Bimodal distribution: Difference between revisions
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[[File:Fundamental parallelogram.png|thumb|The [[parallelogram]] is the general primitive cell for the plane.]] | |||
[[File:Parallelepiped 2013-11-29.svg|thumb|A [[parallelepiped]] is a general primitive cell for 3-dimensional space.]] | |||
A '''primitive cell''' is a [[unit cell]] built on the primitive basis of the direct [[lattice]], namely a crystallographic basis of the vector lattice L such that every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c. | |||
Used predominantly in [[geometry]], [[solid state physics]], and [[mineralogy]], particularly in describing [[crystal structure]], a primitive cell is a minimum volume cell corresponding to a single [[lattice point]] of a structure with [[translational symmetry]] in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its ''primitive cell''. | |||
The primitive cell is a [[fundamental domain]] with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller. | |||
A [[crystal]] can be categorized by its lattice and the atoms that lie in a primitive cell (the ''basis''). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations. | |||
''Primitive translation vectors'' are used to define a crystal translation vector, <math> \vec T </math>, and also gives a lattice cell of smallest volume for a particular lattice. The ''lattice'' and translation vectors <math> \vec a_1 </math>, <math> \vec a_2 </math>, and <math> \vec a_3 </math> are ''primitive'' if the atoms look the same from any lattice points using integers <math> u_1 </math>, <math> u_2 </math>, and <math> u_3 </math>. | |||
:<math> \vec T = u_1\vec a_1 + u_2\vec a_2 + u_3\vec a_3 </math> | |||
The primitive cell is defined by the primitive axes (vectors) <math> \vec a_1 </math>, <math> \vec a_2 </math>, and <math> \vec a_3 </math>. The volume, <math> V_p </math>, of the primitive cell is given by the [[parallelepiped]] from the above axes as | |||
:<math> V_p = | \vec a_1 \cdot ( \vec a_2 \times \vec a_3 ) |.</math> | |||
== See also == | |||
*[[Bravais lattice]] | |||
*[[Wallpaper group]] | |||
*[[Space group]] | |||
[[Category:Condensed matter physics]] | |||
[[Category:Crystallography]] | |||
[[Category:Mineralogy]] |
Revision as of 22:56, 29 January 2014
A primitive cell is a unit cell built on the primitive basis of the direct lattice, namely a crystallographic basis of the vector lattice L such that every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c.
Used predominantly in geometry, solid state physics, and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum volume cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.
The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.
A crystal can be categorized by its lattice and the atoms that lie in a primitive cell (the basis). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations.
Primitive translation vectors are used to define a crystal translation vector, , and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors , , and are primitive if the atoms look the same from any lattice points using integers , , and .
The primitive cell is defined by the primitive axes (vectors) , , and . The volume, , of the primitive cell is given by the parallelepiped from the above axes as