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Shell model. Extreme independent particle model!!! Does the core really remain inert?. 1d 3/2. ?. 1p 1/2. l  pairing . 2s 1/2. 1d 5/2. Shell model. 20. Core. Extreme independent particle model  only 23 rd neutron. More complete shell model  all three “valence” nucleons.

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Shell model

Shell model

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

1d3/2

?

1p1/2

l  pairing 

2s1/2

1d5/2

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model1

Shell model

20

Core

  • Extreme independent particle model  only 23rd neutron.

  • More complete shell model  all three “valence” nucleons.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model2

Shell model

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

HW 26

and 43Sc, 43Ti.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model3

Shell model

Dipole Magnetic Moment

HW 27Show that

and examineEqs. 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 -?

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model4

Shell model

In the xy-plane: Q - r2.

Electric Quadrupole Moment

Refined QM 

Extremes

Single particle: n = 1  - ive Q

Single hole: n = 2j +ive Q

Number of protons in a subshell

Examine Table 5.1 and Fig.5.10 in Krane

<r2> for a uniformly charged sphere

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model5

Shell model

Validity

A < 150

190 < A < 220

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Collective model

Collective model

  • Large quadrupole moments  nucleus as a collective body (Liquid drop model).

  • Interactions between outer nucleons and closed shells cause permanent deformation.

  • Single-particle state calculated in a non-spherical potential  complicated.

  • Spacing between energy levels depends on size of distortion.

  • Doubly magic  1st excited state away from GS.

  • Near closure  single-particle states.

  • Further away from closure  collective motion of the core  excited states.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Collective model1

A net nuclear potential due to filled core shells exists.

Collective model combines both liquid drop model and shell model.

Two major types of collective motion:

Rotations: Rotation of a deformed shape.

Vibrations: Surface oscillations.

Collective model

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Collective model2

Collective model

  • Rotational motion observed for non-spherical nuclei.

  • Deformed nuclei are mainly 150 < A < 190 and A > 220.

  • Ellipsoid of surface:

Difference between semimajor and semiminor axes.

Deformation parameter.

Rav

HW 28 Problems 5.11 and 5.12 in Krane.

Discuss effect on quadrupole moment.

 < 0

 > 0

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Collective model3

Collective model

Symmetry axis

Rotational

States

GS (even-even) 0+

Symmetry  only even I

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Collective model4

Collective model

HW 29compare measured energies of the states of the ground state rotational band to the calculations.

Rigid body or liquid drop? Intermediate  Short range and saturation of nuclear force.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Collective model5

Collective model

HW 29 (continued)

Higher angular momentum  centrifugal stretching  higher moment of inertia  lower energy than expected  additional evidence for lack of rigidity.

164Er

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Collective model

Odd-A

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Collective model

Vibrational States

Average

shape

Instantaneous

shape

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Collective model

Symmetry

Instantaneous

coordinate

Spherical

harmonics

r0A1/3

Amplitude

http://wwwnsg.nuclear.lu.se/basics/excitations.asp?runAnimation=beta10

  • = 1

    dipole

  • = 2

    quadrupole

  • = 3

    octupole

  • = 0

    monopole

.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Collective model

  • = 0

    monopole

R(t) = Ravr +00Y00

  • = 1

    dipole

Both monopole and dipole excitations require high energy.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Collective model

  • = 2

    quadrupole

  • Quantization of quadrupole vibration is called a quadrupole phonon.

  • A phonon carries two units of angular momentum and even parity (-12).

  • This mode is dominant. For most even-even nuclei, a low lying state with Jπ=2+ exists.

  • Octupole phonon.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Collective model

l = 4  = +4, +3, +2, +1, 0, -1, -2, -3, -4

l = 2  = +2, +1, 0, -1, -2

l = 0  = 0

Triplet

0+, 2+, 4+

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Collective model

Two-phonon triplet at twice the energy of the single phonon state.

HW 30

Krane 5.10

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Nuclear Reactions

  • X(a,b)Y

  • First in 1919 by Rutherford:

  • 4He + 14N  17O + 1H

  • 14N(,p)17O

  • Incident particle may: change direction, lose energy, completely be absorbed by the target……

  • Target may: transmute, recoil……

  • b =   Capture reaction.

  • If B.E. permits  fission (comparable masses).

  • Different exit channels a + X  Y1 + b1

  •  Y2 + b2

  •  Y3 + b3…….

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Nuclear Reactions

  • Recoil nucleus Y could be unstable   or  emission.

  • One should think about:

    • Reaction dynamics and conservation lawsi.e. conditions necessary for the reaction to be energetically possible.

    • Reaction mechanism and theories which explain the reaction.

    • Reaction cross sectioni.e. rate or probability.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Nuclear Reactions

  • Conservation Laws

  • Charge, Baryon number, total energy, linear momentum, angular momentum, parity, (isospin??) …….

b

pb

a

pa

X

+ve Q-value  exoergic reaction.

-ve Q-value  endoergic reaction.

pY

Y

+ve Q-value  reaction possible if Ta 0.

-ve Q-value  reaction not possible if Ta 0. (Is Ta > |Q| sufficient?).

Conservation of momentum ……

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Nuclear Reactions

HW 31

  • Conservation of momentum.

  • We usually do not detect Y.

  • Show that:

  • The threshold energy (for Ta): (the condition occurs for  = 0º).

  • +ve Q-value  reaction possible if Ta 0.

  • Coulomb barriers…….!!!

  • -ve Q-value  reaction possible if Ta> TTh.

double valued !?

solve for Q

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


Shell model

Nuclear Reactions

HW 31(continued)

  • The double valued situation occurs between TTh and the upper limit Ta\.

  • Double-valued in a forward cone.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).


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