deformation

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Lecture 8
Shell Model
dipole moments
Excited states,
deformation
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The Shell Model successfully predicts the
ground-state (GS) spin and parity of nuclei with-
a single unpaired neutron or proton
5He, n in 1p3/2 shell I 3/2- 5Li p in 1p3/2
shell I 3/2- 7Li n in 1p1/2 shell I ½- 13C n
in 1p1/2 shell I ½- 15O n in 1p1/2 shell I
½- 17O n in d5/2 shell I 5/2 17F p in d5/2
shell I 5/2
odd N odd Z nuclei 6Li, 14N
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What about excited states?
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A case you should be able to guess
14N Z7, N7 IGS 1
E1st
2.2 MeV
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Some simple excited state configurations in 17O
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Some simple excited state configurations in 17O
I 0
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Some simple excited state configurations in 17O
And similar couplings based on the proton with
mj3/2 give further states in 17O
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Excited states in 18O
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Excited states in 18O
NO! Pauli forbids it, since they would have same
set of quantum numbers j5/2, s1/2, tz1/2
What are the possible couplings of j for these 2
neutrons?
For GS they coupled to 0
Could they couple to 5?
Not if mj 3/2 or ½ for both neutrons.
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j
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j
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j
mj4 (1) mj3 (1) mj2 (2) mj1
(2) mj0 (3) mj-1 (2) mj-2 (2) mj--3
(1) mj-4 (1)
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mj4 (1) mj3 (1) mj2 (2) mj1
(2) mj0 (3) mj-1 (2) mj-2 (2) mj--3
(1) mj-4 (1)
j 4 (9 projections of mj) j 2 (5 projections
of mj) j 0 (1 projection of mj
Coupling 2 identical d5/2 nucleons can lead to 3
states with I 0, 2, 4
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So the two j5/2 neutrons can ONLY couple to I
0, 2, 4
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The Shell Model and spin and parity of nuclear
states
The Shell model accounts fairly well for Ground
state spin and parity for nuclei with even N
and Z. for nuclei with a single unpaired nucleon
Some excited states for nuclei with an unpaired
nucleon e.g. 17O
Some excited states of nuclei with even N or
Z e.g.18O
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Magnetic dipole moments of Nuclear GS
The magnetic moment of a current I circulating an
area A is ? iA
If current due to charge e moving with speed v
in radius r, i.e. with a period T 2?r/v
The measurable Mag. Mom. depends on the
projection of l on the Z axis
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Orbital Magnetic dipole moment
For an atom, m is the electron mass and the term
is called the Bohr magneton (?B) ?B 5.7884 x
10-5 ev/T.
For the nucleus, m is the proton mass. The
nuclear magneton has a value ?N 3.1525 x 10-8
ev/T.
where g is the g factor!!
For protons g 1 for neutrons g 0, (since
they have no charge.)
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The spin contribution to magnetic dipole moment
Nucleons have an intrinsic spin (s1/2) as well
as their orbital AM (l). This also produces a
magnetic field spin magnetic moment
?s gs s ?N
?s ½ gs ?N
For a spin s ½ point particle, such as an
electron, quantum electrodynamics gives a value
of gs 2, and experiment confirms this.
For nucleons the value is far from the
point-particle prediction
Proton gs 5.5856912? ?s 2.7928456
?N neutron gs - 3.8260837??s -1.9130418 ?N
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Although s and l produce ?, ?int is the
projection onto j.
Projection of s and l onto j
Only j is a good quantum number (has fixed
projection onto z-axis), so ?obs is projection of
?int onto z-axis.
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There are 2 ways l and s can combine to give j

• gj ½ gs (j- ½gl) ? gj j/(j1)( -½gs
  (j 3/2 ) gl
• (in units of Nm)
• These limits are called the Schmitt limits.

Examples
17O j 5/2 due to d5/2 neutron for neutron gl
0 gs -3.82 thus ? - 1.91 NM (meas -1.89)
19F j ½ proton in s1/2 calc 2.79 NM
(meas. 2.63)
13C j ½- due to p1/2 neutron for neutron gl
0 gs -3.82 thus ? .63 NM (meas 0.7) 39K
j 3/2 proton in d3/2 calc 0.16 NM (meas
0.39)
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Measured Magnetic moments
Not too good!
Why? (1) Assumption of spherical nuclei is not
always true. (2) Value of gs, INSIDE a nucleus
is not that of free nucleon. (3) The
wavefunctions are never a simple as we assume.
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Shell Model and Deformation
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Single proton outside closed shell
Q is negative (zero for core and negative for
single proton)
Approximating the nucleus as a uniformly charged
sphere ltr2gt 3/5R2 3/5 ro2A2/3 (ro 1.2 fm)
Q -ltr2gt
e.g. 19F Calc. -0.036 Meas. -0.12
Single proton missing from closed shell
Q is ve (zero for core and positive for missing
charge)
  e.g. 27Al (28Si p) Calc 0.036 Meas 0.14
Q ltr2gt but ve
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Predictions for Q for neutrons
For a single neutron outside a closed core,
expect Q0, since neutron has no charge.
NOT TRUE the neutron attracts the protons and
makes a ridge
So Q is ve for a single neutron as for a single
proton!
e.g. 17O (16O n) calc 0.038 meas. 0.026
Conversely a neutron hole leads to ve Q
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At best we can account for sign, and approximate
size for nuclei near closed shells..not really
encouraging
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Electric Quadrupole moments
Shell model can account for Q if N or Z near
within /-1 of magic numbers
The further away from magic numbers the worse the
agreement
Some nuclei have extreme deformations and we need
another model
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The shell Model so far
How good is it???
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The shell Model so far
How good is it???
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The shell Model so far
How good is it???
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The shell Model so far
How good is it???
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The shell Model so far
How good is it???