1. the nuclear force & the shell model
Lectures 4&5
the nuclear force & the shell model
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

2. 4.1 Overview 4.2 Shortcomings of the SEMF 4.3 The nuclear shell model
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
4.3.1 making a shell model
choosing a potential
L*S couplings
4.3.2 predictions from the shell model
magic numbers
spins and parities of ground state nuclei
“simple” excited states in mirror nuclei
collective excitations
4.3.3 Excited Nuclei
odd and even A mirror nuclei
4.4 The collective model
11 Nov 2004, Lecture 4&5

3. 4.2 Shortcomings of the SEMF
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

4. 4.2 Shortcomings of the SEMF (magic numbers in Ebind/A)
SEMF does not apply for A<20
There are deviations from SEMF for A>20
(2,2)
2*(2,2)
= Be(4,4)
Ea-a=94keV
(6,6)
(8,8)
(10,10)
(N,Z)

5. Nuclear Physics Lectures, Dr. Armin ReicholdZ
Neutron Magic
Numbers
4.2 Shortcomings of the SEMF (magic numbers in numbers of stable isotopes and isotones)
Proton
Magic
Numbers
Magic Proton Numbers (stable isotopes)
Magic Neutron Numbers (stable isotones)
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

6. 4.2 Shortcomings of the SEMF (magic numbers in separation energies)
Neutron separation energies
saw tooth from pairing term
step down when N goes across magic number at 82
Ba Neutron separation energy in MeV
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

7. 4.2 Shortcomings of the SEMF (abundances of elements in the solar system)
Z=50
N=82
Z=82
N=126
iron mountain
Complex plot due to dynamics of creation, see lecture on nucleosynthesis
no A=5 or 8

8. 4.2 Shortcomings of the SEMF (other evidence for magic numbers)
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!]
208Pb
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

9. 4.2 Shortcomings of the SEMF (others)
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

10. The nuclear shell model
How to get to a quantum mechanical model of the nucleus?
Can’t just solve the n-body problem because:
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

11. 4.3 The nuclear shell model
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

12. 4.3.1 Making a shell model (Assumptions)
Each nucleon moves in an averaged potential
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)
Each nucleon moves in single particle orbit corresponding to potential
 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

13. 4.3.1 Making a shell model (without thinking, just compute)
Try some potentials; motto: “Eat what you know”
desired
magic
numbers
Coulomb
infin. square
harmonic
126 82 50 28 20 8 2

14. 4.3.1 Making a shell model (with thinking)
We know how potential should look like!
It must be of finite depth and …
If we have short range nucl.-nucl. potential
Average potential must be like the density
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

15. 4.3.1 Making a shell model (what to expect when rounding off a potential well)
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 (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
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!!
Radial Wavefunction U(r)=R(r)*r for the finite square well

16. 4.3.1 Making a shell model (with thinking)
The “well improvement program”
4.3.1 Making a shell model (with thinking)
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

17. 4.3.1 Making a shell model (spin orbit terms)
Q: how does the spin orbit term look like?
Spin S and orbit L are that of single nucleon in average potential
strongest in non symmetric environment 
Dimension: L2
compensate 1/r * d/dr
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

18. 4.3.1 Making a shell model (spin orbit terms)
Good quantum numbers without LS term :
l, lz & s=½ , sz from operators L2, Lz, S2, Sz with Eigenvalues of l(l+1)ħ2, s(s+1)ħ2, lzħ, szħ
With LS term need operators commuting with new H
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 ]
½[j(j+1)-l(l+1)-s(s+1)]ħ2
So potential becomes:
V(r) + ½l ħ2 W(r) for j=l+½
V(r) - ½(l+1) ħ2 W(r) for j=l -½
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

19. 4.3.1 Making a shell model (fine print)
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

20. 4.3.2 predictions from the shell model
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

21. Nuclear Physics Lectures, Dr. Armin Reichold
4.3.2 Predictions from the shell model (total nuclear “spin” in groundstates)
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

22. Nuclear Physics Lectures, Dr. Armin Reichold
4.3.2 Predictions from the shell model (nuclear parity in groundstates)
Parity of a compound system (nucleus):
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

23. 4.3.2 Predictions from the shell model (magnetic dipole moments)
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

24. 4.3.2 Predictions from the shell model (magnetic dipole moments)
Units: Nuclear Magneton mnucl
nucleons have intrinsic magnetic moment from spin
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

25. 4.3.2 Predictions from the shell model (magnetic dipole moments)
Angular momentum also gives magnetic moment for net-charged particle (protons only, gln=0)
Total contribution from each unpaired nucleon mj
Extract from Cottingham & Greenwood about
nuclear moment in terms of j instead of l and s
Schmidt Values

26. 4.3.2 Predictions from the shell model (magnetic dipole moments)
Q: So how does this compare to reality? (odd-A)
max Schmidt Value
min Schmidt Value
A: Can just about determine L of unpaired nucleon
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

27. 4.3.2 Predictions from the shell model (magnetic dipole moments)
Predictions are “pretty bad”! Why?
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” (p± 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

28. 4.3.3 Excited nuclei (simplest ones = odd-A mirror nuclei)J
5/2-
7/2-
1/2-
3/2-
MeV
7.21
6.73
4.57
0.43
0.00
7.46
6.68
4.63
0.48
73Li Be
73Li = 3p + 4n p n
74Li = 4p + 3n 4 4 3 3 2 2 1 1
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

29. 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 Y between n and p
 can not be occupied by (n,n) or (p,p), would violate Pauli principle
4.3.3 Excited nuclei (simple ones = even-A mirror nuclei)
isovector multiplet
MeV 3.357
1.275
0.000
JP 4+ 2+ 0+
MeV 4.071
1.528
0.000
JP 4+ 5+ 3+
MeV 3.308
1.246
0.000
JP 4+ 2+ 0+
2210Ne
10p+12n p n
2212Na
11p+11n
2211Mg
12p+10n
mirror
7 more states without analogue in Ne or Mg 2 6 8 14
accumulated
occupancy
per well
Note: It is often easier to create a hole in an already closed shell then to move say an unpaired nucleon higher and higher up the ladder. If there is an unpaired nucleon then this can be paired off with the nucleon that leaves the closed shell.  it is very difficult to use the single particle shell model pro predict excitation sequences.
b-decay between states of nuclei that are members of an isospin doublet are particularly interesting. All wave functions in the mother and daughter nuclei are nearly identical and the transition rate is particularly high. These are called super allowed transitions
The two nucleon system is a special case of an isospin doublet
if two nucleons have L=0 between them they can be in either the 1S0 or the 3S1 states (spin singlet or triplet)
The spin-triplet is space-spin symmetric  no good for nn or pp  this is an isospin-singlet
The spin-singlet is space spin anti-symmetric  good for nn, np, pp  this is an isospin-triplet
But: Nuclear force binds spin triplet states better then spin singlet states  binding not strong enough to bind spin singlet state so the only chance to bind two nucleons is the spin triplet S=J=1 and L=0 deuteron and it has no excited state.
Keen students: read about how to derive the nucleon-nucleon force from scattering experiments and see that deuteron is not pure spin triplet but has O(4%) contribution from J=2 and is also not spherical.
2210 Ne Na Mg

30. 4.4 The collective model (vibrations)
From liquid drop model might expect collective vibrations of nuclei
Classify them by multipolarity of mode and by:
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
Dipole mode:
isovector only!
large electric
dipole moment.
E0≈77MeV/A1/3
Quadrupole mode:
fundamental, leads
to fission instability
E0≈63MeV/A1/3
Octupole mode:
E0≈32MeV/A1/3
11 Nov 2004, Lecture 4&5
Nuclear Physics Lectures, Dr. Armin Reichold

31. 4.4 The collective model (rotations)
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?
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

32. 4.4 The collective model (rotations)
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:
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