{"id":39236,"date":"2025-06-27T04:55:29","date_gmt":"2025-06-27T04:55:29","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=39236"},"modified":"2025-06-27T04:55:30","modified_gmt":"2025-06-27T04:55:30","slug":"sketch-the-111-direction-and-calculate-the-planar-density-in-the-111-plane-for-bcc-and-fcc","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/sketch-the-111-direction-and-calculate-the-planar-density-in-the-111-plane-for-bcc-and-fcc\/","title":{"rendered":"Sketch the [111] direction and calculate the planar density in the (111) plane for BCC and FCC."},"content":{"rendered":"\n<p>Sketch the [111] direction and calculate the planar density in the (111) plane for BCC and FCC.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">[111] Direction and Planar Density in the (111) Plane for BCC and FCC<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Sketch of the [111] Direction:<\/h4>\n\n\n\n<p>The [111] direction in a cubic crystal passes through the body diagonal of the unit cell. It connects one corner of the cube to the opposite corner, passing through the center.<\/p>\n\n\n\n<p>In both Body-Centered Cubic (BCC) and Face-Centered Cubic (FCC) structures, the [111] direction passes through atoms at fractional coordinates:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From (0,0,0) to (1,1,1)<\/li>\n<\/ul>\n\n\n\n<p>This direction intersects several atoms depending on the crystal structure.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Planar Density in the (111) Plane<\/h3>\n\n\n\n<p><strong>1. FCC Structure:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Atoms per unit area:\n<ul class=\"wp-block-list\">\n<li>The (111) plane in FCC is densely packed.<\/li>\n\n\n\n<li>One (111) plane contains 6 atoms:\n<ul class=\"wp-block-list\">\n<li>3 atoms at the corners contribute 1\/6 each (0.5 atom total)<\/li>\n\n\n\n<li>3 atoms at the face centers contribute 1\/2 each (1.5 atoms total)<\/li>\n\n\n\n<li>3 shared at edges in neighboring planes contribute 1\/3 each (1 atom total)<\/li>\n\n\n\n<li>Effective atoms = 3 atoms per (111) plane<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Area of the (111) plane:\n<ul class=\"wp-block-list\">\n<li>The (111) plane is an equilateral triangle with side \u221a2 * a<\/li>\n\n\n\n<li>Area = (\u221a3 \/ 4) * (\u221a2 * a)\u00b2 = (\u221a3 \/ 4) * 2a\u00b2 = (\u221a3 \/ 2) * a\u00b2<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Planar density = Number of atoms \/ Area<br>= 3 \/ ((\u221a3 \/ 2) * a\u00b2)<br>= (6 \/ \u221a3) * (1 \/ a\u00b2)<br>= (2\u221a3 \/ a\u00b2) atoms per square angstrom<\/li>\n<\/ul>\n\n\n\n<p><strong>2. BCC Structure:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Atoms per unit area:\n<ul class=\"wp-block-list\">\n<li>The (111) plane in BCC is not as densely packed as in FCC.<\/li>\n\n\n\n<li>Atoms at 3 corners contribute 1\/6 each (0.5 atom)<\/li>\n\n\n\n<li>One atom at the body center contributes to the plane (if intersected)<\/li>\n\n\n\n<li>Effective atoms \u2248 1.5 atoms<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Area of the (111) plane = same as FCC<br>= (\u221a3 \/ 2) * a\u00b2<\/li>\n\n\n\n<li>Planar density = 1.5 \/ ((\u221a3 \/ 2) * a\u00b2)<br>= (3 \/ \u221a3) * (1 \/ a\u00b2)<br>= (\u221a3 \/ a\u00b2) atoms per square angstrom<\/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\">Summary:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>FCC (111) planar density = (2\u221a3 \/ a\u00b2) atoms \/ \u00c5\u00b2<\/li>\n\n\n\n<li>BCC (111) planar density = (\u221a3 \/ a\u00b2) atoms \/ \u00c5\u00b2<\/li>\n<\/ul>\n\n\n\n<p>The FCC (111) plane is more densely packed due to the atomic arrangement, resulting in a higher planar density compared to the BCC (111) plane.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1143.jpeg\" alt=\"\" class=\"wp-image-39237\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1143.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1143-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1143-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Sketch the [111] direction and calculate the planar density in the (111) plane for BCC and FCC. The Correct Answer and Explanation is: [111] Direction and Planar Density in the (111) Plane for BCC and FCC Sketch of the [111] Direction: The [111] direction in a cubic crystal passes through the body diagonal of the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-39236","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39236","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=39236"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39236\/revisions"}],"predecessor-version":[{"id":39238,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39236\/revisions\/39238"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=39236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=39236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=39236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}