Single Particle and Collective Modes in Nuclei

Report
Shell Model and Collective Models
in Nuclei
Experimental and theoretical perspectives
R. F. Casten
WNSL, Yale Univ.
RIKEN, January, 2010
Lecture 1
Introduction to Nuclear Structure
and the Independent Particle Model
(IPM)
The scope of Nuclear Structure Physics
The Four Frontiers
1. Proton Rich Nuclei
2. Neutron Rich Nuclei
3. Heaviest Nuclei
4. Evolution of structure within
these boundaries
Terra incognita — huge gene pool of new nuclei
We can customize our system – fabricate “designer” nuclei
to isolate and amplify specific physics or interactions
Themes and challenges of Modern Science
•Complexity out of simplicity -- Microscopic
How the world, with all its apparent complexity and diversity can be
constructed out of a few elementary building blocks and their interactions
•Simplicity out of complexity – Macroscopic
How the world of complex systems can display such remarkable regularity
and simplicity
Simple Observables - Even-Even (cift-cift)
. . Nuclei
4+
1000

1

1
E (4 )
R4 / 2 
E (2 )
B ( E2; 41  21 )
2+
400
B ( E2; 21  01 )
Masses
0+
0
Jπ
E (keV)
B ( E 2; J i  J f ) 
1
2J i  1
 i E2 
2
f
Survey: Empirical evolution of
structure
• Magic numbers, shell gaps, and shell
structure
• 2-particle spectra
• Emergence of collective features –
Vibrations, deformation, and rotation
Energy required to remove two neutrons from nuclei
(2-neutron binding energies = 2-neutron “separation” energies)
N = 82
25
23
21
N = 126
S(2n) MeV
19
17
15
13
Sm
11
Hf
9
Ba
N = 84
7
Pb
Sn
5
52
56
60
64
68
72
76
80
84
88
92
96
100
Neutron Number
104
108
112
116
120
124
128
132
2+
0+
B(E2:
0+
1
2+
1)

2+
1
E20+
2
1
2+
0+
The empirical magic numbers
near stability
• 2, 8, 20, 28, (40), 50, (64), 82, 126
• These are the famous magic numbers that
have been benchmarks of structure for 60
years. Recently, with studies of exotic nuclei
we are now beginning to realize that they are
not as robust as we have thought. Studies in
this area are one of the major thrusts of
nuclear structure research today.
Shell Structure
Mottelson (Nobel Prize for the Unified Model, 1975)
– 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).
“Magic plus 2”: Characteristic spectra

1

1
E (4 )
R4 / 2 
~ 1.3 -ish
E (2 )
What happens with both valence neutrons
and protons? Case of few valence nucleons:
Lowering of energies, development of
multiplets. R4/2  ~2-2.4
Spherical
vibrational
nuclei
Vibrator (H.O.)
E(I) = n ( 0 )
R4/2= 2.0
n = 0,1,2,3,4,5 !!
n = phonon No.
Lots of valence nucleons of both types:
emergence of deformation and therefore rotation
(nuclei live in the lab frame)
R4/2  ~3.33
Deformed nuclei – rotational spectra
Rotor
E(I)  ( ħ2/2I )I(I+1)
R4/2= 3.33
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
Ba
Ce
Nd
Sm
Gd
Dy
Er
Yb
Def.
3.2
3.0
R4/2
2.8
2.6
2.4
2.2
2.0
Sph.
1.8
1.6
84
86
88
90
92
94
N
Onset of deformation 
96
Think about the striking regularities in these data.
Take a nucleus with A ~100-200. The summed volume of all
the nucleons is ~ 60 % the volume of the nucleus, and they
orbit the nucleus ~ 1021 times per second!
Instead of utter chaos, the result is very regular behavior,
reflecting ordered, coherent, motions of these nucleons.
This should astonish you.
How can this happen??!!!!
Much of understanding nuclei is understanding the relation
between nucleonic motions and collective behavior
B(E2; 2+  0+ )
Ab initio calculations: One on-going success story
But we won’t go that way – too complicated for
any but the lightest nuclei.
We will make some simple models –
microscopic and macroscopic
Let’s start with the former, the Independent
particle model and its daughter, the shell model
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
-
=
Nuclei are 3-dimensional
• What is new in 3 dimensions?
– Angular momentum
– 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.
Raising one, lowering the other can give
similar energies – “level clustering”:
H.O:
E = ħ (2n+l)
E (n,l) = E (n-1, l+2)
e.g., E (2s) = E (1d)
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
nlj: Pauli Prin. 2j + 1 nucleons
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
The nuclear potential: a
rounded square well (WoodSaxon shape) works quite
well in reproducing the magic
numbers provided we add in
a spin-orbit force* that lowers
the energies of the
j=l+½
orbits and raises those with
j=l–½
* Maria Goeppert Mayer, Haxel,
Jensen, and Suess, 1948,
Nobel Prize 1963
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
Independent Particle Model
• Put nucleons (protons and neutrons separately) into orbits.
• Key question – how do we figure out the total angular momentum of a
nucleus with more than one particle? Need to do vector combinations of
angular momenta subject to the Pauli Principal. More on that later.
However, there is one trivial yet critical case.
• Put 2j + 1 identical nucleons (fermions) in an orbit with angular momentum
j. Each one MUST go into a different magnetic substate. Remember,
angular momenta add vectorially but projections (m values) add
algebraically.
• So, total M is sum of m’s
M = j + (j – 1) + (j – 2) + …+ 1/2 + (-1/2) + … + [ - (j – 2)] + [ - (j – 1)] + (-j) = 0
M = 0.
So, if the only possible M is 0, then J= 0
Thus, a full shell of nucleons always has total angular momentum 0.
This simplifies things enormously !!!
a)
Hence J = 0
Let’s do 91 40Zr51
Ignore protons (magic), consider 51 neutrons
Independent Particle Model
• Some great successes (for nuclei that are “doubly magic
plus 1”).
• Clearly inapplicable for nuclei with more than one particle
outside a doubly magic “core”. In fact, in such nuclei, it is
not even defined. Thus, as is, it is applicable to only a
couple % of nuclei.
• Residual interactions and angular momentum coupling to
the rescue.
Independent Particle Model – Uh –oh !!!
Trouble shows up

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