shell model and collective models in nuclei experimental and theoretical perspectives r. f. casten...
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Shell Model and Collective Models in Nuclei
Experimental and theoretical perspectives
R. F. Casten
WNSL, Yale Univ.
RIKEN, January, 2010
We can customize our system – fabricate “designer” nuclei to isolate and amplify specific physics or interactions
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
The scope of Nuclear Structure Physics
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
1 12 4 2( ; )B E
1000 4+
2+
0
400
0+
E (keV) Jπ
Sim
ple
Ob
serv
able
s -
Eve
n-E
ven
(ci
ft-c
ift)
Nu
clei
1 12 2 0( ; )B E
212 2
2 1( ; )i f i f
i
B E J J EJ
. .
)2(
)4(
1
12/4
E
ER
Masses
Survey: Empirical evolution of structure
• Magic numbers, shell gaps, and shell structure
• 2-particle spectra
• Emergence of collective features –Vibrations, deformation, and rotation
Sn
Ba
SmHf
Pb
5
7
9
11
13
15
17
19
21
23
25
52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132
Neutron Number
S(2
n)
Me
VEnergy required to remove two neutrons from nuclei
(2-neutron binding energies = 2-neutron “separation” energies)
N = 82
N = 84
N = 126
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.
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).
Shell Structure
What happens with both valence neutrons and protons? Case of few valence nucleons:
Lowering of energies, development of multiplets. R4/2 ~2-2.4
Vibrator (H.O.)
E(I) = n ( 0 )
R4/2= 2.0
Spherical vibrational
nuclei
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
Sudden changes in R4/2 signify changes in structure, usually from spherical to deformed structure
Onset of deformation
Sph.
Def.
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
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
Vij
r
Uir = |ri - rj|
Nucleon-nucleon force – very
complex
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.
First:
3
2
1
Energy ~ 1 / wave length
n = 1,2,3 is principal quantum number
E up with n because wave length is shorter
Particles in a “box” or “potential”
well
Confinement is origin of
quantized energies levels
Second key ingredient: Quantum mechanics
22 2
2 2( ) ( 1)( ) ( ) 0
2 2nl
nl nld R rh h l lE U r R r
m mdr r
Radial Schroedinger
wave function
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)
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
Third key ingredient
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 !!!
The nuclear potential: a rounded square well (Wood-
Saxon 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 !!!
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.