結晶系、晶系

WordNet

(physical chemistry) a sample of matter in which substances in different phases are in equilibrium; "in a static system oil cannot be replaced by water on a surface"; "a system generating hydrogen peroxide"
instrumentality that combines interrelated interacting artifacts designed to work as a coherent entity; "he bought a new stereo system"; "the system consists of a motor and a small computer"
a group of independent but interrelated elements comprising a unified whole; "a vast system of production and distribution and consumption keep the country going" (同)scheme
a complex of methods or rules governing behavior; "they have to operate under a system they oppose"; "that language has a complex system for indicating gender" (同)system of rules
a group of physiologically or anatomically related organs or parts; "the body has a system of organs for digestion"
a procedure or process for obtaining an objective; "they had to devise a system that did not depend on cooperation"
the living body considered as made up of interdependent components forming a unified whole; "exercise helped him get the alcohol out of his system"
a solid formed by the solidification of a chemical and having a highly regular atomic structure
a protective cover that protects the face of a watch (同)watch crystal, watch_glass
a crystalline element used as a component in various electronic devices
glassware made of quartz
a rock formed by the solidification of a substance; has regularly repeating internal structure; external plane faces (同)crystallization

PrepTutorEJDIC

〈C〉(関連した部分から成る)『体系』,系統,組織[網],装置 / 〈C〉(教育・政治などの)『制度』,機構;《the~》体制 / 〈C〉(思想・学問などの)『体系』,学説 / 〈C〉(…の)『方法』,方式,やり方《+of doing》 / 〈U〉正しい方針(筋道,順序) / 〈U〉《the~》(身体の)組織,系統 / 〈U〉《the~,one's~》身体,全身
〈U〉『水晶』 / 〈C〉『水晶製品』,水晶玉 / 〈U〉クリスタルグラス(透明度の高い鉛ガラス);《集合的に》クリスタルグラス製食器類(特にコップ類) / 〈C〉『結晶[体]』 / 〈C〉《米》(時計の)ガラスぶた / 『水晶[製]の』;クリスタルグラス製の;透き通った

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/01/03 02:19:14」(JST)

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The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated 2-atom pattern.

In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.

Crystal systems, crystal families, and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".

Space groups and crystals are divided into 7 crystal systems according to their point groups, and into 7 lattice systems according to their Bravais lattices. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.

Overview

Hexagonal hanksite crystal, with three-fold c-axis symmetry

A lattice system is a class of lattices with the same point group. In three dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. The lattice system of a crystal or space group is determined by its lattice but not always by its point group.

A crystal system is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system, and in these cases the crystal system corresponds to a lattice system and is given the same name. However, for the five point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.

A crystal family also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.

In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called oblique, rectangular, square, and hexagonal.

The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table:

Crystal family	Crystal system	Required symmetries of point group	Point groups	Space groups	Bravais lattices	Lattice system
Triclinic		None	2	2	1	Triclinic
Monoclinic		1 twofold axis of rotation or 1 mirror plane	3	13	2	Monoclinic
Orthorhombic		3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.	3	59	4	Orthorhombic
Tetragonal		1 fourfold axis of rotation	7	68	2	Tetragonal
Hexagonal	Trigonal	1 threefold axis of rotation	5	7	1	Rhombohedral
	Trigonal	1 threefold axis of rotation	5	18	1	Hexagonal
	Hexagonal	1 sixfold axis of rotation	7	27	1	Hexagonal
Cubic		4 threefold axes of rotation	5	36	3	Cubic
Total: 6	7		32	230	14	7

Crystal classes

The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table.

crystal family	crystal system	point group / crystal class	Schönflies	Hermann-Mauguin	Orbifold	Coxeter	Point symmetry	Order	Group structure
triclinic		triclinic-pedial	C₁	1	11	[ ]⁺	enantiomorphic polar	1	trivial
triclinic		triclinic-pinacoidal	C_i	1	1x	[2,1⁺]	centrosymmetric	2	cyclic
monoclinic		monoclinic-sphenoidal	C₂	2	22	[2,2]⁺	enantiomorphic polar	2	cyclic
		monoclinic-domatic	C_s	m	*11	[ ]	polar	2	cyclic
		monoclinic-prismatic	C_2h	2/m	2*	[2,2⁺]	centrosymmetric	4	2×cyclic
orthorhombic		orthorhombic-sphenoidal	D₂	222	222	[2,2]⁺	enantiomorphic	4	dihedral
		orthorhombic-pyramidal	C_2v	mm2	*22	[2]	polar	4	dihedral
		orthorhombic-bipyramidal	D_2h	mmm	*222	[2,2]	centrosymmetric	8	2×dihedral
tetragonal		tetragonal-pyramidal	C₄	4	44	[4]⁺	enantiomorphic polar	4	cyclic
		tetragonal-disphenoidal	S₄	4	2x	[2⁺,2]	non-centrosymmetric	4	cyclic
		tetragonal-dipyramidal	C_4h	4/m	4*	[2,4⁺]	centrosymmetric	8	2×cyclic
		tetragonal-trapezoidal	D₄	422	422	[2,4]⁺	enantiomorphic	8	dihedral
		ditetragonal-pyramidal	C_4v	4mm	*44	[4]	polar	8	dihedral
		tetragonal-scalenoidal	D_2d	42m or 4m2	2*2	[2⁺,4]	non-centrosymmetric	8	dihedral
		ditetragonal-dipyramidal	D_4h	4/mmm	*422	[2,4]	centrosymmetric	16	2×dihedral
hexagonal	trigonal	trigonal-pyramidal	C₃	3	33	[3]⁺	enantiomorphic polar	3	cyclic
		rhombohedral	S₆ (C_3i)	3	3x	[2⁺,3⁺]	centrosymmetric	6	cyclic
		trigonal-trapezoidal	D₃	32 or 321 or 312	322	[3,2]⁺	enantiomorphic	6	dihedral
		ditrigonal-pyramidal	C_3v	3m or 3m1 or 31m	*33	[3]	polar	6	dihedral
		ditrigonal-scalahedral	D_3d	3m or 3m1 or 31m	2*3	[2⁺,6]	centrosymmetric	12	dihedral
	hexagonal	hexagonal-pyramidal	C₆	6	66	[6]⁺	enantiomorphic polar	6	cyclic
		trigonal-dipyramidal	C_3h	6	3*	[2,3⁺]	non-centrosymmetric	6	cyclic
		hexagonal-dipyramidal	C_6h	6/m	6*	[2,6⁺]	centrosymmetric	12	2×cyclic
		hexagonal-trapezoidal	D₆	622	622	[2,6]⁺	enantiomorphic	12	dihedral
		dihexagonal-pyramidal	C_6v	6mm	*66	[6]	polar	12	dihedral
		ditrigonal-dipyramidal	D_3h	6m2 or 62m	*322	[2,3]	non-centrosymmetric	12	dihedral
		dihexagonal-dipyramidal	D_6h	6/mmm	*622	[2,6]	centrosymmetric	24	2×dihedral
cubic		tetrahedral	T	23	332	[3,3]⁺	enantiomorphic	12	alternating
		hextetrahedral	T_d	43m	*332	[3,3]	non-centrosymmetric	24	symmetric
		diploidal	T_h	m3	3*2	[3⁺,4]	centrosymmetric	24	2×alternating
		gyroidal	O	432	432	[4,3]⁺	enantiomorphic	24	symmetric
		hexoctahedral	O_h	m3m	*432	[4,3]	centrosymmetric	48	2×symmetric

Point symmetry can be thought of in the following fashion: consider the coordinates which make up the lattice, and project them all through a single point, so that (x,y,z) becomes (-x,-y,-z). This is the 'reciprocal lattice.' If the lattice and reciprocal lattice are identical, then the crystal is centrosymmetric. If the reciprocal lattice can be rotated to align with the lattice, then the crystal is non-centrosymmetric. If the reciprocal lattice can't be rotated to align with the lattice, that is, with some elements which are a mirror image of the lattice, then the crystal is enantiomorphic.^[1] If rotation of the original lattice reveals an axis where the two ends are different, then the crystal is polar. H₂O is a common example of a polar molecule.

The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 11 enantiomorphic point groups (biological molecules are frequently chiral). The protein assemblies themselves may have symmetries other than those given above, because they are not intrinsically restricted by the Crystallographic restriction theorem. For example the Rad52 DNA binding protein has an 11-fold rotational symmetry (in human), however, it must form crystals in one of the 11 enantiomorphic point groups given above.

Lattice systems

The distribution of the 14 Bravais lattice types into 7 lattice systems is given in the following table.

The 7 lattice systems	The 14 Bravais Lattices
triclinic (parallelepiped)
monoclinic (right prism with parallelogram base; here seen from above)	simple	base-centered

orthorhombic (cuboid)	simple	base-centered	body-centered	face-centered
orthorhombic (cuboid)
tetragonal (square cuboid)	simple	body-centered
tetragonal (square cuboid)
rhombohedral (trigonal trapezohedron)
hexagonal (centered regular hexagon)
cubic (isometric; cube)	simple	body-centered	face-centered
cubic (isometric; cube)

In geometry and crystallography, a Bravais lattice is a category of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices.

Such symmetry groups consist of translations by vectors of the form

where n₁, n₂, and n₃ are integers and a₁, a₂, and a₃ are three non-coplanar vectors, called primitive vectors.

These lattices are classified by space group of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.

All crystalline materials must, by definition fit in one of these arrangements (not including quasicrystals).

For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3 or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.

The Bravais lattices were studied by Moritz Ludwig Frankenheim (1801–1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

Notes

^ Howard D. Flack (2003). "Chiral and Achiral Crystal Structures". Helvetica Chimica Acta 86: 905–921. doi:10.1002/hlca.200390109.

References

Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7. http://it.iucr.org/A/.

External links

Overview of the 32 groups
Mineral galleries – Symmetry
all cubic crystal classes, forms and stereographic projections (interactive java applet)
Crystal system at the Online Dictionary of Crystallography
Crystal family at the Online Dictionary of Crystallography
Lattice system at the Online Dictionary of Crystallography
Conversion Primitive to Standard Conventional for VASP input files
Learning Crystallography

UpToDate Contents

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English Journal

Structure and Strength of Iron-Copper-Carbon Nanotube Nanocomposites.

Boshko O1, Dashevskyi M2, Mykhaliuk O3, Ivanenko K4, Hamamda S5, Revo S6.
Nanoscale research letters.Nanoscale Res Lett.2016 Dec;11(1):78. doi: 10.1186/s11671-016-1298-8. Epub 2016 Feb 9.
Nanocomposite materials of the Fe-Cu system with/without small addition of carbon nanotubes have been synthesized by mechanochemical activation of elemental Fe and Cu powders in a high-energy planetary ball mill and have been examined by the X-ray diffraction method, SEM and the thermopower methods;
PMID 26858160

Concentration- and Temperature-Induced Phase Transitions in PrAlO3-SrTiO3 System.

Vasylechko L1, Stepchuk R2, Prots Y3, Rosner H4.
Nanoscale research letters.Nanoscale Res Lett.2016 Dec;11(1):17. doi: 10.1186/s11671-015-1225-4. Epub 2016 Jan 13.
Single-phase mixed aluminates-titanates Pr1-x Sr x Al1-x Ti x O3 (x = 0.1, 0.2, 0.3, 0.5, 0.7) with rhombohedral perovskite structure were prepared by solid-state reaction technique at 1823-1873 K. Morphotropic rhombohedral-to-cubic phase transition in Pr1-x Sr x Al1-x Ti x O3 series is predict
PMID 26759353

Defining Multiple Characteristic Raman Bands of α-Amino Acids as Biomarkers for Planetary Missions Using a Statistical Method.

Rolfe SM1, Patel MR2,3, Gilmour I2, Olsson-Francis K4, Ringrose TJ2.
Origins of life and evolution of the biosphere : the journal of the International Society for the Study of the Origin of Life.Orig Life Evol Biosph.2016 Jun;46(2-3):323-46. doi: 10.1007/s11084-015-9477-7. Epub 2016 Jan 7.
Biomarker molecules, such as amino acids, are key to discovering whether life exists elsewhere in the Solar System. Raman spectroscopy, a technique capable of detecting biomarkers, will be on board future planetary missions including the ExoMars rover. Generally, the position of the strongest band i
PMID 26744263

Japanese Journal

Relationship between crystal structure and oxide-ion conduction in Ln?Zr?O? (Ln = Eu, Nd and La) system deduced by neutron and X-ray diffraction

HAGIWARA Takeshi,YAMAMURA Hiroshi,NOMURA Katsuhiro
Journal of the Ceramic Society of Japan = 日本セラミックス協会学術論文誌 121(1410), 205-210, 2013-02
NAID 40019560046

Green Fluorescence of Mn-Doped Zn2SiO4 Thin Film Enhanced by Surface Plasmon of Nano-Silver Coating

Peng Kun-Cheng,Lin Yi-Chung,Liu Shiu-Jen,Hsieh Keng-Lin,Tseng Chiun-An,Lin Jing-Chie
Jpn J Appl Phys 52(1), 1-1, 2013-01-25
… One thin film of Mn-doped ZnO was deposited on a quartz glass in a sputtering system where RF magnetron and DC were employed as power sources. … This annealed specimen was identified as Mn-doped Zn2SiO4 (MZS) crystal. …
NAID 150000106300