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THE SHELL MODEL 22.02 Introduction To Applied Nuclear Physics Spring 2012 Atomic Shell Model • Chemical properties show a periodicity • Periodic table of the elements • Add electrons into shell structure 2 Atomic Radius 0.30 Ê Ê Ê 0.25 Radius @nmD Ê 0.20 Ê 0.15 Ê Ê 0.10 Ê Ê Ê 0.05 Ê ‡ 0 Ê Ê Ê Ê ‡ Ê Ê Ê Ê ‡ Ê Ê Ê Ê ‡ Ê Ê Ê Ê ‡ ÊÊ Ê Ê Ê Ê Ê ÊÊ Ê Ê ÊÊÊÊ ‡ Ê Ê ÊÊ Ê Ê ÊÊ ÊÊ ÊÊ ‡ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ ÊÊ ‡ Ê Ê Ê Ê Ê Ê Ê ‡ 20 40 60 Z 3 80 Ionization Energy Ionization Energy (similar to B per nucleon) Ê ‡ Ê ‡ kJ per Mole 2000 Ê ‡ Ê ‡ 1500 Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ 1000 ʇ ‡ Ê Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ ʇ ‡ Ê Ê ‡ Ê ‡ Ê ‡ Ê ‡ 20 Ê ‡ ʇ ‡ Ê Ê ‡ Ê ‡ Ê ‡ Ê Ê‡ ‡ Ê ‡ Ê ‡ Ê ‡ ʇ ‡ Ê Ê‡ Ê ‡ Ê ‡ Ê Ê‡ ‡ Ê ‡‡ Ê Ê ‡ Ê Ê‡ ‡ ʇ Ê ‡ ʇ ‡ Ê Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ ʇ ʇ ‡ Ê Ê ‡ Ê Ê‡ ‡ Ê ‡ Ê ‡ 40 Ê ‡ Z 4 Ê Ê‡ ʇ ʇ ‡ ʇ ʇ ʇ 60 Ê ‡ Ê Ê‡ ʇ ‡ Ê Ê‡ ʇ ‡ Ê Ê‡ ʇ ʇ ‡ ʇ ‡ ʇ ʇ ʇ Ê Ê ‡ Ê ‡ 80 ʇ ‡ Ê Ê ‡ Ê ‡‡ Ê 100 ATOMIC STRUCTURE The atomic wavefunction is written as | i = |n, l, mi = Rn,l (r)Ylm (#,' ) where the labels indicate : n : principal quantum number l : orbital (or azimuthal) quantum number m: magnetic quantum number The degeneracy is 2 D(l) = 2(2l + 1) ! D(n) = 2n 5 AUFBAU PRINCIPLE The orbitals (or shells) are then given by the n-levels (?) l Spectroscopic notation D(l) n 1 2 3 D(n) 2 6 18 e 0 s 1 p 2 d 3 f 2 6 10 14 historic structure in shell 2 8 28 6 4 g 5 h 6 i 18 22 26 heavy nuclei ATOMIC PERIODIC TABLE 7 AUFBAU PRINCIPLE The orbitals (or shells) are then given by close-by energy-levels l Spectroscopic notation D(l) n 1 2 3 D(n) 2 6 18 e 0 s 1 p 2 d 3 f 2 6 10 14 historic structure in shell 2 8 28 4 g 5 h 6 i 18 22 26 heavy nuclei 3s+3p form one level with # 10 4s is filled before 3d 8 Nuclear Shell Model • Picture of adding particles to an external potential is no longer good: each nucleon contributes to the potential • Still many evidences of a shell structure 9 Separation Energy -5 4 50 3 1 0 38Ar -1 184W 132Te 14C 4 3 2 1 28 50 20 82 Hf Cd Ni -4 -5 0 Pb Ca -3 8 5 Pt U Kr -2 126 Dy Ce 0 -1 -5 126 8 5 102Mo 86Kr -2 -4 114Ca S2n(MeV) S2p(MeV) 64Ni 82 20 208Pb 2 -3 28 50 100 150 O 0 Nucleon number 25 50 75 100 125 150 Nucleon number PROTON NEUTRON Image by MIT OpenCourseWare. After Krane. 10 (binding energy per nucleon) B/A B/A: JUMPS 9 8 7 6 5 4 3 A 0 50 100 150 200 (Mass number) 250 “Jumps” in Binding energy from experimental data 11 CHART of NUCLIDES (Z/A vs. A) Z/A 0.55 0.50 0.45 0.40 0.35 A 50 100 150 12 200 250 CHART OF NUCLIDES http://www.nndc.bnl.gov/chart/ “Periodic”, more complex properties → nuclear structure © Brookhaven National Laboratory. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. 13 NUCLEAR POTENTIAL Vp = r 2 ✓ ✓ V0 R02 V0 (Z 1)e 2R03 ◆ 2 3 (Z 1)e 2 R0 2 ◆ Vn = r 2 ✓ V0 R02 ◆ (V0 ) Harmonic potential Steeper for neutrons 14 NUCLEAR POTENTIAL Vp = r 2 ✓ ✓ V0 R02 V0 (Z 1)e 2R03 ◆ 2 3 (Z 1)e 2 R0 2 ◆ Harmonic potential + well depth 15 Vn = r 2 ✓ V0 R02 ◆ Steeper and Deeper for neutrons (V0 ) Shell Mode Harmonic oscillator: solve (part of) the radial equation including the angular momentum (centrifugal force term) we obtain the usual principal quantum number n = (N-l)/2+1 16 Spin-Orbit Coupling • The spin-orbit interaction is given by VSO 1 ˆ ˆ ~ = 2 Vso (r)l · ~s ~ • We can calculate the dot product D E 1 2 ~ ~ˆl · ~sˆ = (~ˆj 2 ~ˆl2 ~sˆ2 ) = [j(j + 1) l(l + 1) 2 2 1 • Because of the addition rules, j = l ± 2 ( 2 D E ~ 1 for j=l+ 2 l2 ~ˆl · ~sˆ = ~2 1 (l + 1) 2 for j=l- 2 17 3 ] 4 Spin-Orbit Coupling • when the spin is aligned with the angular momentum 1 j =l+ 2 the potential becomes more negative, i.e. the well is deeper and the state more tightly bound. • when spin and angular momentum are anti-aligned j = l the system's energy is higher. Vso E= (2l + 1) 2 • The difference in energy is Thus it increases with l . 18 1 2 Example • 3N level, with l=3 (1f level) j=7/2 or j=5/2 • Level is pushed so down that it forms its own shell 3N 2p 2p1/2 1f5/2 1f 2p3/2 1f7/2 2N 19 20 5 4 3 4s 3d 2g 1i 1 3p 3 2f 5 1h 0 3s 2 2d 4 1g 1 2p 3 1f ... 6 0 2 4 6 1i11/2 1i13/2 2f5/2 2f7/2 3s1/2 1g7/2 1g9/2 1f5/2 3p1/2 3p3/2 1 2s 2 1d 2s1/2 1 1p 1p1/2 44 126 32 82 22 50 8 28 12 20 6 8 2 2 1h11/2 2d3/2 2d5/2 2p1/2 2p3/2 1d3/2 0 184 1h9/2 1f7/2 2 58 1d5/2 1p3/2 0 0 1s 1s1/2 21 MIT OpenCourseWare http://ocw.mit.edu 22.02 Introduction to Applied Nuclear Physics Spring 2012 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
