Crystal system
Crystal systems provide a symmetry -related classification scheme for crystalline solids. In crystallography, the crystals to be described with the help of the crystal system are classified in three dimensions. The crystal systems date back to Christian Samuel Weiss ( 1780-1856 ). Today, seven crystal systems are distinguished: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic.
Crystal systems are mainly used in mineralogy, solid state chemistry and solid state physics.
Definition
The highly symmetric point group ( crystal class ) of a crystal system is referred to as so-called holohedry ( "fully " or " full- form " ) and the corresponding crystal body as Holoeder ( " Vollflächner "). Its form, according to the maximum number of crystal faces of all kinds and in its crystal system symmetry elements are present.
If the point group of a crystal is the same requirements for the grid, such as a holohedry, then a part of the crystal structure of the respective crystal system. In general, the symmetry of the crystal structure is less than the symmetry of the grating and it may even occur that the crystal belongs to a lower symmetrical crystal system, as its gate. For example, a crystal with point group " 4" forced a grid that corresponds at least to the point group 4/mmm, and therefore he is assigned to the tetragonal crystal system. This assignment would also be true if the crystal had a cubic lattice.
Assignment of the crystallographic point groups to the crystal systems
Coordinate systems
It makes sense not a Cartesian coordinate system, but is matched to the crystal system coordinate system is used in the description of crystals and crystal structures mostly. Thus, for example, all the rotation matrices of the symmetry operations of integral matrices. These coordinate systems meet certain conditions:
- Triclinic crystal system: Uses the three smallest possible primitive basis vectors. There are no conditions with respect to the angles and lengths of the basis vectors.
- Monoclinic crystal system: A basis vector (usually the y -axis) is placed in the two-fold axis of rotation. This results in two 90 -degree angle, but no limitation with respect to the axis lengths.
- Orthorhombic crystal system: The basis vectors are placed in the two -fold rotation axis. This results in three 90 ° angles can be calculated (hence ortho), but no limitation with respect to the axis lengths.
- Hexagonal crystal system: a base vector ( normally the z axis ) is placed in the 6- fold rotation axis, the other two in the perpendicular two -fold rotation axis. The third axis gives two equally long axes in a plane with 120 ° angle, perpendicular to it.
- Tetragonal crystal system: a base vector ( normally the z axis ) is placed in the four - fold rotation axis, the other two in the perpendicular two -fold rotation axis. This gives two equal axes and three 90 -degree angle.
- Trigonal crystal system: For this crystal system coordinates two statements are used: either three equal basis vectors and three equal angles ( rhombohedral coordinate system), or a statement as in the hexagonal crystal system.
- Cubic crystal system: The basis vectors are placed in the 4 -fold axes. This gives three equal axes and three 90 ° angle
The given conditions are necessary, but not sufficient: it is possible that the axes of a triclinic crystal are equal in length and include 90 °. It does not follow that the crystal is cubic.
Please note that you will get no more primitive basis by these symmetry- related coordinates formation under certain circumstances. It is therefore necessary, in addition to the crystal system or to provide the centering, so that the 14 Bravais lattices are obtained.
Other classifications
The above classification corresponds to that of the International Tables for Crystallography. In the literature there are still others in the American and Russian the trigonal and hexagonal crystal system are combined into one. In the French literature, and partly in German, there is an eighth with the rhombohedral crystal system. This the trigonal space groups are associated with rhombohedral centering. The classification of the International Tables for Crystallography, however, is most consistent and, therefore, is becoming more prevalent through.
History
Crystal systems were initially defined as axis systems. End of the 18th century Haiiy had his theory of the structure of the crystals of the smallest units ( " Molecules constituantes " ) published. C. S. Weiss translated textbooks Haüys. Already in the first edition of his translation he added a supplement that was titled Dynamic Views for crystallization. His view that the external shape of the crystals should be understood as an expression of a system of internal forces, led to the idea to make the system of forces on an analysis of the arrangement particularly striking directions of the crystals, the axes can be described mathematically. He defined an axis as follows:
" Axis linea vero est omnis figurae dominatrix, circa quam sunt omnia aequiabiliter disposita. "
" An axis but is in fact a dominant straight the whole figure around which everything is evenly distributed. "
In this " uniform distribution " around the axis already suggests the idea of rotational symmetry, but which was specifically formulated later by Franks home and Hessel.
White led the axis systems in crystallography. First, he distinguished by the arrangement of the axes four large " departments " of the crystal forms, which he later expanded to three divisions, so that he could a total of six associate " Crystal tralisation systems," the crystal forms. The concept of crystal systems was born. With the help of axes White was the location of all crystal faces by numbers (indices) in the form of first characterize [ ma: pc: nb ]. The numbers m, n, p - the " white 's coefficient " - the axis portions where the respective surface intersects the axis. He received so following systems (in brackets are the modern names of the corresponding crystal systems specified):
Weiss claimed that the location of each surface and each direction can be described by the rectangular crystal systems proposed by him. He tried to describe oblique ( monoclinic and triclinic ) crystals in a rectangular system. Despite the difficulties that arose from the increasing accuracy of the measurement of crystal faces, White held his life firmly in the " Orthogonalitätsdogma " of the crystal axes.
Friedrich Mohs developed about the same time, but independently of white, a concept of crystal systems. According to his own statement had Mohs ( rhombohedral, pyramidal, prismatic and tessular ) developed a classification into four systems already from 1812 to 1814. The concept was oblique axes in principle to, but made Mohs only hint in this direction. Only Mohs ' pupil Carl Friedrich Naumann and Franconia home and Justus Günther Grassmann established the oblique axis systems.
The nomenclature was initially anything but uniform. Traugott Lebrecht Hasse was in 1848 a historical overview of the crystal systems in orthogonal Description:
The hexagonal crystal has long been treated as a family system. William Hallowes Miller six different systems, which he defined as follows:
The rhombohedral system used Miller while also describing hexagonal crystals (which is easily possible ). Up here the crystal systems capable of crystal surfaces in the room was used exclusively for the description of crystal forms, ie. Only with the establishment of the concept of translation lattices through Franconia home and later Auguste Bravais it was useful to distinguish between a hexagonal and rhombohedral lattice one.
1866 different Bravais seven classes of symmetry compounds ( " assemblages symétriques " ) - no longer on the basis of axial ratios, but according to the maximum combined rotary axes. This classification corresponds exactly to the seven modern crystal systems ( shown in parentheses):
Nevertheless, it remained common until the 20th century, assembling them in a trigonal and the hexagonal crystal system. All trigonal and hexagonal crystals can be described with hexagonal and also with rhombohedral axes. Friedrich Klockmann delivered in the third edition of his textbook on mineralogy (1903 ) evidence that "you can get by with 6 Axenkreuzen or 6 Krystallsystemen " (p. vii ). He gave the following definition of a crystal system:
"Those symmetry classes or crystalline forms, which nevertheless can be related to analog Axenkreuze regardless of their different degree of symmetry are called the same Krystallsystem belonging, or in short form a Krystallsystem. There are therefore six Krystallsysteme. "
In the following derivation, although he distinguished seven axis systems, including the rhombohedral and hexagonal, but then stated:
" Since the rhombohedral system displays peculiar geometric relations with the hexagonal system and all forms of the same on a hexagonal Axenkreuz and vice versa can be obtained, so it has become common to combine both into a single Krystallsystem and mainly to the hexagonal system, reducing the number the crystal systems is reduced to 6. "
Only in the later 20th century, the concepts were strictly separated from each other, so that today there is a distinction between crystal system, crystal family, and lattice system, which differ ultimately only by the division of the trigonal / hexagonal systems.