in the Spin-Orbit Coupling Model.

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(1)Spin-orbit potential PH. YSI CAL REVIEW. VOLUME. Nuclear Configurations. 78, NUMBER. Flat bottom A. 1. in the Spin-Orbit Coupling Model.. I. Empirical. PRIL. 1, 1950. 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 harmonic oscillator and a levels of a single particle in a potential between that of a three-dimensional 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 value 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. j. predicted.. "UCLEI. containing 2, 8, 20, 28, 50, 82 or 126 stable. ' neutrons or protons are particularly These closed shells have been explained in diBerent It has also been pointed out that the "magic ways. numbers" can be explained on the basis of a single particle picture with the assumption of strong spin-orbit coupling. 4 The detailed evidence supporting this point of view mill be discussed in this paper.. ". I. THE. SHELLS. The single particle orbits for the neutrons and protons in a nucleus are determined by a potential energy which has a shape somewhat between that of a square well isotropic oscillator. The level and a three-dimensional order for these two potentials is closely related. In Table I the order of the levels is given. The 6rst line contains the oscillator quantum number. The levels which would be degenerate for the. oscillator are grouped together. The eigenfunctions of any such group have the same parity. The order of levels in each group is that calculated for the square well. The major quantum number in the second line counts the number of spherical nodes. The third line contains the number of neutrons or protons which completely 611 all states up to each oscillator level. For the square-well potential, the grouping of energy levels indicated above exists only and explains the for the low lying eigenfunctions stability of the numbers 2, 8, and 20. Beyond n= 2, the grouping is no longer pronounced; 3s and 1h, and also. 3p and 1i, have approximately the same energy in a square well calculation. The grouping of oscillator levels does not explain the occurrence of the higher magic numbers; it is apparent that their stability must be due to a diferent cause. Assume that there exists a strong spin-orbit coupling such that the orbits with higher total angular momenta, j=l+~, have a higher binding energy. Since this coupling should depend on the orbital angular momentum, I, and be higher for large / values, it is greatest for the 6rst level in each of the oscillator groups. For this level, the state with j=l+-, will be lowered in energy towards the group with lower n, the state with = l ——, raised, so that a gap occurs at this point. Table II contains the order of levels obtained from those of the square well by spin-orbit coupling. It is seen that the shells so obtained correspond exactly to the magic numbers. In the middle of the shell, spin-orbit coupling may give rise to crossing of some levels.. j. '. II. ASSUMPTIONS. κ is negative!. As will be shown in the detailed discussion, the following assumptions are adequate to explain the observed facts in all but a very few exceptional cases:. (1) The succession of energies of single particle orbits is that of a square well with strong spin orbit coupling giving rise to inverted doublets. l+ ~ has invariably lower (1a) For given f, the level -, energy and will be 6lled before that for TAM. E I. Order of energy levels for a potential somewhat between that of a square well and of a three-dimensional isotropic one shell, which arise Pairs levels within of spin (1b) osci0ator. from adjacent orbital levels in the square well in such a way that spin-orbit coupling tends to bring their 6 @=0 1 2 4 5 3 ii 2g 3d 4s energy closer together can, and very often will, cross. 1s ip id 2s 1f 2p ig 2d 3s ih 2f 3P 112 70 S or Z= 2 8 20 40 Examples»«3/2 and ~1/2 f5/2 pa/2 pl/2 g9/2 g7/2 /f$/2 sg/2 k/1/g pl/2 A)3 /T2hese level pairs may cross to energy order reversed from that of Table II. '%. Elsasser, J. de phys. et rad. 5, 625 {1934);M. G. Mayer, have their even number of identical nucleons in any (2) An Phys. Rev. 74, 235 {1948). ~ E. Feenberg and K. C. Hammack, Phys. .Rev. 75, 1877 {1949). orbit with total angular momentum quantum number L. W. Nordheim, Phys. Rev. 75, 1894 {1949). will always couple to give a spin zero and no con4 Haxel, Jensen, and Suess, Phys. Rev. 75, 1766 {1949);M. G. tribution to the magnetic moment. Mayer, Phys. Rev. 75, 1969 {1949).. j= The value of spin-orbit strength κ j=l—'. cannot be derived from a simple Thomas precession, as incorrectly stated in Jackson (next slide). j. orbits with higher angular momentum shifted down!.

(2) Jackson, Classical Electrodynamics, Sec. 11.5.

(3) N=6!. 4s! 3d! 2g! 1i!. N=5!. 126!. 3p! 2f! 1h!. 82!. 3s!. N=4!. 2d! 1g!. s1/2! d3/2! g7/2! d5/2! j15/2! i11/2! g9/2! p1/2! f5/2! i13/2! p3/2! h9/2! f7/2! d3/2! s1/2! h11/2! g7/2! d5/2!. 50! g9/2!. Harmonic! +flat bottom! +spin-orbit! oscillator!.

(4) 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?.

(5) Average one-body Hamiltonian (characteristic features) 120Sn. Coulomb ! barrier!. Unbound! states! Discrete! (bound)! states!. 0!. εF. εF Surface! region!. • The nucleus is a self-bound system • The potential is not infinite: the nucleus is an open system • The potential is self-consistent • The potential depends on both spin and isospin. n Flat ! bottom!. p.

(6) Nuclear shell model potential 114Sn.

(7) Shell effects and classical periodic orbits. Shells • Typical time scale: babyseconds (10-22s) • Closed orbits and s.p. quantum numbers. dN density of states (number of states per energy interval) dε Trace formula, Gutzwiller, g(ε ) = g˜(ε ) + ∑ Aγ (ε ) cos Sγ (ε ) /  − α γ J. Math. Phys. 8 (1967) g (ε ) =. γ. smooth part.   Sγ (ε ) = ∫ p dq γ. [. ]. 1979. oscillating part (shell effects) The action integral for the periodic (closed) orbit γ.

(8) Lissajous curves (complex harmonic motion). The appearance of the figure is highly sensitive to the ratio a/b. For a/b=1, the figure is an ellipse, with special cases including circles (A = B, δ=π/2) and lines (δ=0). Another simple Lissajous figure is the parabola (a/b=2, δ=π/4). Other ratios produce more complicated curves, which are closed only if a/b is rational.. 2D. 3D http://en.wikipedia.org/wiki/Lissajous_curve.

(9) Pronounced shell structure. Shell structure absent. (quantum numbers). shell! gap! shell! gap! shell!. closed trajectory! (regular motion)!. trajectory does not close !.

(10) experiment!. P. Moller et al. experiment!. 0!. -10!. 20 28. discrepancy!. 0! -10!. Nuclei!. theory!. 0!. -10!. Shells in mesoscopic systems. theory!. 50. 82. 126. S. Frauendorf et al.. diff.! 20!. 1! experiment!. 60! 100! Number of Neutrons!. Sodium Clusters! • Jahn-Teller Effect (1936) • Symmetry breaking and deformed mean-field. Shell Energy (eV)!. Shell Energy (MeV)!. 10!. 0! -1!. 58. 1! theory!. 92. 138. 198. spherical! clusters!. 0! deformed! clusters!. -1! 50!. 100! 150! 200! Number of Electrons!.

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