面心立方格子

WordNet

of or belonging to neither the right nor the left politically or intellectually
a building dedicated to a particular activity; "they were raising money to build a new center for research" (同)centre
(American football) the position of the player on the line of scrimmage who puts the ball in play; "it is a centers responsibility to get the football to the quarterback"
a position on a basketball team of the player who participates in the jump that starts the game
the position on a hockey team of the player who participates in the face off at the beginning of the game
a place where some particular activity is concentrated; "they received messages from several centers" (同)centre
the sweet central portion of a piece of candy that is enclosed in chocolate or some other covering (同)centre
the object upon which interest and attention focuses; "his stories made him the center of the party" (同)centre, center of attention, centre of attention
an area that is approximately central within some larger region; "it is in the center of town"; "they ran forward into the heart of the struggle"; "they were in the eye of the storm" (同)centre, middle, heart, eye
a point equidistant from the ends of a line or the extremities of a figure (同)centre, midpoint
a cluster of nerve cells governing a specific bodily process; "in most people the speech center is in the left hemisphere" (同)centre, nerve_center, nerve centre
(football) the person who plays center on the line of scrimmage and snaps the ball to the quarterback; "the center fumbled the handoff" (同)snapper
(basketball) the person who plays center on a basketball team
(ice hockey) the person who plays center on a hockey team
politically moderate persons; centrists
the middle of a military or naval formation; "they had to reinforce the center"
move into the center; "That vase in the picture is not centered" (同)centre
equally distant from the extremes (同)halfway, middle, midway
turn so as to face; turn the face in a certain direction; "Turn and face your partner now"
a vertical surface of a building or cliff
the general outward appearance of something; "the face of the city is changing"
the side upon which the use of a thing depends (usually the most prominent surface of an object); "he dealt the cards face down"
the striking or working surface of an implement
the front of the human head from the forehead to the chin and ear to ear; "he washed his face"; "I wish I had seen the look on his face when he got the news" (同)human face
a part of a person that is used to refer to a person; "he looked out at a roomful of faces"; "when he returned to work he met many new faces"
status in the eyes of others; "he lost face"
the part of an animal corresponding to the human face
be opposite; "the facing page"; "the two sofas face each other"
cover the front or surface of; "The building was faced with beautiful stones"
line the edge (of a garment) with a different material; "face the lapels of the jacket"
turn so as to expose the face; "face a playing card"
give birth to cubs; "bears cub every year"
the young of certain carnivorous mammals such as the bear or wolf or lion (同)young carnivore
an awkward and inexperienced youth (同)greenhorn, rookie
a male child (a familiar term of address to a boy) (同)lad, laddie, sonny, sonny boy
a protective covering that protects the outside of a building (同)cladding
an ornamental coating to a building (同)veneer
a lining applied to the edge of a garment for ornamentation or strengthening
framework consisting of an ornamental design made of strips of wood or metal (同)latticework, fretwork
an arrangement of points or particles or objects in a regular periodic pattern in 2 or 3 dimensions

PrepTutorEJDIC

〈C〉『顔』 / 〈C〉表情,顔つき / 〈C〉(物の)『表面』(surface),(建物の)正面(front),(貨幣・カードなどの)表《+『of』+『名』》 / 〈C〉(…の)外観,様子《+『of』+『名』》 / 〈U〉面目,面子(めんつ) / 〈C〉(多角面体の)面 / 〈C〉(活字・版の)面(印刷する部分);(活字の)書体 / 〈C〉(鉱山の)採掘現場 / …‘に'『面する』,‘の'ほうを向く / (…のほうに)…‘を'向ける《+『名』+『toward』+『名』》 / 〈人が〉〈危険など〉‘に'直面する,立ち向かう,対抗する;〈危険などが〉〈人〉‘に'迫る / 〈事実・現実など〉‘を'直視する,認める / (…で)〈壁など〉‘に'上塗り(上張り)をする《+『名』+『with』+『名』》 / (ある場所・方角に)面する,向く《+『on』(『to』,『toward』)+『名』》
(キツネ・クマ・ライオンなど)野獣の子 / (しつけの悪い)子ども,小僧 / 新米,見習い / =cub scout
〈U〉(壁などの)表面仕上げ,化粧張り / 〈U〉(着年の)へり取り,へり飾り / 《複数形で》(軍服のえりやそで口につけ,兵種・職種を示す)縫い取り,定色
格子(こうし) / 格子窓,格子の戸(門) / …‘を'格子模様(造り)にする / …‘に'格子を取り付ける

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/07/19 16:45:28」(JST)

wiki en

A rock containing three crystals of pyrite (FeS₂). The crystal structure of pyrite is primitive cubic, and this is reflected in the cubic symmetry of its natural crystal facets.

A network model of a primitive cubic system.

The primitive and cubic close-packed (also known as face-centered cubic) unit cells.

In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.

There are three main varieties of these crystals:

Primitive cubic (abbreviated cP^[1] and alternatively called simple cubic)
Body-centered cubic (abbreviated cI^[1] or bcc),
Face-centered cubic (abbreviated cF^[1] or fcc, and alternatively called cubic close-packed or ccp)

Each is subdivided into other variants listed below. Note that although the unit cell in these crystals is conventionally taken to be a cube, the primitive unit cell often is not. This is related to the fact that in most cubic crystal systems, there is more than one atom per cubic unit cell.

Cubic space groups[edit]

The three Bravais lattices which form cubic crystal systems are:

Cubic Bravais lattices
Name	Primitive cubic	Body-centered cubic	Face-centered cubic
Pearson symbol	cP	cI	cF
Unit cell

The primitive cubic system (cP) consists of one lattice point on each corner of the cube. Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom (¹⁄₈ × 8).
The body-centered cubic system (cI) has one lattice point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell (¹⁄₈ × 8 + 1).
The face-centered cubic system (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell (¹⁄₈ × 8 from the corners plus ¹⁄₂ × 6 from the faces).

The face-centered cubic system is closely related to the hexagonal close packed system, and the two systems differ only in the relative placements of their hexagonal layers. The [111] plane of a face-centered cubic system is a hexagonal grid.

Attempting to create a C-centered cubic crystal system (i.e., putting an extra lattice point in the center of each horizontal face) would result in a simple tetragonal Bravais lattice.

Crystal classes[edit]

The isometric crystal system class names, examples, Schönflies notation, Hermann-Mauguin notation, point groups, International Tables for Crystallography space group number,^[2] orbifold, type, and space groups are listed in the table below. There are a total 36 cubic space groups.

#	Point group					Example	Type	Space groups
#	Class^[3]	Schönflies	Intl	Orbifold	Coxeter	Example	Type	Space groups
195-199	Tetartoidal	T	23	332	[3,3]⁺	Ullmannite	enantiomorphic	P23	F23	I23	P2₁3	I2₁3
200-206	Diploidal	T_h	2/m3 (m3)	3*2	[3⁺,4]	Pyrite	centrosymmetric	Pm3	Pn3	Fm3	Fd3	I3	Pa3	Ia3
207-214	Gyroidal	O	432	432	[3,4]⁺	Petzite	enantiomorphic	P432	P4₂32	F432	F4₁32	I432	P4₃32	P4₁32	I4₁32
215-220	Hextetrahedral	T_d	43m	*332	[3,3]	Sphalerite		P43m	F43m	I43m	P43n	F43c	I43d
221-230	Hexoctahedral	O_h	4/m32/m (m3m)	*432	[3,4]	Galena	centrosymmetric	Pm3m	Pn3n	Pm3n	Pn3m	Fm3m	Fm3c	Fd3m	Fd3c	Im3m	Ia3d

Other terms for hexoctahedral are: normal class, holohedral, ditesseral central class, galena type.

Voids in the unit cell[edit]

A simple cubic unit cell has a single cubic void in the center.

A body-centered cubic unit cell has six octahedral voids located at the center of each face of the unit cell, for a total of three net octahedral voids. Additionally, there are 36 tetrahedral voids located in an octahedral spacing around each octahedral void, for a total of eighteen net tetrahedral voids. These tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells.

A face-centered cubic unit cell has eight tetrahedral voids located slightly towards the center from each corner of the unit cell, for a total of eight net tetrahedral voids. Additionally, there are twelve octahedral voids located at the center of edge of the unit cell as well as one octahedral hole in the very center, for a total of four net octahedral voids.

Atomic packing factors and examples[edit]

One important characteristic of a crystalline structure is its atomic packing factor. This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts the next. The atomic packing factor is the proportion of space filled by these spheres.

Assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be ^a⁄₂ and the atomic packing factor turns out to be about 0.524 (which is quite low). Similarly, in a bcc lattice, the atomic packing factor is 0.680, and in fcc it is 0.740. The fcc value is the highest theoretically possible value for any lattice, although there are other lattices which also achieve the same value, such as hexagonal close packed and one version of tetrahedral bcc.

As a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common. (Loosely packed arrangements do occur, though, for example if the orbital hybridization demands certain bond angles.) Accordingly, the primitive-cubic structure, with especially low atomic packing factor, is rare in nature, but is found in polonium.^[4] The bcc and fcc, with their higher densities, are both quite common in nature. Examples of bcc include iron, chromium, tungsten, and niobium. Examples of fcc include aluminium, copper, gold and silver.

Multi-element compounds[edit]

Compounds that consist of more than one element (e.g. binary compounds) often have crystal structures based on a cubic crystal system. Some of the more common ones are listed here.

Interpenetrating primitive cubic (caesium chloride) structure[edit]

A caesium chloride unit cell. The two colors of spheres represent the two types of atoms.

One structure is the "interpenetrating primitive cubic" structure, also called the "caesium chloride" structure. Each of the two atom types forms a separate primitive cubic lattice, with an atom of one type at the center of each cube of the other type. Altogether, the arrangement of atoms is the same as body-centered cubic, but with alternating types of atoms at the different lattice sites. (See picture here.) Alternately, one could view this lattice as a simple cubic structure with a secondary atom in its cubic void.

Examples of compounds with this structure include caesium chloride itself, as well as certain other alkali halides when prepared at low temperatures or high pressures.^[5] More generally, this structure is more likely to be formed from two elements whose ions are of roughly the same size (for example, ionic radius of Cs⁺ = 167 pm, and Cl⁻ = 181 pm) .

The space group of this structure is called Pm3m (in Hermann–Mauguin notation), or "221" (in the International Tables for Crystallography). The Strukturbericht designation is "B2".^[6]

The rock-salt crystal structure. Each atom has six nearest neighbors, with octahedral geometry.

The coordination number of each atom in the structure is 8: the central cation is coordinated to 8 anions on the corners of a cube as shown, and similarly, the central anion is coordinated to 8 cations on the corners of a cube.

Rock-salt structure[edit]

Another structure is the "rock salt" or "sodium chloride" (halite) structure. In this, each of the two atom types forms a separate face-centered cubic lattice, with the two lattices interpenetrating so as to form a 3D checkerboard pattern. (See picture here.) Alternately, one could view this structure as a face-centered cubic structure with secondary atoms in its octahedral holes.

Examples of compounds with this structure include sodium chloride itself, along with almost all other alkali halides, and "many divalent metal oxides, sulfides, selenides, and tellurides".^[5] More generally, this structure is more likely to be formed if the cation is slightly smaller than the anion (a cation/anion radius ratio of 0.414 to 0.732).

The space group of this structure is called Fm3m (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht designation is "B1".^[7]

The coordination number of each atom in this structure is 6: each cation is coordinated to 6 anions at the vertices of an octahedron, and similarly, each anion is coordinated to 6 cations at the vertices of an octahedron.

The interatomic distance (distance between cation and anion, or half the unit cell length a) in some rock-salt-structure crystals are: 2.3 Å (2.3 × 10⁻¹⁰ m) for NaF,^[8] 2.8 Å for NaCl,^[9] and 3.2 Å for SnTe.^[10]

Zincblende structure[edit]

A zincblende unit cell

Another common structure is the "zincblende" structure (also spelled "zinc blende"), named after the mineral zincblende (sphalerite). As in the rock-salt structure, the two atom types form two interpenetrating face-centered cubic lattices. However, it differs from rock-salt structure in how the two lattices are positioned relative to one another. The zincblende structure has tetrahedral coordination: Each atom's nearest neighbors consist of four atoms of the opposite type, positioned like the four vertices of a regular tetrahedron. Altogether, the arrangement of atoms in zincblende structure is the same as diamond cubic structure, but with alternating types of atoms at the different lattice sites. (See picture here.)

Examples of compounds with this structure include zincblende itself, lead(II) nitrate, many compound semiconductors (such as gallium arsenide and cadmium telluride), and a wide array of other binary compounds.

Additionally, if one were to take the two atoms to both be carbon, the zincblende structure is equivalent to the structure of diamond.

The space group of this structure is called F43m (in Hermann–Mauguin notation), or "#216" in the International Tables for Crystallography.^[11]^[12] The Strukturbericht designation is "B3".^[13]

References[edit]

^ ^a ^b ^c International Tables for Crystallography (2006). Vol. A, Section 2.1.3, pp. 14–16. Link
^ Prince, E., ed. (2006). International Tables for Crystallography. International Union of Crystallography. doi:10.1107/97809553602060000001. ISBN 978-1-4020-4969-9.
^ Crystallography and Minerals Arranged by Crystal Form, Webmineral
^ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth–Heinemann. ISBN 0080379419. . The original discovery was in J. Chem. Phys. 14, 569 (1946), web link. Also see this more recent non-technical article.
^ ^a ^b Seitz, Modern Theory of Solids (1940), p.49
^ The CsCl (B2) Structure
^ The NaCl (B1) Structure
^ Sundquist, J. J.; Lin, C. C. (1981). "Electronic structure of the F centre in a sodium fluoride crystal". Journal of Physics C: Solid State Physics 14 (32): 4797–4805. Bibcode:1981JPhC...14.4797S. doi:10.1088/0022-3719/14/32/016. edit
^ Abrahams, S. C.; Bernstein, J. L. (1965). "Accuracy of an automatic diffractometer. Measurement of the sodium chloride structure factors". Acta Cryst. 18 (5): 926–932. doi:10.1107/S0365110X65002244.
^ Kao, W.; Peretti, E. (1970). "The ternary subsystem Sn4As3-SnAs-SnTe". Journal of the Less Common Metals 22: 39–50. doi:10.1016/0022-5088(70)90174-8. edit
^ L. Kantorovich (2004). Quantum Theory of the Solid State. Springer. p. 32. ISBN 1-4020-2153-4.
^ Birkbeck College, University of London
^ The Zincblende (B3) Structure

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. 網膜剥離 retinal detachment
2. 分娩時の顔位および額位 face and brow presentations in labor
3. 医師と患者の関係における患者中心の考え方 a patient centered view of the clinician patient relationship
4. 手掌および足底の色素性病変のダーモスコピー dermoscopy of pigmented lesions of the palms and soles
5. インフォームドコンセント informed consent

English Journal

Structural Evolution and Stability of Hydrogenated Li(n) (n = 1-30) Clusters: A Density Functional Study.

Gautam S, Dharamvir K, Goel N.SourceDepartment of Physics & Center of Advanced Studies in Physics and ‡Department of Chemistry & Center of Advanced Studies in Chemistry, Panjab University , Chandigarh-160014, India.
The journal of physical chemistry. A.J Phys Chem A.2011 Jun 23;115(24):6383-9. Epub 2011 May 27.
The structure and electronic and optical properties of hydrogenated lithium clusters Li(n)H(m) (n = 1-30, m ? n) have been investigated by density functional theory (DFT). The structural optimizations are performed with the Becke 3 Lee-Yang-Parr (B3LYP) exchange-correlation functional with 6-311G+
PMID 21568308

Helium-vacancy cluster in a single bcc iron crystal lattice.

Gao N, Victoria M, Chen J, Van Swygenhoven H.SourceNES-High Temperature Materials, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland. Ecole Polytechnique Federale de Lausanne (EPFL), Institute of Materials (IMX), CH-1015 Lausanne, Switzerland.
Journal of physics. Condensed matter : an Institute of Physics journal.J Phys Condens Matter.2011 Jun 22;23(24):245403. Epub 2011 Jun 1.
The properties of the cluster He(n)V, an iron vacancy with an increasing number of He atoms, is studied with molecular statics and molecular dynamics simulations. A study of the binding energy of the self-interstitial atom?(SIA) and the He, shows that from n = 6 the He(n)V cluster is stable and can
PMID 21628782

Japanese Journal

垂直磁気記録媒体のための擬似六方晶薄膜の開発(媒体,一般)

斉藤伸,高橋研
映像情報メディア学会技術報告 34(26), 17-24, 2010-07-08
実験室設置規模のX線回折装置により擬似六方晶薄膜中の積層欠陥の定量評価法を提案した.擬似六方晶構造とは(111)面配向面心立方構造,c面配向六法晶構造(以下fcc,hcpと略記),ならびにこれらの中間構造で配向面に平行に積層欠陥が導入された構造を指す.擬似六方晶薄膜のインプレーンX線回折プロファイルには,2本の回折線(高角側:D_H,低角側:D_L)が現れる.構造因子によるとD_Hは結晶粒内の最密 …
NAID 110007701219

垂直磁気記録媒体のための擬似六方晶薄膜の開発(媒体,一般)

斉藤伸,高橋研
電子情報通信学会技術研究報告. MR, 磁気記録 110(112), 17-24, 2010-07-01
実験室設置規模のX線回折装置により擬似六方晶薄膜中の積層欠陥の定量評価法を提案した.擬似六方晶構造とは(111)面配向面心立方構造,c面配向六法晶構造(以下fcc, hcpと略記),ならびにこれらの中間構造で配向面に平行に積層欠陥が導入された構造を指す.擬似六方晶薄膜のインプレーンX線回折プロファイルには,2本の回折線(高角側:D_H,低角側:D_L)が現れる.構造因子によるとD_Hは結晶粒内の最 …
NAID 110008094865