cell translations in a standard installation are selected for the coordinate system. Depending on the ratio between the lengths of these broadcasts and the angles between them There are six different syngonies, which fall into three categories depending on the number of equal lengths of broadcasts:
- Lower category (all broadcasts are not equal to each other)
- Triclinic : ,
- Monoclinic : ,
- Rhombic : ,
- Middle category (two of the three broadcasts are equal)
- Tetragonal : ,
- Hexagonal : ,
- Highest category (all broadcasts are equal)
- Cubic : ,
Crystal System
The division into crystalline systems is performed depending on the set of symmetry elements describing the crystal . Such a division leads to seven crystalline systems, two of which — trigonal (with one axis of the 3rd order) and hexagonal (with one axis of the 6th order) —are an elementary cell of the same shape and therefore belong to the same hexagonal syngony. It is sometimes said that hexagonal syngony is divided into two subingony [5] or hyposingonia. [6]
Crystal systems are also divided into three categories, depending on the number of higher order axes (higher than second order axes).
Possible in three-dimensional space crystalline systems with symmetry elements defining them, that is, symmetry elements, the presence of which is necessary for classifying a crystal or a point group to a specific crystalline system:
- Lower category (no higher order axes)
- Triclinic : no symmetry or only center of inversion
- Monoclinic : one axis th order and / or plane of symmetry
- Rhombic : three mutually perpendicular axes th order and / or plane of symmetry (the direction of the plane of symmetry is considered perpendicular to it)
- Middle category (one axis of higher order)
- Tetragonal : one axis th order or
- Trigonal : one axis th order
- Hexagonal : one axis th order or
- Highest category (several higher order axes)
- Cubic : four axes th order
The crystal system of a space group is determined by the system of the corresponding point group. For example, the groups Pbca, Cmcm, Immm, Fddd ( class mmm) belong to the rhombic system.
The modern definition of the crystal system (applicable not only to ordinary three-dimensional groups, but also for spaces of any dimensions) assigns point groups (and space groups derived from them) to the same crystal system, if these groups can be combined with the same Bravais lattices. For example, the mm2 and 222 groups both belong to the rhombic system, since for each of them there are space groups with all types of rhombic lattices (Pmm2, Cmm2, Imm2, Fmm2 and P222, C222, I222, F222), at the same time groups 32 and 6 does not belong to the same crystal system, since for group 32, primitive and double-centered hexagonal cells (groups P321 and R32) are permissible, while group 6 is combined only with a primitive hexagonal cell (there is a group P 6 , but R 6 does not exist).
Lattice system
Describes the types of crystal lattices. In short: gratings are of the same type, if their point symmetry groups (when treating gratings as geometric objects) are the same. Such point groups, which describe the symmetry of the lattice, are called holohedry . [7]
In total, there are seven lattice systems, which, similar to the previous classifications (syngony and crystal system), are divided into three categories.
- Lower category (all broadcasts are not equal to each other)
- Triclinic : ,
- Monoclinic : ,
- Rhombic : ,
- Middle category
- Tetragonal : ,
- Hexagonal : ,
- Rhombohedral : ,
- Highest category (all broadcasts are equal)
- Cubic : ,
Do not confuse the rhombohedral lattice system with the trigonal crystal system. Crystals of the rhombohedral lattice system always belong to the trigonal crystal system, but trigonal crystals can belong to both rhombohedral and hexagonal lattice systems. For example, the groups R 3 and P321 (both of the trigonal crystal system) belong to different lattice systems (rhombohedral and hexagonal, respectively).
General definition applicable to spaces of any dimensions - Lattices are of the same type if they are combined with the same point groups. For example, all rhombic lattices (rhombic P, rhombic C, rhombic I, and rhombic F) are of the same type, since they are combined with the point groups 222, mm2 and mmm, forming space groups P222, Pmm2, Pmmm; C222, Cmm2, Cmmm; I222, Imm2, Immm; F222, Fmm2, Fmmm. At the same time, the cells of the hexagonal system (primitive P and double-centered R) correspond to different lattice systems: both are combined with point groups of the trigonal crystal system, but only the primitive cell is combined with the groups of the hexagonal system (there are groups P6, P 6 , P6 / m, P622, P6mm, P 6 m2, P6 / mmm, but there are no groups R6, R 6 , R6 / m, R622, R6mm, R 6 m2, R6 / mmm).
The relationship between the syngony, the crystal system, and the lattice system in three-dimensional space is given in the following table:
Syngony | Crystal system | Point groups | The number of space groups | Grate Bravais [8] | Grid system | Golohedria |
---|---|---|---|---|---|---|
Triclinic | 1, 1 | 2 | aP | Triclinic | one | |
Monoclinic | 2, m, 2 / m | 13 | mP, mS | Monoclinic | 2 / m | |
Rhombic | 222, mm2, mmm | 59 | oP, oS, oI, oF | Rhombic | mmm | |
Tetragonal | 4, 4 , 422, 4mm, 4 2m, 4 / m, 4 / mmm | 68 | tP, tI | Tetragonal | 4 / mmm | |
Hexagonal | Trigonal | 3, 3 , 32, 3m, 3 m | 7 | hR | Rhombohedral | 3 m |
18 | hP | Hexagonal | 6 / mmm | |||
Hexagonal | 6, 6 , 622, 6mm, 6 m2, 6 / m, 6 / mmm | 27 | ||||
Cubic | 23, m 3 , 4 3m, 432, m 3 m | 36 | cP, cI, cF | Cubic | m 3 m | |
Total: 6 | 7 | 32 | 230 | 14 | 7 |
Point groups overview
Crystal system | point group / symmetry class | Schoenflies symbol | International symbol | Shubnikov symbol | Type of |
---|---|---|---|---|---|
triclinic | monohedral | C 1 | enantiomorphic polar | ||
pinacoid | C i | centrosymmetric | |||
monoclinic | dihedral axial | C 2 | enantiomorphic polar | ||
non-axial dihedral (domatic) | C s | polar | |||
prismatic | C 2h | centrosymmetric | |||
Rhombic | rhombo-tetrahedral | D 2 | enantiomorphic | ||
rhombo- pyramidal | C 2v | polar | |||
rhombic bipyramidal | D 2h | centrosymmetric | |||
Tetragonal | tetragonal pyramidal | C 4 | enantiomorphic polar | ||
tetragonal tetrahedral | S 4 | ||||
tetragonal-dipyramidal | C 4h | centrosymmetric | |||
tetragonal trapezoidal | D 4 | enantiomorphic | |||
dietragonal-pyramidal | C 4v | polar | |||
tetragonal-scalohedral | D 2d | or | |||
dititragonal-dipyramidal | D 4h | centrosymmetric | |||
Trigonal | trigonal pyramidal | C 3 | enantiomorphic polar | ||
rhombohedral | S 6 (C 3i ) | centrosymmetric | |||
trigonal trapezoidal | D 3 | or or | enantiomorphic | ||
ditrigonal pyramidal | C 3v | or or | polar | ||
di-trigonal scaleno | D 3d | or or | centrosymmetric | ||
Hexagonal | hexagonal-pyramidal | C 6 | enantiomorphic polar | ||
trigonal dipyramidal | C 3h | ||||
hexagonal dipyramidal | C 6h | centrosymmetric | |||
hexagonal trapezoidal | D 6 | enantiomorphic | |||
dihexagonal-pyramidal | C 6v | polar | |||
ditrigonal dipyramidal | D 3h | or | |||
dihexagonal-dipyramidal | D 6h | centrosymmetric | |||
Cubic | tritetrahedral | T | {\ displaystyle 23 \} | enantiomorphic | |
didodecahedral | T h | centrosymmetric | |||
hexa tetrahedral | T d | ||||
trioctahedral | O | enantiomorphic | |||
hexactahedral | O h | centrosymmetric |
Lattice classification
Syngony | Type of centering cell Brava | ||||
---|---|---|---|---|---|
primitive | base centered | volume centered | facet centered | twice volume centered | |
Triclinic ( parallelepiped ) | |||||
Monoclinic ( prism with parallelogram at the base) | |||||
Rhombic ( rectangular parallelepiped ) | |||||
Tetragonal ( rectangular box with a square at the base) | |||||
Hexagonal ( prism with the base of a regular centered hexagon) | |||||
Cubic ( cube ) |
History
The first geometric classification of crystals was given independently by Christian Weiss and Friedrich Moos at the beginning of the 19th century. Both scientists classified crystals according to the symmetry of their external shape (faceting). In this case, Weiss actually introduces the concept of a crystallographic axis (axis of symmetry). According to Weiss, “The axis is the line that dominates the entire figure of the crystal, since around it all the parts are arranged in a similar way and relative to it they correspond to each other mutually” [12] . In his work “A visual representation of the natural divisions of crystallization systems,” Weiss classified crystals according to the presence of axes into four large sections of crystalline forms, “crystallization systems” corresponding to the modern concept of syngony [13] . In brackets are the modern names.
- Section 1 - “correct”, “spherohedral”, “equiaxial”, “equal” (cubic) system: three dimensions are the same, forming right angles between themselves.
- subsection homospheric system (crystals of symmetry m 3 m)
- subsection of hemisphere-hedron system (crystals of symmetry 432, 43m and m 3 )
- Section 2 - “four-membered” (tetragonal) system: the axes form between themselves right angles, the two axes are equal to each other and not the third.
- Section 3 - “two-member” system: all three axes are unequal and form right angles between themselves.
- subdivision "two-and-two-member" (rhombic) system
- subdivision “two-and-term” (monoclinic) system
- subsection "one-and-one-term" (triclinic) system
- Section 4 - one unequal axis is perpendicular to three equal axes forming angles of 120 ° between each other.
- subsection "six-membered" (hexagonal) system:
- subdivision "three-and-trichinas" or "rhombohedral" (trigonal) system:
For the monoclinic and triclinic syngony, Weiss used a rectangular coordinate system (modern crystallographic coordinate systems for these syngonies are oblique).
Around the same time, Friedrich Moos developed the concept of crystalline systems [14] . Each system is characterized by the simplest, “basic form” of faces, from which all other forms of this system can be derived. Thus, Moos obtained the following four systems:
- 1. Rhombohedral system (hexagonal syngony). The main form is a rhombohedron.
- 2. Pyramidal system (tetragonal syngony). The main form is the tetragonal bipyramid.
- 3. The dessular system (cubic syngony). The main forms are a cube and an octahedron.
- 4. Prismatic system (rhombic syngony). The main form is the rhombic bipyramid.
- Hemiprimatic subsystem (monoclinic syngony)
- Tetaropismatic subsystem (triclinic syngony)
In both classifications, Weiss and Moos distinguishes only four systems, although all six syngonies are listed, they consider only the monoclinic and triclinic syngonies as rhombic subsystems. According to his own statement, Mohs developed this concept in the years 1812-14, which was the subject of a dispute with Weiss about the priority of the discovery of crystalline systems. Unlike Weiss, Mohs pointed out the need for an oblique axis system for monoclinic and triclinic crystals.
His pupil Karl Friedrich Naumann finally developed and introduced kosoo-angled systems into crystallography. Nauman based the classification of crystallographic axes and angles between them, thus highlighting for the first time all six syngonies [15] [16] . Interestingly, as early as 1830, Nauman uses the names of syngonies that are identical or close to modern (the names tetragonal , hexagonal and rhombic were originally proposed by Breitgaupt).
- 1. Tesseral (from tessera - cube) - all three angles between the coordinate axes are straight, all three axes are equal.
- 2. Tetragonal - all three corners are right, two axes are equal.
- 3. Hexagonal - the only four-axis system: one unequal axis is perpendicular to three equal axes forming angles of 60 ° between themselves.
- 4. Rhombic - all three corners are straight, all axes are unequal.
- 5. Monoclinohedral - two right angles and one oblique.
- 6. Diclinhedral - two oblique angles and one straight.
- 7. Triclinohedral — all three angles are oblique.
Since at that time the theory of symmetry was only developing, an unusual diclinohedral (diclin) system appeared in the list of systems. Such a crystal system is in principle impossible in three-dimensional space, since the presence of the axis of symmetry always guarantees the presence of translations perpendicular to the axis, chosen for the coordinate axes. Diklinnaya system existed in crystallography for about half a century (although already in 1856 Dufrenois showed that this is only a special case of the triclinic system). In 1880, Dana in her famous book “The System of Mineralogy” [17] mentions “the so-called diklinnuyu system”, but at the same time notes that not a single natural or artificial crystal belonging to this system is known, and moreover, it has been mathematically proven that there are only six crystalline systems. Until the end of his life, Nauman himself believed in a diklinny syngony, and in the ninth edition of The Foundations of Mineralogy [18] , posthumously published in 1874, this syngony is still on the list, although Nauman observes that this system is found only in a few artificial salts, and further does not consider it.
The names of crystallographic syngonies from the authors of the XIX century
Author | Cubic | Tetragonal | Hexagonal | Rhombic | Monoclinic | Triclinic |
---|---|---|---|---|---|---|
Weiss | Regular, Spherical, Ball, Spheronomic, Equal-Axis, Equal | Quadruple, Dual and Uniaxial | Six-membered, three-and-uniaxial | Two-and-two-threaded, Single-and-uniaxial | Double-and-term | One and one term |
Moos | Teselyarnaya, Tesselyarnaya | Pyramidal | Rhombohedral | Prismatic Orthotype | Hemiprimatic, hemiortypical | Tetaroprimatic, anorotype |
Breitgaupt | Tetragonal | Hexagonal | Rhombic | Hemirmbic | Tetartorhombic | |
Nauman | Tesseral | Tetragonal | Hexagonal | Rhombic, Anisometric | Monoclinhedral, Klinorombic | Triclinohedral, triclinometric |
Gausman | Isometric | Monodimetric | Monotrimetric | Trimetric, Orthorhombic | Klinorombicheskaya, Orthorhombic | Klinoromboidicheskaya |
Miller 1839 | Octahedral | Pyramidal | Rhombohedral | Prismatic | Oblique prismatic | Double-oblique-prismatic |
Gadolin | Correct | Square | Hexagonal | Rhombic | Monoclinohedral | Triclinohedral |
Other authors | Tetrahedral (Bedan), Cubic (Dufrenois) | Dimetric | Twofold (Quenstedt) | Monoclinometric (Frankenheim), Augitovaya (Geidinger) | Triclinic (Frankenheim), Anortic (Geidinger) |
For the first time, the division into seven crystallographic systems was given in 1850 in the work of Auguste Brava “Memoir on systems of points correctly distributed on a plane or in space” [19] . In fact, this is the first division based on symmetry elements, and not on coordinate systems. Therefore, all previous classifications correspond to today's definition of a syngony, while the Bravais classification is a classification according to crystalline systems (strictly speaking, lattice systems).
Depending on their symmetry, Bravais divides lattices into 7 systems (classes of aggregates).
- 1. Three-Quad (Cubic System)
- 2. Six (hexagonal system)
- 3. Quadruple (tetragonal system)
- 4. Triple (rhombohedral system)
- 5. Triple (rhombic system)
- 6. Dual (monoclinic system)
- 7. Asymmetric (triclinic system)
At the same time, Bravais himself notes that Haüy also divided the lattices of the hexagonal system (according to Naumann’s classification) “into crystals generated by a regular hexagonal prism, and crystals generated by a rombohedron nucleus”.
Classification of groups in multidimensional spaces
In the second half of the 20th century, crystallographic groups were studied and classified in four-dimensional, five-dimensional and six-dimensional spaces. With increasing dimension, the number of groups and classes increases significantly [20] . The number of enantiomorphic pairs is indicated in parentheses.
Space dimension: | one | 2 | 3 | four | five | 6 |
---|---|---|---|---|---|---|
Number of syngonies | one | four | 6 | 23 (+6) | 32 | 91 |
Number of lattice systems | one | four | 7 | 33 (+7) | 57 | 220 |
The number of crystalline systems | one | four | 7 | 33 (+7) | 59 | 251 |
The number of lattices Brava | one | five | 14 | 64 (+10) | 189 | 841 |
The number of point groups | 2 | ten | 32 | 227 (+44) | 955 | 7103 |
The number of space groups | 2 | 17 | 219 (+11) | 4783 (+111) | 222018 (+79) | 28927915 (+?) [21] |
In four-dimensional space, the unit cell is defined by four sides ( ) and six corners between them ( ). The following relationships between them define 23 syngonies:
- Hexaclinic:
- Triclinic:
- Diklinnaya:
- Monoclinic:
- Orthogonal:
- Tetragonal monoclinic:
- Hexagonal monoclinic:
- Dietragonal diklinnaya:
- Ditrigonal Diklinnaya:
- Tetragonal orthogonal:
- Hexagonal orthogonal:
- Dietragonal monoclinic:
- Ditrigonal Monoclinic:
- Dietragonal orthogonal:
- Hexagonal tetragonal:
- Dihexagonal orthogonal:
- Cubic orthogonal:
- Octagonal:
- Decagonal:
- Dodecagonal:
- Di-isohexagonal orthogonal:
- Ikosagonalnaya:
- Hypercubic:
The relationship between the syngony, the crystal system, and the lattice system in four-dimensional space is given in the following table [22] [23] . Asterisks indicate enantiomorphic systems. The number of enantiomorphic groups (or lattices) is indicated in parentheses.
room syngony | Syngony | Crystal system | room systems | The number of point groups | The number of space groups | The number of lattices Brava | Grid system |
---|---|---|---|---|---|---|---|
I | Hexaclinic | one | 2 | 2 | one | Hexaclinic P | |
II | Triclinic | 2 | 3 | 13 | 2 | Triclinic P, S | |
III | Diklinnaya | 3 | 2 | 12 | 3 | Diklinnaya P, S, D | |
IV | Monoclinic | four | four | 207 | 6 | Monoclinic P, S, S, I, D, F | |
V | Orthogonal | Axisless orthogonal | five | 2 | 2 | one | Orthogonal KU |
112 | eight | Orthogonal P, S, I, Z, D, F, G, U | |||||
Axial orthogonal | 6 | 3 | 887 | ||||
VI | Tetragonal monoclinic | 7 | 7 | 88 | 2 | Tetragonal monoclinic P, I | |
VII | Hexagonal Monoclinic | Trigonal Monoclinic | eight | five | 9 | one | Hexagonal Monoclinic R |
15 | one | Hexagonal Monoclinic P | |||||
Hexagonal Monoclinic | 9 | 7 | 25 | ||||
Viii | Dietragonal diklinnaya * | ten | 1 (+1) | 1 (+1) | 1 (+1) | Dietragonal diklinnaya P * | |
Ix | Ditrigonal Diklinnaya * | eleven | 2 (+2) | 2 (+2) | 1 (+1) | Ditrigonal Diclinic P * | |
X | Tetragonal orthogonal | Inversion tetragonal orthogonal | 12 | five | 7 | one | Tetragonal orthogonal KG |
351 | five | Tetragonal orthogonal P, S, I, Z, G | |||||
Rotary tetragonal orthogonal | 13 | ten | 1312 | ||||
Xi | Hexagonal orthogonal | Trigonal orthogonal | 14 | ten | 81 | 2 | Hexagonal orthogonal R, RS |
150 | 2 | Hexagonal orthogonal P, S | |||||
Hexagonal orthogonal | 15 | 12 | 240 | ||||
Xii | Dietragonal monoclinic * | sixteen | 1 (+1) | 6 (+6) | 3 (+3) | Dietragonal monoclinic P *, S *, D * | |
XIII | Ditrigonal monoclinic * | 17 | 2 (+2) | 5 (+5) | 2 (+2) | Ditrigonal monoclinic P *, RR * | |
Xiv | Dietragonal orthogonal | Cryptodietragonal orthogonal | 18 | five | ten | one | Dietragonal orthogonal D |
165 (+2) | 2 | Dietragonal orthogonal P, Z | |||||
Dietragonal orthogonal | nineteen | 6 | 127 | ||||
Xv | Hexagonal tetragonal | 20 | 22 | 108 | one | Hexagonal Tetragonal P | |
XVI | Dihexagonal orthogonal | Crypto-ditrigonal orthogonal * | 21 | 4 (+4) | 5 (+5) | 1 (+1) | Dihexagonal orthogonal G * |
5 (+5) | one | Dihexagonal orthogonal P | |||||
Dihexagonal orthogonal | 23 | eleven | 20 | ||||
Ditrigonal Orthogonal | 22 | eleven | 41 | ||||
sixteen | one | Dihexagonal orthogonal RR | |||||
XVII | Cubic orthogonal | Simple cubic orthogonal | 24 | five | 9 | one | Cubic orthogonal KU |
96 | five | Cubic orthogonal P, I, Z, F, U | |||||
Complex cubic orthogonal | 25 | eleven | 366 | ||||
XVIII | Octagon * | 26 | 2 (+2) | 3 (+3) | 1 (+1) | Octagonal P * | |
XIX | Decagonal | 27 | four | five | one | Decagonal P | |
Xx | Dodecagonal * | 28 | 2 (+2) | 2 (+2) | 1 (+1) | Dodecagonal P * | |
XXI | Di-isohexagonal orthogonal | Simple di-isohexagonal orthogonal | 29 | 9 (+2) | 19 (+5) | one | Di-isohexagonal orthogonal RR |
19 (+3) | one | Di-isohexagonal orthogonal P | |||||
Difficult di-isohexagonal orthogonal | thirty | 13 (+8) | 15 (+9) | ||||
XXII | Ikosagonalnaya | 31 | 7 | 20 | 2 | Icosal P, SN | |
XXIII | Hypercubic | Octagonal hypercubic | 32 | 21 (+8) | 73 (+15) | one | Hypercubic P |
107 (+28) | one | Hypercubic Z | |||||
Dodecagonal hypercubic | 33 | 16 (+12) | 25 (+20) | ||||
Total: | 23 (+6) | 33 (+7) | 227 (+44) | 4783 (+111) | 64 (+10) | 33 (+7) |
See also
- Crystal structure
- Dot group
- Crystallographic point symmetry group
- Pearson symbol
Notes
- ↑ Crystal family - Online Dictionary of Crystallography . Archived March 21, 2013.
- ↑ Crystal system - Online Dictionary of Crystallography . Archived March 21, 2013.
- ↑ Lattice system - Online Dictionary of Crystallography . Archived April 29, 2013.
- ↑ Shubnikov A. V., Boky G. B., Flint E. E., Basics of Crystallography, Publishing House of the Academy of Sciences of the USSR, 1940
- ↑ "Yu. K. Egorov-Tismenko, G. P. Litvinskaya, Theory of Crystal Symmetry, GEOS, 2000. Chapter III. Coordinate Systems, Categories, Syngonies." Archived April 29, 2013.
- ↑ Fedorov E.S., Course of crystallography. Ed. 3rd, 1901 online
- ↑ Holohedry - Online Dictionary of Crystallography . Archived March 21, 2013.
- ↑ de Wolff et al., Nomenclature for crystal families, Bravais-lattice types and arithmetic classes, Acta Cryst. (1985). A41,278-280. online
- ↑ Weinstein B.K. Modern crystallography. Volume 1. Symmetry of crystals, methods of structural crystallography. Science, Moscow, 1979.
- ↑ Sirotin Yu.I., Shaskolskaya MP Basics of crystal physics. Science, Moscow, 1979.
- ↑ Flint E.E. A practical guide to geometric crystallography. 3rd ed, perarab. and add., Gosgeoltehizdat, Moscow, 1956.
- ↑ CS Weiss De indagando formarum crystallinarum charactere geometrico principali dissertatio. Lipsiae [Leipzig] 1809
- ↑ CS Weiss : Ueber die natürlichen Abtheilungen der Crystallisations Systeme. Abhandl. k. Akad. Wiss., Berlin 1814–1815, S. 290–336.
- ↑ Friedrich Mohs : Grund-Riß der Mineralogie. Erster Theil. Terminologie, Systematik, Nomenklatur, Charakteristik. Dresden 1822
- ↑ Carl Friedrich Naumann , Lehrbuch der Mineralogie Mineralogie, 1828 online
- ↑ Carl Friedrich Naumann , Lehrbuch der reinen und angewandten Krystallographie, 1830 online
- ↑ Edward Salisbury Dana, James Dwight Dana, A text-book of mineralogy, 1880 online
- ↑ Carl Friedrich Naumann, Elemente der mineralogie, 1874 online
- ↑ Bravais, A. (1850) Mémoire sur les systèmes formés par les points distribute régulièrement sur planes ou dans l'espace. Journal de L'Ecole Polytechnique.
- ↑ B. Souvignier: "Enantiomorphism of crystallographic groups". Acta Crystallographica Section A, vol.59, pp.210-220, 2003.
- ↑ The CARAT Homepage Undefeated . The circulation date was May 5, 2015. Part of the calculations in Souvignier (2003) for the six-dimensional space was based on the erroneous version of the CARAT program.
- J EJW Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.
- ↑ H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978.
Links
- Glossary of terms on the website of the International Union of Crystallographers