Crystal system
Crystal systems offer a symmetry -related classification scheme for crystalline solids . In crystallography , crystals are classified three-dimensionally using the crystal system.
Seven crystal systems can be distinguished, each of which can be related to the same axis cross that intersects the crystal body in the center: triclinic , monoclinic , orthorhombic , tetragonal , trigonal , hexagonal and cubic . The decisive factor for assigning a crystalline substance to one of the systems is the symmetry of the substance, which in turn gives rise to certain requirements for the length of the axes and the angles at which they intersect. The idea of crystal systems goes back to Christian Samuel Weiss (1780–1856).
In contrast to this, amorphous material has no ordered structures and thus no crystal system, which means that its atoms or molecules form an irregular pattern.
Crystal systems are mainly used in mineralogy , solid-state chemistry and solid-state physics .
definition
The most symmetrical point group (crystal class) of a crystal system is called holoedry ("full surface" or "full form") and the corresponding crystal body is called holohedron ("full surface"). Its shape shows the highest possible number of crystal faces and all symmetry elements possible in its crystal system are present.
If the point group of a crystal makes the same demands on the lattice as a holoedry, then the crystal structure belongs to the relevant crystal system. As a rule, the symmetry of the crystal structure is lower than the symmetry of the lattice, and it can even happen that the crystal belongs to a lower symmetrical crystal system than its lattice. For example, a crystal with the point group “4” is forced to have a lattice that corresponds at least to the point group 4 / mmm , and therefore it is assigned to the tetragonal crystal system. This assignment would also apply if the crystal had a cubic lattice.
Assignment of the crystallographic point groups to the crystal systems
Crystal system | Holoedry | Point groups ( Hermann Mauguin short symbols ) |
---|---|---|
triclinic | 1 | 1 , 1 |
monoclinic | 2 / m | 2 / m , m , 2 |
orthorhombic | mmm | mmm , mm 2, 222 |
tetragonal | 4 / mmm | 4 / mmm , 4 2 m , 4 mm , 422, 4 / m , 4 , 4 |
trigonal | 3 m | 3 m , 3 m , 32, 3 , 3 |
hexagonal | 6 / mmm | 6 / mmm , 6 2 m , 6 mm , 622, 6 / m , 6 , 6 |
cubic | m 3 m | m 3 m , 4 3 m , 432, m 3 , 23 |
Coordinate systems
When describing crystals and crystal structures, it makes sense not to use a Cartesian coordinate system , but rather a coordinate system that is adapted to the crystal system. In this way, for example, all rotation matrices of the symmetry operations become integral matrices. These coordinate systems meet certain conditions:
- Triclinic crystal system : The three smallest possible primitive basis vectors are used. There are no constraints on the angles and lengths of the basis vectors.
- Monoclinic crystal system : A basis vector (usually the y-axis) is placed in the twofold axis of rotation. This results in two 90 ° angles, but no restriction on the axis lengths.
- Orthorhombic crystal system : The base vectors are placed in the 2-fold axes of rotation. This results in three 90 ° angles (therefore ortho ), but no restriction on the axis lengths.
- Hexagonal crystal system : A basic vector (usually the z-axis) is placed in the 6-fold axis of rotation, the other two in the 2-fold axis of rotation perpendicular to it. Two axes of equal length are obtained in a plane with an angle of 120 °, the third axis perpendicular to it.
- Tetragonal crystal system : A basic vector (usually the z-axis) is placed in the 4-fold axis of rotation, the other two in the 2-fold rotation axes perpendicular to it. You get two axes of equal length and three 90 ° angles.
- Trigonal crystal system : Two coordinate arrangements are common for this crystal system: either three base vectors of the same length and three equal angles (rhombohedral coordinate system) or one arrangement as in the hexagonal crystal system.
- Cubic crystal system : The basic vectors are placed in the 4-fold axes. You get three axes of equal length and three 90 ° angles
The given conditions are necessary, but not sufficient: It is possible that the axes of a triclinic crystal are of equal length and each include 90 °. It does not follow from this that the crystal is cubic.
It should be noted that this symmetry-related co-ordinate listing may no longer give a primitive basis. It is therefore necessary to indicate the centering in addition to the crystal system, whereby the 14 Bravais lattices are obtained.
Other classifications
The classification given above corresponds to that from the International Tables for Crystallography . There are others in the literature: In the American and the Russian, the trigonal and the hexagonal crystal system are combined into one. In French literature, and partly also in German, there is an eighth crystal system with the rhombohedral. The trigonal space groups with rhombohedral centering are assigned to this. The classification of the International Tables for Crystallography is the most consistent and is therefore becoming more and more popular.
history
Crystal systems were initially defined as axis systems. At the end of the 18th century, Haüy published his theory of the structure of crystals from the smallest building units ("molécules constituantes"). CS Weiss translated Haüy's textbooks. In the first edition of his translation, he added a supplement entitled Dynamic Views on Crystallization . His view that the outer shape of a system of internal forces of the crystals should be understood as the expression, led to the idea of the system of forces about an analysis of the arrangement particularly conspicuous directions of the crystals, the axes to make described mathematically. He defined an axis as follows:
"Axis vero linea est omnis figurae dominatrix, circa quam omnia aequiabiliter sunt disposita."
"But an axis is actually a straight line that dominates the whole figure, around which everything is evenly distributed."
This “even distribution” around the axis already suggests the idea of rotational symmetry , which was only later formulated in concrete terms by Frankenheim and Hessel .
Weiss introduced the axis systems into crystallography. First, he differentiated four large “departments” of the crystal forms according to the arrangement of the axes, which he later expanded to include three sub-departments so that he could assign the crystal forms to a total of six “crystallization systems”. The concept of crystal systems was born. With the help of the axes, Weiss was able to characterize the position of all crystal faces for the first time using numbers (indices) in the form [ma: nb: pc]. The numbers m, n, p - the "Weiss coefficients" - are the intercepts at which the respective surface intersects the axes. He received the following systems (the modern names of the corresponding crystal systems are given in brackets):
- Division: the "regular" (cubic) system: a = b = c, α = β = γ = 90 °
- Division: the "four-part" (tetragonal) system: a = b ≠ c, α = β = γ = 90 °
- Division: the "two-part" system: a, b, c different in pairs, α = β = γ = 90 °
- Subdivision: the "two and two-part" (orthorhombic) system
- Subdivision: the "two and one-part" (monoclinic) system
- Subdivision: the “one and one-part” (trikline) system
- Division: the "three or six-membered" (tri- / hexagonal) system: three equal axes intersect at 60 ° and the fourth unequal axis at 90 °
Weiss claimed that the right-angled crystal systems proposed by him could describe the position of any surface and any direction. He also tried to describe obliquely angled (monoclinic and triclinic) crystals in a right-angled system. Despite the difficulties that resulted from the increasing accuracy of the measurement of crystal surfaces, Weiss stuck to the "orthogonality dogma" of the crystal axes throughout his life.
Friedrich Mohs developed a concept of crystal systems around the same time, but independently of Weiss. According to his own statement, Mohs had developed a division into four systems ( rhombohedral, pyramidal, prismatic and tessular ) as early as 1812–1814. The concept allowed skewed axes in principle, but Mohs only made a hint in this direction. It was only Mohs' pupil Carl Friedrich Naumann as well as Frankenheim and Justus Günther Graßmann who established the oblique axis systems.
Initially, the nomenclature was anything but uniform. In 1848, Traugott Leberecht Hasse gave a historical overview of the crystal systems in an orthogonal description:
Weiss 1815 | Mohs 1822 (1824) | Naumann 1824 (1826, 1830) | |
---|---|---|---|
Tesseral (cubic) | tessular, regular, spheroidal, equal system | tessular (also tessular) system | tesseral or isometric system |
Tetragonal | four-part or two-u. single axis system | pyramidal system | tetragonal or monodimetric system |
Hexagonal | six-part, three- u. tripartite system | rhombohedral system | hexagonal or monometric system |
Rhombic | two- and two-part, also two- and one-part, one- and two-part and one- and one-part system | prismatic system | the rhombic or clinorhombic system |
The hexagonal crystal family has long been treated as a system. William Hallowes Miller distinguished six systems, which he defined as follows:
Miller's definition | Designation (1839) | Designation (1863) |
---|---|---|
Axes at right angles, all parameters a, b, c the same | Octahedral system | Cubic system |
Axes perpendicular, two parameters a, b equal | Pyramidal system | Pyramidal system |
the axes form the same angle, all parameters are the same | Rhombohedral system | Rhombohedral system |
Axes at right angles | Prismatic System | Prismatic System |
one axis is perpendicular to the other two | Oblique Prismatic System | Oblique system |
The form {hkl} has two parallel surfaces (hkl), ( h k l ) | Doubly-Oblique Prismatic System | Anorthic system |
Miller also used the rhombohedral system to describe hexagonal crystals (which is easily possible). Up to this point, the crystal systems were used exclusively to describe crystal forms, i.e. the position of crystal faces in space. It was only with the establishment of the concept of translation grids by Frankenheim and later Auguste Bravais that it made sense to distinguish between a hexagonal and a rhombohedral grid.
In 1866 Bravais distinguished between seven classes of symmetry connections ("assemblages symétriques") - no longer on the basis of the axial relationships, but according to the maximum number of combinable axes of rotation. This classification corresponds exactly to the seven modern crystal systems (given in brackets):
- Assemblages terquaternaires : 3 fourfold, 4 threefold, 6 twofold rotary axes (cubic)
- Assemblages sénaires : 1 six- digit , 6 two- digit rotary axes (hexagonal)
- Assemblages quaternaires : 1 fourfold, 4 twofold rotation axes (tetragonal)
- Assemblages ternaires : 1 threefold, 3 twofold rotary axes (trigonal)
- Assemblages terbinaires : 3 twofold axes of rotation (orthorhombic)
- Assemblages binaires : 1 twofold axis of rotation (monoclinic)
- Assemblages asymétriques : no axes of rotation (triclinic)
Nevertheless, it remained common until the 20th century to combine the trigonal and the hexagonal crystal system into one. All trigonal and hexagonal crystals can be described with hexagonal and also with rhombohedral axes. In the 3rd edition of his textbook on mineralogy (1903), Friedrich Klockmann provided evidence “that one can get by with 6 ax crosses or 6 crystal systems” (p. Vii). He gave the following definition of the term crystal system :
“Those classes of symmetry or crystal forms that, regardless of their different degrees of symmetry, can be related to analogous ax crosses are referred to as belonging to the same crystal system, or in short form a crystal system . There are therefore six crystal systems. "
In the following derivation, he distinguished between seven axis systems, including the rhombohedral and the hexagonal, but then explained:
“Since the rhombohedral system shows peculiar geometrical relationships to the hexagonal system and all forms of the same can be related to a hexagonal axis cross and vice versa, it has become customary to unite both to a single crystal system, mostly to the hexagonal system, whereby the number the crystal systems are reduced to 6. "
It was not until the later 20th century that the concepts were more strictly delimited from one another, so that today there is a distinction between crystal system, crystal family and lattice system , which ultimately only differ in the subdivision of the trigonal / hexagonal systems.
Individual evidence
- ↑ JJ Burckhardt: The symmetry of the crystals . Birkhäuser Verlag, Basel 1988, ISBN 3-7643-1918-6 , p. 31-47 .
- ↑ Rudolf Graubner: Lexicon of Geology, Minerals and Rocks . Emil Vollmer Verlag GmbH, Munich 1980, ISBN 3-87876-327-1 .
- ↑ CS Weiss : Coup d'œil dynamique sur la crystallisation. Annal. de Chemie 52 (1804) pp. 308-339
- ↑ CS Weiss : De indagando formarum crystallinarum charactere geometrico principali dissertatio. Lipsiae [Leipzig] 1809
- ^ CS Weiss : About the natural divisions of the crystallization systems. Treatise k. Akad. Wiss., Berlin 1814–1815, pp. 290–336.
- ^ Friedrich Mohs : Basic rift of mineralogy. First part. Terminology, systematics, nomenclature, characteristics. Dresden 1822
- ↑ TL Hasse : Memorandum to commemorate the merits of KS Bergrath's Werner, who died in Dresden on June 30, 1817, and the progress made at the Bergakademie zu Freiberg. Dresden and Leipzig 1848
- ↑ William Hallowes Miller : A treatise on crystallography . Deighton, Cambridge 1839, LCCN 04-030688 , OCLC 4083997 (English, full text in the Google book search).
- ^ William Hallowes Miller : A Tract on Crystallography . Deighton, Cambridge 1863 (English, full text in the Google book search).
- ↑ Auguste Bravais : Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans l'espace. submitted to the Paris Academy on December 11, 1848, published in: J. Ecole Polytech. 19, 1850 pp. 1-18
- ↑ Auguste Bravais : Études Cristallographiques . Gauthier Villars, Paris 1866 (French, full text in the Google book search).
- ^ Friedrich Klockmann : Textbook of Mineralogy . 3. Edition. Ferdinand Enke, Stuttgart 1903.
Web links
- IUCr: Crystal system (official definition)
- Mineral Atlas - Crystal System