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Shell model

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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).

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).

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).

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).

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).

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

Validity

A < 150

190 < A < 220

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

- 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).

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.

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

- 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).

Symmetry axis

Rotational

States

GS (even-even) 0+

Symmetry only even I

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

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).

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).

Collective model

Odd-A

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

Collective model

Vibrational States

Average

shape

Instantaneous

shape

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

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).

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).

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).

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).

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).

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).

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).

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).

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).

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).