Nuclear Term Schemata and Magic Nucleon Numbers

* Structures in Micro Objects * Nuclear Term Schema The nuclear term schema for an arbitrary nuclide is obtained by solving the Schrödinger equation for a modified 3-dimensional harmonic oscillator potential. Taking into account the spin orbit interaction of angular momentum, Maria Goeppert-Mayer succeeded in analyzing the structure of the nuclear shells, interpreting the magic occupation numbers, and showing that protons and neutrons have slightly different higher terms. For any nucleon state, we denote by n ... the principal quantum number (oscillator shells with n = 1,2,...), l ... the orbital angular momentum (orbitals s,p,... with l = 0,...,n), s ... the spin quantum number (s = -l/2 or +1/2), j ... the total angular momentum. Boldface state numbers denote closed neutron shells and are called nuclear magic numbers. The schema for protons is similar up to occupation number 82 but differs in higher regions. # of states s (l = 0) p (l = 1) d (l = 2) f (l = 3) g (l = 4) h (l = 5) i (l = 6) j (l = 7) k (l = 8) 184 ___3d(3/2)4 180 ___4s(1/2)2 178 ___2g(7/2)8 170 ___1j(15/2)16 154 ___1i(11/2)12 142 ___3d(5/2)6 136 ___2g(9/2)10 126 ___2p(1/2)2 124 ___2p(3/2)4 120 ___1i(13/2)14 106 ___2f(5/2)6 100 ___2f(7/2)8 92 ___1h(9/2)10 82 ___3s(1/2)2 80 ___2d(3/2)4 76 ___1h(11/2)12 64 ___2d(5/2)6 58 ___1g(7/2)8 50 ___1g(9/2)10 40 ___2p(1/2)2 38 ___1f(5/2)6 32 ___2p(3/2)4 28 ___1f(7/2)8 20 ___1d(3/2)4 16 ___2s(1/2)2 14 ___1d(5/2)6 8 ___1p(1/2)2 6 ___1p(3/2)4 2 ___1s(1/2)2 Magic Nucleon Numbers The table below lists the few stable nuclides with doubly magic nucleon numbers. 2 He 4 8 O 16 20 Ca 40 20 Ca 48 82 Pb 208 114 Uuq 298 164 Uhq 482 References: Rod Nave, Georgia State University Jim Branson, University of California, San Diego