WordNet

a triangular lyre of ancient Greece and Rome

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/03/04 17:26:20」(JST)

wiki en

Example trigonal crystals (quartz)	Example trigonal rhombohedral crystals (dolomite)
Hexagonal lattice cell	Hexagonal rhombohedral unit cell

In crystallography, the trigonal crystal system is one of the seven crystal systems. Sometimes the term rhombohedral lattice system is used as an exact synonym, whereas it is a subset. Crystals in the rhombohedral lattice system are always in the trigonal crystal system, but some crystals such as quartz are in the trigonal crystal system but not in the rhombohedral lattice system. The rhombohedral lattice system consists of the rhombohedral lattice, while the trigonal crystal system consists of the five point groups of the seven space groups with a rhombohedral lattice. There are 25 space groups whose 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 of the 45 space groups associated with the hexagonal lattice system.

The trigonal point group describes the symmetry of an object produced by stretching a cube along the body diagonal. The resultant rhombohedral (or trigonal) Bravais lattice is generated by three primitive vectors of equal length that make equal angles with one another.^[1]

"Rhombohedral crystal system" is an ambiguous term that confuses the trigonal crystal system with the rhombohedral lattice system and may mean either of them (or even the hexagonal crystal family).

In the classification into 6 crystal families, the trigonal crystal system is combined with the hexagonal crystal system and grouped into a larger hexagonal family.^[2]

Rhombohedral lattice system

There are two descriptions (settings) of the rhombohedral lattice system.

Hexagonal axes. The unit cell is a = b ≠ c; α = β = 90°, γ = 120°. Two additional lattice points occupy space diagonal of the unit cell and have 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.
Rhombohedral axes. The unit cell is a rhombohedron (which gives the name for rhombohedral lattice system). This is a primitive unit cell (no additional lattice points inside unit cell) with parameters a = b = c; α = β = γ ≠ 90°.

In practice, the hexagonal description is more commonly used because it is easier to deal with coordinate system with two 90° angles. However, the rhombohedral axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. The relation between two settings is given below.

The rhombohedral lattice system is combined with the hexagonal lattice system and grouped into a larger hexagonal family. These lattice systems belong to hexagonal family because the same (hexagonal) unit cell can be used for both of them.

Crystal classes

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.)^[3]^[2]^[4]

#	Point group					Examples	Space group
#	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, quartz, 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

References

^ Ashcroft, Neil W.; Mermin, N. David, 1976, "Solid State Physics," 1st ed., p. 119 ISBN 0-03-083993-9
^ ^a ^b Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., pp. 78–89 ISBN 0-471-80580-7
^ Frederick H. Pough, Roger Tory Peterson (1998). A Field Guide to Rocks and Minerals. Houghton Mifflin Harcourt. p. 62. ISBN 0-395-91096-X.
^ Crystallography and Minerals Arranged by Crystal Form, Webmineral

External links

Mineralogy Database

UpToDate Contents

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

1. 非神経性の下部尿路症状の治療におけるボツリヌス毒素の使用 use of botulinum toxin for treatment of non neurogenic lower urinary tract conditions
2. 尿管瘤 ureterocele
3. 尿路上皮膀胱癌に対する根治的膀胱全摘除術および膀胱温存治療 radical cystectomy and bladder sparing treatments for urothelial bladder cancer

English Journal

Entrapped elemental selenium nanoparticles affect physicochemical properties of selenium fed activated sludge.

Jain R1, Seder-Colomina M2, Jordan N3, Dessi P4, Cosmidis J5, van Hullebusch ED6, Weiss S3, Farges F5, Lens PN4.
Journal of hazardous materials.J Hazard Mater.2015 Sep 15;295:193-200. doi: 10.1016/j.jhazmat.2015.03.043. Epub 2015 Mar 23.
Selenite containing wastewaters can be treated in activated sludge systems, where the total selenium is removed from the wastewater by the formation of elemental selenium nanoparticles, which are trapped in the biomass. No studies have been carried out so far on the characterization of selenium fed
PMID 25919502
PMID 25835377

Fluorescent mixed ligand copper(ii) complexes of anthracene-appended Schiff bases: studies on DNA binding, nuclease activity and cytotoxicity.

Jaividhya P1, Ganeshpandian M, Dhivya R, Akbarsha MA, Palaniandavar M.
Dalton transactions (Cambridge, England : 2003).Dalton Trans.2015 Jul 14;44(26):11997-2010. doi: 10.1039/c5dt00899a. Epub 2015 Jun 15.
A series of mixed ligand copper(ii) complexes of the type [Cu(L)(phen)(ACN)](ClO4)21-5, where L is a bidentate Schiff base ligand (N(1)-(anthracen-10-ylmethylene)-N(2)-methylethane-1,2-diamine (L1), N(1)-(anthracen-10-ylmethylene)-N(2),N(2)-dimethylethane-1,2-diamine (L2), N(1)-(anthracen-10-yl-meth
PMID 26076117

Japanese Journal

Flexibility of the coordination geometry at the N-site of Cu(II)2 human serum-transferrin induced by the different orientations of Arg124.

, , , , , , , ,
Biological and Pharmaceutical Bulletin advpub(0), 2015
… In conformations I and II, the residue of Arg124 at the N-site is located either far from or near the copper-binding site, respectively, and in both cases the coordination geometry of the cupric ions at the N-site has changed from a flattened tetrahedron to a trigonal bipyramid. …
NAID 130004872292

Flexibility of the Coordination Geometry at the N-Site of Cu(II)2 Human Serum-Transferrin Induced by the Different Orientations of Arg124

, , , , , , , ,
Biological and Pharmaceutical Bulletin 38(3), 358-364, 2015
… In conformations I and II, the residue of Arg124 at the N-site is located either far from or near the copper-binding site, respectively, and in both cases the coordination geometry of the cupric ions at the N-site has changed from a flattened tetrahedron to a trigonal bipyramid. …
NAID 130004872273

Resonance enhancement of first- and second-order coherent phonons in metallic single-walled carbon nanotubes

Sato K.,Tahara K.,Minami Y.,Katayama I.,Kawai H.,Yanagi K.,Takeda J.
PHYSICAL REVIEW B 90(23), 2014-12-29
… These results indicate that the enhancement originates from a Stokes-stimulatedRaman-scattering process at van Hove singularities and that efficient resonance enhancement occurs for the 2Dmode, possibly through double resonance due to the trigonal warping effect and strong electron-phonon couplingdue to the Kohn anomaly. …
NAID 120005549812