Transcript Single Particle and Collective Modes in Nuclei

Surrey Mini-School Lecture Series
Single Particle and Collective Modes in
Nuclei
R. F. Casten
WNSL, Yale
June, 2009
Wright Nuclear Structure Laboratory
TINSTAASQ
While I don’t mind hearing myself talk, these lectures are actually for
YOU
So, please ask questions if stuff isn’t clear.
A confluence of advances leading to a great opportunity for nuclear
science
Why we live in such cool
times in nuclear physics
(and are so lucky if we are at the
beginnings of our careers)
Breaching the technological wall
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
Remember, the nuclei are always right. It is us that
have troubles and uncertainties about them.
Moral: Never force an interpretation on a nucleus. The
nucleus is talking to you trying to give you hints. Listen
to it. Never do an experiment to prove that XXXX. Do
an experiments to find out YYYY.
Having said that, nuclei have spoken and given us some basic
ideas about how they behave. Much the rest of these lectures
will discuss a series of models that describe a lot of the data.
However, they are exactly that, a series of models, not a single
coherent unified framework Discovering that framework and
developing a comprehensive understanding of nuclei will be your
job.
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
Outline
•
Introduction, survey of data – what nuclei do
•
Independent particle model and Residual interactions
– Particles in orbits in the nucleus
– Residual interactions: results and simple physical interpretation
– Multipole decomposition
– Seniority – the best thing since buffalo mozzarella
•
Collective models -- Geometrical
– Vibrational models
– Deformed rotors
– Axially asymmetric rotors
– Quantum phase transitions
•
Linking the microscopic and macroscopic – p-n interactions
•
The Interacting Boson Approximation (IBA) model
Simple Observables - Even-Even (cift-cift)
. . Nuclei
4+
1000

1

1
E (4 )
R4 / 2 
E (2 )
B ( E 2; 41  21 )
2+
400
B ( E 2; 21  01 )
Masses
0+
0
Jπ
E (keV)
B ( E 2; J i  J f ) 
1
2J i  1
 i E2 
2
f
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 numbers, and a couple of R4/2
values, are the only things I will ask
you to memorize.
“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 world, not in their own solipsistic enclaves)
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
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 behaviour,
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
R4/2
Sudden changes in R4/2
signify changes in
structure, usually from
spherical to deformed
structure
Observable
E2
Nucleon number, Z or N
3.4
R4/2
2.8
2.6
2.4
2.2
0,012
Z=56
Z=58
Z=60
Z=62
Z=64
Z=66
Z=68
1/E2
0,010
+
3.0
0,014
1/E(21 )
Ba
Ce
Nd
Sm
Gd
Dy
Er
Yb
Def.
3.2
0,008
0,006
0,004
2.0
Sph.
1.8
0,002
1.6
84
86
88
90
92
94
N
Onset of deformation
96
0,000
84
86
88
90
92
Neutron Number
94
96
Broad perspective on structural evolution:
R4/2
Note the characteristic, repeated patterns
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
OK, I lied, I
want you to
memorize
this notation
also if you
don’t know it
already
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
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
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