Single Particle and Collective
Modes in Nuclei
Lecture Series
R. F. Casten
WNSL, Yale
Sept., 2008
TINSTAASQ
You disagree?
So, an example of a really really stupid
question that leads to a useful discussion:
Are nuclei blue?
Sizes and forces
Uncertainty Principle:
DE Dt > h

Dm Dx/c > h
Nuclear force mediated by pion exchange: m ~ 140 MeV

Range of nuclear force / nuclear sizes ~ fermis
--------------------------------------------------------------------------------Uncertainty Principle:
Dx D p > h
 Characteristic nuclear energies are 105 times atomic
energies: 10 ev  1 MeV
Probes and “probees”
E=h/
Energy of probe correlated with sizes of probee and production devices
Atoms – lasers – table top
Nuclei – tandems, cyclotrons, etc – room size
Quarks, gluons – LHC – city size
Overview of nuclear structure
also
Some preliminaries
Independent particle model
and clustering in simple potentials
Concept of collectivity
(Note: many slides are VG images – and contain typos I can’t easily correct)
Simple Observables - Even-Even
.
. Nuclei
4+
1000

1

1
E (4 )
R4 / 2 
E (2 )
B ( E 2; 41  21 )
2+
400
Masses
B ( E 2; 21  01 )
0+
0
Jπ
E (keV)
B ( E 2; J i  J f ) 
1
2J i  1
 i E2 
2
f
Evolution of structure – First, the
data
• Magic numbers, shell gaps, and shell
structure
• 2-particle spectra
• Emergence of collective features,
deformation and rotation
The magic numbers:” special benchmark numbers of nucleons
B(E2:
0+
1
2+
1)

2+
1
E20+
2
1
2+
0+
Be astonished by this: Nuclei with 100’s of nucleons
orbiting 1021 times/s, not colliding, and acting in concert !!!
The empirical magic numbers
near stability
• 2, 8, 20, 28, (40), 50, (64), 82, 126
• This is the only thing I ask you to
memorize.
“Magic plus 2”: Characteristic spectra

1

1
E (4 )
R4 / 2 
< 2.0
E (2 )
What happens with both valence neutrons
and protons? Case of few valence nucleons:
Lowering of energies, development of
multiplets. R4/2  ~2
Spherical
vibrational
nuclei
Vibrator (H.O.)
E(I) = n ( 0 )
R4/2= 2.0
n = 0,1,2,3,4,5 !!
n = phonon No.
(Z = 52)
Neutron number
68
70
72
74
76
78
80
82
Val. Neutr. number 14
12
10
8
6
4
2
0
Lots of valence nucleons of both types
R4/2  ~3.33
Deformed nuclei – rotational spectra
Rotor
E(I)  ( ħ2/2I )I(I+1)
R4/2= 3.33
BTW, note value of
paradigm in
spotting physics
(otherwise invisible)
from deviations
8+
6+
4+
2++
0
Broad perspective on structural evolution:
R4/2
Note the characteristic, repeated patterns
Sudden changes in R4/2 signify changes in structure,
usually from spherical to deformed structure
3.4
3.4
3.2
Def.
3.0
R4/2
2.8
2.6
2.4
2.2
Sph.
2.0
3.2
2.8
2.6
2.4
2.0
1.8
1.6
1.6
86
88
90
Sph.
2.2
1.8
84
N=84
N=86
N=88
N=90
N=92
N=94
N=96
Def.
3.0
R4/2
Ba
Ce
Nd
Sm
Gd
Dy
Er
Yb
92
94
N
Onset of deformation
96
56
58
60
62
64
66
68
70
Z
Onset of deformation
as a phase transition
Another, simpler observable
E2, or 1/E2,
R4/2
is among the first
pieces of data
obtainable in nuclei far
from stability. Can we
use just this quantity
alone?
Observable
E2
1/E2 –
Note
similarity
to R4/2
Nucleon number, Z or N
0,014
Z=56
Z=58
Z=60
Z=62
Z=64
Z=66
Z=68
0,012
0,008
+
1/E(21 )
+
1/E(21 )
0,010
0,006
0,012
0,010
0,008
0,006
0,004
0,004
0,002
0,002
0,000
84
0,000
56
86
88
90
92
Neutron Number
94
96
N=84
N=86
N=88
N=90
N=92
N=94
N=96
0,014
58
60
62
64
Proton Number
66
68
B(E2; 2+  0+ )
Basic Models
• (Ab initio calculations using free nucleon forces, up to A ~ 12)
• (Microscopic approaches, such as Density Functional Theory)
• Independent Particle Model  Shell Model
and its extensions to weakly bound nuclei
• Collective Models – vibrator, transitional, rotor
• Algebraic Models – IBA
One on-going success story
Independent particle model: magic numbers,
shell structure, valence nucleons.
Three key ingredients
First:
Vij
r = |ri - rj|
Nucleon-nucleon
force – very
complex
Ui

~
r
One-body potential –
very simple: Particle
in a box
This extreme approximation cannot be the full story.
Will need “residual” interactions. But it works
surprisingly well in special cases.
Second key ingredient:
Particles in
a “box” or
“potential”
well
Quantum mechanics
Confinement is
origin of
quantized
energies levels
3
1
2
Energy
~ 1 / wave length
n = 1,2,3 is principal quantum number
E
up with n because wave length is shorter
-
=
But nuclei are 3- dimensional. What’s new in 3-dimensions?
Angular momentum, hence centrifugal effects.
Radial Schroedinger
wave function
2
2

h 2 d R nl (r )
h
l (l  1)
 E nl  U ( r ) 

2m
2m r 2

dr 2


 R nl ( r )  0


Higher Ang Mom: potential well is raised
and squeezed. Wave functions have
smaller wave lengths. Energies rise
Energies also rise with principal
quantum number, n.
Hence raising one and lowering the
other can lead to similar energies and
to “level clustering”:
H.O:
E = ħ (2n+l)
E (n,l) = E (n-1, l+2)
e.g., E (2s) = E (1d)
Add spin-orbit force
nlj: Pauli Prin. 2j + 1 nucleons
Too low by 14
Too low by 12
Too low by 10
We can see how to
improve the
potential by looking
at nuclear Binding
Energies.
The plot gives B.E.s
PER nucleon.
Note that they
saturate. What does
this tell us?
Consider the simplest possible
model of nuclear binding.
Assume that each nucleon
interacts with n others. Assume
all such interactions are equal.
Look at the resulting binding as
a function of n and A. Compare
this with the B.E./A plot.
Each nucleon interacts
with 10 or so others.
Nuclear force is short
range – shorter range than
the size of heavy nuclei !!!
~
Compared to SHO, will mostly affect orbits
at large radii – higher angular momentum states
So, modify Harm.
Osc. By squaring off
the outer edge.
Then, add in a spinorbit force that lowers
the energies of the
j=l+½
orbits and raises
those with
j=l–½
Third key ingredient
Pauli Principle
• Two fermions, like protons or neutrons, can NOT be in
the same place at the same time: can NOT occupy the
same orbit.
• Orbit with total Ang Mom, j, has 2j + 1 substates, hence
can only contain 2j + 1 neutrons or protons.
This, plus the clustering of levels in simple
potentials, gives nuclear SHELL STRUCTURE
Clusters of levels + Pauli Principle 
magic numbers, inert cores
Concept of valence nucleons – key to
structure. Many-body  few-body: each
body counts.
Addition of 2 neutrons in a nucleus with
150 can drastically alter structure
a)
Hence J = 0
Applying the Independent Particle
Model to real Nuclei
• Some great successes (for nuclei that are “doubly magic plus 1”).
• Clearly fails totally with more than a single particle outside a doubly
magic “core”. In fact, in such nuclei, it is not even defined.
• Residual interactions to the rescue. (We will discuss extensively.)
• Further from closed shells, collective phenomena emerge (as a result
of residual interactions). What are these interactions? Many models.
Shell model too crude. Need to add in extra
interactions among valence nucleons outside closed
shells.
These dominate the evolution of Structure
• Residual interactions
– Pairing – coupling of two identical nucleons to angular
momentum zero. No preferred direction in space, therefore
drives nucleus towards spherical shapes
– p-n interactions – generate configuration mixing, unequal
magnetic state occupations, therefore drive towards
collective structures and deformation
– Monopole component of p-n interactions generates changes
in single particle energies and shell structure
Independent Particle Model – Uh –oh !!!
Trouble shows up
Shell Structure
Mottelson – ANL, Sept. 2006
Shell gaps, magic numbers, and shell structure are not
merely details but are fundamental to our
understanding of one of the most basic features of
nuclei – independent particle motion. If we don’t
understand the basic quantum levels of nucleons in the
nucleus, we don’t understand nuclei. Moreover,
perhaps counter-intuitively, the emergence of nuclear
collectivity itself depends on independent particle
motion (and the Pauli Principle).
Backups
So, we will have a Hamiltonian
H = H0 + Hresid.
where H0 is that of the Ind. Part. Model
The eigenstates of H will therefore be
mixtures of those of H0
Wave fcts: