Crystal system | Trigonal | Hexagonal | |
---|---|---|---|
Lattice system | Rhombohedral |
Hexagonal | |
Example | Dolomite |
α-Quartz |
Beryl |
In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see section crystal systems below).^{[1]} In particular, there are crystals with trigonal symmetry but belong to the hexagonal lattice (such as α-Quartz).
The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system.^{[2]} There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral.
Lattice systems
The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice.
Bravais lattice | Hexagonal | Rhombohedral |
---|---|---|
Pearson symbol | hP | hR |
Hexagonal unit cell |
||
Rhombohedral unit cell |
In the hexagonal family, the crystal is conventionally described by a right rhombic prism unit cell with two equal axes (a by a), an included angle of 120° (γ) and a height (c, which can be different from a) perpendicular to the two base axes.
The hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell. There are two ways to do this, which can be thought of as two notations which represent the same structure. In the usual so-called obverse setting, the additional lattice points are at coordinates (2⁄3, 1⁄3, 1⁄3) and (1⁄3, 2⁄3, 2⁄3), whereas in the alternative reverse setting they are at the coordinates (1⁄3,2⁄3,1⁄3) and (2⁄3,1⁄3,2⁄3).^{[3]} In either case, there are 3 lattice points per unit cell in total and the lattice is non-primitive.
The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes.^{[4]}^{[5]} The unit cell is a rhombohedron (which gives the name for the rhombohedral lattice). This is a unit cell with parameters a = b = c; α = β = γ ≠ 90°.^{[6]} In practice, the hexagonal description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are often shown (for the rhombohedral lattice) in textbooks because this cell reveals 3m symmetry of crystal lattice.
The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered^{[7]} cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates (1⁄3, 1⁄3, 1⁄3) and (2⁄3, 2⁄3, 2⁄3). However, such a description is rarely used.
Crystal systems
Crystal system | Required symmetries of point group | Point groups | Space groups | Bravais lattices | Lattice system |
---|---|---|---|---|---|
Trigonal | 1 threefold axis of rotation | 5 | 7 | 1 | Rhombohedral |
18 | 1 | Hexagonal | |||
Hexagonal | 1 sixfold axis of rotation | 7 | 27 |
The hexagonal crystal family consists of two crystal systems: trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system (see table in Crystal system#Crystal classes).
The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis, which includes space groups 143 to 167. These 5 point groups have 7 corresponding space groups (denoted by R) assigned to the rhombohedral lattice system and 18 corresponding space groups (denoted by P) assigned to the hexagonal lattice system. Hence, the trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups.
The hexagonal crystal system consists of the 7 point groups that have a single six-fold rotation axis. These 7 point groups have 27 space groups (168 to 194), all of which are assigned to the hexagonal lattice system.
Trigonal crystal system
The 5 point groups in this crystal system are listed below, with their international number and notation, their space groups in name and example crystals.^{[8]}^{[9]}^{[10]}
Space group no. | Point group | Type | Examples | Space groups | |||||
---|---|---|---|---|---|---|---|---|---|
Name^{[11]} | Intl | Schoen. | Orb. | Cox. | Hexagonal | Rhombohedral | |||
143–146 | Trigonal pyramidal | 3 | C_{3} | 33 | [3]^{+} | enantiomorphic polar | carlinite, jarosite | P3, P3_{1}, P3_{2} | R3 |
147–148 | Rhombohedral | 3 | C_{3i} (S_{6}) | 3× | [2^{+},6^{+}] | centrosymmetric | dolomite, ilmenite | P3 | R3 |
149–155 | Trigonal trapezohedral | 32 | D_{3} | 223 | [2,3]^{+} | enantiomorphic | abhurite, alpha-quartz (152, 154), cinnabar | P312, P321, P3_{1}12, P3_{1}21, P3_{2}12, P3_{2}21 | R32 |
156–161 | Ditrigonal pyramidal | 3m | C_{3v} | *33 | [3] | polar | schorl, cerite, tourmaline, alunite, lithium tantalate | P3m1, P31m, P3c1, P31c | R3m, R3c |
162–167 | Ditrigonal scalenohedral | 3m | D_{3d} | 2*3 | [2^{+},6] | centrosymmetric | antimony, hematite, corundum, calcite, bismuth | P31m, P31c, P3m1, P3c1 | R3m, R3c |
Hexagonal crystal system
The 7 point groups (crystal classes) in this crystal system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation, and mineral examples, if they exist.^{[2]}^{[12]}
Space group no. | Point group | Type | Examples | Space groups | ||||
---|---|---|---|---|---|---|---|---|
Name^{[11]} | Intl | Schoen. | Orb. | Cox. | ||||
168–173 | Hexagonal pyramidal | 6 | C_{6} | 66 | [6]^{+} | enantiomorphic polar | nepheline, cancrinite | P6, P6_{1}, P6_{5}, P6_{2}, P6_{4}, P6_{3} |
174 | Trigonal dipyramidal | 6 | C_{3h} | 3* | [2,3^{+}] | laurelite and boric acid | P6 | |
175–176 | Hexagonal dipyramidal | 6/m | C_{6h} | 6* | [2,6^{+}] | centrosymmetric | apatite, vanadinite | P6/m, P6_{3}/m |
177–182 | Hexagonal trapezohedral | 622 | D_{6} | 226 | [2,6]^{+} | enantiomorphic | kalsilite and high quartz | P622, P6_{1}22, P6_{5}22, P6_{2}22, P6_{4}22, P6_{3}22 |
183–186 | Dihexagonal pyramidal | 6mm | C_{6v} | *66 | [6] | polar | greenockite, wurtzite^{[13]} | P6mm, P6cc, P6_{3}cm, P6_{3}mc |
187–190 | Ditrigonal dipyramidal | 6m2 | D_{3h} | *223 | [2,3] | benitoite | P6m2, P6c2, P62m, P62c | |
191–194 | Dihexagonal dipyramidal | 6/mmm | D_{6h} | *226 | [2,6] | centrosymmetric | beryl | P6/mmm, P6/mcc, P6_{3}/mcm, P6_{3}/mmc |
Hexagonal close packed
Hexagonal close packed (hcp) is one of the two simple types of atomic packing with the highest density, the other being the face-centered cubic (fcc). However, unlike the fcc, it is not a Bravais lattice, as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravais lattice by using a two-atom motif (the additional atom at about (2⁄3, 1⁄3, 1⁄2)) associated with each lattice point.^{[14]}
In two dimensions
There is only one hexagonal Bravais lattice in two dimensions: the hexagonal lattice.
Bravais lattice | Hexagonal |
---|---|
Pearson symbol | hp |
Unit cell |
See also
References
- ^ Hahn (2002)
- ^ ^{a} ^{b} Dana, James Dwight; Hurlbut, Cornelius Searle (1959). Dana's Manual of Mineralogy (17th ed.). New York: Chapman Hall. pp. 78–89.
- ^ Edward Prince (2004). Mathematical Techniques in Crystallography and Materials Science. Springer Science & Business Media. p. 41.
- ^ "Page not found - QuantumWise". quantumwise.com.
{{cite web}}
: Cite uses generic title (help) - ^ "Medium-Resolution Space Group Diagrams and Tables". img.chem.ucl.ac.uk.
- ^ Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics (1st ed.). p. 119. ISBN 0-03-083993-9.
- ^ Hahn (2002), p. 73
- ^ Pough, Frederick H.; Peterson, Roger Tory (1998). A Field Guide to Rocks and Minerals. Houghton Mifflin Harcourt. p. 62. ISBN 0-395-91096-X.
- ^ Hurlbut, Cornelius S.; Klein, Cornelis (1985). Manual of Mineralogy (20th ed.). pp. 78–89. ISBN 0-471-80580-7.
- ^ "Crystallography and Minerals Arranged by Crystal Form". Webmineral.
- ^ ^{a} ^{b} Hahn (2002), p. 794
- ^ "Crystallography". Webmineral.com. Retrieved 2014-08-03.
- ^ "Minerals in the Hexagonal crystal system, Dihexagonal Pyramidal class (6mm)". Mindat.org. Retrieved 2014-08-03.
- ^ Jaswon, Maurice Aaron (1965-01-01). An introduction to mathematical crystallography. American Elsevier Pub. Co.
Further reading
- Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. International Tables for Crystallography. Vol. A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7.
External links
- Media related to Hexagonal lattices at Wikimedia Commons
- Mineralogy database