Physics:Orthorhombic crystal system
In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.
Bravais lattices
Two-dimensional
In two dimensions there are two orthorhombic Bravais lattices: primitive rectangular and centered rectangular.
Bravais lattice | Rectangular | Centered rectangular |
---|---|---|
Pearson symbol | op | oc |
Standard unit cell | ||
Rhombic unit cell |
The primitive rectangular lattice can also be described by a centered rhombic unit cell, while the centered rectangular lattice can also be described by a primitive rhombic unit cell. Note that the length [math]\displaystyle{ a }[/math] in the lower row is not the same as in the upper row. For the first column above, [math]\displaystyle{ a }[/math] of the second row equals [math]\displaystyle{ \sqrt{a^2+b^2} }[/math] of the first row, and for the second column it equals half of this.
Three-dimensional
In three dimensions, there are four orthorhombic Bravais lattices: primitive orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic.
Bravais lattice | Primitive orthorhombic |
Base-centered orthorhombic |
Body-centered orthorhombic |
Face-centered orthorhombic |
---|---|---|---|---|
Pearson symbol | oP | oS | oI | oF |
Standard unit cell | ||||
Right rhombic prism unit cell |
In the orthorhombic system there is a rarely used second choice of crystal axes that results in a unit cell with the shape of a right rhombic prism;^{[1]} it can be constructed because the rectangular two-dimensional base layer can also be described with rhombic axes. In this axis setting, the primitive and base-centered lattices swap in centering type, while the same thing happens with the body-centered and face-centered lattices. Note that the length [math]\displaystyle{ a }[/math] in the lower row is not the same as in the upper row, as can be seen in the figure in the section on two-dimensional lattices. For the first and third column above, [math]\displaystyle{ a }[/math] of the second row equals [math]\displaystyle{ \sqrt{a^2+b^2} }[/math] of the first row, and for the second and fourth column it equals half of this.
Crystal classes
The orthorhombic crystal system class names, examples, Schönflies notation, Hermann-Mauguin notation, point groups, International Tables for Crystallography space group number,^{[2]} orbifold notation, type, and space groups are listed in the table below.
№ | Point group | Type | Example | Space groups | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Name^{[3]} | Schön. | Intl | Orb. | Cox. | Primitive | Base-centered | Face-centered | Body-centered | |||
16–24 | Rhombic disphenoidal | D_{2} (V) | 222 | 222 | [2,2]^{+} | Enantiomorphic | Epsomite | P222, P222_{1}, P2_{1}2_{1}2, P2_{1}2_{1}2_{1} | C222_{1}, C222 | F222 | I222, I2_{1}2_{1}2_{1} |
25–46 | Rhombic pyramidal | C_{2v} | mm2 | *22 | [2] | Polar | Hemimorphite, bertrandite | Pmm2, Pmc2_{1}, Pcc2, Pma2, Pca2_{1}, Pnc2, Pmn2_{1}, Pba2, Pna2_{1}, Pnn2 | Cmm2, Cmc2_{1}, Ccc2 Amm2, Aem2, Ama2, Aea2 |
Fmm2, Fdd2 | Imm2, Iba2, Ima2 |
47–74 | Rhombic dipyramidal | D_{2h} (V_{h}) | mmm | *222 | [2,2] | Centrosymmetric | Olivine, aragonite, marcasite | Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma | Cmcm, Cmca, Cmmm, Cccm, Cmme, Ccce | Fmmm, Fddd | Immm, Ibam, Ibca, Imma |
See also
- Crystal structure
- Overview of all space groups
References
- ↑ See Hahn (2002), p. 746, row oC, column Primitive, where the cell parameters are given as a1 = a2, α = β = 90°
- ↑ Prince, E., ed (2006). International Tables for Crystallography. International Union of Crystallography. doi:10.1107/97809553602060000001. ISBN 978-1-4020-4969-9.
- ↑ "The 32 crystal classes". https://www.tulane.edu/~sanelson/eens211/32crystalclass.htm.
Further reading
- Hurlbut, Cornelius S.; Klein, Cornelis (1985). Manual of Mineralogy (20th ed.). pp. 69–73. ISBN 0-471-80580-7. https://archive.org/details/manualofmineralo00klei/page/69.
- Hahn, Theo, ed (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. International Tables for Crystallography. A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7. http://it.iucr.org/A/.
Original source: https://en.wikipedia.org/wiki/ Orthorhombic crystal system.
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