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having threefold symmetry (同)trigonal
a parallelepiped bounded by six similar faces (either rhombuses or parallelograms)

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/08/28 10:22:43」(JST)

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Crystal system	Trigonal		Hexagonal
Lattice system	Rhombohedral	Hexagonal
Example	Dolomite	Quartz	Beryl

In crystallography, the hexagonal crystal family is one of the 6 crystal families. 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 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.^[1] There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral.

1 Lattice systems
2 Crystal systems
3 Comparison
- 3.1 Rhombohedral lattice system vs Trigonal crystal system
- 3.2 Hexagonal lattice system vs Hexagonal crystal system
4 Hexagonal Close Packed
5 Crystal classes
- 5.1 Trigonal crystal system
- 5.2 Hexagonal crystal system
6 See also
7 References
8 External links

Lattice systems

The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice.

Relation between the two settings for the rhombohedral Bravais lattice

Hexagonal crystal family
Bravais lattice	Hexagonal	Rhombohedral
Pearson symbol	hP	hR
Conventional hexagonal cell
Rhombohedral cell

The conventional cell for the rhombohedral Bravais lattice is the rhombohedrally-centered (R-centered) hexagonal cell, consisting of two additional lattice points which occupy the longest body diagonal of the unit cell with coordinates ( ²⁄₃, ¹⁄₃, ¹⁄₃) and ( ¹⁄₃, ²⁄₃, ²⁄₃). Hence, 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. The unit cell is a rhombohedron (which gives the name for the rhombohedral lattice system). This is a unit cell with parameters a = b = c; α = β = γ ≠ 90°.^[2] 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 system) in textbooks because this cell reveals 3m symmetry of crystal lattice.

The unit cell for the hexagonal Bravais lattice in rhombohedral axes, consisting of two additional lattice points which occupy the longest body diagonal of the unit cell with coordinates ( ¹⁄₃, ¹⁄₃, ¹⁄₃) and ( ²⁄₃, ²⁄₃, ²⁄₃), is called the hexagonally-centered (D-centered)^[3] rhombohedral cell. However, such a description is rarely used.

Crystal systems

Quartz mineral embedded in limestone (top right of the sample), easily identifiable by its hexagonal form.

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 associated with a lattice system.

The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis (see table in Crystal system#Crystal classes). Crystals in the rhombohedral lattice system are always in the trigonal crystal system, but some crystals such as alpha-quartz are in the trigonal crystal system but not in the rhombohedral lattice system (alpha-quartz is in the hexagonal lattice system). There are 25 space groups (143 to 167) whose corresponding point groups are one of the five in the trigonal crystal system, consisting of the seven space groups associated with the rhombohedral lattice system together with 18 associated with the hexagonal lattice system. The crystal structures of alpha-quartz in the previous example are described by two of those 18 space groups (152 and 154) associated with the hexagonal lattice system. (In total there are 45 space groups associated with the hexagonal lattice system, the remaining 27 space groups being part of the hexagonal crystal system to which another modification of quartz (β-quartz) belongs.)

The hexagonal crystal system consists of the seven point groups such that all their space groups have the hexagonal lattice as underlying lattice. Graphite is an example of a crystal that crystallizes in the hexagonal crystal system.

Comparison

Crystal system	Required symmetries of point group	Point groups	Space groups	Lattice system
Trigonal	1 threefold axis of rotation	5	7	Rhombohedral
Trigonal	1 threefold axis of rotation	5	18	Hexagonal
Hexagonal	1 sixfold axis of rotation	7	27	Hexagonal

Rhombohedral lattice system vs Trigonal crystal system

The rhombohedral lattice system consists of the rhombohedral lattice, while the trigonal crystal system consists of the five point groups that have seven corresponding space groups associated with the rhombohedral lattice system (and 18 corresponding space groups associated with the hexagonal lattice system). An additional source of confusion is that all members of the trigonal crystal system with assigned rhombohedral lattice system (space groups 146, 148, 155, 160, 161, 166, and 167), can be represented with an equivalent hexagonal lattice with so called R-centering (rhombohedral-centering); there is a choice of using a R-centered hexagonal or a primitive rhombohedral setting for the lattice.^[4]^[5]

Hexagonal lattice system vs Hexagonal crystal system

The hexagonal lattice system is associated with 45 space groups whose underlying lattice has point group of order 24, while the hexagonal crystal system consists of the 27 space groups such that all space groups with the same point group are in the hexagonal lattice system.

Hexagonal Close Packed

Note that the atom in the body center of the HCP unit cell in the hexagonal lattice system does not appear in the unit cell of the hexagonal lattice. It is part of the two atom motif (the additional atom at ( ²⁄₃, ¹⁄₃, ¹⁄₂)) associated with each lattice point in the underlying lattice.^[6]

Crystal classes

Trigonal crystal system

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 and rhombohedral lattices both appear.

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. (All these point groups are also associated with some space groups not in the rhombohedral lattice system.)^[7]^[8]^[9]

Space group no.	Point group					Examples	Space group
Space group no.	Class	Intl	Schoen.	Orb.	Cox.	Examples	Space group
143–146	Pyramidal rhombohedral tetartohedral	3	C₃	33	[3]⁺	carlinite, jarosite	P3, P3₁, P3₂ R3
147–148	Rhombohedral rhombohedral tetartohedral	3	S₆	3×	[2⁺,6⁺]	dolomite, ilmenite	P3 R3
149–155	Trapezohedral	32	D₃	223	[2,3]⁺	abhurite, alpha-quartz (152, 154), cinnabar	P312, P321, P3₁12, P3₁21, P3₂12, P3₂21 R32
156–161	Ditrigonal pyramidal rhombohedral hemimorphic	3m	C_3v	*33	[3]	schorl, cerite, tourmaline, alunite, lithium tantalate	P3m1, P31m, P3c1, P31c R3m, R3c
162–167	Hexagonal scalenohedral rhombohedral holohedral	3m	D_3d	2*3	[2⁺,6]	antimony, hematite, corundum, calcite	P31m, P31c, P3m1, P3c1 R3m, R3c

Hexagonal crystal system

The 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.^[1]^[10]

No.	Point group					Example	Space groups
No.	Class	Intl	Schoen.	Orb.	Cox.	Example	Space groups
168–173	Hexagonal pyramidal	6	C₆	66	[6]⁺	nepheline, cancrinite	P6, P6₁, P6₅, P6₂, P6₄, P6₃
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⁺]	apatite, vanadinite	P6/m, P6₃/m
177–182	Hexagonal trapezohedral	622	D₆	226	[2,6]⁺	kalsilite and high quartz	P622, P6₁22, P6₅22, P6₂22, P6₄22, P6₃22
183–186	Dihexagonal pyramidal	6mm	C_6v	*66	[6]	greenockite, wurtzite ^[11]	P6mm, P6cc, P6₃cm, P6₃mc
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]	beryl	P6/mmm, P6/mcc, P6₃/mcm, P6₃/mmc

References

^ ^a ^b Dana, James Dwight; Hurlbut, Cornelius Searle (1959). Dana's Manual of Mineralogy (17th ed.). New York: Chapman Hall. pp. 78–89.
^ Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics (1st ed.). p. 119. ISBN 0-03-083993-9.
^ Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. A (5th ed.). Berlin, New York: Springer-Verlag. p. 73. ISBN 978-0-7923-6590-7. doi:10.1107/97809553602060000100.
^ "Page not found - QuantumWise". quantumwise.com.
^ "Medium-Resolution Space Group Diagrams and Tables". img.chem.ucl.ac.uk.
^ Jaswon, Maurice Aaron (1965-01-01). An introduction to mathematical crystallography. American Elsevier Pub. Co.
^ 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.
^ "Crystallography". Webmineral.com. Retrieved 2014-08-03.
^ "Minerals in the Hexagonal crystal system, Dihexagonal Pyramidal class (6mm)". Mindat.org. Retrieved 2014-08-03.

External links

Mineralogy database

English Journal

Calcite microneedle arrays produced by inorganic ion-assisted anisotropic dissolution of bulk calcite crystal.

Long X1, Meng R, Wu W, Ma Y, Yang D, Qi L.Author information 1Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, College of Chemistry, Peking University (PKU), Beijing 100871 (P.R. China), Fax: (+86) 1062751708.AbstractBesides studies on the mineralization process, research on the demineralization of minerals provides another way to understand the crystallization mechanism of biominerals and fabricate crystals with complicated morphologies. The formation of ordered arrays of c-axis-oriented calcite microneedles with a tri-symmetric structure and lengths of more than 20 μm was realized on a large scale for the first time through anisotropic dissolution of calcite substrates in undersaturated aqueous solution in the presence of ammonium salts. The lengths and the aspect ratios of the calcite microneedles can be tuned by simply changing the concentrations of the ammonium salts and the dissolution time. The shape of the transverse cross sections of the calcite microneedles obtained in the presence of NH4 Cl and NH4 Ac is almost regularly triangular. The tri-symmetric transverse cross-section geometry of the calcite microneedles could be attributed to the tri-symmetric feature of rhombohedral calcite atomic structures, the synergetic interactions between electrostatic interaction of ammonium ions and dangling surface carbonate groups, and the ion incorporation of halide ions.
Chemistry (Weinheim an der Bergstrasse, Germany).Chemistry.2014 Apr 7;20(15):4264-72. doi: 10.1002/chem.201304229. Epub 2014 Mar 18.
Besides studies on the mineralization process, research on the demineralization of minerals provides another way to understand the crystallization mechanism of biominerals and fabricate crystals with complicated morphologies. The formation of ordered arrays of c-axis-oriented calcite microneedles wi
PMID 24644237

Soft ferromagnetism in mixed valence Sr1-xLaxTi0.5Mn0.5O3 perovskites.

Qasim I1, Blanchard PE, Kennedy BJ, Ling CD, Jang LY, Kamiyama T, Miao P, Torii S.Author information 1School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia. B.Kennedy@chem.usyd.edu.au.AbstractThe structural, magnetic and electrical properties of the mixed Ti-Mn oxides Sr1-xLaxTi0.5Mn0.5O3 (0 ≤ x ≤ 0.5) are reported. At room temperature the oxides have a cubic structure in space group Pm3[combining macron]m for x ≤ 0.25 and rhombohedral in R3[combining macron]c for 0.3 ≤ x ≤ 0.50. X-ray absorption spectroscopic measurements demonstrate the addition of La3+ is compensated by the partial reduction of Mn4+ to Mn3+. Variable temperature neutron diffraction measurements show that cooling Sr0.6La0.4Ti0.5Mn0.5O3 results in a first order transition from rhombohedra to an orthorhombic structure in Imma. Complex magnetic behaviour is observed. The magnetic behaviour of the mixed valent (Mn3+/4+) examples is dominated by ferromagnetic interactions, although cation disorder frustrates long range magnetic ordering.
Dalton transactions (Cambridge, England : 2003).Dalton Trans.2014 Mar 25. [Epub ahead of print]
The structural, magnetic and electrical properties of the mixed Ti-Mn oxides Sr1-xLaxTi0.5Mn0.5O3 (0 ≤ x ≤ 0.5) are reported. At room temperature the oxides have a cubic structure in space group Pm3[combining macron]m for x ≤ 0.25 and rhombohedral in R3[combining macron]c for 0.3 ≤ x ≤ 0.5
PMID 24667811

New lead-free piezoelectric ceramics based on (K0.48Na0.52)(Nb0.95Ta0.05)O3-Bi0.5(Na0.7K0.2Li0.1)0.5ZrO3.

Cheng X1, Wu J, Wang X, Zhang B, Zhu J, Xiao D, Wang X, Lou X.Author information 1Department of Materials Science, Sichuan University, Chengdu 610064, P. R. China. msewujg@scu.edu.cn wujiagang0208@163.com nic0402@scu.edu.cn.Abstract(1 - x)(K0.48Na0.52)(Nb0.95Ta0.05)O3-xBi0.5(Na0.7K0.2Li0.1)0.5ZrO3 lead-free piezoelectric ceramics with a new type of phase boundary have been designed and fabricated. This phase boundary lies in the compositional range of 0.04 ≤ x ≤ 0.05, and is formed by the coexistence of the rhombohedral, orthorhombic, and tetragonal phases. Interestingly, we found that the ferroelectric, dielectric, and piezoelectric properties of the ceramics with compositions near the phase boundary are significantly enhanced. In particular, the ceramic with x = 0.045 shows the best piezoelectric behavior of d33 ∼ 290 pC/N and kp ∼ 0.42 among all the compositions studied in this work, and it also exhibits a good thermal stability at annealing temperatures of ≤270 °C. All these results indicate that such a material system is a good candidate for lead-free piezoelectric applications in the future.
Dalton transactions (Cambridge, England : 2003).Dalton Trans.2014 Mar 7;43(9):3434-42. doi: 10.1039/c3dt52603h.
(1 - x)(K0.48Na0.52)(Nb0.95Ta0.05)O3-xBi0.5(Na0.7K0.2Li0.1)0.5ZrO3 lead-free piezoelectric ceramics with a new type of phase boundary have been designed and fabricated. This phase boundary lies in the compositional range of 0.04 ≤ x ≤ 0.05, and is formed by the coexistence of the rhombohedral, o
PMID 24452129

Japanese Journal

Evidence of rhombohedral structure within BiFeO

Bae In-Tae,Naganuma Hiroshi
Appl. Phys. Express 8(3), 031501, 2015-02-24
… Nano-beam electron diffraction patterns combined with structure factor calculations and high-resolution transmission electron microscopy images unambiguously revealed that the BiFeO<inf>3</inf>thin layer grew with a rhombohedral structure identical to its bulk form. …
NAID 150000110224

Influence of rhenium on the structural and optical properties of molybdenum disulfide

Sigiro Mula,Huang Ying-Sheng,Ho Ching-Hwa,Lin Yung-Chang,Suenaga Kazu
Jpn. J. Appl. Phys. 54(4S), 04DH05, 2015-02-10
… The Re dopant clearly caused a structural change in MoS<inf>2</inf>from a two-layer hexagonal (2H) structure to a three-layer rhombohedral (3R) structure, which has been clearly verified and identified herein. …
NAID 150000110334

AE法によるTi-Ni合金の相変態の発生状態の評価（MD法を用いた構造解析結果との比較）

佐藤知広,宅間正則,齋藤賢一,中野雄亮,森上篤
日本機械学会論文集 advpub(0), 2015
… And, the alloy has two kinds of the phase transformations of rhombohedral (R) and martensitic (M) phase. …
NAID 130005064529