Transcript LiCAS - University of Oxford

Lectures 4&5
the nuclear force & the shell
model
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
1
4.1 Overview

4.2 Shortcomings of the SEMF
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magic numbers for N and Z
spin & parity of nuclei unexplained
magnetic moments of nuclei
value of nuclear density
values of the SEMF coefficients
4.3 The nuclear shell model
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4.3.1 making a shell model
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4.3.2 predictions from the shell model
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magic numbers
spins and parities of ground state nuclei
“simple” excited states in mirror nuclei
collective excitations
4.3.3 Excited Nuclei
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choosing a potential
L*S couplings
odd and even A mirror nuclei
4.4 The collective model
11 Nov 2004, Lecture 4&5
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4.2 Shortcomings of the SEMF
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
3
4.2 Shortcomings of the SEMF
(magic numbers in Ebind/A)
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SEMF does not apply for A<20
There are deviations from SEMF for A>20
(N,Z)
(10,10)
(6,6)
(2,2)
(8,8)
2*(2,2)
= Be(4,4)
Ea-a=94keV
4
Z
4.2 Shortcomings
of the SEMF
Neutron Magic
Numbers
(magic numbers in numbers of
stable isotopes and isotones)
Proton
Magic
Numbers
• Magic Proton Numbers
(stable isotopes)
• Magic Neutron Numbers
(stable isotones)
N
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
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4.2 Shortcomings of the SEMF
(magic numbers in separation energies)

Neutron separation
energies
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saw tooth from
pairing term
step down when N
goes across magic
number at 82
11 Nov 2004, Lecture 4&5
Ba Neutron separation energy in MeV
Nuclear Physics Lectures, Dr. Armin Reichold
6
4.2 Shortcomings of the SEMF
(abundances of elements in the solar system)

Complex plot due to dynamics of creation, see
lecture on nucleosynthesis
Z=82
N=126
N=82
Z=50
N=50
iron mountain
no A=5 or 8
7
4.2 Shortcomings of the SEMF
(other evidence for magic numbers)
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Nuclei with N=magic have abnormally small n-capture cross
sections (they don’t like n’s) First excitation energy
Close to magic numbers
nuclei can have “long lived”
excited states (tg>O(10-6 s)
called “isomers”. One speaks
of “islands of isomerism”
[Don’t make hollidays there!]
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
208Pb
8
4.2 Shortcomings of the SEMF
(others)
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spin & parity of nuclei do not fit into a drop
model
magnetic moments of nuclei are incompatible
with drops
value of nuclear density is unpredicted
values of the SEMF coefficients are completely
empirical
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
9
The nuclear shell model
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How to get to a quantum mechanical model of
the nucleus?
Can’t just solve the n-body problem because:
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we don’t know the two body potentials
and if we did, we could not even solve a three
body problem
But we can solve a two body problem!
Need simplifying assumptions
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
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4.3 The nuclear shell model
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
11
4.3.1 Making a shell model
(Assumptions)
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Assumptions:
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Each nucleon moves in an averaged potential
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Each nucleon moves in single particle orbit corresponding to potential
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neutron see average of all nucleon-nucleon nuclear interactions
protons see same as neutrons plus proton-proton electric repulsion
the two potentials are wells of some form (nucleons are bound)
 We are making a single particle shell model
Q: why does this make sense if nucleus full of nucleons and typical mean
free paths of nuclear scattering projectiles = O(2fm)
A: Because nucleons are fermions and stack up. They can not loose energy
in collisions since there is no state to drop into after collision
Use Schroedinger Equation to compute Energies (i.e. non-relativistic),
justified by simple infinite square well Energies
Aim to get the correct magic numbers (shell closures) and be content
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
12
4.3.1 Making a shell model
(without thinking, just compute)

Try some potentials; motto: “Eat what you know”
desired
magic
numbers
126
82
50
28
20
8
2
Coulomb
infin. square
harmonic
13
4.3.1 Making a shell model
(with thinking)
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We know how potential should look like!
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It must be of finite depth and …
If we have short range nucl.-nucl. potential
Average potential must be like the density
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flat in the middle (you don’t know where the middle is if you are
surrounded by nucleons)
steep at the edge (due to short range nucl.-nucl. potential)
R ≈ Nuclear Radius
d ≈ width of the edge
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
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4.3.1 Making a shell model
(what to expect when rounding off a potential well)
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Higher L solutions get larger “angular momentum barrier 
Higher L wave functions are “localised” at larger r and thus closer to “edge”
Clipping the edge
Radial Wavefunction U(r)=R(r)*r for the finite square well
(finite size and
rounded) affects high L
states most because
they are closer to the
edge then low L ones.
High L states drop in
energy because
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can now spill out
across the “edge”
this reduces their
curvature
which reduces their
energy
So high L states drop
when clipping and
rounding the well!!
15
4.3.1 Making a
shell model
The “well improvement program”
(with thinking)
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Harmonic is bad
Even realistic well
does not match
magic numbers
Need more shift of
high L states
Include spin-orbit
coupling a’la atomic
magnetic coupling
much too weak and
wrong sign
Two-nucleon
potential has nuclear
spin orbit term
deep in nucleus it
averages away
at the edge it has
biggest effect
the higher L the
bigger the shift
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4.3.1 Making a shell model
(spin orbit terms)

Q: how does the spin orbit term look like?
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Spin S and orbit L are that of single nucleon in average
potential
1 dV (r )
W
(
r
)
:
strongest in non symmetric environment 
r dr
V (r )  V (r )  W (r )L gS
2
 h  1 dV (r )
Dimension: L2
where: W (r ) = - V LS 

compensate 1/r * d/dr
m
c
   r dr
1
and VLS = VLS (E nucleon ) and V (r ) =
(Woods-Saxon)
r - a 
1  exp 

 d 
with a = R 0 A
1
`3
11 Nov 2004, Lecture 4&5
and R 0  1.2 fm and d  0.75 fm = "thickness of edge"
Nuclear Physics Lectures, Dr. Armin Reichold
17
4.3.1 Making a shell model
(spin orbit terms)
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Good quantum numbers without LS term :
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With LS term need operators commuting with new H
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J=L+S & Jz=Lz+Sz with quantum numbers j, jz, l, s
Since s=½ one gets j=l+½ or j=l-½ (l≠0)
Giving eigenvalues of LS [ LS=(L+S)2-L2-S2 ]
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l, lz & s=½ , sz from operators L2, Lz, S2, Sz with
Eigenvalues of l(l+1)ħ2, s(s+1)ħ2, lzħ, szħ
½[j(j+1)-l(l+1)-s(s+1)]ħ2
So potential becomes:
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V(r) + ½l ħ2 W(r)
V(r) - ½(l+1) ħ2 W(r)
11 Nov 2004, Lecture 4&5
for j=l+½
for j=l -½
Nuclear Physics Lectures, Dr. Armin Reichold
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4.3.1 Making a shell model
(fine print)
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There are of course two wells with different
potentials for n and p
The shape of the well depends on the size of the
nucleus and this will shift energy levels as one adds
more nucleons
This is too long winded for us though perfectly
doable
So lets not use this model to precisely predict exact
energy levels but to make magic numbers and …
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
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4.3.2 predictions from the shell model
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
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4.3.2 Predictions from the shell model
(total nuclear “spin” in groundstates)
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Total nuclear angular momentum is called nuclear
spin = Jtot
Just a few empirical rules on how to add up all
nucleon J’s to give Jtot of the whole nucleus
Two identical nucleons occupying same level (same
n,j,l) couple their J’s to give J(pair)=0
Jtot(even-even ground states) = 0
Jtot(odd-A; i.e. one unpaired nucleon) = J(unpaired
nucleon) Carefull: Need to know which level nucleon
occupies. I.e. accurate shell model wanted!
|Junpaired-n-Junpaired-p|<Jtot(odd-odd)< Junpaired-n+Junpaired-p
there is no rule on how to combine the two unpaired J’s
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
21
4.3.2 Predictions from the shell model
(nuclear parity in groundstates)

Parity of a compound system (nucleus):
P (A -nucleon system) =
Pintrinsic (nucleoni )  P i (nucleoni )

i
A
i
A
=1,
=1,
where P  i = (-1)l i and Pintrinsic (nucleoni ) = 1 
P (A-nucleon system) =
li
(
1)

i =1, A
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P(even-even groundstates) = +1 because all
levels occupied by two nucleons
P(odd-A groundstates) = P(unpaired nucleon)
No prediction for parity of odd-odd nuclei
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
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4.3.2 Predictions from the shell model
(magnetic dipole moments)
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The truth: Nobody can really predict nuclear magnetic
moments!
But: we should at least find out what a single particle shell
model would predict because …
Nuclear magnetic moments very important in (amongst other
things) Nuclear Magnetic Resonance (Imaging) NMR
Q: What is special about nuclear magnetic moments
compared to atomic magnetic moments?
A: Nuclei don’t collide with each other and are shielded by
electrons
 Precession of magnetic moments in external B-fields
(excited by RF pulses) are nearly undamped Q=108
 Even smallest frequency shifts give information about
chemical surroundings of magnetic moment
See Minor Option on Medical & Environmental Physics
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
23
4.3.2 Predictions from the shell model
(magnetic dipole moments)
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Units: Nuclear Magneton mnucl
mnucl

eh
eh
=
analogue to atomic Bohr Magneton mBohr = 2M p
2M e
nucleons have intrinsic magnetic moment from
ms
spin
= g s where:
s
mnucl
ms
( proton ) = 2.7928
mnucl
ms
(neutron ) = -1.9130
mnucl
s = 1  gsp = 2.7928  2 and gns = -1.9130  2
2
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
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4.3.2 Predictions from the shell model
(magnetic dipole moments)
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Angular momentum also gives magnetic moment for
net-charged particle (protons only, gln=0)
Total contribution from each unpaired nucleon mj
mj
= g j j where:
mnucl
j ( j  1) - l (l  1)  s (s  1)
j ( j  1)  l (l  1) - s (s  1)
g j = gs
 gl
2 j ( j  1)
2 j ( j  1)
if j = l  s and s = 1
2
mj
= gl l  gs 1
2
mnucl
Schmidt Values
if j = l - s and s = 1
mj
= gl
mnucl

1
l  2 l -1

2


2

g l -1
s
2




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4.3.2 Predictions from the shell model
(magnetic dipole moments)

Q: So how does this compare to reality? (odd-A)
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A: Can just about determine L of unpaired nucleon
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
26
4.3.2 Predictions from the shell model
(magnetic dipole moments)

Predictions are “pretty bad”! Why?
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intrinsic nucleon magnetic moment influenced by
environment (compound nucleons)
Configuration mixing: The pairing of spins is not
exact. Many configurations possible with slightly
“unpaired” nucleons to give one effective unpaired
nucleon
Nucleon-Nucleon interaction has “charged
component” (± exchange) giving extra currents!
nuclei are not really spherical as assumed (see
collective model)
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
27
4.3.3 Excited nuclei
(simplest ones = odd-A mirror nuclei)
p
7
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n
n
3Li = 3p + 4n
7
p
4Li = 4p + 3n
MeV
J
MeV
7.46
5/2-
7.21
6.68
5/2-
6.73
4
3
2
4.63
7/2-
4.57
2
1
0.48
1/2-
0.43
1
0.00
3/2-
0.00
4
3
7
3Li
7
4Be
Energy levels very similar  charge independence of nuclear force
Spin in ground state and first exited state is easy: One unpaired nucleon!
Predicted first excitations
Second excitation is only reconstructable not predictable
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4.3.3 Excited nuclei
(simple ones = even-A mirror nuclei)
• Non analogues states in Na called
isoscalar singlet.
• Unpaired nucleons in spin-space
symmetric  between n and p
•  can not be occupied by (n,n) or
(p,p), would violate Pauli principle
isovector multiplet
MeV
3.357
mirror
p
n
p
n
n
JP
4+
p
8
6
10Ne
10p+12n
12Na
22
11Mg
11p+11n 12p+10n
accumulated
occupancy
per well
2
22
MeV
3.308
JP
4+
1.246
2+
0.000
0+
7 more states
without
analogue in
Ne or Mg
14
22
JP
4+
MeV
4.071
1.275
0.000
22
2+
0+
10 Ne
1.984
1.952
1.937
3+
2+
1+
1.528
5+
0.891
0.657
0.583
4+
0+
1+
0.000
3+
22
11Na
22
12Mg
29
4.4 The collective model
(vibrations)
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From liquid drop model might expect collective vibrations of nuclei
Classify them by multipolarity of mode and by:
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isoscalar (n’s move with p’s) or
isovector (n’s move against p’s)
Breathing or Monopole
Mode: compresses
nuclear matter  high
excitation energy.
E0≈80MeV/A1/3
11 Nov 2004, Lecture 4&5
Dipole mode:
isovector only!
large electric
dipole moment.
E0≈77MeV/A1/3
Quadrupole mode: Octupole mode:
fundamental, leads E0≈32MeV/A1/3
to fission instability
E0≈63MeV/A1/3
Nuclear Physics Lectures, Dr. Armin Reichold
30
4.4 The collective model
(rotations)

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Need non-spherical nuclei to excite rotations!
Observation: asphericity (electric quadrupole
moment) largest when many nucleons far away from
shell closure (150<A<190 & A>220)
How does this happen?
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some non-closed shell nucleons have non spherical
wavefunctions
these can distort the potential well for the complete
nucleus
E(distorted nucleus) < E(undistorted nucleus) distortion
will happen
Most large A distortions are prolate!
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
31
4.4 The collective model
(rotations)
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Many nucleons participate in rotations
 can treat them quasi classically
Classical energies: E=I2/2J where

J = moment of inertia and

I = angular momentum
Quantum mechanical:
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I2=j(j+1)ħ2 but: look only at even-even
nuclei  j=0,2,4,6
Fits observed even-even states
well
But: most cases are more
complex. Combinations of rot. &
vib. & single particle excitations.
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold
32