Lattice 1 8 12

Return to site

Lattice 1 8 12

Lattice 1 8 12 Download
1.8 Download
Lattice 1 8 12 5 4
 (Redirected from Hexagonal lattice system)
Crystal systemTrigonalHexagonal
Lattice system
Rhombohedral
Hexagonal
Example
Dolomite
Cinnabar
Beryl
1.8: 1/12/2015: ZIP: 477.7 KB: SDR SDRAM Controller - Documentation RD1174: 1.1: 3/1/2014: PDF: 1.4 MB: SD Flash Controller Using SD Bus - Source Code. (OLB file type) for all Lattice products. This.zip file also includes a.xls worksheet with a list of the contents of the OLB. These symbols can be used to help with OrCAD schematic designs.
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).
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.
Lattice systems[edit]Lattice multiplication is also known as Italian multiplication, Gelosia multiplication, sieve multiplication, shabakh, Venetian squares, or the Hindu lattice.1 It uses a grid with diagonal lines to help the student break up a multiplication problem into smaller steps. Lattice works with People teams across the globe to turn managers into leaders, employees into high-performers, and companies into the best places to work. See open positions. It's more than words. It's people and results: 100%. 96% recommended to a friend.
The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral/trigonal[2]. Each lattice system consists of one Bravais lattice.
Relation between the two settings for the rhombohedral lattice
Hexagonal crystal familyBravais latticeHexagonalRhombohedral
Pearson symbolhPhR
Hexagonal
unit cell
Rhombohedral
unit cell
In the hexagonal family, the crystal is conventionally described by a right rhombicprism 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). Chameleon 0 95. However, such a description is rarely used.
Crystal systems[edit]Crystal systemRequired symmetries of point groupPoint groupsSpace groupsLattice system
Trigonal1 threefold axis of rotation57Rhombohedral
18Hexagonal
Hexagonal1 sixfold axis of rotation727
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.
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. Graphite is an example of a crystal that crystallizes in the hexagonal crystal system.
Crystal classes[edit]Trigonal crystal system[edit]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.[8][9][10]
Space group no.Point groupTypeExamplesSpace groups
Name[11]IntlSchoen.Orb.Cox.HexagonalRhombohedral
143–146Trigonal pyramidal3C333[3]+carlinite, jarositeP3, P31, P32R3
147–148Rhombohedral3C3i (S6)3×[2+,6+]centrosymmetricdolomite, ilmeniteP3R3
149–155Trigonal trapezohedral32D3223[2,3]+enantiomorphicabhurite, alpha-quartz (152, 154), cinnabarP312, P321, P3112, P3121, P3212, P3221R32
156–161Ditrigonal pyramidal3mC3v*33[3]polarschorl, cerite, tourmaline, alunite, lithium tantalateP3m1, P31m, P3c1, P31cR3m, R3c
162–167Ditrigonal scalenohedral3mD3d2*3[2+,6]centrosymmetricantimony, hematite, corundum, calcite, bismuthP31m, P31c, P3m1, P3c1R3m, R3c
Hexagonal crystal system[edit]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][12]
Space group no.Point groupTypeExamplesSpace groups
Name[11]IntlSchoen.Orb.Cox.
168–173Hexagonal pyramidal6C666[6]+nepheline, cancriniteP6, P61, P65, P62, P64, P63
174Trigonal dipyramidal6C3h3*[2,3+]laurelite and boric acidP6
175–176Hexagonal dipyramidal6/mC6h6*[2,6+]centrosymmetricapatite, vanadiniteP6/m, P63/m
177–182Hexagonal trapezohedral622D6226[2,6]+enantiomorphickalsilite and high quartzP622, P6122, P6522, P6222, P6422, P6322
183–186Dihexagonal pyramidal6mmC6v*66[6]polargreenockite, wurtzite[13]P6mm, P6cc, P63cm, P63mc
187–190Ditrigonal dipyramidal6m2D3h*223[2,3]benitoiteP6m2, P6c2, P62m, P62c
191–194Dihexagonal dipyramidal6/mmmD6h*226[2,6]centrosymmetricberylP6/mmm, P6/mcc, P63/mcm, P63/mmc
Lattice 1 8 12 DownloadHexagonal close packed[edit]Hexagonal close packed (hcp) unit cell
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]
Rhombohedral lattice angle[edit]The lattice angles and the lengths of the lattice vectors are all the same for both the cubic and rhombohedral lattice systems. The lattice angles for simple cubic, face-centered cubic, and body-centered cubic lattices are π/2 radians, π/3 radians, and arccos(−1/3) radians, respectively.[15] A rhombohedral lattice will result from lattice angles other than these.
See also[edit]Close-packing (hcp)
References[edit]^ abDana, James Dwight; Hurlbut, Cornelius Searle (1959). Dana's Manual of Mineralogy (17th ed.). New York: Chapman Hall. pp. 78–89.
^https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Book%3A_Physical_Methods_in_Chemistry_and_Nano_Science_(Barron)/07%3A_Molecular_and_Solid_State_Structure/7.01%3A_Crystal_Structure
^Edward Prince (2004). Mathematical Techniques in Crystallography and Materials Science. Springer Science & Business Media. p. 41.
^'Page not found - QuantumWise'. quantumwise.com.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. ISBN0-03-083993-9.
^Hahn (2002), p. 73
^Pough, Frederick H.; Peterson, Roger Tory (1998). A Field Guide to Rocks and Minerals. Houghton Mifflin Harcourt. p. 62. ISBN0-395-91096-X.
^Hurlbut, Cornelius S.; Klein, Cornelis (1985). Manual of Mineralogy (20th ed.). pp. 78–89. ISBN0-471-80580-7.
^'Crystallography and Minerals Arranged by Crystal Form'. Webmineral.
^ abHahn (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.
^Hahn (2002), p. 747
Further reading[edit]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. ISBN978-0-7923-6590-7.CS1 maint: ref=harv (link)
External links[edit] Media related to Hexagonal lattices at Wikimedia Commons
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Hexagonal_crystal_family&oldid=982586015'
 (Redirected from Hexagonal lattice system)
Crystal systemTrigonalHexagonal
Lattice system
Rhombohedral
Hexagonal
Example
Dolomite
Cinnabar
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).
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.
Lattice systems[edit]The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral/trigonal[2]. Each lattice system consists of one Bravais lattice.
Relation between the two settings for the rhombohedral lattice
Hexagonal crystal familyBravais latticeHexagonalRhombohedral
Pearson symbolhPhR
Hexagonal
unit cell
Rhombohedral
unit cell
In the hexagonal family, the crystal is conventionally described by a right rhombicprism 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. Cleanmymac x 4 4 6 x 4. 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.

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

Bravais lattice	Hexagonal	Rhombohedral
Pearson symbol	hP	hR
Hexagonal unit cell
Rhombohedral unit cell

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

Space group no.	Point group	Type	Examples	Space groups
Name^[11]	Intl	Schoen.	Orb.	Cox.	Hexagonal	Rhombohedral
143–146	Trigonal pyramidal	3	C₃	33	[3]⁺	carlinite, jarosite	P3, P3₁, P3₂	R3
147–148	Rhombohedral	3	C_3i (S₆)	3×	[2⁺,6⁺]	centrosymmetric	dolomite, ilmenite	P3	R3
149–155	Trigonal trapezohedral	32	D₃	223	[2,3]⁺	enantiomorphic	abhurite, alpha-quartz (152, 154), cinnabar	P312, P321, P3₁12, P3₁21, P3₂12, P3₂21	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

Space group no.	Point group	Type	Examples	Space groups
Name^[11]	Intl	Schoen.	Orb.	Cox.
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⁺]	centrosymmetric	apatite, vanadinite	P6/m, P6₃/m
177–182	Hexagonal trapezohedral	622	D₆	226	[2,6]⁺	enantiomorphic	kalsilite and high quartz	P622, P6₁22, P6₅22, P6₂22, P6₄22, P6₃22
183–186	Dihexagonal pyramidal	6mm	C_6v	*66	[6]	polar	greenockite, wurtzite^[13]	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]	centrosymmetric	beryl	P6/mmm, P6/mcc, P6₃/mcm, P6₃/mmc

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

Bravais lattice	Hexagonal	Rhombohedral
Pearson symbol	hP	hR
Hexagonal unit cell
Rhombohedral unit cell

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[edit]Crystal systemRequired symmetries of point groupPoint groupsSpace groupsLattice system
Trigonal1 threefold axis of rotation57Rhombohedral
18Hexagonal
Hexagonal1 sixfold axis of rotation727
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.
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. Graphite is an example of a crystal that crystallizes in the hexagonal crystal system.
1.8 DownloadCrystal classes[edit]Trigonal crystal system[edit]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.
How to crack autocad 2017. 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 groupTypeExamplesSpace groups
Name[11]IntlSchoen.Orb.Cox.HexagonalRhombohedral
143–146Trigonal pyramidal3C333[3]+carlinite, jarositeP3, P31, P32R3
147–148Rhombohedral3C3i (S6)3×[2+,6+]centrosymmetricdolomite, ilmeniteP3R3
149–155Trigonal trapezohedral32D3223[2,3]+enantiomorphicabhurite, alpha-quartz (152, 154), cinnabarP312, P321, P3112, P3121, P3212, P3221R32
156–161Ditrigonal pyramidal3mC3v*33[3]polarschorl, cerite, tourmaline, alunite, lithium tantalateP3m1, P31m, P3c1, P31cR3m, R3c
162–167Ditrigonal scalenohedral3mD3d2*3[2+,6]centrosymmetricantimony, hematite, corundum, calcite, bismuthP31m, P31c, P3m1, P3c1R3m, R3c
Hexagonal crystal system[edit]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][12]
Space group no.Point groupTypeExamplesSpace groups
Name[11]IntlSchoen.Orb.Cox.
168–173Hexagonal pyramidal6C666[6]+nepheline, cancriniteP6, P61, P65, P62, P64, P63
174Trigonal dipyramidal6C3h3*[2,3+]laurelite and boric acidP6
175–176Hexagonal dipyramidal6/mC6h6*[2,6+]centrosymmetricapatite, vanadiniteP6/m, P63/m
177–182Hexagonal trapezohedral622D6226[2,6]+enantiomorphickalsilite and high quartzP622, P6122, P6522, P6222, P6422, P6322
183–186Dihexagonal pyramidal6mmC6v*66[6]polargreenockite, wurtzite[13]P6mm, P6cc, P63cm, P63mc
187–190Ditrigonal dipyramidal6m2D3h*223[2,3]benitoiteP6m2, P6c2, P62m, P62c
191–194Dihexagonal dipyramidal6/mmmD6h*226[2,6]centrosymmetricberylP6/mmm, P6/mcc, P63/mcm, P63/mmc
Hexagonal close packed[edit]Hexagonal close packed (hcp) unit cell
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]
Rhombohedral lattice angle[edit]The lattice angles and the lengths of the lattice vectors are all the same for both the cubic and rhombohedral lattice systems. The lattice angles for simple cubic, face-centered cubic, and body-centered cubic lattices are π/2 radians, π/3 radians, and arccos(−1/3) radians, respectively.[15] A rhombohedral lattice will result from lattice angles other than these.
See also[edit]Close-packing (hcp)
References[edit]^ abDana, James Dwight; Hurlbut, Cornelius Searle (1959). Dana's Manual of Mineralogy (17th ed.). New York: Chapman Hall. pp. 78–89.
^https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Book%3A_Physical_Methods_in_Chemistry_and_Nano_Science_(Barron)/07%3A_Molecular_and_Solid_State_Structure/7.01%3A_Crystal_Structure
^Edward Prince (2004). Mathematical Techniques in Crystallography and Materials Science. Springer Science & Business Media. p. 41.
^'Page not found - QuantumWise'. quantumwise.com.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. ISBN0-03-083993-9.
^Hahn (2002), p. 73
^Pough, Frederick H.; Peterson, Roger Tory (1998). A Field Guide to Rocks and Minerals. Houghton Mifflin Harcourt. p. 62. ISBN0-395-91096-X.
^Hurlbut, Cornelius S.; Klein, Cornelis (1985). Manual of Mineralogy (20th ed.). pp. 78–89. ISBN0-471-80580-7.
^'Crystallography and Minerals Arranged by Crystal Form'. Webmineral.
^ abHahn (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.
^Hahn (2002), p. 747
Lattice 1 8 12 5 4Further reading[edit]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. ISBN978-0-7923-6590-7.CS1 maint: ref=harv (link)
External links[edit] Media related to Hexagonal lattices at Wikimedia Commons
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Hexagonal_crystal_family&oldid=982586015'