Hexagonal crystal system
The hexagonal crystal system is one of the seven crystal systems crystallography. It includes all point groups with a six-fold rotation or axis of rotary inversion. The hexagonal crystal system is closely related to the trigonal crystal system, and together with him the hexagonal crystal family.
- 3.1 Historical Notes
The hexagonal point groups
The hexagonal crystal system includes the point groups. These are all the point groups of the hexagonal crystal family, in which there is no space group rhombohedral centering. The space group of the hexagonal crystal system can all be described by the primitive hexagonal axis system. The hexagonal point groups have no cubic supergroup. Thus, the hexagonal holohedry is the highly symmetric crystallographic point group together with the cubic.
The hexagonal axis system
In the hexagonal crystal family, there are the hexagonal and the trigonal crystal system, as well as the hexagonal and rhombohedral lattice the system. The classification into crystal systems based on the symmetry of the crystals, the division into grid systems refers to the metric of the grid. While in the five other crystal families and crystal systems perform these different perspectives on the same schedule, this is not so in the hexagonal crystal family. In addition, this takes place, the division into grid systems not on the basis of point groups, but the space groups. Since the conditions are relatively complicated, they are described in more detail at this point.
The hexagonal axis system in crystallography
As in all time farmers crystal systems, the axis of rotation is placed with the highest multiplicity in the direction of the c - axis lattice. The plane perpendicular thereto is defined by two equally long axes a1 and a2, which are each at an angle of 120 °. This yields the following metric: and. The unit cell formed by these basis vectors is shown in Figure 1.
The hexagonal axis system in other disciplines
In mineralogy and particularly in the metal buyer, it is customary to have an additional axis a3 in (a1, a2) level to use (see Figure 3). This has the same length as A1 and is at an angle of 120 ° to both A1 and A2 to. Miller indices are added to the index i to the so-called Miller - Bravais indices and have four components: (h, k, i, l). Here, the index i is redundant, since the following applies: i = - (h k). Similarly, in the metallurgy are also directions by four-membered symbols [ uvtw ], the Weber indices shown.
Often in the literature is the hexagonal cell is represented as hexagonal prism (see Figure 2). Since this prism is not a parallelepiped, but it is not is a unit cell. This prism consists of three hexagonal unit cells.
The a1 -a2 - level
Figure 3 shows the a1 -a2 plane of the hexagonal axis system dar. Specifically:
- Points: the grid points of the hexagonal axis system in the a1 -a2 plane part with coordinates x, y, 0
- Grey points: points ± with an index 2
- Bold lines: the base of the hexagonal unit cell.
- Black line: the base of the hexagonal prism is often used for illustration of the hexagonal lattice system.
- Red arrows: the lattice vectors of the hexagonal lattice, thin: the in mineralogy usual 3 a- axis.
- Blue arrows: direction of view of the third space group symbol by Hermann- Mauguin according to the International Tables for Crystallography 3rd edition.
- Green: Area of orthohexagonalen cell. (See below)
- Green arrows: lattice vectors of orthohexagonalen cell. ( The third lattice vector of the hexagonal c- vector)
The rhombohedral centering
When considering possible centering it comes in this axis system to a special feature. If you add additional grid points so that the full symmetry of the hexagonal lattice is maintained, then there are only point grid, which can be described (in other lattice constants ) by a primitive hexagonal lattice.
Adds one but additional lattice points at the points and, respectively, and a, we obtain a new grid, but no longer has the full symmetry of the hexagonal lattice point but the lower symmetry.
This grid system can also be described with a primitive unit cell. For the metric of this cell is considered: and. This unit cell is in the form of a rhombohedron, along the spatial diagonal of a distorted cube. This unit cell is indeed primitive, but not conventional, because the threefold axis is not in the direction of the grating vector, but in the direction of the body diagonal. This grid system is called rhombohedral, has the holohedry and is independent of the site (hexagonal or rhombohedral axes) referred to as R- lattice.
The position of the rhombohedral to hexagonal axes depends on which of the two alternatives is used for centering of the hexagonal cell. In the first case, ie, the formation of the axes obvers, reverse in the second case. In the first edition of the International Tables of 1935, the reverse line-up was used in the subsequent obverse the. The difference between the two constellations, there is a rotation of the hexagonal to rhombohedral axes by 60 °, 180 ° or 300 °.
Since this system does not have the full grid of the hexagonal symmetry, it does not happen in all the point groups of the hexagonal crystal family.
Use in the trigonal and hexagonal crystal system
The hexagonal system axis is used to describe all the point groups hexagonal crystal family. Point groups whose space groups can be described only with the primitive hexagonal lattice, form the hexagonal crystal system. All groups of points in which the rhombohedral and centered lattice occurs form the trigonal crystal system. Also in this system, all the off-center area groups will be described with the hexagonal system axis. A description of these groups with the space rhombohedral lattice system is not possible, even if they are counted for the holohedry rhombohedral lattice system. Only in the centric space groups (symbol R) you have the choice between the hexagonal and the rhombohedral axis system.
Rhombohedral or hexagonal axes
In contrast to the hexagonal cell rhombohedral cell is a conventional cell, so the hexagonal axle system is usually used. The structural data of the minerals the rhombohedral system plays only a minor role.
The rhombohedron is a distorted in the direction of the body diagonal cube. Therefore, the use of this preparation is provided in the cases where a cubic and a rhombohedral structure are compared with each other, since in this case they do not need to change the axis system.
The orthohexagonale system
Since the hexagonal axis system is not orthogonal system, its metric is more complicated. One of the approaches to deal with it is to be described by an orthorhombic lattice system, the so-called orthohexagonale system. This is a C-centered orthorhombic cell. The base of this system is a square with the side length b: a ratio of. It is shown in figure 3 green. The third axis corresponds to the hexagonal c-axis.
The advantage of this installation is the simpler metric, the disadvantage is the loss of an explicit three - or sixfold axis.
More centered cells
In the description of the upper or lower groups is three times larger hexagonal cell, the so-called H- cell is used in the International Tables.
It is also possible to describe the hexagonal lattice with six centered rhombohedral cells. These cells are called D- cells. For a description of structures they are not used.
Historical Notes
The classification of crystals in crystal systems was originally based on morphology. In the trigonal or hexagonal system all the crystals were combined, whose crystal form is indicative of the presence of a three - or six-fold axis of rotation. However, since the six-fold axis of rotary inversion causes a three-fold crystalline form, the point groups ( trigonal dipyramidal- ) and ( - ditrigonal dipyramidal ) were first counted on the trigonal crystal system, such as one today does in the designations for the crystal forms.
The hexagonal crystal classes
For a description of the hexagonal crystal classes in Hermann- Mauguin symbols of symmetry operations is presented in respect of predetermined directions in the grid system.
Axis in the hexagonal system: the first symbol in the direction of c- axis ( <001 > ). Second symbol in the direction of a- axis (<100 > ). 3.Find in a direction perpendicular to a and the c-axis (<120 > ). The generally non-equivalent direction < 210 > is also often specified for the third direction. Even though this is especially used to indicate the position of the symmetry elements does not matter, this claim does not comply with the conventions.
Characteristic of all space groups of the hexagonal crystal system is the 6 (or 6) at the 1st digit of the space group symbol.
For information on the physical properties means: - Due to the symmetry forbidden. Means: Due to the symmetry allowed. About the magnitude of the effect can be made due to the symmetry no statement, but one can assume that this effect is never exactly disappear.
More hexagonally crystallized chemical substances, see Category: Hexagonal crystal system
Hexagonal crystal forms
- Image examples of hexagonal crystal forms
Hexagonal prism
Combining prism - Pyramid
Combining prism with pyramidal base
Trigonal Dipyramid (crystal class 6 )
The hexagonal close packing
The hexagonal close packing can be described as follows: The connection of adjacent atoms form a body with a hexagonal base and top surface. In the middle of these two areas, there is one further atom. Between the base and top surface, three additional atoms have space. With the assumption of equal -sized spheres, this corresponds to a close packing of spheres, their bulkiness is approximately 74.05 %. The stacking sequence can be described by ABA. So you can find here also the name hexagonal- close-packed ( hcp, Eng. Hcp ). Here, the aspect ratio is in the ideal case.
A unit cell with hexagonal close -packed ( hcp ) consists of two diamond-shaped base areas. Atoms are within the unit cell of the crystallographic layers 1 /3, 2 /3, 1 /4 and 2/3, 1/3, 3/4 ( a center of symmetry of the structure is then convention, 0, 0, 0).
Many metals crystallize in a hexagonal close packing of spheres: Be, Mg, Sc, Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Lu, Hf, Re, Os, Tl and some lanthanides. As a prominent representative shall magnesium, which is why this type of structure is also called magnesium type.