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</nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Cubique centré</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 7 langues." > <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-7" 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">7 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en badge-Q70894304 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Body-centered_cubic" title="Body-centered cubic – anglais" lang="en" hreflang="en" data-title="Body-centered cubic" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estructura_c%C3%BAbica_centrada" title="Estructura cúbica centrada – espagnol" lang="es" hreflang="es" data-title="Estructura cúbica centrada" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E6%A7%8B%E9%80%A0" title="体心立方格子構造 – japonais" lang="ja" hreflang="ja" data-title="体心立方格子構造" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%B4%EC%8B%AC_%EC%9E%85%EB%B0%A9_%EA%B2%A9%EC%9E%90" title="체심 입방 격자 – coréen" lang="ko" hreflang="ko" data-title="체심 입방 격자" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kubisch_ruimtelijk_gecentreerd" title="Kubisch ruimtelijk gecentreerd – néerlandais" lang="nl" hreflang="nl" data-title="Kubisch ruimtelijk gecentreerd" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B1%27%D1%94%D0%BC%D0%BD%D0%BE%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B0_%D0%BA%D1%83%D0%B1%D1%96%D1%87%D0%BD%D0%B0_%D2%91%D1%80%D0%B0%D1%82%D0%BA%D0%B0" title="Об'ємноцентрована кубічна ґратка – ukrainien" lang="uk" hreflang="uk" data-title="Об'ємноцентрована кубічна ґратка" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li 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class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Structure_cubique_centr%C3%A9e.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Structure_cubique_centr%C3%A9e.svg/220px-Structure_cubique_centr%C3%A9e.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Structure_cubique_centr%C3%A9e.svg/330px-Structure_cubique_centr%C3%A9e.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Structure_cubique_centr%C3%A9e.svg/440px-Structure_cubique_centr%C3%A9e.svg.png 2x" data-file-width="1567" data-file-height="1550" /></a><figcaption>Structure cubique centrée composée d'un seul type d'atome</figcaption></figure> <p>La structure <b>cubique centrée</b> (cc), dite aussi <b>cubique à corps centré</b> (ccc), est un type de <a href="/wiki/Structure_cristalline" title="Structure cristalline">structure cristalline</a>. Elle se rencontre dans tous les <a href="/wiki/M%C3%A9tal_alcalin" title="Métal alcalin">métaux alcalins</a> et dans les métaux <a href="/wiki/Baryum" title="Baryum">baryum</a>, <a href="/wiki/Radium" title="Radium">radium</a>, <a href="/wiki/Vanadium" title="Vanadium">vanadium</a>, <a href="/wiki/Niobium" title="Niobium">niobium</a>, <a href="/wiki/Tantale_(chimie)" title="Tantale (chimie)">tantale</a>, <a href="/wiki/Chrome" title="Chrome">chrome</a>, <a href="/wiki/Molybd%C3%A8ne" title="Molybdène">molybdène</a>, et <a href="/wiki/Tungst%C3%A8ne" title="Tungstène">tungstène</a>, ainsi que le <a href="/wiki/Fer" title="Fer">fer</a> à l'<a href="/wiki/Ferrite_(acier)" title="Ferrite (acier)">état alpha</a> (stable aux températures ordinaires et jusqu'à <span title="1 673,6 °F ou 1 185,2 K">912 </span><abbr class="abbr" title="degré Celsius">°C</abbr>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Structure">Structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cubique_centr%C3%A9&veaction=edit&section=1" title="Modifier la section : Structure" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cubique_centr%C3%A9&action=edit&section=1" title="Modifier le code source de la section : Structure"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans cette structure, les atomes sont situés : </p> <ul><li>aux 8 sommets d'un <a href="/wiki/Cube" title="Cube">cube</a> ;</li> <li>au centre du cube.</li></ul> <p>L'image ci-dessus indique que l'atome au centre possède huit (8) voisins et alors l'<a href="/wiki/Coordinence" title="Coordinence">indice de coordination</a> est de 8 (chaque <a href="/wiki/Atome" title="Atome">atome</a> au sommet possède aussi huit voisins si on suppose que le dessin continue jusqu'à une distance infinie dans toutes les directions). </p><p>Chaque atome dans les coins ne comptant que pour 1/8 (chaque atome est partagé par 8 mailles différents) et l'atome central pour 1, la <a href="/wiki/Maille_(cristallographie)" title="Maille (cristallographie)">maille conventionnelle</a> de cette structure comporte 2 atomes. Elle est entièrement déterminée par un unique <a href="/wiki/Param%C3%A8tre_cristallin" title="Paramètre cristallin">paramètre de maille</a>, la longueur de l'arête <i>a</i> : </p> <ul><li><i>a</i> = <i>b</i> = <i>c</i> ;</li> <li>α = β = γ = 90°.</li></ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fichier:Lattic_simple_cubic.svg" class="mw-file-description" title="Cubique primitif"><img alt="Cubique primitif" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Lattic_simple_cubic.svg/120px-Lattic_simple_cubic.svg.png" decoding="async" width="120" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Lattic_simple_cubic.svg/180px-Lattic_simple_cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Lattic_simple_cubic.svg/240px-Lattic_simple_cubic.svg.png 2x" data-file-width="406" data-file-height="360" /></a></span></div> <div class="gallerytext"><a href="/wiki/Syst%C3%A8me_r%C3%A9ticulaire_cubique" title="Système réticulaire cubique">Cubique primitif</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fichier:Lattice_body_centered_cubic.svg" class="mw-file-description" title="Cubique centré"><img alt="Cubique centré" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Lattice_body_centered_cubic.svg/120px-Lattice_body_centered_cubic.svg.png" decoding="async" width="120" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Lattice_body_centered_cubic.svg/180px-Lattice_body_centered_cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Lattice_body_centered_cubic.svg/240px-Lattice_body_centered_cubic.svg.png 2x" data-file-width="403" data-file-height="354" /></a></span></div> <div class="gallerytext">Cubique centré</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fichier:Lattice_face_centered_cubic.svg" class="mw-file-description" title="Cubique à faces centrées"><img alt="Cubique à faces centrées" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Lattice_face_centered_cubic.svg/120px-Lattice_face_centered_cubic.svg.png" decoding="async" width="120" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Lattice_face_centered_cubic.svg/180px-Lattice_face_centered_cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Lattice_face_centered_cubic.svg/240px-Lattice_face_centered_cubic.svg.png 2x" data-file-width="399" data-file-height="359" /></a></span></div> <div class="gallerytext"><a href="/wiki/Cubique_%C3%A0_faces_centr%C3%A9es" title="Cubique à faces centrées">Cubique à faces centrées</a></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Paramètre_de_maille"><span id="Param.C3.A8tre_de_maille"></span>Paramètre de maille</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cubique_centr%C3%A9&veaction=edit&section=2" title="Modifier la section : Paramètre de maille" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cubique_centr%C3%A9&action=edit&section=2" title="Modifier le code source de la section : Paramètre de maille"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fichier:%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E3%81%AE%E5%8E%9F%E5%AD%90%E5%8D%8A%E5%BE%84%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E3%81%AE%E5%8E%9F%E5%AD%90%E5%8D%8A%E5%BE%84%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9.svg/400px-%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E3%81%AE%E5%8E%9F%E5%AD%90%E5%8D%8A%E5%BE%84%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9.svg.png" decoding="async" width="400" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E3%81%AE%E5%8E%9F%E5%AD%90%E5%8D%8A%E5%BE%84%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9.svg/600px-%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E3%81%AE%E5%8E%9F%E5%AD%90%E5%8D%8A%E5%BE%84%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E3%81%AE%E5%8E%9F%E5%AD%90%E5%8D%8A%E5%BE%84%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9.svg/800px-%E4%BD%93%E5%BF%83%E7%AB%8B%E6%96%B9%E6%A0%BC%E5%AD%90%E3%81%AE%E5%8E%9F%E5%AD%90%E5%8D%8A%E5%BE%84%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9.svg.png 2x" data-file-width="500" data-file-height="250" /></a><figcaption></figcaption></figure> <p>Dans le cas d'un cristal composé d'un seul type d'atome, on peut utiliser le modèle des <a href="/wiki/Sph%C3%A8res_%C3%A9lastiques_infiniment_dures" title="Sphères élastiques infiniment dures">sphères dures</a> : les atomes sont considérés comme étant des boules indéformables de rayon <i>R</i> qui sont au contact. </p><p>La distance entre deux coins opposés du cube est égale à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}"></span> fois le paramètre de <a href="/wiki/Maille_(cristallographie)" title="Maille (cristallographie)">maille</a> <i>a</i>. Mais, dans le cas d'une structure cubique centrée, cette distance est deux fois le diamètre atomique ou quatre fois le rayon = 4R, de sorte que : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {4}{\sqrt {3}}}\cdot R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {4}{\sqrt {3}}}\cdot R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4984b794a86030ade53a8dec1a4e91035a0bb86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.706ex; height:6.176ex;" alt="{\displaystyle a={\frac {4}{\sqrt {3}}}\cdot R}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Compacité"><span id="Compacit.C3.A9"></span>Compacité</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cubique_centr%C3%A9&veaction=edit&section=3" title="Modifier la section : Compacité" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cubique_centr%C3%A9&action=edit&section=3" title="Modifier le code source de la section : Compacité"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Compacit%C3%A9_(cristallographie)" title="Compacité (cristallographie)">compacité</a>, proportion d'espaces occupés par les atomes de la maille cubique centrée est de : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {\mathrm {Volume~occup{\acute {e}}} }{\text{Volume total}}}={\frac {2\times V_{atome}}{a^{3}}}={\frac {2\times ({\frac {4}{3}}\cdot \pi R^{3})}{({\frac {4}{\sqrt {3}}}\cdot R)^{3}}}={\frac {\pi \cdot {\sqrt {3}}}{8}}\simeq 0,68}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">e</mi> <mo>´</mo> </mover> </mrow> </mrow> </mrow> <mtext>Volume total</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>×</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>m</mi> <mi>e</mi> </mrow> </msub> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mi>R</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>≃</mo> <mn>0</mn> <mo>,</mo> <mn>68</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {\mathrm {Volume~occup{\acute {e}}} }{\text{Volume total}}}={\frac {2\times V_{atome}}{a^{3}}}={\frac {2\times ({\frac {4}{3}}\cdot \pi R^{3})}{({\frac {4}{\sqrt {3}}}\cdot R)^{3}}}={\frac {\pi \cdot {\sqrt {3}}}{8}}\simeq 0,68}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383374f2e0a729f7189f05365e88d265d1170f54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:69.203ex; height:8.676ex;" alt="{\displaystyle c={\frac {\mathrm {Volume~occup{\acute {e}}} }{\text{Volume total}}}={\frac {2\times V_{atome}}{a^{3}}}={\frac {2\times ({\frac {4}{3}}\cdot \pi R^{3})}{({\frac {4}{\sqrt {3}}}\cdot R)^{3}}}={\frac {\pi \cdot {\sqrt {3}}}{8}}\simeq 0,68}"></span>. </p><p>Cette valeur est inférieure à celles des structures <a href="/wiki/Cubique_%C3%A0_faces_centr%C3%A9es" title="Cubique à faces centrées">cubique à faces centrées</a> et <a href="/wiki/Empilement_compact" title="Empilement compact">hexagonale compacte</a> qui sont toutes deux de 0,74. On dit que ces deux dernières correspondent à l'<a href="/wiki/Empilement_compact" title="Empilement compact">empilement compact</a> ou de compacité maximale, tandis que la structure cubique centrée est une structure non-compacte au sens strict parce que sa compacité n'est pas maximale. </p> <div class="mw-heading mw-heading2"><h2 id="Sites_interstitiels">Sites interstitiels</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cubique_centr%C3%A9&veaction=edit&section=4" title="Modifier la section : Sites interstitiels" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cubique_centr%C3%A9&action=edit&section=4" title="Modifier le code source de la section : Sites interstitiels"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Sites_interstitiels_cubique_centre.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Sites_interstitiels_cubique_centre.svg/220px-Sites_interstitiels_cubique_centre.svg.png" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Sites_interstitiels_cubique_centre.svg/330px-Sites_interstitiels_cubique_centre.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Sites_interstitiels_cubique_centre.svg/440px-Sites_interstitiels_cubique_centre.svg.png 2x" data-file-width="152" data-file-height="158" /></a><figcaption>Sites interstitiels cubique centré</figcaption></figure> <ul><li>Sites tétraédriques dans un empilement Cubique Centré</li></ul> <p>Situés aux 1/4 et 3/4 des médiatrices des arêtes : 4 sites par face conjointe à 2 mailles : 6 × 4/2 = 12 sites par maille. </p><p>Soit au total <i>12 sites tétraédriques par maille</i>. </p> <ul><li>Sites octaédriques dans un empilement Cubique Centré <ul><li>Centre des faces : 6 faces conjointes à 2 mailles : 6/2 = 3 sites par maille.</li> <li>Milieu des arêtes : 12 arêtes conjointes à 4 mailles : 12/4 = 3 sites par maille.</li></ul></li></ul> <p>Soit au total <i>6 sites octaédriques par maille</i>. </p><p><i>Remarque</i> : par une vision géométrique des sites interstitiels, on peut déterminer le rayon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db421291be9d0103404ced7495b363437b67b6b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle R_{i}}"></span> de l'atome en insertion sans qu'il n'y ait déformation de la maille. On a par exemple pour les sites octaédriques : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i}=0.07a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0.07</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}=0.07a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d783bbd093ece53b969bcddb07909223fd3ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.026ex; height:2.509ex;" alt="{\displaystyle R_{i}=0.07a}"></span> , (avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> le paramètre de maille). </p> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cubique_centr%C3%A9&veaction=edit&section=5" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cubique_centr%C3%A9&action=edit&section=5" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cubique_centr%C3%A9&veaction=edit&section=6" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cubique_centr%C3%A9&action=edit&section=6" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Structure_cristalline" title="Structure cristalline">Structure cristalline</a></li> <li><a href="/wiki/R%C3%A9seau_de_Bravais" title="Réseau de Bravais">Réseau de Bravais</a></li> <li><a href="/wiki/Cubique_%C3%A0_faces_centr%C3%A9es" title="Cubique à faces centrées">Cubique à faces centrées</a></li> <li><a href="/wiki/Atomium" title="Atomium">Atomium</a>, un monument à Bruxelles qui représente la maille cubique centrée du <a href="/wiki/Fer" title="Fer">fer</a>, agrandie 165 milliards de fois.</li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Syst%C3%A8mes_r%C3%A9ticulaires" title="Modèle:Palette Systèmes réticulaires"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Syst%C3%A8mes_r%C3%A9ticulaires&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Syst%C3%A8me_r%C3%A9ticulaire" title="Système réticulaire">Systèmes réticulaires</a></div></th> </tr> <tr> <td class="navbox-list" style="text-align:center;" colspan="2"><div class="liste-horizontale"> <ul><li><a href="/wiki/Syst%C3%A8me_r%C3%A9ticulaire_triclinique" title="Système réticulaire triclinique">triclinique</a> <span typeof="mw:File"><a href="/wiki/Fichier: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/Syst%C3%A8me_r%C3%A9ticulaire_monoclinique" title="Système réticulaire monoclinique">monoclinique</a> <span typeof="mw:File"><a href="/wiki/Fichier: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/Syst%C3%A8me_r%C3%A9ticulaire_orthorhombique" title="Système réticulaire orthorhombique">orthorhombique</a> <span typeof="mw:File"><a href="/wiki/Fichier: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/Syst%C3%A8me_r%C3%A9ticulaire_t%C3%A9tragonal" title="Système réticulaire tétragonal">tétragonal</a> <span typeof="mw:File"><a href="/wiki/Fichier: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 href="/wiki/Syst%C3%A8me_r%C3%A9ticulaire_rhombo%C3%A9drique" title="Système réticulaire rhomboédrique">rhomboédrique</a> <span typeof="mw:File"><a href="/wiki/Fichier: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></li> <li><a href="/wiki/Syst%C3%A8me_r%C3%A9ticulaire_hexagonal" title="Système réticulaire hexagonal">hexagonal</a> <span typeof="mw:File"><a href="/wiki/Fichier: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/Syst%C3%A8me_r%C3%A9ticulaire_cubique" title="Système réticulaire cubique">cubique</a> <span typeof="mw:File"><a href="/wiki/Fichier: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> </tbody></table> </div><p>. </p><ul 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