The semi-empirical mass formula, based on the liquid drop model, compared to the data

Transcription

1 Nucleonic Shells

2 The semi-empirical mass formula, based on the liquid drop model, compared to the data

3 E shell = E total E LD (Z=82, N=126) (Z=28, N=50) Nature 449, 411 (2007) Magic numbers at Z or N= 2, 8, 20, 28, 50, 82,126

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5 1912 electronic shells of the atom 1949 nucleonic shells of the nucleus Nobel Prize 1922 Bohr s picture still serves as an elucidation of the physical and chemical properties of the elements. noble gases (closed shells) Xe 5p 4d 5s Kr 4p 3d 4s Ar 3p 3s Nobel Prize 1963 magic nuclei (closed shells) p 1/2 2f 5/2 1i 13/2 3p 3/2 1h 9/2 2f 7/2 2d 3/2 1h 11/2 3s 1/2 10 Ne 1g 7/2 2d 5/2 Krypton Atom 2p 2s We know now that this picture is very incomplete 50 1g 9/2

6 For each shell, the allowed orbital angular momenta are: ĥ = ˆt + mω 2 0r 2 2 Since each nucleon has an intrinsic spin s=1/2, the maximum number of " ε N = N + 3 % $ '!ω 0 # 2 & l = N, N 2,,1, or 0, j = l ± 1 2 nucleons in a HO shell is: D N = 2( 2l +1) = N +1 The total number of states is: Spherical Harmonic Oscillator N l ( )( N + 2) ( N '+1) ( N '+ 2) = 1 3 N +1 N '=0 N>>1 1 ( 3 N + 2 ) 3 # N + 3 & % N>>1 $ 2 ( ' ( )( N + 2) ( N + 3) Dimension of orbits: r 2 = "! N + 3 $ # & Nl mω 0 " 2 %!ω 0 41 (MeV) 1/3 A 2

7 Spin-orbit potential PH YSI CAL REVIEW VOLUME 78, NUMBER 1 A P R I L 1, 1950 Flat bottom Nuclear Configurations in the Spin-Orbit Coupling Model. I. Empirical Evidence MARIA GOEPPERT MAYER Argonne Eationa/ Laboratory, Chicago, I/linois {Received December 7, 1949) An extreme one particle model of the nucleus is proposed. The model is based on the succession of energy levels of a single particle in a potential between that of a three-dimensional harmonic oscillator and a square well. {1)Strong spin orbit coupling leading to inverted doublets is assumed. {2) An even number of identical nucleons are assumed to couple to zero angular momentum, and, {3) an odd number to the angular momentum of the single odd particle. {4)A {negative) pairing energy, increasing with the jvalue of the orbit is assumed. With these four assumptions all but 2 of the 64 known spins of odd nuclei are satisfactorily explained, and all but 1 of the 46 known magnetic moments. The two spin discrepancies are probably due to failure of rule {3).The magnetic moments of the Gve known odd-odd nuclei are also in agreement with the model. The existence, and region in the periodic table, of nuclear isomerism is correctly predicted. κ is negative! The value of spin-orbit strength κ cannot be derived from a simple Thomas precession, as incorrectly stated in Jackson (next slide) orbits with higher angular momentum shifted down!

8 Jackson, Classical Electrodynamics, Sec. 11.5

9 N=6 N=5 4s 3d 2g 1i 3p 2f 126 s 1/2 d 3/2 g 7/2 d j 5/2 15/2 i 11/2 g 9/2 p 1/2 f 5/2 i 13/2 p 3/2 h 9/2 1h 3s N=4 2d 1g f 7/2 d 3/2 s 1/2 h 11/2 g 7/2 d 5/2 g 9/2 Harmonic oscillator +flat bottom +spin-orbit

10 HW: Extend the nuclear shell model scheme beyond Z=82, N=126. What should be the next neutron and proton magic numbers in superheavy/ hyperheavy nuclei?