Experimental evidence for closed nuclear shells

2828 50

50

82

82126

NeutronProton

Deviations from Bethe-Weizsäcker mass formula:

mass number A

B/A

(M

eV p

er n

ucl

eon

)

242 He

8168O

204020Ca

284820Ca

12620882 Pb

very stable:

Shell structure from masses

• Deviations from Weizsäcker mass formula:

Energy required to remove two neutrons from nuclei(2-neutron binding energies = 2-neutron “separation” energies)

Sn

Ba

SmHf

Pb

5

7

9

11

13

15

17

19

21

23

25

52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132

Neutron Number

S(2

n)

Me

V

N = 82

N = 84

N = 126

Shell structure from Ex(21) and B(E2;2+→0+)

high energy of first 2+ states

low reduced transition probabilities B(E2)

The three faces of the shell model

Average nuclear potential well: Woods-Saxon

aRrVrV /exp1/ 00

02

22

rrV

m

smm XY

r

rur ,

A

jiji

A

i i

i rrVm

pH ,ˆ

2

ˆˆ1

2

A

ji

A

iiji

A

ii

i

i rVrrVrVm

pH

11

2

ˆ,ˆˆ2

ˆˆ

Woods-Saxon potential

Woods-Saxon gives proper magic numbers (2, 8, 20, 28, 50, 82, 126) Meyer und Jensen (1949): strong spin-orbit interaction

02

22

rsrVrV

m s

01

~ mitdr

dV

rrV s

dr

rdV

rV r

Spin-orbit term has its origin in the relativistic description of the single-particle motion in the nucleus.

Woods-Saxon potential (jj-coupling)

2

2222

1112

12

1

ssjj

sjssj

2/12

jforVrV s

The nuclear potential with the spin-orbit term is

spin-orbit interaction leads to a large splitting for large ℓ.

2/12

1

jforVrV s

2/1j

2/1j

2/1j

sV 2/1

sV 2/

Woods-Saxon potential

The spin-orbit term

reduces the energy of states with spin oriented parallel to the orbital angular momentum j = ℓ+1/2 (Intruder states) reproduces the magic numbers large energy gaps → very stable nucleiss VE

2

2

1221

21Important consequences: Reduced orbitals from higher lying N+1 shell have different parities than orbitals from the N shell

Strong interaction preserves their parity. The reduced orbitals with different parity are rather pure states and do not mix within the shell.

Shell model – mass dependence of single-particle energies

Mass dependence of the neutron energies:

Number of neutrons in each level: 122

2~ RE

½ Nobel price in physics 1963: The nuclear shell model

Experimental single-particle energies

208Pb → 209Bi Elab = 5 MeV/u

1 h9/2

2 f7/2

1 i13/2 1609 keV

896 keV

0 keV

γ-spectrumsingle-particle energies

12620983 Bi

Experimental single-particle energies

208Pb → 207Pb Elab = 5 MeV/u

γ-spectrum

single-hole energies

3 p1/2

2 f5/2

3 p3/2 898 keV

570 keV

0 keV

12520782 Pb

Experimental single-particle energies

209Pb209Bi

207Pb207Tl

)2()()( 2/9208209 gEPbBEPbBE

)3()()( 2/1208207 pEPbBEPbBE

energy of shell closure:

432.3

)(2)()()3(2 2082072092/12/9

PbBEPbBEPbBEpEgE

)1()()( 2/9208209 hEPbBEBiBE

)3()()( 2/1208207 sEPbBETlBE

MeV

PbBETlBEBiBEsEhE

211.4

)(2)()()3(1 2082072092/12/9

1 h9/2

2 f7/2

1 i13/21609 keV

896 keV

0 keV

12620882 Pb

particle states

hole states

proton

Level scheme of 210Pb

0.0 keV

779 keV

1423 keV

1558 keV

2202 keV

2846 keV

-1304 keV (pairing energy)

M. Rejmund Z.Phys. A359 (1997), 243

12720982 Pb

Level scheme of 206Hg

0.0 keV

997 keV

1348 keV

2345 keV

12/5

12/1

ds

12/5

12/3

dd

B. Fornal et al., Phys.Rev.Lett. 87 (2001) 212501

126207

81Tl

Success of the extreme single-particle model

Ground state spin and parity:

Every orbit has 2j+1 magnetic sub-states, fully occupied orbitals have spin J=0, they do not contribute to the nuclear spin.

For a nucleus with one nucleon outside a completely occupied orbit the nuclear spin is given by the single nucleon.

n ℓ j → J (-)ℓ = π

Success of the extreme single-particle model

magnetic moments: The g-factor gj is given by:

with

Simple relation for the g-factor of single-particle states

jgsgg jsj

2222 2 ssjjsj

2222 2

jjjs

j

jj

jjgjjg sj

12

4/3114/311

2/1

12

jfor

gggg s

KernK

j

j

j

jsgg sj

ssj ggjj

ssggg

1

11

2

1

2

1

Success of the extreme single-particle model

magnetic moments:

g-faktor of nucleons:proton: gℓ = 1; gs = +5.585 neutron: gℓ = 0; gs = -3.82

proton:

neutron:

2/1

2

1

2

3

1

2/12

1

2

1

jfürgjgj

j

jfürgjg

Ks

Ks

z

2/1

1293.2

2/1293.2

jfürj

jj

jfürj

K

K

z

2/1

191.1

2/191.1

jfürj

jjfür

K

K

z

Magnetic moments: Schmidt lines

magnetic moments: neutron

magnetic moments: proton