Nuclear Models

Nuclear Physics, JU, Second Semester, 2010-2011(Saed Dababneh).

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Nuclear Models• Nuclear force is not yet fully understood.• No absolutely satisfying model, but models.• Specific experimental data specific model.• Model success in a certain range.• Some are: Individual particle model. (Energy. states, static properties, …).

Liquid drop model. (Strong force, B.E., Fission, …).

Collective model. -particle model. Optical model. Fermi Gas model. Statistical model. others …..


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Shell model• Electron configuration….

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 ….• AtomicAtomic Electron magic numbers: 2, 10, 18, 36, 54, …

Common center of “external” attraction. Well understood Coulomb force. One kind of particles. Clear meaning for electron orbits. …

• NuclearNuclear magic numbers: 2, 8, 20, 28, 50, 82,126, …(for Z or N).

Chemistry!


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Shell model

Evidence:Evidence:1) End of radioactive series:

thorium series 208Pb

uranium series 206Pb

actinium series 207Pb

neptunium series 209Bi

2) At Z and N mn’s there are relatively large numbers of isotopes and isotones.


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Shell model


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Shell model


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Shell model

3) Natural abundances.


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Shell model

NEUTRON NUMBER

NE

UT

RO

N C

AP

TU

RE

C

RO

SS

SE

CT

ION

4) Neutron capture cross section.


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Shell model5) Binding energy of the last neutron

(Separation Energy). (The measured values are plotted relative to the calculations without ).


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Shell model

Pb (even-A) isotopes.

6) Excited states.


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Shell model


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Shell model

• Nucleons are in definite states of energy and angular momentum.• Nucleon orbit ?? Continuous scattering expected ..!!• No vacancy for scattering at low energy levels.• Nuclear potential?• Infinite square well:

• Harmonic oscillator:

Rr

RrV

0

22

2

1rmV


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Shell model

Infinite spherical well(R=8F)

Harmonic oscillator

???

2(2l + 1)accounts correctly for the number of nucleons in each each levellevel.But what about magic magic numbersnumbers?

)( 23E

)2( 21 lnEnl


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Shell model

MeVVe

VrV

aRr57~

1)( 0/

0

• More realistic! (Can it solve the problem?)• Finite square well potential:

• Rounded well potential:

• Correction for asymmetry and Coulomb repulsion.

Rr

RrVV

00

Adjusted by the separation energies.


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Shell model


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Shell model

)(27 MeVA

ZNVas

Coulomb repulsion? Vc(r) = ??


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Shell model

Transition probability?

Nuclear reactions?


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Shell model• Separation of variables:

• For a given spherically symmetric potential V(r), the bound-state energy levels can be calculated from radial wave equation for a particular orbital angular momentum l. • Notice the important centrifugal potential.

),()()()()(),,( mlYrRrRr


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Shell model


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1s 1p 1d 2s 1f 2p 1g 2d 3s

2(2l +1) 2 6 10 2 14 6 18 10 2

Total 2 8 18 20 34 40 58 68 70

ml

ms

• 2, 8, 20 ok.• What about other magic numbers?• Situation does not improve with other potentials.• Something very fundamental about the single-particle interaction picture is missing in the description…..!!!!!• Spin-orbit coupling.


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Shell model

• So far, 2(2l + 1) accounts correctly for the number of nucleons in each level, since we already considered both orbital angular momentum, and spin, but still not for closed shells.

sl ms

mlsl Ymsml ,,,

Spherical Harmonics,

Eigenfunctions of L2 and Lz.

smsmS

sssS

smss

msz

ms

ms

ss

ss

21)1( 22

But this representation does not solve the

problem.

Separate.


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Shell model

Spin-Orbit CouplingSpin-Orbit Coupling• M. G. Mayer and independently Haxel, Jensen, and Suess.• Spin-Orbit term added to the Hamiltonian:

Central, attractive

No longer

Spherically symmetric

Orientation

LSrVrVm

pH SO .)()(

2

2


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Shell model

LLantiparallel

ULparallel

L SJ

LSJLSJLS 2/)(. 222

21.)1(

,....2,1,0,)1(

,

,)1(

22

22

22

slsjmsslsjmS

llsjmlllsjmL

jmjlsjmmlsjmJ

sljsllsjmjjlsjmJ

jj

jj

jjjjz

jj


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Shell model

2(2x3 + 1) = 14

2j+1

1f7/2

First time

l = 3

j


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Shell model

2)]1()1()1([2

1. sslljjSL

HW 7HW 7 0,)12(2

1 2 llgap

LSrVrVm

pH SO .)()(

2

2


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Shell model

Notes: 1. The shell model is most useful when applied to closed-shell or near closed-shell nuclei. 2. Simple versions of the shell model do not take into account pairing forces, the effects of which are to make two like-nucleons combine to give zero orbital angular momentum. The pairing force increases with l.3. Away from closed-shell nuclei collective models taking into account the rotation and vibration of the nucleus are more appropriate. 4. Shell model does not treat distortion effects (deformed nuclei) due to the attraction between one or more outer nucleons and the closed-shell core.


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Shell model

Ground state: (near closed shells)

1. Angular momentum of odd-A nuclei is determined by the angular momentum of the last nucleon that is odd. 2. Even-even nuclei have zero ground-state spin, because the net angular momentum associated with even N and even Z is zero, and even parity. 3. In odd-odd nuclei the last neutron couples to the last proton with their intrinsic spins in parallel orientation.

Provided that the ordering is known….!!

A < 150190 < A < 220


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Shell model

Near valley of

stability

No spin-orbit

coupling

Harmonic oscillator

Near drip line


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Shell model


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Shell model

• 17 p, 21 n.• p in 1d3/2 l s = +• n in 1f7/2 l s = -• Rule 3 sp sn lp ln

• ½ + ½ + 3 – 2 = 2

total = -


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Shell model

Excited states:


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Extreme independent particle model!!! Does the core really remain inert?

Shell model

1d5/2

2s1/2

1d3/2

1p1/2?

?

l pairing

1p3/2 ?

Core


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Shell model

• Extreme independent particle model only 23rd neutron.• More complete shell model all three “valence” nucleons.

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Shell model

HW 8HW 8

and 43Sc, 43Ti.

Discuss the energy levels of nuclei with odd number of nucleons in the 1f7/2 shell.


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Shell model

)1(2

)1()1()1(

)1(2

)1()1()1(

jj

sslljjg

jj

sslljjgg lsj

Dipole Magnetic Moment

Nj jg HW 9HW 9 Show that

and examineexamine Eqs. 5.9 in Krane. In addition, work out problem 5.8 in Krane Conclusion?Proton: gs(free) = 5.5856912 ? gl = 1 ?Neutron: gs(free) = -3.8260837 ? gl = 0 ?What about + and -?


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Shell model

Electric Quadrupole MomentRefined QM

12

121

)1(2

12 322

053

j

nAr

j

jQ

<r2> for a uniformly charged sphere(0.03 – 0.3 b)

jn 21 Number of protons in

a subshell

ExtremesSingle particle: n = 1 - ive QSingle hole: n = 2j +ive Q

Examine Table 5.1 and Fig.5.10 in

Krane

In the xy-plane: Q - r2.


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Shell model

ValidityA < 150

190 < A < 220

Nuclide Q (b)2H (D) +0.00288

17O -0.0257859Co +0.4063Cu -0.209133Cs -0.003

161Dy +2.4176Lu +8.0

209Bi -0.37

Nuclear Models

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