{"id":217852,"date":"2025-05-22T15:10:17","date_gmt":"2025-05-22T15:10:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217852"},"modified":"2025-05-22T15:10:20","modified_gmt":"2025-05-22T15:10:20","slug":"calculate-the-number-of-spheres-in-the-simple-cubic-body-centered-cubic-and-face-centered-cubic-cells","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/calculate-the-number-of-spheres-in-the-simple-cubic-body-centered-cubic-and-face-centered-cubic-cells\/","title":{"rendered":"Calculate the number of spheres in the simple cubic, body-centered cubic, and face-centered cubic cells."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><br>Calculate the number of spheres in the simple cubic, body-centered cubic, and face-centered cubic cells. Also, calculate the packing efficiency of each type of cell. simple cubic cells: spheres packing efficiency % body-centered cubic cells: spheres packing efficiency face-centered cubic cells: spheres packing efficiency % None<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s break down the calculation of the <strong>number of spheres per unit cell<\/strong> and <strong>packing efficiency<\/strong> for each of the three types of cubic crystal systems:<\/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. Simple Cubic (SC) Cell<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Number of spheres per unit cell:<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A simple cubic unit cell has <strong>1 atom at each of its 8 corners<\/strong>.<\/li>\n\n\n\n<li>Each corner atom is <strong>shared among 8 adjacent unit cells<\/strong>.<\/li>\n\n\n\n<li>Contribution per corner atom = 1\/8.<\/li>\n\n\n\n<li>Total atoms per unit cell = 8 corners \u00d7 (1\/8) = <strong>1 atom<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Packing efficiency:<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Assume each sphere (atom) has radius <strong>r<\/strong>.<\/li>\n\n\n\n<li>The edge length <strong>a<\/strong> of the unit cell = 2r.<\/li>\n\n\n\n<li>Volume of unit cell = a\u00b3 = (2r)\u00b3 = <strong>8r\u00b3<\/strong>.<\/li>\n\n\n\n<li>Volume of one sphere = (4\/3)\u03c0r\u00b3.<\/li>\n\n\n\n<li>Packing efficiency = [Volume of sphere \/ Volume of unit cell] \u00d7 100<br>= [(4\/3)\u03c0r\u00b3 \/ 8r\u00b3] \u00d7 100<br>= (\u03c0\/6) \u00d7 100 \u2248 <strong>52.4%<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. Body-Centered Cubic (BCC) Cell<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Number of spheres per unit cell:<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>8 corner atoms \u00d7 (1\/8 each) = 1 atom.<\/li>\n\n\n\n<li>1 atom completely inside the body = 1 atom.<\/li>\n\n\n\n<li>Total = <strong>1 + 1 = 2 atoms<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Packing efficiency:<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In BCC, the atoms touch along the <strong>body diagonal<\/strong>.<\/li>\n\n\n\n<li>Body diagonal = \u221a3 \u00d7 a = 4r \u21d2 a = 4r\/\u221a3.<\/li>\n\n\n\n<li>Volume of unit cell = a\u00b3 = (4r\/\u221a3)\u00b3 = <strong>64r\u00b3 \/ 3\u221a3<\/strong>.<\/li>\n\n\n\n<li>Total volume of 2 spheres = 2 \u00d7 (4\/3)\u03c0r\u00b3 = <strong>(8\/3)\u03c0r\u00b3<\/strong>.<\/li>\n\n\n\n<li>Packing efficiency = [(8\/3)\u03c0r\u00b3 \/ (64r\u00b3 \/ 3\u221a3)] \u00d7 100<br>= (\u03c0\u221a3 \/ 8) \u00d7 100 \u2248 <strong>68.0%<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>3. Face-Centered Cubic (FCC) Cell<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Number of spheres per unit cell:<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>8 corner atoms \u00d7 (1\/8) = 1 atom.<\/li>\n\n\n\n<li>6 face atoms \u00d7 (1\/2) = 3 atoms.<\/li>\n\n\n\n<li>Total = <strong>1 + 3 = 4 atoms<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Packing efficiency:<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Atoms touch along the <strong>face diagonal<\/strong>.<\/li>\n\n\n\n<li>Face diagonal = \u221a2 \u00d7 a = 4r \u21d2 a = 4r\/\u221a2 = 2\u221a2r.<\/li>\n\n\n\n<li>Volume of unit cell = a\u00b3 = (2\u221a2r)\u00b3 = <strong>16\u221a2r\u00b3<\/strong>.<\/li>\n\n\n\n<li>Total volume of 4 spheres = 4 \u00d7 (4\/3)\u03c0r\u00b3 = <strong>(16\/3)\u03c0r\u00b3<\/strong>.<\/li>\n\n\n\n<li>Packing efficiency = [(16\/3)\u03c0r\u00b3 \/ 16\u221a2r\u00b3] \u00d7 100<br>= (\u03c0 \/ \u221a2) \u00d7 100 \u2248 <strong>74.0%<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Summary Table:<\/strong><\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th><strong>Lattice Type<\/strong><\/th><th><strong>Atoms per Cell<\/strong><\/th><th><strong>Packing Efficiency<\/strong><\/th><\/tr><\/thead><tbody><tr><td>Simple Cubic (SC)<\/td><td>1<\/td><td>\u2248 52.4%<\/td><\/tr><tr><td>Body-Centered Cubic<\/td><td>2<\/td><td>\u2248 68.0%<\/td><\/tr><tr><td>Face-Centered Cubic<\/td><td>4<\/td><td>\u2248 74.0%<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (300+ words):<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Crystals form in highly ordered arrangements called <strong>unit cells<\/strong>, which are the smallest repeating units in a crystal lattice. In <strong>simple cubic (SC)<\/strong>, <strong>body-centered cubic (BCC)<\/strong>, and <strong>face-centered cubic (FCC)<\/strong> structures, atoms are placed in different positions within the cube, affecting how densely packed they are\u2014this is referred to as <strong>packing efficiency<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In an <strong>SC structure<\/strong>, atoms are only at the cube&#8217;s corners. Each corner atom is shared among 8 neighboring unit cells, so only 1\/8 of it belongs to a single cell. Therefore, the unit cell contains a total of 1 atom. The atoms touch each other along the cube\u2019s edges. Its low packing efficiency of about <strong>52.4%<\/strong> means nearly half of the space is empty\u2014this structure is rare in nature due to its inefficiency.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>BCC structure<\/strong> adds one atom in the center of the cube, in addition to the 8 corner atoms. The central atom is entirely within the unit cell. Atoms touch along the cube\u2019s body diagonal, allowing more efficient packing. This yields 2 atoms per cell and a higher efficiency of about <strong>68.0%<\/strong>. This arrangement is found in metals like iron and chromium.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The most efficient of the three is the <strong>FCC structure<\/strong>, which has atoms at each corner and at the center of each face. Atoms touch along face diagonals, and each face-centered atom contributes half to the unit cell. This results in 4 atoms per cell and the highest packing efficiency of about <strong>74.0%<\/strong>. FCC structures are common in ductile metals like aluminum, copper, and gold due to their high density and close packing.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In conclusion, while SC is structurally simple, FCC is the most space-efficient and mechanically favorable structure.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-87.png\" alt=\"\" class=\"wp-image-217853\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the number of spheres in the simple cubic, body-centered cubic, and face-centered cubic cells. Also, calculate the packing efficiency of each type of cell. simple cubic cells: spheres packing efficiency % body-centered cubic cells: spheres packing efficiency face-centered cubic cells: spheres packing efficiency % None The Correct Answer and Explanation is: Let&#8217;s break down [&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-217852","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217852","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=217852"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217852\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}