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class="vector-toc-numb">2</span> <span>Crystal systems</span> </div> </a> <button aria-controls="toc-Crystal_systems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Crystal systems subsection</span> </button> <ul id="toc-Crystal_systems-sublist" class="vector-toc-list"> <li id="toc-Trigonal_crystal_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trigonal_crystal_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Trigonal crystal system</span> </div> </a> <ul id="toc-Trigonal_crystal_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hexagonal_crystal_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hexagonal_crystal_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Hexagonal crystal system</span> 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Multi-element structures subsection</span> </button> <ul id="toc-Multi-element_structures-sublist" class="vector-toc-list"> <li id="toc-Wurtzite_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wurtzite_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Wurtzite structure</span> </div> </a> <ul id="toc-Wurtzite_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nickel_arsenide_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nickel_arsenide_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Nickel arsenide structure</span> </div> </a> <ul id="toc-Nickel_arsenide_structure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_two_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_two_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In two dimensions</span> </div> </a> <ul id="toc-In_two_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " 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Available in 32 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-32" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">32 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Heksagonale_digpakking" title="Heksagonale digpakking – Afrikaans" lang="af" hreflang="af" data-title="Heksagonale digpakking" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%A8%D9%84%D9%88%D8%B1%D9%8A_%D8%B3%D8%AF%D8%A7%D8%B3%D9%8A" title="نظام بلوري سداسي – Arabic" lang="ar" hreflang="ar" data-title="نظام بلوري سداسي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_hexagonal" title="Sistema hexagonal – Catalan" lang="ca" hreflang="ca" data-title="Sistema hexagonal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C5%A0estere%C4%8Dn%C3%A1_soustava" title="Šesterečná soustava – Czech" lang="cs" hreflang="cs" data-title="Šesterečná soustava" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hexagonales_Kristallsystem" title="Hexagonales Kristallsystem – German" lang="de" hreflang="de" data-title="Hexagonales Kristallsystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Heksagonaalne_s%C3%BCngoonia" title="Heksagonaalne süngoonia – Estonian" lang="et" hreflang="et" data-title="Heksagonaalne süngoonia" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_cristalino_hexagonal" title="Sistema cristalino hexagonal – Spanish" lang="es" hreflang="es" data-title="Sistema cristalino hexagonal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Heksagona_kristalsistemo" title="Heksagona kristalsistemo – Esperanto" lang="eo" hreflang="eo" data-title="Heksagona kristalsistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kristal-sistema_hexagonal" title="Kristal-sistema hexagonal – Basque" lang="eu" hreflang="eu" data-title="Kristal-sistema hexagonal" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D8%A8%D9%84%D9%88%D8%B1%DB%8C_%D8%B4%D8%B4%E2%80%8C%DA%AF%D9%88%D8%B4%D9%87" title="دستگاه بلوری شش‌گوشه – Persian" lang="fa" hreflang="fa" data-title="دستگاه بلوری شش‌گوشه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_cristallin_hexagonal" title="Système cristallin hexagonal – French" lang="fr" hreflang="fr" data-title="Système cristallin hexagonal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A1%EB%B0%A9%EC%A0%95%EA%B3%84" title="육방정계 – Korean" lang="ko" hreflang="ko" data-title="육방정계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/%C5%A0esterokutna_kristalna_re%C5%A1etka" title="Šesterokutna kristalna rešetka – Croatian" lang="hr" hreflang="hr" data-title="Šesterokutna kristalna rešetka" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Keluarga_kristal_heksagonal" title="Keluarga kristal heksagonal – Indonesian" lang="id" hreflang="id" data-title="Keluarga kristal heksagonal" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_esagonale" title="Sistema esagonale – Italian" lang="it" hreflang="it" data-title="Sistema esagonale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%94%D7%92%D7%91%D7%99%D7%A9%D7%99%D7%AA_%D7%94%D7%94%D7%A7%D7%A1%D7%92%D7%95%D7%A0%D7%9C%D7%99%D7%AA" title="המערכת הגבישית ההקסגונלית – Hebrew" lang="he" hreflang="he" data-title="המערכת הגבישית ההקסגונלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BA%D1%81%D0%B0%D0%B3%D0%BE%D0%BD%D0%B4%D1%8B%D2%9B_%D1%81%D0%B8%D0%BD%D0%B3%D0%BE%D0%BD%D0%B8%D1%8F" title="Гексагондық сингония – Kazakh" lang="kk" hreflang="kk" data-title="Гексагондық сингония" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Heksagon%C4%81l%C4%81_singonija" title="Heksagonālā singonija – Latvian" lang="lv" hreflang="lv" data-title="Heksagonālā singonija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hexagonaal_kristalstelsel" title="Hexagonaal kristalstelsel – Dutch" lang="nl" hreflang="nl" data-title="Hexagonaal kristalstelsel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_heksagonalny" title="Układ heksagonalny – Polish" lang="pl" hreflang="pl" data-title="Układ heksagonalny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_cristalino_hexagonal" title="Sistema cristalino hexagonal – Portuguese" lang="pt" hreflang="pt" data-title="Sistema cristalino hexagonal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_cristalin_hexagonal" title="Sistem cristalin hexagonal – Romanian" lang="ro" hreflang="ro" data-title="Sistem cristalin hexagonal" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BA%D1%81%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D0%BD%D0%B3%D0%BE%D0%BD%D0%B8%D1%8F" title="Гексагональная сингония – Russian" lang="ru" hreflang="ru" data-title="Гексагональная сингония" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hexagon%C3%A1lna_s%C3%BAstava" title="Hexagonálna sústava – Slovak" lang="sk" hreflang="sk" data-title="Hexagonálna sústava" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Heksagonalni_kristalni_sistem" title="Heksagonalni kristalni sistem – Slovenian" lang="sl" hreflang="sl" data-title="Heksagonalni kristalni sistem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B5%D0%BA%D1%81%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%B0_%D0%BA%D1%80%D0%B8%D1%81%D1%82%D0%B0%D0%BB%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Хексагонална кристална система – Serbian" lang="sr" hreflang="sr" data-title="Хексагонална кристална система" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Heksagonalna_kristalna_sistema" title="Heksagonalna kristalna sistema – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Heksagonalna kristalna sistema" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hexagonala_kristallsystemet" title="Hexagonala kristallsystemet – Swedish" lang="sv" hreflang="sv" data-title="Hexagonala kristallsystemet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BA%D1%81%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D1%81%D0%B8%D0%BD%D0%B3%D0%BE%D0%BD%D1%96%D1%8F" title="Гексагональна сингонія – Ukrainian" lang="uk" hreflang="uk" data-title="Гексагональна сингонія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_tinh_th%E1%BB%83_l%E1%BB%A5c_ph%C6%B0%C6%A1ng" title="Hệ tinh thể lục phương – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ tinh thể lục phương" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%85%AD%E6%96%B9%E6%99%B6%E7%B3%BB" title="六方晶系 – Wu" lang="wuu" hreflang="wuu" data-title="六方晶系" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%85%AD%E6%96%B9%E6%99%B6%E7%B3%BB" title="六方晶系 – Chinese" lang="zh" hreflang="zh" data-title="六方晶系" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Hexagonal_crystal_system&redirect=no" class="mw-redirect" title="Hexagonal crystal system">Hexagonal crystal system</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Union of crystal groups with related structures and lattices</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Hexagonal_lattice" title="Hexagonal lattice">Hexagonal lattice</a>.</div> <table class="wikitable" align="right" width="300"> <tbody><tr valign="top" align="center"> <th>Crystal system </th> <td colspan="2">Trigonal </td> <td>Hexagonal </td></tr> <tr valign="top" align="center"> <th>Lattice system </th> <td><span typeof="mw:File"><a href="/wiki/File:Rhombohedral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/100px-Rhombohedral.svg.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/150px-Rhombohedral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/200px-Rhombohedral.svg.png 2x" data-file-width="139" data-file-height="141" /></a></span><br />Rhombohedral </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Hexagonal_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/100px-Hexagonal_lattice.svg.png" decoding="async" width="100" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/150px-Hexagonal_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/200px-Hexagonal_lattice.svg.png 2x" data-file-width="160" data-file-height="186" /></a></span><br />Hexagonal </td></tr> <tr valign="top" align="center"> <th>Example </th> <td><span typeof="mw:File"><a href="/wiki/File:Dolomite_sur_mim%C3%A9tite_(Maroc).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dolomite_sur_mim%C3%A9tite_%28Maroc%29.jpg/140px-Dolomite_sur_mim%C3%A9tite_%28Maroc%29.jpg" decoding="async" width="140" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dolomite_sur_mim%C3%A9tite_%28Maroc%29.jpg/210px-Dolomite_sur_mim%C3%A9tite_%28Maroc%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dolomite_sur_mim%C3%A9tite_%28Maroc%29.jpg/280px-Dolomite_sur_mim%C3%A9tite_%28Maroc%29.jpg 2x" data-file-width="3008" data-file-height="2000" /></a></span><br /><a href="/wiki/Dolomite_(mineral)" title="Dolomite (mineral)">Dolomite (white)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Kwarc,_Madagaskar.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Kwarc%2C_Madagaskar.jpg/150px-Kwarc%2C_Madagaskar.jpg" decoding="async" width="150" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Kwarc%2C_Madagaskar.jpg/225px-Kwarc%2C_Madagaskar.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Kwarc%2C_Madagaskar.jpg/300px-Kwarc%2C_Madagaskar.jpg 2x" data-file-width="650" data-file-height="450" /></a></span><br />α-<a href="/wiki/Quartz" title="Quartz">Quartz</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Berillo.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Berillo.jpg/150px-Berillo.jpg" decoding="async" width="150" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Berillo.jpg/225px-Berillo.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Berillo.jpg/300px-Berillo.jpg 2x" data-file-width="2592" data-file-height="1944" /></a></span><br /><a href="/wiki/Beryl" title="Beryl">Beryl</a> </td></tr></tbody></table> <p>In <a href="/wiki/Crystallography" title="Crystallography">crystallography</a>, the <b>hexagonal crystal family</b> is one of the six <a href="/wiki/Crystal_family" class="mw-redirect" title="Crystal family">crystal families</a>, which includes two crystal systems (hexagonal and <b>trigonal</b>) and two lattice systems (hexagonal and <b>rhombohedral</b>). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see section <a class="mw-selflink-fragment" href="#Crystal_systems">crystal systems</a> below).<sup id="cite_ref-hahn_1-0" class="reference"><a href="#cite_note-hahn-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice (such as α-<a href="/wiki/Quartz" title="Quartz">quartz</a>). </p><p>The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system.<sup id="cite_ref-Hurlbut_2-0" class="reference"><a href="#cite_note-Hurlbut-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> There are 52 space groups associated with it, which are exactly those whose <a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattice</a> is either hexagonal or rhombohedral. </p> <meta property="mw:PageProp/toc" /> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Lattice_systems">Lattice systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=1" title="Edit section: Lattice systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The hexagonal crystal family consists of two <a href="/wiki/Lattice_system" class="mw-redirect" title="Lattice system">lattice systems</a>: hexagonal and rhombohedral. Each lattice system consists of one <a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattice</a>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:RhombohedralR.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/RhombohedralR.svg/200px-RhombohedralR.svg.png" decoding="async" width="200" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/RhombohedralR.svg/300px-RhombohedralR.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/RhombohedralR.svg/400px-RhombohedralR.svg.png 2x" data-file-width="153" data-file-height="176" /></a><figcaption>Relation between the two settings for the rhombohedral lattice</figcaption></figure> <table class="wikitable"> <caption>Hexagonal crystal family </caption> <tbody><tr> <th>Bravais lattice </th> <th>Hexagonal </th> <th>Rhombohedral </th></tr> <tr align="center"> <th><a href="/wiki/Pearson_symbol" title="Pearson symbol">Pearson symbol</a> </th> <td>hP </td> <td>hR </td></tr> <tr align="center"> <th><a href="/wiki/Hexagon" title="Hexagon">Hexagonal</a><br />unit cell </th> <td><span typeof="mw:File"><a href="/wiki/File:Hexagonal_latticeFRONT.svg" class="mw-file-description" title="Hexagonal, primitive"><img alt="Hexagonal, primitive" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Hexagonal_latticeFRONT.svg/90px-Hexagonal_latticeFRONT.svg.png" decoding="async" width="90" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Hexagonal_latticeFRONT.svg/135px-Hexagonal_latticeFRONT.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Hexagonal_latticeFRONT.svg/180px-Hexagonal_latticeFRONT.svg.png 2x" data-file-width="160" data-file-height="206" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Hexagonal_latticeR.svg" class="mw-file-description" title="Hexagonal, R-centered"><img alt="Hexagonal, R-centered" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Hexagonal_latticeR.svg/90px-Hexagonal_latticeR.svg.png" decoding="async" width="90" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Hexagonal_latticeR.svg/135px-Hexagonal_latticeR.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Hexagonal_latticeR.svg/180px-Hexagonal_latticeR.svg.png 2x" data-file-width="160" data-file-height="206" /></a></span> </td></tr> <tr align="center"> <th><a href="/wiki/Rhombohedron" title="Rhombohedron">Rhombohedral</a><br />unit cell </th> <td><span typeof="mw:File"><a href="/wiki/File:RhombohedralD.svg" class="mw-file-description" title="Rhombohedral, D-centered"><img alt="Rhombohedral, D-centered" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/RhombohedralD.svg/90px-RhombohedralD.svg.png" decoding="async" width="90" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/RhombohedralD.svg/135px-RhombohedralD.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/RhombohedralD.svg/180px-RhombohedralD.svg.png 2x" data-file-width="139" data-file-height="141" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rhombohedral.svg" class="mw-file-description" title="Rhombohedral, primitive"><img alt="Rhombohedral, primitive" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/90px-Rhombohedral.svg.png" decoding="async" width="90" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/135px-Rhombohedral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/180px-Rhombohedral.svg.png 2x" data-file-width="139" data-file-height="141" /></a></span> </td></tr></tbody></table> <p>In the hexagonal family, the crystal is conventionally described by a right <a href="/wiki/Rhombus" title="Rhombus">rhombic</a> <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prism</a> unit cell with two equal axes (<i>a</i> by <i>a</i>), an included angle of 120° (<i>γ</i>) and a height (<i>c</i>, which can be different from <i>a</i>) perpendicular to the two base axes. </p><p>The hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell. There are two ways to do this, which can be thought of as two notations which represent the same structure. In the usual so-called obverse setting, the additional lattice points are at coordinates (<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>) and (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>), whereas in the alternative reverse setting they are at the coordinates (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>) and (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>).<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> In either case, there are 3 lattice points per unit cell in total and the lattice is non-primitive. </p><p>The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes.<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> The unit cell is a <a href="/wiki/Rhombohedron" title="Rhombohedron">rhombohedron</a> (which gives the name for the rhombohedral lattice). This is a unit cell with parameters <i>a</i> = <i>b</i> = <i>c</i>; <i>α</i> = <i>β</i> = <i>γ</i> ≠ 90°.<sup id="cite_ref-Ashcroft_5-0" class="reference"><a href="#cite_note-Ashcroft-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In practice, the hexagonal description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are often shown (for the rhombohedral lattice) in textbooks because this cell reveals the <i><span style="text-decoration:overline;">3</span>m</i> symmetry of the crystal lattice. </p><p>The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered<sup id="cite_ref-hahn_1-1" class="reference"><a href="#cite_note-hahn-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>) and (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>). However, such a description is rarely used. </p> <div class="mw-heading mw-heading2"><h2 id="Crystal_systems">Crystal systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=2" title="Edit section: Crystal systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="margin: 1em auto; text-align: center;"> <tbody><tr> <th><a href="/wiki/Crystal_system" title="Crystal system">Crystal system</a> </th> <th>Required symmetries of point group </th> <th><a href="/wiki/Crystallographic_point_group" title="Crystallographic point group">Point groups</a> </th> <th><a href="/wiki/Space_group" title="Space group">Space groups</a> </th> <th><a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattices</a> </th> <th><a href="/wiki/Lattice_system" class="mw-redirect" title="Lattice system">Lattice system</a> </th></tr> <tr> <th rowspan="2">Trigonal </th> <td rowspan="2">1 threefold axis of rotation </td> <td rowspan="2">5 </td> <td>7 </td> <td>1 </td> <td>Rhombohedral </td></tr> <tr> <td>18 </td> <td rowspan="2">1 </td> <td rowspan="2">Hexagonal </td></tr> <tr> <th>Hexagonal </th> <td>1 sixfold axis of rotation </td> <td>7 </td> <td>27 </td></tr></tbody></table> <p>The hexagonal crystal family consists of two <a href="/wiki/Crystal_system" title="Crystal system">crystal systems</a>: trigonal and hexagonal. A crystal system is a set of <a href="/wiki/Point_group" title="Point group">point groups</a> in which the point groups themselves and their corresponding <a href="/wiki/Space_group" title="Space group">space groups</a> are assigned to a <a href="/wiki/Lattice_system" class="mw-redirect" title="Lattice system">lattice system</a> (see table in <a href="/wiki/Crystal_system#Crystal_classes" title="Crystal system">Crystal system#Crystal classes</a>). </p><p>The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis, which includes space groups 143 to 167. These 5 point groups have 7 corresponding space groups (denoted by R) assigned to the rhombohedral lattice system and 18 corresponding space groups (denoted by P) assigned to the hexagonal lattice system. Hence, the trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups. </p><p>The hexagonal crystal system consists of the 7 point groups that have a single six-fold rotation axis. These 7 point groups have 27 space groups (168 to 194), all of which are assigned to the hexagonal lattice system. </p> <div class="mw-heading mw-heading3"><h3 id="Trigonal_crystal_system">Trigonal crystal system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=3" title="Edit section: Trigonal crystal system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 5 point groups in this crystal system are listed below, with their international number and notation, their space groups in name and example crystals.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="font-size: 100%; width:100%; text-align: center"> <tbody><tr> <th rowspan="2">Space group no. </th> <th colspan="5">Point group </th> <th rowspan="2">Type </th> <th rowspan="2">Examples </th> <th colspan="2">Space groups </th></tr> <tr> <th>Name<sup id="cite_ref-hahn_1-2" class="reference"><a href="#cite_note-hahn-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </th> <th><a href="/wiki/Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Intl</a> </th> <th><a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schoen.</a> </th> <th><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orb.</a> </th> <th><a href="/wiki/Coxeter_notation" title="Coxeter notation">Cox.</a> </th> <th>Hexagonal </th> <th>Rhombohedral </th></tr> <tr align="center"> <th>143–146 </th> <td>Trigonal pyramidal </td> <td>3 </td> <td>C<sub>3</sub> </td> <td>33 </td> <td>[3]<sup>+</sup> </td> <td><a href="/wiki/Enantiomorphic" class="mw-redirect" title="Enantiomorphic">enantiomorphic</a> <a href="/wiki/Polar_point_group" title="Polar point group">polar</a> </td> <td><a href="/w/index.php?title=Carlinite&action=edit&redlink=1" class="new" title="Carlinite (page does not exist)">carlinite</a>, <a href="/wiki/Jarosite" title="Jarosite">jarosite</a> </td> <td align="left">P3, P3<sub>1</sub>, P3<sub>2</sub> </td> <td>R3 </td></tr> <tr align="center"> <th>147–148 </th> <td>Rhombohedral </td> <td><span style="text-decoration:overline;">3</span> </td> <td>C<sub>3i</sub> (S<sub>6</sub>) </td> <td>3× </td> <td>[2<sup>+</sup>,6<sup>+</sup>] </td> <td><a href="/wiki/Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a> </td> <td><a href="/wiki/Dolomite_(mineral)" title="Dolomite (mineral)">dolomite</a>, <a href="/wiki/Ilmenite" title="Ilmenite">ilmenite</a> </td> <td align="left">P<span style="text-decoration:overline;">3</span> </td> <td>R<span style="text-decoration:overline;">3</span> </td></tr> <tr align="center"> <th>149–155 </th> <td>Trigonal trapezohedral </td> <td>32 </td> <td>D<sub>3</sub> </td> <td>223 </td> <td>[2,3]<sup>+</sup> </td> <td><a href="/wiki/Enantiomorphic" class="mw-redirect" title="Enantiomorphic">enantiomorphic</a> </td> <td><a href="/wiki/Abhurite" title="Abhurite">abhurite</a>, alpha-<a href="/wiki/Quartz" title="Quartz">quartz</a> (152, 154), <a href="/wiki/Cinnabar" title="Cinnabar">cinnabar</a> </td> <td align="left">P312, P321, P3<sub>1</sub>12, P3<sub>1</sub>21, P3<sub>2</sub>12, P3<sub>2</sub>21 </td> <td>R32 </td></tr> <tr align="center"> <th>156–161 </th> <td><a href="/wiki/Ditrigonal" class="mw-redirect" title="Ditrigonal">Ditrigonal</a> pyramidal </td> <td>3m </td> <td>C<sub>3v</sub> </td> <td>*33 </td> <td>[3] </td> <td><a href="/wiki/Polar_point_group" title="Polar point group">polar</a> </td> <td><a href="/wiki/Schorl" class="mw-redirect" title="Schorl">schorl</a>, <a href="/wiki/Cerite" title="Cerite">cerite</a>, <a href="/wiki/Tourmaline" title="Tourmaline">tourmaline</a>, <a href="/wiki/Alunite" title="Alunite">alunite</a>, <a href="/wiki/Lithium_tantalate" title="Lithium tantalate">lithium tantalate</a> </td> <td align="left">P3m1, P31m, P3c1, P31c </td> <td>R3m, R3c </td></tr> <tr align="center"> <th>162–167 </th> <td>Ditrigonal scalenohedral </td> <td><span style="text-decoration:overline;">3</span>m </td> <td>D<sub>3d</sub> </td> <td>2*3 </td> <td>[2<sup>+</sup>,6] </td> <td><a href="/wiki/Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a> </td> <td><a href="/wiki/Antimony" title="Antimony">antimony</a>, <a href="/wiki/Hematite" title="Hematite">hematite</a>, <a href="/wiki/Corundum" title="Corundum">corundum</a>, <a href="/wiki/Calcite" title="Calcite">calcite</a>, <a href="/wiki/Bismuth" title="Bismuth">bismuth</a> </td> <td align="left">P<span style="text-decoration:overline;">3</span>1m, P<span style="text-decoration:overline;">3</span>1c, P<span style="text-decoration:overline;">3</span>m1, P<span style="text-decoration:overline;">3</span>c1 </td> <td>R<span style="text-decoration:overline;">3</span>m, R<span style="text-decoration:overline;">3</span>c </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Hexagonal_crystal_system">Hexagonal crystal system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=4" title="Edit section: Hexagonal crystal system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 7 <a href="/wiki/Crystallographic_point_group" title="Crystallographic point group">point groups</a> (<i>crystal classes</i>) in this crystal system are listed below, followed by their representations in <a href="/wiki/Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Hermann–Mauguin</a> or international notation and <a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schoenflies notation</a>, and <a href="/wiki/Mineral" title="Mineral">mineral</a> examples, if they exist.<sup id="cite_ref-Hurlbut_2-1" class="reference"><a href="#cite_note-Hurlbut-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Space group no. </th> <th colspan="5">Point group </th> <th rowspan="2">Type </th> <th rowspan="2">Examples </th> <th rowspan="2">Space groups </th></tr> <tr> <th>Name<sup id="cite_ref-hahn_1-3" class="reference"><a href="#cite_note-hahn-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </th> <th><a href="/wiki/Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Intl</a> </th> <th><a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schoen.</a> </th> <th><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orb.</a> </th> <th><a href="/wiki/Coxeter_notation" title="Coxeter notation">Cox.</a> </th></tr> <tr align="center"> <th>168–173 </th> <td>Hexagonal pyramidal </td> <td>6 </td> <td>C<sub>6</sub> </td> <td>66 </td> <td>[6]<sup>+</sup> </td> <td><a href="/wiki/Enantiomorphic" class="mw-redirect" title="Enantiomorphic">enantiomorphic</a> <a href="/wiki/Polar_point_group" title="Polar point group">polar</a> </td> <td><a href="/wiki/Nepheline" title="Nepheline">nepheline</a>, <a href="/wiki/Cancrinite" title="Cancrinite">cancrinite</a> </td> <td align="left">P6, P6<sub>1</sub>, P6<sub>5</sub>, P6<sub>2</sub>, P6<sub>4</sub>, P6<sub>3</sub> </td></tr> <tr align="center"> <th>174 </th> <td>Trigonal dipyramidal </td> <td><span style="text-decoration:overline;">6</span> </td> <td>C<sub>3h</sub> </td> <td>3* </td> <td>[2,3<sup>+</sup>] </td> <td> </td> <td><a href="/wiki/Cesanite" title="Cesanite">cesanite</a>, <a href="/w/index.php?title=Laurelite&action=edit&redlink=1" class="new" title="Laurelite (page does not exist)">laurelite</a> </td> <td align="left">P<span style="text-decoration:overline;">6</span> </td></tr> <tr align="center"> <th>175–176 </th> <td>Hexagonal dipyramidal </td> <td>6/m </td> <td>C<sub>6h</sub> </td> <td>6* </td> <td>[2,6<sup>+</sup>] </td> <td><a href="/wiki/Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a> </td> <td><a href="/wiki/Apatite" title="Apatite">apatite</a>, <a href="/wiki/Vanadinite" title="Vanadinite">vanadinite</a> </td> <td align="left">P6/m, P6<sub>3</sub>/m </td></tr> <tr align="center"> <th>177–182 </th> <td>Hexagonal trapezohedral </td> <td>622 </td> <td>D<sub>6</sub> </td> <td>226 </td> <td>[2,6]<sup>+</sup> </td> <td><a href="/wiki/Enantiomorphic" class="mw-redirect" title="Enantiomorphic">enantiomorphic</a> </td> <td><a href="/wiki/Kalsilite" title="Kalsilite">kalsilite</a>, beta-<a href="/wiki/Quartz" title="Quartz">quartz</a> </td> <td align="left">P622, P6<sub>1</sub>22, P6<sub>5</sub>22, P6<sub>2</sub>22, P6<sub>4</sub>22, P6<sub>3</sub>22 </td></tr> <tr align="center"> <th>183–186 </th> <td>Dihexagonal pyramidal </td> <td>6mm </td> <td>C<sub>6v</sub> </td> <td>*66 </td> <td>[6] </td> <td><a href="/wiki/Polar_point_group" title="Polar point group">polar</a> </td> <td><a href="/wiki/Greenockite" title="Greenockite">greenockite</a>, <a href="/wiki/Wurtzite" title="Wurtzite">wurtzite</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td> <td align="left">P6mm, P6cc, P6<sub>3</sub>cm, P6<sub>3</sub>mc </td></tr> <tr align="center"> <th>187–190 </th> <td>Ditrigonal dipyramidal </td> <td><span style="text-decoration:overline;">6</span>m2 </td> <td>D<sub>3h</sub> </td> <td>*223 </td> <td>[2,3] </td> <td> </td> <td><a href="/wiki/Benitoite" title="Benitoite">benitoite</a> </td> <td align="left">P<span style="text-decoration:overline;">6</span>m2, P<span style="text-decoration:overline;">6</span>c2, P<span style="text-decoration:overline;">6</span>2m, P<span style="text-decoration:overline;">6</span>2c </td></tr> <tr align="center"> <th>191–194 </th> <td>Dihexagonal dipyramidal </td> <td>6/mmm </td> <td>D<sub>6h</sub> </td> <td>*226 </td> <td>[2,6] </td> <td><a href="/wiki/Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a> </td> <td><a href="/wiki/Beryl" title="Beryl">beryl</a> </td> <td align="left">P6/mmm, P6/mcc, P6<sub>3</sub>/mcm, P6<sub>3</sub>/mmc </td></tr></tbody></table> <p>The unit cell volume is given by <i>a</i><sup>2</sup><i>c</i>•sin(60°) </p> <div class="mw-heading mw-heading2"><h2 id="Hexagonal_close_packed">Hexagonal close packed</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=5" title="Edit section: Hexagonal close packed"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Close-packing_of_equal_spheres" title="Close-packing of equal spheres">Close-packing of equal spheres</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hexagonal_close_packed.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Hexagonal_close_packed.svg/150px-Hexagonal_close_packed.svg.png" decoding="async" width="150" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Hexagonal_close_packed.svg/225px-Hexagonal_close_packed.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Hexagonal_close_packed.svg/300px-Hexagonal_close_packed.svg.png 2x" data-file-width="311" data-file-height="307" /></a><figcaption>Hexagonal close packed (hcp) unit cell</figcaption></figure> <p>Hexagonal close packed (hcp) is one of the two simple types of atomic packing with the highest density, the other being the face-centered cubic (fcc). However, unlike the fcc, it is not a Bravais lattice, as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravais lattice by using a two-atom motif (the additional atom at about (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>)) associated with each lattice point.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Multi-element_structures">Multi-element structures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=6" title="Edit section: Multi-element structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Compounds that consist of more than one element (e.g. <a href="/wiki/Binary_compound" class="mw-redirect" title="Binary compound">binary compounds</a>) often have crystal structures based on the hexagonal crystal family. Some of the more common ones are listed here. These structures can be viewed as two or more interpenetrating sublattices where each sublattice occupies the <a href="/wiki/Interstitial_site" title="Interstitial site">interstitial sites</a> of the others. </p> <div class="mw-heading mw-heading3"><h3 id="Wurtzite_structure">Wurtzite structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=7" title="Edit section: Wurtzite structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable selfref">See also: <a href="/wiki/Category:Wurtzite_structure_type" title="Category:Wurtzite structure type">Category:Wurtzite structure type</a></div> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Wurtzite_cellGIF.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Wurtzite_cellGIF.gif/220px-Wurtzite_cellGIF.gif" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Wurtzite_cellGIF.gif/330px-Wurtzite_cellGIF.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Wurtzite_cellGIF.gif/440px-Wurtzite_cellGIF.gif 2x" data-file-width="1280" data-file-height="980" /></a><figcaption>Wurtzite unit cell as described by symmetry operators of the space group.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wurtzite-unit-cell-3D-balls.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wurtzite-unit-cell-3D-balls.png/150px-Wurtzite-unit-cell-3D-balls.png" decoding="async" width="150" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wurtzite-unit-cell-3D-balls.png/225px-Wurtzite-unit-cell-3D-balls.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wurtzite-unit-cell-3D-balls.png/300px-Wurtzite-unit-cell-3D-balls.png 2x" data-file-width="990" data-file-height="1100" /></a><figcaption>Another representation of the wurtzite <a href="/wiki/Unit_cell" title="Unit cell">unit cell</a><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2018)">citation needed</span></a></i>]</sup></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wurtzite_polyhedra.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Wurtzite_polyhedra.png/150px-Wurtzite_polyhedra.png" decoding="async" width="150" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Wurtzite_polyhedra.png/225px-Wurtzite_polyhedra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Wurtzite_polyhedra.png/300px-Wurtzite_polyhedra.png 2x" data-file-width="1417" data-file-height="1266" /></a><figcaption>Another representation of the wurtzite structure<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2018)">citation needed</span></a></i>]</sup></figcaption></figure> <p>The wurtzite crystal structure is referred to by the <a href="/wiki/Strukturbericht_designation" title="Strukturbericht designation">Strukturbericht designation</a> B4 and the <a href="/wiki/Pearson_symbol" title="Pearson symbol">Pearson symbol</a> hP4. The corresponding <a href="/wiki/Space_group#Notation" title="Space group">space group</a> is <a href="/wiki/List_of_space_groups" title="List of space groups">No. 186</a> (in International Union of Crystallography classification) or <b>P6<sub>3</sub>mc</b> (in <a href="/wiki/Hermann%E2%80%93Mauguin_notation" title="Hermann–Mauguin notation">Hermann–Mauguin notation</a>). The Hermann-Mauguin symbols in <b>P6<sub>3</sub>mc</b> can be read as follows:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <ul><li><b>6<sub>3</sub>..</b> : a six fold screw rotation around the c-axis</li> <li><b>.m.</b> : a mirror plane with normal {100}</li> <li><b>..c</b> : glide plane in the <b>c</b>-directions with normal {120}.</li></ul> <p>Among the compounds that can take the wurtzite structure are <a href="/wiki/Wurtzite" title="Wurtzite">wurtzite</a> itself (<a href="/wiki/ZnS" class="mw-redirect" title="ZnS">ZnS</a> with up to 8% <a href="/wiki/Iron" title="Iron">iron</a> instead of <a href="/wiki/Zinc" title="Zinc">zinc</a>), <a href="/wiki/Silver_iodide" title="Silver iodide">silver iodide</a> (AgI), <a href="/wiki/Zinc_oxide" title="Zinc oxide">zinc oxide</a> (ZnO), <a href="/wiki/Cadmium_sulfide" title="Cadmium sulfide">cadmium sulfide</a> (CdS), <a href="/wiki/Cadmium_selenide" title="Cadmium selenide">cadmium selenide</a> (CdSe), <a href="/wiki/Silicon_carbide" title="Silicon carbide">silicon carbide</a> (α-SiC), <a href="/wiki/Gallium_nitride" title="Gallium nitride">gallium nitride</a> (GaN), <a href="/wiki/Aluminium_nitride" title="Aluminium nitride">aluminium nitride</a> (AlN), <a href="/wiki/Boron_nitride#Wurtzite_form_(w-BN)" title="Boron nitride">boron nitride</a> (w-BN) and other <a href="/wiki/Semiconductors" class="mw-redirect" title="Semiconductors">semiconductors</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In most of these compounds, wurtzite is not the favored form of the bulk crystal, but the structure can be favored in some nanocrystal forms of the material. </p><p>In materials with more than one crystal structure, the prefix "w-" is sometimes added to the <a href="/wiki/Empirical_formula" title="Empirical formula">empirical formula</a> to denote the wurtzite crystal structure, as in <a href="/wiki/W-BN" class="mw-redirect" title="W-BN">w-BN</a>. </p><p>Each of the two individual atom types forms a sublattice which is <a href="/wiki/Hexagonal_close-packed" class="mw-redirect" title="Hexagonal close-packed">hexagonal close-packed</a> (HCP-type). When viewed all together, the atomic positions are the same as in <a href="/wiki/Lonsdaleite" title="Lonsdaleite">lonsdaleite</a> (hexagonal <a href="/wiki/Diamond" title="Diamond">diamond</a>). Each atom is <a href="/wiki/Tetrahedral_molecular_geometry" title="Tetrahedral molecular geometry">tetrahedrally</a> coordinated. The structure can also be described as an HCP lattice of zinc with sulfur atoms occupying half of the <a href="/wiki/Interstitial_site" title="Interstitial site">tetrahedral voids</a> or vice versa. </p><p>The wurtzite structure is <a href="/wiki/Centrosymmetry" title="Centrosymmetry">non-centrosymmetric</a> (i.e., lacks <a href="/wiki/Inversion_symmetry" class="mw-redirect" title="Inversion symmetry">inversion symmetry</a>). Due to this, wurtzite crystals can (and generally do) have properties such as <a href="/wiki/Piezoelectricity" title="Piezoelectricity">piezoelectricity</a> and <a href="/wiki/Pyroelectricity" title="Pyroelectricity">pyroelectricity</a>, which centrosymmetric crystals lack.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2012)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Nickel_arsenide_structure">Nickel arsenide structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=8" title="Edit section: Nickel arsenide structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable selfref">See also: <a href="/wiki/Category:Nickel_arsenide_structure_type" title="Category:Nickel arsenide structure type">Category:Nickel arsenide structure type</a></div> <p>The nickel arsenide structure consists of two interpenetrating sublattices: a primitive hexagonal nickel sublattice and a <a href="/wiki/Hexagonal_close-packed" class="mw-redirect" title="Hexagonal close-packed">hexagonal close-packed</a> arsenic sublattice. Each nickel atom is <a href="/wiki/Octahedral_molecular_geometry" title="Octahedral molecular geometry">octahedrally</a> <a href="/wiki/Coordination_number" title="Coordination number">coordinated</a> to six arsenic atoms, while each arsenic atom is <a href="/wiki/Trigonal_prismatic_molecular_geometry" title="Trigonal prismatic molecular geometry">trigonal prismatically</a> coordinated to six nickel atoms.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The structure can also be described as an HCP lattice of arsenic with nickel occupying each <a href="/wiki/Interstitial_site" title="Interstitial site">octahedral void</a>. </p><p>Compounds adopting the NiAs structure are generally the <a href="/wiki/Chalcogen" title="Chalcogen">chalcogenides</a>, <a href="/wiki/Arsenide" title="Arsenide">arsenides</a>, <a href="/wiki/Antimony" title="Antimony">antimonides</a> and <a href="/wiki/Bismuth" title="Bismuth">bismuthides</a> of <a href="/wiki/Transition_metal" title="Transition metal">transition metals</a>. <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2019)">citation needed</span></a></i>]</sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Nickel-arsenide-3D-unit-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Nickel-arsenide-3D-unit-cell.png/220px-Nickel-arsenide-3D-unit-cell.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Nickel-arsenide-3D-unit-cell.png/330px-Nickel-arsenide-3D-unit-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Nickel-arsenide-3D-unit-cell.png/440px-Nickel-arsenide-3D-unit-cell.png 2x" data-file-width="1100" data-file-height="1075" /></a><figcaption>The unit cell of nickeline</figcaption></figure> <p>The following are the members of the nickeline group:<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Achavalite" class="mw-redirect" title="Achavalite">Achavalite</a>: <style data-mw-deduplicate="TemplateStyles:r1123817410">.mw-parser-output .template-chem2-su{display:inline-block;font-size:80%;line-height:1;vertical-align:-0.35em}.mw-parser-output .template-chem2-su>span{display:block;text-align:left}.mw-parser-output sub.template-chem2-sub{font-size:80%;vertical-align:-0.35em}.mw-parser-output sup.template-chem2-sup{font-size:80%;vertical-align:0.65em}</style><span class="chemf nowrap">FeSe</span></li> <li><a href="/wiki/Breithauptite" title="Breithauptite">Breithauptite</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">NiSb</span></li> <li>Freboldite: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">CoSe</span></li> <li>Kotulskite: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">Pd(Te,Bi)</span></li> <li>Langistite: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">(Co,Ni)As</span></li> <li>Nickeline: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">NiAs</span></li> <li>Sobolevskite: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">Pd(Bi,Te)</span></li> <li>Sudburyite: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">(Pd,Ni)Sb</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="In_two_dimensions">In two dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=9" title="Edit section: In two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hexagonal_lattice" title="Hexagonal lattice">Hexagonal lattice</a></div> <p>There is only one hexagonal Bravais lattice in two dimensions: the hexagonal lattice. </p> <table class="wikitable"> <tbody><tr> <th>Bravais lattice </th> <th>Hexagonal </th></tr> <tr align="center"> <th><a href="/wiki/Pearson_symbol" title="Pearson symbol">Pearson symbol</a> </th> <td>hp </td></tr> <tr> <th><a href="/wiki/Unit_cell" title="Unit cell">Unit cell</a> </th> <td><span typeof="mw:File"><a href="/wiki/File:2d_hp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/2d_hp.svg/100px-2d_hp.svg.png" decoding="async" width="100" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/2d_hp.svg/150px-2d_hp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/2d_hp.svg/200px-2d_hp.svg.png 2x" data-file-width="290" data-file-height="280" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Close-packing_of_equal_spheres" title="Close-packing of equal spheres">Close-packing</a></li> <li><a href="/wiki/Crystal_structure" title="Crystal structure">Crystal structure</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-hahn-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-hahn_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hahn_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-hahn_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-hahn_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHahn2005" class="citation book cs1">Hahn, Theo, ed. (2005). <i>International tables for crystallography</i> (5th ed.). Dordrecht, Netherlands: Published for the International Union of Crystallography by Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-6590-7" title="Special:BookSources/978-0-7923-6590-7"><bdi>978-0-7923-6590-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=International+tables+for+crystallography&rft.place=Dordrecht%2C+Netherlands&rft.edition=5th&rft.pub=Published+for+the+International+Union+of+Crystallography+by+Springer&rft.date=2005&rft.isbn=978-0-7923-6590-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexagonal+crystal+family" class="Z3988"></span></span> </li> <li id="cite_note-Hurlbut-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hurlbut_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hurlbut_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDanaHurlbut1959" class="citation book cs1">Dana, James Dwight; Hurlbut, Cornelius Searle (1959). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e7MSMwEACAAJ"><i>Dana's Manual of Mineralogy</i></a> (17th ed.). New York: Chapman Hall. pp. <span class="nowrap">78–</span>89.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dana%27s+Manual+of+Mineralogy&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E78-%3C%2Fspan%3E89&rft.edition=17th&rft.pub=Chapman+Hall&rft.date=1959&rft.aulast=Dana&rft.aufirst=James+Dwight&rft.au=Hurlbut%2C+Cornelius+Searle&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De7MSMwEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexagonal+crystal+family" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdward_Prince2004" class="citation book cs1">Edward Prince (2004). <i>Mathematical Techniques in Crystallography and Materials Science</i>. 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Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+mathematical+crystallography&rft.pub=American+Elsevier+Pub.+Co.&rft.date=1965-01-01&rft.aulast=Jaswon&rft.aufirst=Maurice+Aaron&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlTRRAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexagonal+crystal+family" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_GraefMcHenry2012" class="citation book cs1">De Graef, Marc; McHenry, Michael E. (2012). <a rel="nofollow" class="external text" href="http://som.web.cmu.edu/StructuresAppendix2.pdf"><i>Structure of Materials; An introduction to Crystallography, Diffraction and Symmetry</i></a> <span class="cs1-format">(PDF)</span>. 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"Distributions of phonon lifetimes in Brillouin zones". <i>Physical Review B</i>. <b>91</b> (9): 094306. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1501.00691">1501.00691</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PhRvB..91i4306T">2015PhRvB..91i4306T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.91.094306">10.1103/PhysRevB.91.094306</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118851924">118851924</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Distributions+of+phonon+lifetimes+in+Brillouin+zones&rft.volume=91&rft.issue=9&rft.pages=094306&rft.date=2015-03-20&rft_id=info%3Aarxiv%2F1501.00691&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118851924%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.91.094306&rft_id=info%3Abibcode%2F2015PhRvB..91i4306T&rft.aulast=Togo&rft.aufirst=Atsushi&rft.au=Chaput%2C+Laurent&rft.au=Tanaka%2C+Isao&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexagonal+crystal+family" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Inorganic Chemistry by Duward Shriver and Peter Atkins, 3rd Edition, W.H. Freeman and Company, 1999, pp.47,48.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.mindat.org/min-2901.html">http://www.mindat.org/min-2901.html</a> Mindat.org</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hexagonal_crystal_family&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" 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template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Crystal_systems&action=edit&redlink=1" class="new" title="Template talk:Crystal systems (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Crystal_systems" title="Special:EditPage/Template:Crystal systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Crystal_systems104" style="font-size:114%;margin:0 4em"><a href="/wiki/Crystal_system" title="Crystal system">Crystal systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattice</a></li> <li><a href="/wiki/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="/wiki/Triclinic_crystal_system" title="Triclinic crystal system">triclinic</a> (anorthic) <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Triclinic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/30px-Triclinic.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/45px-Triclinic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/60px-Triclinic.svg.png 2x" data-file-width="129" data-file-height="149" /></a></span></li> <li><a href="/wiki/Monoclinic_crystal_system" title="Monoclinic crystal system">monoclinic</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Monoclinic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/30px-Monoclinic.svg.png" decoding="async" width="30" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/45px-Monoclinic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/60px-Monoclinic.svg.png 2x" data-file-width="114" data-file-height="143" /></a></span></li> <li><a href="/wiki/Orthorhombic_crystal_system" title="Orthorhombic crystal system">orthorhombic</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Orthorhombic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/30px-Orthorhombic.svg.png" decoding="async" width="30" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/45px-Orthorhombic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/60px-Orthorhombic.svg.png 2x" data-file-width="108" data-file-height="142" /></a></span></li> <li><a href="/wiki/Tetragonal_crystal_system" title="Tetragonal crystal system">tetragonal</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Tetragonal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/30px-Tetragonal.svg.png" decoding="async" width="30" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/45px-Tetragonal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/60px-Tetragonal.svg.png 2x" data-file-width="108" data-file-height="165" /></a></span></li> <li><a class="mw-selflink selflink">trigonal & hexagonal</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Rhombohedral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/30px-Rhombohedral.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/45px-Rhombohedral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/60px-Rhombohedral.svg.png 2x" data-file-width="139" data-file-height="141" /></a></span><span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Hexagonal_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/30px-Hexagonal_lattice.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/45px-Hexagonal_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/60px-Hexagonal_lattice.svg.png 2x" data-file-width="160" data-file-height="186" /></a></span></li> <li><a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic</a> (isometric) <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Cubic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/30px-Cubic.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/45px-Cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/60px-Cubic.svg.png 2x" data-file-width="109" data-file-height="127" /></a></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="/wiki/Oblique_lattice" title="Oblique lattice">oblique</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:2d_mp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/30px-2d_mp.svg.png" decoding="async" width="30" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/45px-2d_mp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/60px-2d_mp.svg.png 2x" data-file-width="140" data-file-height="170" /></a></span></li> <li><a href="/wiki/Rectangular_lattice" title="Rectangular lattice">rectangular</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:2d_op_rectangular.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/30px-2d_op_rectangular.svg.png" decoding="async" width="30" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/45px-2d_op_rectangular.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/60px-2d_op_rectangular.svg.png 2x" data-file-width="210" data-file-height="170" /></a></span></li> <li><a href="/wiki/Square_lattice" title="Square lattice">square</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:2d_tp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2d_tp.svg/30px-2d_tp.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2d_tp.svg/45px-2d_tp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2d_tp.svg/60px-2d_tp.svg.png 2x" data-file-width="170" data-file-height="170" /></a></span></li> <li><a href="/wiki/Hexagonal_lattice" title="Hexagonal lattice">hexagonal</a> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:2d_hp.svg" class="mw-file-description"><img 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