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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Tetragonal crystal system</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">"Tetragonal" redirects here. For other uses, see <a href="Tetragon" class="mw-redirect" title="Tetragon">tetragon</a>.</div>
<p>In <a href="Crystallography" title="Crystallography">crystallography</a>, the <b>tetragonal crystal system</b> is one of the seven <a href="Crystal_system" title="Crystal system">crystal systems</a>. Tetragonal <a href="Crystal_lattice" class="mw-redirect" title="Crystal lattice">crystal lattices</a> result from stretching a cubic lattice along one of its lattice vectors, so that the <a href="Cube_(geometry)" class="mw-redirect" title="Cube (geometry)">cube</a> becomes a rectangular <a href="Prism_(geometry)" title="Prism (geometry)">prism</a> with a square base (<i>a</i> by <i>a</i>) and height (<i>c</i>, which is different from <i>a</i>).
</p>
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<div class="mw-heading mw-heading2"><h2 id="Bravais_lattices">Bravais lattices</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Bravais_lattice" title="Bravais lattice">Bravais lattice</a></div>
<p>There are two tetragonal Bravais lattices: the primitive tetragonal and the body-centered tetragonal.
</p>
<table class="wikitable">
<tbody><tr>
<th>Bravais lattice
</th>
<th>Primitive<br>tetragonal
</th>
<th>Body-centered<br>tetragonal
</th></tr>
<tr align="center">
<th><a href="Pearson_symbol" title="Pearson symbol">Pearson symbol</a>
</th>
<td>tP
</td>
<td>tI
</td></tr>
<tr>
<th><a href="Unit_cell" title="Unit cell">Unit cell</a>
</th>
<td><span class="skin-invert-image" typeof="mw:File"></span>
</td>
<td><span class="skin-invert-image" typeof="mw:File"></span>
</td></tr></tbody></table>
<p>The body-centered tetragonal lattice is equivalent to the primitive tetragonal lattice with a smaller unit cell, while the face-centered tetragonal lattice is equivalent to the body-centered tetragonal lattice with a smaller unit cell.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Crystal_classes">Crystal classes</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Crystallographic_point_group" title="Crystallographic point group">Crystallographic point group</a></div>
<p>The point groups that fall under this crystal system are listed below, followed by their representations in international notation, <a href="Schoenflies_notation" title="Schoenflies notation">Schoenflies notation</a>, <a href="Orbifold_notation" title="Orbifold notation">orbifold notation</a>, <a href="Coxeter_notation" title="Coxeter notation">Coxeter notation</a> and <a href="Mineral" title="Mineral">mineral</a> examples.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<table class="wikitable">
<tbody><tr>
<th rowspan="2" width="60">#
</th>
<th colspan="5">Point group
</th>
<th rowspan="2">Type
</th>
<th rowspan="2">Example
</th>
<th colspan="2"><a href="Space_group" title="Space group">Space groups</a>
</th></tr>
<tr>
<th>Name<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
</th>
<th><a href="Hermann%E2%80%93Mauguin_notation" title="Hermann–Mauguin notation">Intl</a>
</th>
<th><a href="Schoenflies_notation" title="Schoenflies notation">Schoen.</a>
</th>
<th><a href="Orbifold" title="Orbifold">Orb.</a>
</th>
<th><a href="Coxeter_notation" title="Coxeter notation">Cox.</a>
</th>
<th>Primitive
</th>
<th>Body-centered
</th></tr>
<tr align="center">
<th>75–80
</th>
<td>Tetragonal pyramidal
</td>
<td>4
</td>
<td><i>C</i><sub>4</sub>
</td>
<td>44
</td>
<td>[4]<sup>+</sup>
</td>
<td><a href="Enantiomorphic" class="mw-redirect" title="Enantiomorphic">enantiomorphic</a> <a href="Polar_point_group" title="Polar point group">polar</a>
</td>
<td><a href="Pinnoite" title="Pinnoite">pinnoite</a>, <br><a href="Piypite" title="Piypite">piypite</a>
</td>
<td align="left">P4, P4<sub>1</sub>, P4<sub>2</sub>, P4<sub>3</sub>
</td>
<td>I4, I4<sub>1</sub>
</td></tr>
<tr align="center">
<th>81–82
</th>
<td>Tetragonal disphenoidal
</td>
<td><span style="text-decoration:overline;">4</span>
</td>
<td><i>S</i><sub>4</sub>
</td>
<td>2×
</td>
<td>[2<sup>+</sup>,4<sup>+</sup>]
</td>
<td>
</td>
<td><a href="Cahnite" title="Cahnite">cahnite</a>, <a href="Tugtupite" title="Tugtupite">tugtupite</a>
</td>
<td align="left">P<span style="text-decoration:overline;">4</span>
</td>
<td>I<span style="text-decoration:overline;">4</span>
</td></tr>
<tr align="center">
<th>83–88
</th>
<td>Tetragonal dipyramidal
</td>
<td>4/m
</td>
<td><i>C</i><sub>4<i>h</i></sub>
</td>
<td>4*
</td>
<td>[2,4<sup>+</sup>]
</td>
<td><a href="Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a>
</td>
<td><a href="Scheelite" title="Scheelite">scheelite</a>, <a href="Wulfenite" title="Wulfenite">wulfenite</a>, <a href="Leucite" title="Leucite">leucite</a>
</td>
<td align="left">P4/m, P4<sub>2</sub>/m, P4/n, P4<sub>2</sub>/n
</td>
<td>I4/m, I4<sub>1</sub>/a
</td></tr>
<tr align="center">
<th>89–98
</th>
<td>Tetragonal trapezohedral
</td>
<td>422
</td>
<td><i>D</i><sub>4</sub>
</td>
<td>224
</td>
<td>[2,4]<sup>+</sup>
</td>
<td><a href="Enantiomorphic" class="mw-redirect" title="Enantiomorphic">enantiomorphic</a>
</td>
<td><a href="Cristobalite" title="Cristobalite">cristobalite</a>, <a href="Wardite" title="Wardite">wardite</a>
</td>
<td align="left">P422, P42<sub>1</sub>2, P4<sub>1</sub>22, P4<sub>1</sub>2<sub>1</sub>2, P4<sub>2</sub>22, P4<sub>2</sub>2<sub>1</sub>2, P4<sub>3</sub>22, P4<sub>3</sub>2<sub>1</sub>2
</td>
<td>I422, I4<sub>1</sub>22
</td></tr>
<tr align="center">
<th>99–110
</th>
<td>Ditetragonal pyramidal
</td>
<td>4mm
</td>
<td><i>C</i><sub>4<i>v</i></sub>
</td>
<td>*44
</td>
<td>[4]
</td>
<td><a href="Polar_point_group" title="Polar point group">polar</a>
</td>
<td><a href="Diaboleite" title="Diaboleite">diaboleite</a>
</td>
<td align="left">P4mm, P4bm, P4<sub>2</sub>cm, P4<sub>2</sub>nm, P4cc, P4nc, P4<sub>2</sub>mc, P4<sub>2</sub>bc
</td>
<td>I4mm, I4cm, I4<sub>1</sub>md, I4<sub>1</sub>cd
</td></tr>
<tr align="center">
<th>111–122
</th>
<td>Tetragonal scalenohedral
</td>
<td><span style="text-decoration:overline;">4</span>2m
</td>
<td><i>D</i><sub>2<i>d</i></sub> (V<sub>d</sub>)
</td>
<td>2*2
</td>
<td>[2<sup>+</sup>,4]
</td>
<td>
</td>
<td><a href="Chalcopyrite" title="Chalcopyrite">chalcopyrite</a>, <a href="Stannite" title="Stannite">stannite</a>
</td>
<td align="left">P<span style="text-decoration:overline;">4</span>2m, P<span style="text-decoration:overline;">4</span>2c, P<span style="text-decoration:overline;">4</span>2<sub>1</sub>m, P<span style="text-decoration:overline;">4</span>2<sub>1</sub>c, P<span style="text-decoration:overline;">4</span>m2, P<span style="text-decoration:overline;">4</span>c2, P<span style="text-decoration:overline;">4</span>b2, P<span style="text-decoration:overline;">4</span>n2
</td>
<td>I<span style="text-decoration:overline;">4</span>m2, I<span style="text-decoration:overline;">4</span>c2, I<span style="text-decoration:overline;">4</span>2m, I<span style="text-decoration:overline;">4</span>2d
</td></tr>
<tr align="center">
<th>123–142
</th>
<td>Ditetragonal dipyramidal
</td>
<td>4/mmm
</td>
<td><i>D</i><sub>4<i>h</i></sub>
</td>
<td>*224
</td>
<td>[2,4]
</td>
<td><a href="Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a>
</td>
<td><a href="Rutile" title="Rutile">rutile</a>, <a href="Pyrolusite" title="Pyrolusite">pyrolusite</a>, <a href="Zircon" title="Zircon">zircon</a>
</td>
<td align="left">P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4<sub>2</sub>/mmc, P4<sub>2</sub>/mcm, P4<sub>2</sub>/nbc, P4<sub>2</sub>/nnm, P4<sub>2</sub>/mbc, P4<sub>2</sub>/mnm, P4<sub>2</sub>/nmc, P4<sub>2</sub>/ncm
</td>
<td>I4/mmm, I4/mcm, I4<sub>1</sub>/amd, I4<sub>1</sub>/acd
</td></tr></tbody></table>
<div class="mw-heading mw-heading2"><h2 id="In_two_dimensions">In two dimensions</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Square_lattice" title="Square lattice">Square lattice</a></div>
<p>There is only one tetragonal <a href="Bravais_lattice" title="Bravais lattice">Bravais lattice</a> in two dimensions: the square lattice.
</p>
<table class="wikitable">
<tbody><tr>
<th>Bravais lattice
</th>
<th>Square
</th></tr>
<tr align="center">
<th><a href="Pearson_symbol" title="Pearson symbol">Pearson symbol</a>
</th>
<td>tp
</td></tr>
<tr>
<th><a href="Unit_cell" title="Unit cell">Unit cell</a>
</th>
<td><span class="skin-invert-image" typeof="mw:File"></span>
</td></tr></tbody></table>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Bravais_lattices" class="mw-redirect" title="Bravais lattices">Bravais lattices</a></li>
<li><a href="Crystal_system" title="Crystal system">Crystal system</a></li>
<li><a href="Crystal_structure" title="Crystal structure">Crystal structure</a></li>
<li><a href="Point_groups" class="mw-redirect" title="Point groups">Point groups</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.aem.umn.edu/people/faculty/shield/hane/tet.html">Cubic-to-Tetragonal Transition</a></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://webmineral.com/crystal/Tetragonal.shtml">Webmineral data</a></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Hurlbut, Cornelius S.; Klein, Cornelis, 1985, <i>Manual of Mineralogy</i>, 20th ed., pp. 73–78, <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-80580-7</bdi> </span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.tulane.edu/~sanelson/eens211/32crystalclass.htm">"The 32 crystal classes"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-06-19</span></span>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><span class="noviewer" typeof="mw:File"></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Tetragonal_lattices" class="extiw external" title="commons:Category:Tetragonal lattices">Tetragonal lattices</a> at Wikimedia Commons</li></ul>
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</style><div id="Crystal_systems104" style="font-size:114%;margin:0 4em"><a href="Crystal_system" title="Crystal system">Crystal systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><a href="Bravais_lattice" title="Bravais lattice">Bravais lattice</a></li>
<li><a href="Crystallographic_point_group" title="Crystallographic point group">Crystallographic point group</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Seven 3D systems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Triclinic_crystal_system" title="Triclinic crystal system">triclinic</a> (anorthic) <span class="skin-invert" typeof="mw:File"></span></li>
<li><a href="Monoclinic_crystal_system" title="Monoclinic crystal system">monoclinic</a> <span class="skin-invert" typeof="mw:File"></span></li>
<li><a href="Orthorhombic_crystal_system" title="Orthorhombic crystal system">orthorhombic</a> <span class="skin-invert" typeof="mw:File"></span></li>
<li> <span class="skin-invert" typeof="mw:File"></span></li>
<li><a href="Hexagonal_crystal_family" title="Hexagonal crystal family">trigonal & hexagonal</a> <span class="skin-invert" typeof="mw:File"></span><span class="skin-invert" typeof="mw:File"></span></li>
<li><a href="Cubic_crystal_system" title="Cubic crystal system">cubic</a> (isometric) <span class="skin-invert" typeof="mw:File"></span></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Four 2D systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Oblique_lattice" title="Oblique lattice">oblique</a> <span class="skin-invert" typeof="mw:File"></span></li>
<li><a href="Rectangular_lattice" title="Rectangular lattice">rectangular</a> <span class="skin-invert" typeof="mw:File"></span></li>
<li><a href="Square_lattice" title="Square lattice">square</a> <span class="skin-invert" typeof="mw:File"></span></li>
<li><a href="Hexagonal_lattice" title="Hexagonal lattice">hexagonal</a> <span class="skin-invert" typeof="mw:File"></span></li></ul>
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