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Unit cell
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<h2 id="primitive-cell">Primitive cell</h2> <p>A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by n cells are counted as 1/n of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain 1/8 of each of them.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> An alternative conceptualization is to consistently pick only one of the n lattice points to belong to the given unit cell (so the other n-1 lattice points belong to adjacent unit cells). </p><p>The <i>primitive translation vectors</i> <i>a</i>→1, <i>a</i>→2, <i>a</i>→3 span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector </p> T → = u 1 a → 1 + u 2 a → 2 + u 3 a → 3 , {\displaystyle {\vec {T}}=u_{1}{\vec {a}}_{1}+u_{2}{\vec {a}}_{2}+u_{3}{\vec {a}}_{3},} <p>where <i>u</i>1, <i>u</i>2, <i>u</i>3 are integers, translation by which leaves the lattice invariant.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> That is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + <i>T</i>→ as from r.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>Since the primitive cell is defined by the primitive axes (vectors) <i>a</i>→1, <i>a</i>→2, <i>a</i>→3, the volume <i>V</i>p of the primitive cell is given by the <a href="/facts/Parallelepiped/Qn7cDgSr">parallelepiped</a> from the above axes as </p> V p = | a → 1 ⋅ ( a → 2 × a → 3 ) | . {\displaystyle V_{\mathrm {p} }=\left|{\vec {a}}_{1}\cdot ({\vec {a}}_{2}\times {\vec {a}}_{3})\right|.} <p>Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h3>Wigner–Seitz cell</h3> <p class="note">Main article: <a href="/facts/Wigner%25E2%2580%2593Seitz_cell/qU6azUs6">Wigner–Seitz cell</a></p> <p>In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of <a href="/facts/Voronoi_cell/AhQY5LD8">Voronoi cell</a>. The Wigner–Seitz cell of the <a href="/facts/Reciprocal_lattice/jYCjcEpH">reciprocal lattice</a> in <a href="/facts/Momentum_space/kxKPCeoI">momentum space</a> is called the <a href="/facts/Brillouin_zone/V8unhAAl">Brillouin zone</a>. </p> <h2 id="conventional-cell">Conventional cell</h2> <p>For each particular lattice, a conventional cell has been chosen on a case-by-case basis by crystallographers based on convenience of calculation.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> These conventional cells may have additional lattice points located in the middle of the faces or body of the unit cell. The number of lattice points, as well as the volume of the conventional cell is an integer multiple (1, 2, 3, or 4) of that of the primitive cell.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="two-dimensions">Two dimensions</h2> <p>For any 2-dimensional lattice, the unit cells are <a href="/facts/Parallelogram/DRuktq60">parallelograms</a>, which in special cases may have orthogonal angles, equal lengths, or both. Four of the five two-dimensional <a href="/facts/Bravais_lattice/7gQKTsXk">Bravais lattices</a> are represented using conventional primitive cells, as shown below. </p> <table><tbody><tr><th>Conventional primitive cell</th><td></td><td></td><td></td><td></td></tr><tr><th>Shape name</th><td><a href="/facts/Parallelogram/DRuktq60">Parallelogram</a></td><td><a href="/facts/Rectangle/mdLfIGKA">Rectangle</a></td><td><a href="/facts/Square/JHJeMvmL">Square</a></td><td><a href="/facts/Rhombus/T4pU50GF">Rhombus</a></td></tr><tr><th>Bravais lattice</th><td>Primitive Oblique</td><td>Primitive Rectangular</td><td>Primitive Square</td><td>Primitive Hexagonal</td></tr></tbody></table> <p>The centered rectangular lattice also has a primitive cell in the shape of a rhombus, but in order to allow easy discrimination on the basis of symmetry, it is represented by a conventional cell which contains two lattice points. </p> <table><tbody><tr><th>Primitive cell</th><td></td></tr><tr><th>Shape name</th><td><a href="/facts/Rhombus/T4pU50GF">Rhombus</a></td></tr><tr><th>Conventional cell</th><td></td></tr><tr><th>Bravais lattice</th><td>Centered Rectangular</td></tr></tbody></table> <h2 id="three-dimensions">Three dimensions</h2> <p>For any 3-dimensional lattice, the conventional unit cells are <a href="/facts/Parallelepiped/Qn7cDgSr">parallelepipeds</a>, which in special cases may have orthogonal angles, or equal lengths, or both. Seven of the fourteen three-dimensional <a href="/facts/Bravais_lattice/7gQKTsXk">Bravais lattices</a> are represented using conventional primitive cells, as shown below. </p> <table><tbody><tr><th>Conventional primitive cell</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Shape name</th><td><a href="/facts/Parallelepiped/Qn7cDgSr">Parallelepiped</a></td><td>Oblique rectangular <a href="/facts/Prism_(geometry)/baIRRkCy">prism</a></td><td>Rectangular <a href="/facts/Cuboid/eLLKX4rh">cuboid</a></td><td>Square <a href="/facts/Cuboid/eLLKX4rh">cuboid</a></td><td><a href="/facts/Trigonal_trapezohedron/w5qPRnR3">Trigonal trapezohedron</a></td><td><a href="/facts/Cube/65lUaAqN">Cube</a></td><td>Right rhombic <a href="/facts/Prism_(geometry)/baIRRkCy">prism</a></td></tr><tr><th>Bravais lattice</th><td>Primitive <a href="/facts/Triclinic_crystal_system/Y6yEdqfN">Triclinic</a></td><td>Primitive <a href="/facts/Monoclinic_crystal_system/5PCwXvzX">Monoclinic</a></td><td>Primitive <a href="/facts/Orthorhombic_crystal_system/3zo4F8P0">Orthorhombic</a></td><td>Primitive <a href="/facts/Tetragonal_crystal_system/12Aop255">Tetragonal</a></td><td>Primitive <a href="/facts/Rhombohedral_lattice_system/sSSok8wo">Rhombohedral</a></td><td>Primitive <a href="/facts/Cubic_crystal_system/92Co7BKM">Cubic</a></td><td>Primitive <a href="/facts/Hexagonal_crystal_system/sSSok8wo">Hexagonal</a></td></tr></tbody></table> <p>The other seven Bravais lattices (known as the centered lattices) also have primitive cells in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are represented by conventional cells which contain more than one lattice point. </p> <table><tbody><tr><th>Primitive cell</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Shape name</th><td>Oblique rhombic <a href="/facts/Prism_(geometry)/baIRRkCy">prism</a></td><td>Right rhombic <a href="/facts/Prism_(geometry)/baIRRkCy">prism</a></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Conventional cell</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Bravais lattice</th><td>Base-centered <a href="/facts/Monoclinic_crystal_system/5PCwXvzX">Monoclinic</a></td><td>Base-centered <a href="/facts/Orthorhombic_crystal_system/3zo4F8P0">Orthorhombic</a></td><td>Body-centered <a href="/facts/Orthorhombic_crystal_system/3zo4F8P0">Orthorhombic</a></td><td>Face-centered <a href="/facts/Orthorhombic_crystal_system/3zo4F8P0">Orthorhombic</a></td><td>Body-centered <a href="/facts/Tetragonal_crystal_system/12Aop255">Tetragonal</a></td><td>Body-centered <a href="/facts/Cubic_crystal_system/92Co7BKM">Cubic</a></td><td>Face-centered <a href="/facts/Cubic_crystal_system/92Co7BKM">Cubic</a></td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Wigner%25E2%2580%2593Seitz_cell/qU6azUs6">Wigner–Seitz cell</a></li> <li><a href="/facts/Bravais_lattice/7gQKTsXk">Bravais lattice</a></li> <li><a href="/facts/Wallpaper_group/egxEaC8q">Wallpaper group</a></li> <li><a href="/facts/Space_group/2XQx0J7M">Space group</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Ashcroft, Neil W. (1976). "Chapter 4". Solid State Physics. W. B. Saunders Company. p. 72. ISBN 0-03-083993-9. <a href="0-03-083993-9" target="_blank">0-03-083993-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Simon, Steven (2013). The Oxford Solid State Physics (1 ed.). Oxford University Press. p. 114. ISBN 978-0-19-968076-4. <a href="978-0-19-968076-4" target="_blank">978-0-19-968076-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"DoITPoMS – TLP Library Crystallography – Unit Cell". Online Materials Science Learning Resources: DoITPoMS. University of Cambridge. Retrieved 21 February 2015. <a href="http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php" target="_blank">http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>In n dimensions the crystal translation vector would be T → = ∑ i = 1 n u i a → i , where u i ∈ Z ∀ i . {\displaystyle {\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.} That is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + T→ as from r. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Kittel, Charles (11 November 2004). Introduction to Solid State Physics (8 ed.). Wiley. p. 4. ISBN 978-0-471-41526-8. <a href="978-0-471-41526-8" target="_blank">978-0-471-41526-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Mehl, Michael J.; Hicks, David; Toher, Cormac; Levy, Ohad; Hanson, Robert M.; Hart, Gus; Curtarolo, Stefano (2017). "The AFLOW Library of Crystallographic Prototypes: Part 1". Computational Materials Science. 136. Elsevier BV: S1 – S828. arXiv:1806.07864. doi:10.1016/j.commatsci.2017.01.017. ISSN 0927-0256. S2CID 119490841. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Aroyo, M. I., ed. (2016-12-31). International Tables for Crystallography. Chester, England: International Union of Crystallography. p. 25. doi:10.1107/97809553602060000114. ISBN 978-0-470-97423-0. <a href="978-0-470-97423-0" target="_blank">978-0-470-97423-0</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Ashcroft, Neil W. (1976). Solid State Physics. W. B. Saunders Company. p. 73. ISBN 0-03-083993-9. <a href="0-03-083993-9" target="_blank">0-03-083993-9</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>