# Hexagonal crystal system

The hexagonal crystal system is one of the seven crystal systems in crystallography . It includes all point groups with a sixfold axis of rotation or rotation inversion. The hexagonal crystal system is closely related to the trigonal crystal system and together with it forms the hexagonal crystal family .

## The hexagonal point groups

The hexagonal crystal system includes the point groups and . These are all the point groups of the hexagonal crystal family in which there is no space group with rhombohedral centering. The space groups of the hexagonal crystal system can all be described with the hexagonal primitive axis system. The hexagonal point groups do not have a cubic parent group . Thus the hexagonal holoedry together with the cubic is the most highly symmetrical crystallographic point group. ${\ displaystyle 6, \, {\ bar {6}}, \, 6 / m, \, 622, \, 6mm, \, {\ bar {6}} m2}$ ${\ displaystyle \ 6 / mmm}$ ## The hexagonal axis system

In the hexagonal crystal family there is the hexagonal and the trigonal crystal system, as well as the hexagonal and the rhombohedral lattice system . The division into crystal systems is based on the symmetry of the crystals, the division into lattice systems relates to the metric of the lattice. While these different perspectives lead to the same classification in the five other crystal families or crystal systems, this is not the case in the hexagonal crystal family. In addition, the division into grid systems is not based on the point groups, but on the room groups. Since the relationships are relatively complicated, they are described in more detail at this point.

### The hexagonal axis system in crystallography

As in all whirling crystal systems , the axis of rotation with the highest number is placed in the direction of the c-lattice axis. The plane perpendicular to this is described by two axes a 1 and a 2 of equal length , which are at an angle of 120 ° to one another. This results in the following metric: and The unit cell formed by these basis vectors is shown in Figure 1. It has a volume of${\ displaystyle a = b \ neq c}$ ${\ displaystyle \ alpha = \ beta = 90 ^ {\ circ}, \, \ gamma = 120 ^ {\ circ}.}$ ${\ displaystyle V = a ^ {2} c \ sin 60 ^ {\ circ} = {\ tfrac {\ sqrt {3}} {2}} a ^ {2} c.}$ ### The hexagonal axis system in other disciplines

In mineralogy and especially in metallurgy, it is common to use an additional axis a 3 in the (a 1 , a 2 ) plane (see Fig. 3). This has the same length as a 1 and is at an angle of 120 ° to both a 1 and a 2 . The Miller indices are extended by the index i to form so-called Miller-Bravais indices and then have four components: (h, k, i, l). The index i is redundant because: i = - (h + k). Similarly, in metallurgy, directions are also represented by four-part symbols [uvtw], the Weber indices .

The hexagonal cell is often represented in the literature as a hexagonal prism (see Fig. 2). Since this prism is not a parallelepiped , it is not a unit cell. This prism consists of three hexagonal unit cells.

### The a 1 -a 2 plane

Figure 3 shows the a 1 -a 2 plane of the hexagonal axis system. In detail:

• Points: grid points of the hexagonal axis system in the a 1 -a 2 plane, partly with coordinates x, y, 0.
• Gray points: points with an index of ± 2.
• Bold lines: the base of the hexagonal unit cell.
• Black lines: the base of the hexagonal prism, which is often used to illustrate the hexagonal grid system.
• Red arrows: the grid vectors of the hexagonal grid, thin: the 3rd a-axis common in mineralogy.
• Blue arrows: Direction of view of the 3rd space group symbol according to Hermann-Mauguin according to the International Tables for Crystallography 3rd Edition.
• Green: base of the ortho-hexagonal cell. (See below)
• Green arrows: ortho-hexagonal cell grid vectors. (The 3rd grid vector is the hexagonal c-vector)

### The rhombohedral centering

When considering possible centerings, there is a special feature in this axis system. If you add additional grid points in such a way that the full symmetry of the hexagonal grid is retained, then only point grids result that can also be described by a primitive hexagonal grid (with other grid constants).

However, if you add additional grid points at the points and or and , a new grid results, which no longer has the full symmetry of the hexagonal point grid, but the lower symmetry . ${\ displaystyle ({\ tfrac {2} {3}}, \, {\ tfrac {1} {3}}, \, {\ tfrac {1} {3}})}$ ${\ displaystyle ({\ tfrac {1} {3}}, \, {\ tfrac {2} {3}}, \, {\ tfrac {2} {3}})}$ ${\ displaystyle ({\ tfrac {1} {3}}, \, {\ tfrac {2} {3}}, \, {\ tfrac {1} {3}})}$ ${\ displaystyle ({\ tfrac {2} {3}}, \, {\ tfrac {1} {3}}, \, {\ tfrac {2} {3}})}$ ${\ displaystyle {\ bar {3}} m}$ This lattice system can also be described with a primitive unit cell . The following applies to the metric of this cell: and . This unit cell has the shape of a rhombohedron , a cube that is distorted along its spatial diagonal. This unit cell is primitive, but not conventional , since the threefold axis is not in the direction of a grid vector, but in the direction of the space diagonal. This grid system is called rhombohedral, has holoedry and is referred to as an R grid regardless of the arrangement (hexagonal or rhombohedral axes). ${\ displaystyle a = b = c}$ ${\ displaystyle \ alpha = \ beta = \ gamma \ neq 90 ^ {\ circ}}$ ${\ displaystyle {\ bar {3}} m}$ The position of the rhombohedral to the hexagonal axes depends on which of the two possibilities was used to center the hexagonal cell. In the first case the arrangement of the axes is called obverse, in the second case reverse. In the first edition of the International Tables of 1935 the reverse table was used, in the subsequent ones the obverse. The difference between the two setups is a rotation of the hexagonal to the rhombohedral axes by 60 °, 180 ° or 300 °.

Since this lattice system does not have the full symmetry of the hexagonal, it does not occur in all point groups of the hexagonal crystal family.

### Use in the trigonal and hexagonal crystal system

The hexagonal axis system is used to describe all point groups of the hexagonal crystal family. Point groups whose space groups can only be described with the primitive hexagonal lattice form the hexagonal crystal system. All point groups, in which the rhombohedral centered lattice occurs, form the trigonal crystal system. In this system, too, all non-centric space groups are described with the hexagonal axis system. A description of these space groups with the rhombohedral grid system is not possible, even if they are included in the holoedry of the rhombohedral grid system. Only with the centric space groups (symbol R) you can choose between the hexagonal and the rhombohedral axis system.

### Rhombohedral or hexagonal axes

In contrast to the rhombohedral cell, the hexagonal cell is a conventional cell, so the hexagonal axis system is usually used. The rhombohedral system only plays a subordinate role in the structural data of the minerals.

The rhombohedron is a cube distorted in the direction of the space diagonals. It is therefore appropriate to use this list in cases in which a cubic and a rhombohedral structure are compared with one another, since the axis system does not have to be changed.

### The orthohexagonal system

Since the hexagonal axis system is not an orthogonal system, its metric is more complicated. One of the approaches to deal with this is the description using an orthorhombic grid system, the so-called orthohexagonal system. It is an orthorhombic C-centered cell. The base of this system is a rectangle with the aspect ratio b: a of . It is shown in green in Figure 3. The third axis corresponds to the hexagonal c-axis. ${\ displaystyle {\ sqrt {3}} \ approx 1 {,} 732}$ The advantage of this list is the simpler metric, the disadvantage is the loss of an explicit three or six-fold axis.

## More centered cells

In the description of the upper or subgroups in the International Tables a three-times magnified hexagonal cell, the so-called H-cell is used.

It is also possible to describe the hexagonal grid with six centered rhombohedral cells. These cells are called D cells. They are not used to describe structures.

### Historical notes

The division of the crystals into crystal systems was originally based on the morphology. In the trigonal or hexagonal system, all those crystals were combined whose crystal shape suggests the presence of a three or six-fold axis of rotation. However, since the six-fold rotational inversion axis produces a threefold crystal form, the point groups (trigonal-dipyramidal) and (ditrigonal-dipyramidal) were initially counted as part of the trigonal crystal system, as can still be seen from the names for the crystal forms today. ${\ displaystyle {\ bar {6}}}$ ${\ displaystyle {\ bar {6}} m2}$ ## Point groups in the hexagonal crystal system and their physical properties

To describe the hexagonal crystal classes in Hermann-Mauguin symbology , the symmetry operations are given in relation to given directions in the lattice system.

In the hexagonal axis system: 1. Symbol in the direction of the c-axis (<001>). 2. Symbol in the direction of an a-axis (<100>). 3. Symbol in a direction perpendicular to an a and the c axis (<120>). The generally non-equivalent direction <210> is also often specified for the 3rd direction. Even if this does not play a role in specifying the position of the symmetry elements, this specification does not correspond to the conventions.

Characteristic for all space groups of the hexagonal crystal system is the 6 (or 6 ) in the 1st position of the space group symbol.

Point group (crystal class) Physical Properties Examples
No. Crystal system Surname Schoenflies icon International symbol
( Hermann-Mauguin )
Tepid class Associated room
groups ( no.)
Enantiomorphism Optical activity Pyroelectricity Piezoelectricity ; SHG effect
Full Short
21st hexagonal hexagonal-pyramidal C 6 6th 6th 6 / m 168-173 + + +  + Nepheline
zinkenite
22nd trigonal-dipyramidal C 3 h 6th 6th 174 - - - + Penfieldite
Laurelite
23 hexagonal-dipyramidal C 6 h 6 / m 6 / m 175-176 - - - - Apatite
zemannite
24 hexagonal-trapezoidal D 6 622 622 6 / mmm 177-182 + + - + High quartz
pseudorutile
25th dihexagonal-pyramidal C 6 v 6 mm 6 mm 183-186 - - +  + Wurtzit
Zinkit
26th ditrigonal-dipyramidal D 3 h 6 m 2 or  6 2 m 6 m 2 187-190 - - - + Bastnasite
benitoite
27 dihexagonal-dipyramidal D 6 h 6 / m 2 / m 2 / m 6 / mmm 191-194 - - - - Graphite
magnesium
1. In the information on the physical properties, " - " means forbidden due to the symmetry and " + " means allowed. No statement can be made about the magnitude of the optical activity, pyro- and piezoelectricity and the SHG effect purely due to the symmetry. However, one can assume that there is always at least a weak expression of the property. For the pyroelectricity, the direction of the pyroelectric vector is given, if available.

For more hexagonal crystallizing chemical substances see category: Hexagonal crystal system

## The hexagonal closest packing of spheres

The hexagonal close packing of spheres (hdp, hcp ) is one of the two possibilities of close packing of spheres . It can be described as follows: Its basic cell is a six-sided prism , the 12 corners of which are each occupied by a sphere of the same size. The ball diameter is equal to the edge length (the 6 balls per ball are in contact). In the middle of each of the 6 balls there is a 7th ball of the same diameter. The height of the prism is such that 3 more balls of the same diameter fit between the 7 upper and 7 lower balls. These 3 balls touch each other and hit one gap each within the 7 upper and 7 lower balls. The aspect ratio of the hexagonal cell (s. Figure 2) is: . ${\ displaystyle \ textstyle {\ frac {c} {a}} = {\ sqrt {\ frac {8} {3}}}}$ The stacking sequence of its three hexagonal spherical layers is written ABA.

A unit cell with hexagonal close packing (hdp) consists of two diamond-shaped bases. The atoms are located within the unit cell on the crystallographic positions 1/3, 2/3, 1/4 and 2/3, 1/3, 3/4 (a symmetry center of the structure is then, according to 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 lanthanoids . Magnesium is the most prominent representative , which is why this structure type is also called the magnesium type.

## literature

• Martin Okrusch, Siegfried Matthes: Mineralogy. 7th edition. Springer Verlag, Berlin 2005, ISBN 3-540-23812-3
• Hans Murawski, Wilhelm Meyer: Geological dictionary . 12th edition. Spektrum Akademischer Verlag, Heidelberg 2010, ISBN 978-3-8274-1810-4 .
• Rüdiger Borchert, Siegfried Turowski: Theory of Symmetry in Crystallography; Models of the 32 crystal classes . Oldenbourg Wissenschaftsverlag GmbH, Munich, Vienna 1999, ISBN 3-486-24648-8 , p. 52-64 .
• Werner Massa: Crystal structure determination . 3. Edition. BG Teubner GmbH, Stuttgart / Leipzig / Wiesbaden 2002, ISBN 3-519-23527-7 .
• Ulrich Müller: Inorganic Structural Chemistry . 4th edition. BG Teubner / GWV Fachverlage GmbH, Wiesbaden 2004, ISBN 3-519-33512-3 , p. 182 .
• Hahn, Theo (Ed.): International Tables for Crystallography Vol. A D. Reidel publishing Company, Dordrecht 1983, ISBN 90-277-1445-2
• Will Kleber, et al. Introduction to crystallography 19th edition Oldenbourg Wissenschaftsverlag, Munich 2010, ISBN 978-3-486-59075-3
• Walter Borchard-Ott Crystallography 7th edition Springer Verlag, Berlin 2009, ISBN 978-3-540-78270-4

## Individual evidence

1. Lothar Spieß, Robert Schwarzer, Gerd Teichert, Herfried Behnken: Modern X-ray diffraction: X-ray diffractometry for materials scientists, physicists and chemists . Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8351-0166-1 , p. 57 .