In an HCP unit cell, there are 6 atoms effectively. The unit cell consists of 12 corner atoms (each shared by 6 adjacent unit cells), 2 atoms on the faces (each shared by 2 adjacent unit cells), and 2 atoms fully contained within the unit cell. The total number of atoms in an HCP unit cell is:<\/p>\n\n\n\n
- \n
- Corner atoms: 12 \u00d7 1\/6 = 2 atoms<\/li>\n\n\n\n
- Face atoms: 2 \u00d7 1\/2 = 1 atom<\/li>\n\n\n\n
- 2 fully contained atoms<\/li>\n<\/ul>\n\n\n\n
Total atoms = 2 + 1 + 2 = 6 atoms<\/p>\n\n\n\n
- \n
- Volume Occupied by Atoms:<\/strong><\/li>\n<\/ol>\n\n\n\n
Each atom is assumed to be spherical. The volume of one atom is given by the formula:Vatom=43\u03c0r3V_{\\text{atom}} = \\frac{4}{3} \\pi r^3Vatom\u200b=34\u200b\u03c0r3<\/p>\n\n\n\n
Where rrr is the radius of the atom. Therefore, the total volume occupied by 6 atoms is:Vatoms=6\u00d7(43\u03c0r3)V_{\\text{atoms}} = 6 \\times \\left(\\frac{4}{3} \\pi r^3 \\right)Vatoms\u200b=6\u00d7(34\u200b\u03c0r3)<\/p>\n\n\n\n
- \n
- Volume of the HCP Unit Cell:<\/strong><\/li>\n<\/ol>\n\n\n\n
The unit cell of HCP is a hexagonal prism. The dimensions of the HCP unit cell are characterized by the base edge length aaa and the height ccc. The relationship between these parameters is given by:c=8\/3\u00d7ac = \\sqrt{8\/3} \\times ac=8\/3\u200b\u00d7a<\/p>\n\n\n\n
The volume of the HCP unit cell is then:Vcell=32a2cV_{\\text{cell}} = \\frac{\\sqrt{3}}{2} a^2 cVcell\u200b=23\u200b\u200ba2c<\/p>\n\n\n\n
Substituting c=8\/3\u00d7ac = \\sqrt{8\/3} \\times ac=8\/3\u200b\u00d7a, we get:Vcell=32a2\u00d78\/3\u00d7a=22a3V_{\\text{cell}} = \\frac{\\sqrt{3}}{2} a^2 \\times \\sqrt{8\/3} \\times a = \\frac{\\sqrt{2}}{2} a^3Vcell\u200b=23\u200b\u200ba2\u00d78\/3\u200b\u00d7a=22\u200b\u200ba3<\/p>\n\n\n\n
- \n
- Atomic Packing Factor:<\/strong><\/li>\n<\/ol>\n\n\n\n
Now, the APF is the ratio of the total volume occupied by the atoms to the volume of the unit cell:APF=VatomsVcellAPF = \\frac{V_{\\text{atoms}}}{V_{\\text{cell}}}APF=Vcell\u200bVatoms\u200b\u200b<\/p>\n\n\n\n
Substituting the expressions for VatomsV_{\\text{atoms}}Vatoms\u200b and VcellV_{\\text{cell}}Vcell\u200b:APF=6\u00d743\u03c0r322a3APF = \\frac{6 \\times \\frac{4}{3} \\pi r^3}{\\frac{\\sqrt{2}}{2} a^3}APF=22\u200b\u200ba36\u00d734\u200b\u03c0r3\u200b<\/p>\n\n\n\n
By substituting the known relationship between aaa and rrr (i.e., a=2ra = 2ra=2r):APF=6\u00d743\u03c0r322(2r)3APF = \\frac{6 \\times \\frac{4}{3} \\pi r^3}{\\frac{\\sqrt{2}}{2} (2r)^3}APF=22\u200b\u200b(2r)36\u00d734\u200b\u03c0r3\u200b<\/p>\n\n\n\n
Simplifying this expression:APF=6\u00d743\u03c0r322\u00d78r3=6\u00d743\u03c022\u00d78APF = \\frac{6 \\times \\frac{4}{3} \\pi r^3}{\\frac{\\sqrt{2}}{2} \\times 8r^3} = \\frac{6 \\times \\frac{4}{3} \\pi}{\\frac{\\sqrt{2}}{2} \\times 8}APF=22\u200b\u200b\u00d78r36\u00d734\u200b\u03c0r3\u200b=22\u200b\u200b\u00d786\u00d734\u200b\u03c0\u200bAPF=24\u03c082=0.74APF = \\frac{24 \\pi}{8 \\sqrt{2}} = 0.74APF=82\u200b24\u03c0\u200b=0.74<\/p>\n\n\n\n
Hence, the Atomic Packing Factor (APF) for HCP is 0.74, which means 74% of the volume of the HCP unit cell is occupied by atoms, and the remaining 26% is empty space.<\/p>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-74.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"Show that the atomic packing factor for HCP is 0.74. Density The Atomic Packing Factor (APF) for Hexagonal Close-Packed (HCP) structures can be calculated by determining the volume occupied by atoms in a unit cell and dividing it by the total volume of the unit cell. In an HCP unit cell, there are 6 atoms […]<\/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-239830","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/239830","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=239830"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/239830\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=239830"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=239830"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=239830"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Atomic Packing Factor:<\/strong><\/li>\n<\/ol>\n\n\n\n
- Volume of the HCP Unit Cell:<\/strong><\/li>\n<\/ol>\n\n\n\n
- Volume Occupied by Atoms:<\/strong><\/li>\n<\/ol>\n\n\n\n