Nucleonic Shells

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

Eshell

= Etotal

� ELD

Nature 449, 411 (2007)

Magic numbers at Z or N= 2, 8, 20, 28, 50, 82,126

(Z=28, N=50)

(Z=82, N=126)

electronic shells of the atom

10

18

36

54

2s 2p

3s 3p

4s

4p 3d

5s

5p 4d

noble gases(closed shells)

nucleonic shells of

the nucleus

magic nuclei(closed shells)

50

82

126

1g9/2

1g7/2 2d5/2

2d3/2

3s1/2

1h11/2

1h9/2

2f5/2

2f7/2

3p3/2

3p1/2

1i13/2

1949

Nobel Prize 1963

1912

Nobel Prize 1922Bohr’s picture still serves as an elucidation of the physical and chemical properties of the elements.

Ne

Ar

Kr

Xe

Krypton Atom

We know now that this picture is very incomplete…

h = t + mω02r2

2⇒ εN = N + 3

2"

#$

%

&'!ω0

For each shell, the allowed orbital angular momenta are: ℓ = N,N − 2,…,1, or 0, j = ℓ± 1

2Since each nucleon has an intrinsic spin s=1/2, the maximum number of nucleons in a HO shell is: DN = 2 2ℓ+1( )

ℓ

∑ = N +1( ) N + 2( ) ≈N>>1

N + 32

#

$%

&

'(2

The total number of states is: N '+1( ) N '+ 2( )N '=0

N

∑ =13N +1( ) N + 2( ) N +3( )

≈N>>1

13N + 2( )3

Dimension of orbits:

r2Nℓ="

mω0

N + 32

!

"#

$

%&

⇒ !ω0 ≈41A1/3

(MeV)

Spherical Harmonic Oscillator

PH YSI CAL REVIEW VOLUME 78, NUMBER 1 A P R I L 1, 1950

Nuclear Configurations in the Spin-Orbit Coupling Model. I. Empirical EvidenceMARIA 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 energylevels of a single particle in a potential between that of a three-dimensional harmonic oscillator and asquare well. {1)Strong spin orbit coupling leading to inverted doublets is assumed. {2) An even numberof identical nucleons are assumed to couple to zero angular momentum, and, {3) an odd number to theangular momentum of the single odd particle. {4)A {negative) pairing energy, increasing with the j value ofthe orbit is assumed. With these four assumptions all but 2 of the 64 known spins of odd nuclei are satis-factorily explained, and all but 1 of the 46 known magnetic moments. The two spin discrepancies areprobably due to failure of rule {3).The magnetic moments of the Gve known odd-odd nuclei are also inagreement with the model. The existence, and region in the periodic table, of nuclear isomerism is correctlypredicted.

"UCLEI containing 2, 8, 20, 28, 50, 82 or 126neutrons or protons are particularly stable. '

These closed shells have been explained in diBerentways. "It has also been pointed out that the "magicnumbers" can be explained on the basis of a singleparticle picture with the assumption of strong spin-orbitcoupling. 4 The detailed evidence supporting this pointof view mill be discussed in this paper.

I. THE SHELLS

The single particle orbits for the neutrons and protonsin a nucleus are determined by a potential energy whichhas a shape somewhat between that of a square welland a three-dimensional isotropic oscillator. The levelorder for these two potentials is closely related. InTable I the order of the levels is given. The 6rst linecontains the oscillator quantum number. The levelswhich would be degenerate for the. oscillator are groupedtogether. The eigenfunctions of any such group havethe same parity. The order of levels in each group is thatcalculated for the square well. The major quantumnumber in the second line counts the number of sphericalnodes. The third line contains the number of neutronsor protons which completely 611 all states up to eachoscillator level. For the square-well potential, thegrouping of energy levels indicated above exists onlyfor the low lying eigenfunctions and explains thestability of the numbers 2, 8, and 20. Beyond n= 2, thegrouping is no longer pronounced; 3s and 1h, and also

TAM.E I. Order of energy levels for a potential somewhat betweenthat of a square well and of a three-dimensional isotropic

osci0ator.

@=0 1 2 3 4 5 61s ip id 2s 1f 2p ig 2d 3s ih 2f 3P ii 2g 3d 4sS or Z= 2 8 20 40 70 112

'%. Elsasser, J. de phys. et rad. 5, 625 {1934);M. G. Mayer,Phys. Rev. 74, 235 {1948).

~ E. Feenberg and K. C. Hammack, Phys. .Rev. 75, 1877 {1949).L. W. Nordheim, Phys. Rev. 75, 1894 {1949).

4 Haxel, Jensen, and Suess, Phys. Rev. 75, 1766 {1949);M. G.Mayer, Phys. Rev. 75, 1969 {1949).

3p and 1i, have approximately the same energy in asquare well calculation.The grouping of oscillator levels does not explain the

occurrence of the higher magic numbers; it is apparentthat their stability must be due to a diferent cause.Assume that there exists a strong spin-orbit couplingsuch that the orbits with higher total angular momenta,j=l+~, have a higher binding energy. Since thiscoupling should depend on the orbital angular mo-mentum, I, and be higher for large / values, it is greatestfor the 6rst level in each of the oscillator groups. Forthis level, the state with j=l+-, will be lowered inenergy towards the group with lower n, the state withj=l——,' raised, so that a gap occurs at this point. TableII contains the order of levels obtained from those ofthe square well by spin-orbit coupling. It is seen thatthe shells so obtained correspond exactly to the magicnumbers.In the middle of the shell, spin-orbit coupling may

give rise to crossing of some levels.

II. ASSUMPTIONSAs will be shown in the detailed discussion, the fol-

lowing assumptions are adequate to explain the ob-served facts in all but a very few exceptional cases:

(1) The succession of energies of single particle orbitsis that of a square well with strong spin orbit couplinggiving rise to inverted doublets.(1a) For given f, the level j=l+ ~ has invariably lower

energy and will be 6lled before that for j=l—-,'.(1b) Pairs of spin levels within one shell, which arise

from adjacent orbital levels in the square well in sucha way that spin-orbit coupling tends to bring theirenergy closer together can, and very often will, cross.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 tohave their energy order reversed from that of Table II.(2) An even number of identical nucleons in any

orbit with total angular momentum quantum numberj will always couple to give a spin zero and no con-tribution to the magnetic moment.

Spin-orbit potential

κ is negative!

Flat bottom

orbits with higher angular momentum shifted down!

The value of spin-orbit strength κ cannot be derived from a simple Thomas precession, as incorrectly stated in Jackson (next slide)

Jackson, Classical Electrodynamics, Sec. 11.5

82

1g

N=6

N=5

N=4 2d3s

1h

2f3p

1i

2g3d4s

g9/2

g7/2d5/2

d3/2s1/2h11/2

j15/2i11/2

d3/2

d5/2

g7/2

g9/2

p3/2h9/2

p1/2

i13/2f5/2

f7/2

s1/2

Harmonicoscillator

+flat bottom +spin-orbit

50

126