Shell model

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

1

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

Shell model

1d5/2

2s1/2

1d3/2

1p1/2

?

l pairing

Core


2

Shell model

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

20


3

Shell model

HW 26HW 26

and 43Sc, 43Ti.

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


4

Shell model

)1(2)1()1()1(

)1(2)1()1()1(

jjsslljjg

jjsslljjgg lsj

Dipole Magnetic Moment

Nj jg HW 27HW 27 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 -?


5

Shell modelElectric Quadrupole MomentRefined QM

12

121)1(2

12 322

053

jnAr

jjQ

<r2> for a uniformly charged sphere

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.


6


7

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


8

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.


9

Collective model

• 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.


10

Collective model• Rotational motion observed for non-spherical nuclei.• Deformed nuclei are mainly 150 < A < 190 and A > 220.• Ellipsoid of surface:

31

31

0

200

534

),(1),(

ArR

YArR

Difference between

semimajor and semiminor axes.Deformation

parameter.

HW 28HW 28 Problems 5.11 and 5.12 in Krane. Problems 5.11 and 5.12 in Krane.Discuss effect on quadrupole moment.

> 0 < 0

Rav


11

Collective modelSymmetry

axis

)1(22

21

22

IIlE

l

E

gg

gg 2

GS (even-even) 0+

Symmetry only even I

)2/(20)4(

23.152/4.91)2/(6)2(

0)0(

2

22

ggg

E

keVkeVE

E


12

Collective model

HW 29HW 29 compare 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.


13

Collective model

Spin parity Emeasured

(keV)E/E(2+) I(I + 1)/6

12+ 10+ 1518.00 16.61 18.338+ 6+ 7.004+ 299.44 3.28 3.332+ 91.4 1.0 1.00+ 0

164Er

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

HW 29 HW 29 (continued)(continued)


14

nucleonEE 2g21

Collective modelOdd-A


15

Collective modelAverageshape

Instantaneousshape


16

Collective model

φ),(θY (t)αRR(t) λμλ

λ

λ μλμav

μ- λ,λμ

Instantaneouscoordinate

Symmetry

Amplitude Sphericalharmonics

r0A1/3

= 0monopole

= 1dipole

= 2quadrupole

= 3octupole

.

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

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


17

Collective model

YαR YαYαYα R

),(YαRR(t)

1010avr

1- 1,1- 1,10101111avr

1μ

1

1 μ1μavr

Both monopole and dipole excitations require high energy.

R(t) = Ravr +00 Y00 = 0monopole

= 1dipole


18

Collective model = 2quadrupole

• 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.

2020avr

2- 2,2- 2,2,-12,-1202021212222avr

2μ

2

2 μ2μavr

YαR YαYαYαYαYα R

),(YαRR(t)


19

Collective model

-2 -1 0 1 2

-2 -4 -3 -2 -1 0

-1 -3 -2 -1 0 +1

0 -2 -1 0 +1 +2

1 -1 0 +1 +2 +3

2 0 +1 +2 +3 +4

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

Triplet0+, 2+, 4+


20

Collective modelTwo-phonon triplet at twice the energy of the single phonon state.

HW 30HW 30Krane 5.10


21

Nuclear Reactions

X(a,b)Y• First in 1919 by Rutherford:

4He + 14N 17O + 1H14N(,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 …….


22

Nuclear Reactions• Recoil nucleus Y could be unstable or emission.

• One should think about: Reaction dynamics and conservation laws i.e. conditions necessary for the reaction to be energetically possible. Reaction mechanism and theories which explain the reaction. Reaction cross section i.e. rate or probability.


23

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

apa X

pY

pb

Y

bQTTcmcm iffi 22

+ve Q-value exoergic reaction. -ve Q-value endoergic reaction.

aYb TQTT +ve Q-value reaction possible if Ta 0. -ve Q-value reaction not possible if Ta 0. (Is Ta > |Q| sufficient?).Conservation of momentum ……


24

Nuclear Reactions• 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.

bY

aaYYbYabaabab mm

TmmQmmmTmmTmmT

])()[(coscos 2

HW 31HW 31

abY

bYTh mmm

mmQT


25

Nuclear Reactions

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

\.

• Double-valued in a forward cone.

aY

Ya mm

mQT

\

aba

aaYYbY

TmmTmmQmmm ])()[(cos max

2

HW 31HW 31 (continued)(continued)

Shell model

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