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The parallelogram is the general primitive cell for the plane.

File:Parallelepiped 2013-11-29.svg

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, $\vec{T}$ , and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors ${\vec{a}}_{1}$ , ${\vec{a}}_{2}$ , and ${\vec{a}}_{3}$ are primitive if the atoms look the same from any lattice points using integers $u_{1}$ , $u_{2}$ , and $u_{3}$ .

\vec{T} = u_{1} {\vec{a}}_{1} + u_{2} {\vec{a}}_{2} + u_{3} {\vec{a}}_{3}

The primitive cell is defined by the primitive axes (vectors) ${\vec{a}}_{1}$ , ${\vec{a}}_{2}$ , and ${\vec{a}}_{3}$ . The volume, $V_{p}$ , of the primitive cell is given by the parallelepiped from the above axes as

V_{p} = | {\vec{a}}_{1} \cdot ({\vec{a}}_{2} \times {\vec{a}}_{3}) | .

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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]]

Bimodal distribution: Difference between revisions

Revision as of 22:56, 29 January 2014

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