{"id":194477,"date":"2025-02-24T06:39:04","date_gmt":"2025-02-24T06:39:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=194477"},"modified":"2025-02-24T06:39:09","modified_gmt":"2025-02-24T06:39:09","slug":"derive-planar-density-expressions-for-bcc-100-and-110-planes-in-terms-of-the-atomic-radius-r","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/24\/derive-planar-density-expressions-for-bcc-100-and-110-planes-in-terms-of-the-atomic-radius-r\/","title":{"rendered":"Derive planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R"},"content":{"rendered":"\n<p>Derive planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Planar Density Derivation for BCC (100) and (110) Planes<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Planar Density Definition<\/strong><\/h4>\n\n\n\n<p>The <strong>planar density (PD)<\/strong> is defined as the number of atoms per unit area in a specific crystallographic plane. Mathematically,<\/p>\n\n\n\n<p>[<br>PD = \\frac{\\text{Number of atoms centered on the plane}}{\\text{Area of the plane}}<br>]<\/p>\n\n\n\n<p>where the number of atoms is determined by the atomic positions in the given plane, and the area is calculated based on the lattice parameters.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>1. Planar Density for BCC (100) Plane<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Step 1: Number of Atoms in the (100) Plane<\/strong><\/h4>\n\n\n\n<p>In the <strong>(100) plane<\/strong> of a <strong>BCC (Body-Centered Cubic) structure<\/strong>, we observe:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Four<\/strong> corner atoms, each contributing <strong>( \\frac{1}{4} ) of an atom<\/strong> (since each corner atom is shared among four adjacent unit cells).<\/li>\n\n\n\n<li><strong>No atoms at the center of the (100) face<\/strong> in a BCC structure.<\/li>\n<\/ul>\n\n\n\n<p>Thus, the total number of atoms in the (100) plane:<\/p>\n\n\n\n<p>[<br>\\text{Total atoms} = 4 \\times \\frac{1}{4} = 1<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Step 2: Area of the (100) Plane<\/strong><\/h4>\n\n\n\n<p>The unit cell edge length <strong>a<\/strong> in terms of the atomic radius ( R ) for BCC is:<\/p>\n\n\n\n<p>[<br>a = \\frac{4R}{\\sqrt{3}}<br>]<\/p>\n\n\n\n<p>Since the (100) plane is a square with side length <strong>a<\/strong>, its area is:<\/p>\n\n\n\n<p>[<br>A_{100} = a^2 = \\left(\\frac{4R}{\\sqrt{3}}\\right)^2 = \\frac{16R^2}{3}<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Step 3: Planar Density Calculation<\/strong><\/h4>\n\n\n\n<p>[<br>PD_{(100)} = \\frac{1}{A_{100}} = \\frac{1}{\\frac{16R^2}{3}} = \\frac{3}{16R^2}<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. Planar Density for BCC (110) Plane<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Step 1: Number of Atoms in the (110) Plane<\/strong><\/h4>\n\n\n\n<p>In the <strong>(110) plane<\/strong> of a <strong>BCC structure<\/strong>, we observe:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Four<\/strong> corner atoms, each contributing <strong>( \\frac{1}{4} ) of an atom<\/strong>.<\/li>\n\n\n\n<li><strong>One full body-centered atom<\/strong> lies entirely within the plane.<\/li>\n<\/ul>\n\n\n\n<p>Thus, the total number of atoms in the (110) plane:<\/p>\n\n\n\n<p>[<br>\\text{Total atoms} = (4 \\times \\frac{1}{4}) + 1 = 2<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Step 2: Area of the (110) Plane<\/strong><\/h4>\n\n\n\n<p>The (110) plane forms a <strong>rectangular<\/strong> surface with dimensions <strong>a<\/strong> (one unit cell edge) and <strong>( \\sqrt{2}a )<\/strong> (diagonal of the square face). The area is:<\/p>\n\n\n\n<p>[<br>A_{110} = a \\times \\sqrt{2}a = a^2\\sqrt{2}<br>]<\/p>\n\n\n\n<p>Substituting ( a = \\frac{4R}{\\sqrt{3}} ):<\/p>\n\n\n\n<p>[<br>A_{110} = \\left(\\frac{4R}{\\sqrt{3}}\\right)^2 \\times \\sqrt{2} = \\frac{16R^2}{3} \\times \\sqrt{2} = \\frac{16\\sqrt{2} R^2}{3}<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Step 3: Planar Density Calculation<\/strong><\/h4>\n\n\n\n<p>[<br>PD_{(110)} = \\frac{2}{A_{110}} = \\frac{2}{\\frac{16\\sqrt{2} R^2}{3}} = \\frac{3}{8\\sqrt{2} R^2}<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Final Expressions<\/strong><\/h3>\n\n\n\n<p>[<br>PD_{(100)} = \\frac{3}{16R^2}<br>]<\/p>\n\n\n\n<p>[<br>PD_{(110)} = \\frac{3}{8\\sqrt{2} R^2}<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation<\/strong><\/h3>\n\n\n\n<p>Planar density provides insight into how closely packed atoms are within a specific crystallographic plane. It is important for understanding <strong>surface energy, slip systems, and material properties<\/strong> such as plastic deformation and diffusion rates.<\/p>\n\n\n\n<p>For <strong>BCC (100) vs. (110):<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>(110) plane has a higher planar density<\/strong> than (100), which means atoms are more closely packed in (110).<\/li>\n\n\n\n<li>The <strong>(110) plane<\/strong> is a common slip plane in BCC metals because of this higher atomic density, allowing for easier dislocation movement.<\/li>\n<\/ul>\n\n\n\n<p>These expressions are fundamental in <strong>materials science<\/strong>, particularly for <strong>mechanical properties and failure mechanisms<\/strong> in BCC metals like iron, tungsten, and molybdenum.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Derive planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R. The Correct Answer and Explanation is : Planar Density Derivation for BCC (100) and (110) Planes Planar Density Definition The planar density (PD) is defined as the number of atoms per unit area in a specific crystallographic plane. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-194477","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194477","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=194477"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194477\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=194477"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=194477"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=194477"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}