{"id":183285,"date":"2025-01-16T07:16:04","date_gmt":"2025-01-16T07:16:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183285"},"modified":"2025-01-16T07:16:06","modified_gmt":"2025-01-16T07:16:06","slug":"assume-that-in-the-shell-model-the-nucleon-energy-levels-are-ordered","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/16\/assume-that-in-the-shell-model-the-nucleon-energy-levels-are-ordered\/","title":{"rendered":"Assume that in the shell model the nucleon energy levels are ordered"},"content":{"rendered":"\n

Assume that in the shell model the nucleon energy levels are ordered as shown in Figure 8.4. Write down the shell model configuration of the nucleus 7 3Li and hence find its spin, parity, and magnetic moment (in nuclear magnetons). Give the two most likely configurations for the first excited state, assuming that only protons are excited.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To determine the shell model configuration of ( ^7_3\\text{Li} ) and its properties, we start by analyzing the given nucleus.<\/p>\n\n\n\n

Ground State Configuration:<\/h3>\n\n\n\n
    \n
  1. ( ^7_3\\text{Li} ) has ( Z = 3 ) (3 protons) and ( N = 4 ) (4 neutrons).<\/li>\n\n\n\n
  2. In the nuclear shell model, nucleons fill energy levels in the sequence ( 1s_{1\/2}, 1p_{3\/2}, 1p_{1\/2}, \\dots ), following the Pauli exclusion principle and spin-orbit coupling.<\/li>\n\n\n\n
  3. The 3 protons fill the levels as follows:<\/li>\n<\/ol>\n\n\n\n
      \n
    • Two protons occupy the ( 1s_{1\/2} ) orbital.<\/li>\n\n\n\n
    • One proton occupies the ( 1p_{3\/2} ) orbital.<\/li>\n<\/ul>\n\n\n\n
        \n
      1. For neutrons (( N = 4 )):<\/li>\n<\/ol>\n\n\n\n
          \n
        • Two neutrons occupy the ( 1s_{1\/2} ) orbital.<\/li>\n\n\n\n
        • Two neutrons occupy the ( 1p_{3\/2} ) orbital.<\/li>\n<\/ul>\n\n\n\n

          The ground-state configuration is:
          [
          (1s_{1\/2})^2_p (1p_{3\/2})^1_p \\quad (1s_{1\/2})^2_n (1p_{3\/2})^2_n
          ]<\/p>\n\n\n\n

          The unpaired proton in the ( 1p_{3\/2} ) orbital determines the spin-parity of the nucleus:
          [
          J^\\pi = \\frac{3}{2}^-
          ]<\/p>\n\n\n\n

          Magnetic Moment:<\/h3>\n\n\n\n

          The magnetic moment is given by:
          [
          \\mu = g_J J
          ]
          For a single proton in ( 1p_{3\/2} ), ( g_J = g_l + \\frac{g_s – g_l}{2l + 1} ), where ( g_l = 1 ) and ( g_s \\approx 5.586 ). For ( l = 1 ) and ( j = \\frac{3}{2} ):
          [
          g_J = 1 + \\frac{5.586 – 1}{3} = 2.528
          ]
          Thus:
          [
          \\mu = 2.528 \\times \\frac{3}{2} = 3.792 \\, \\mu_N
          ]<\/p>\n\n\n\n

          First Excited State:<\/h3>\n\n\n\n

          In the first excited state, a proton is excited from ( 1p_{3\/2} ) to ( 1p_{1\/2} ). The two most likely configurations are:<\/p>\n\n\n\n

            \n
          1. The proton in ( 1p_{3\/2} ) moves to ( 1p_{1\/2} ), giving ( J^\\pi = \\frac{1}{2}^- ).<\/li>\n\n\n\n
          2. Another possibility is the excitation to a higher ( p )-shell orbital (e.g., ( 2s_{1\/2} )), leading to ( J^\\pi = \\frac{1}{2}^+ ).<\/li>\n<\/ol>\n\n\n\n

            This explains the shell model configuration and its predictions for ( ^7_3\\text{Li} ).<\/p>\n","protected":false},"excerpt":{"rendered":"

            Assume that in the shell model the nucleon energy levels are ordered as shown in Figure 8.4. Write down the shell model configuration of the nucleus 7 3Li and hence find its spin, parity, and magnetic moment (in nuclear magnetons). Give the two most likely configurations for the first excited state, assuming that only protons […]<\/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-183285","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183285","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=183285"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183285\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183285"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183285"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183285"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}