Single Particle and Collective Modes in Nuclei

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Single Particle and Collective Modes in Nuclei
Surrey Mini-School Lecture Series

• R. F. Casten
• WNSL, Yale
• June, 2009
• Wright Nuclear Structure Laboratory

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TINSTAASQ
While I dont mind hearing myself talk, these
lectures are actually for YOU So, please ask
questions if stuff isnt clear.
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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)
Production and extraction of exotic nuclei
A new era in the science of nuclei, and
their applications
Breaching the technological wall
New detectors, separators, traps, and
experimental techniques
Advanced computing for data acquisition,
analysis, and theory
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The scope of Nuclear Structure Physics

• The Four Frontiers
• Proton Rich Nuclei
• Neutron Rich Nuclei
• Heaviest Nuclei
• 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
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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.
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• 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

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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

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.
.
Simple Observables - Even-Even (cift-cift) Nuclei
1000
4
2
400
Masses
0
0
Jp
E (keV)
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Empirical evolution of structure

• Magic numbers, shell gaps, and shell structure
• 2-particle spectra
• Emergence of collective features Vibrations,
  deformation, and rotation

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Energy required to remove two neutrons from
nuclei (2-neutron binding energies 2-neutron
separation energies)
N 82
N 126
N 84
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2
0
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B(E2 01 ? 21) ? ? 21 ??E2???01?2
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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.

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Magic plus 2 Characteristic spectra
1.3 -ish
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What happens with both valence neutrons and
protons? Case of few valence nucleons Lowering
of energies, development of multiplets. R4/2 ?
2-2.4
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Spherical vibrational nuclei
Vibrator (H.O.) E(I) n (? ?0 ) R4/2
2.0
n 0,1,2,3,4,5 !! n phonon No.
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Lots of valence nucleons of both typesemergence
of deformation and therefore rotation (nuclei
live in the world, not in their own solipsistic
enclaves)
R4/2 ? 3.33
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Deformed nuclei rotational spectra
Rotor E(I) ? ( h2/2I )I(I1) R4/2 3.33
BTW, note value of paradigm in spotting physics
(otherwise invisible) from deviations
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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
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Sudden changes in R4/2 signify changes in
structure, usually from spherical to deformed
structure
Def.
1/E2
Sph.
Onset of deformation
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Broad perspective on structural evolution R4/2
Note the characteristic, repeated patterns
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B(E2 2 ? 0 )
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Ab initio calculations One on-going success
story
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But we wont go that way too complicated for
any but the lightest nuclei.We will make some
simple models microscopic and
macroscopicLets start with the former, the
Independent particle model and its daughter, the
shell model
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Independent particle model magic numbers, shell
structure, valence nucleons. Three key
ingredients
First
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.
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Second key ingredient Quantum mechanics
Particles in a box or potential well
Confinement is origin of quantized energies levels
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1
2
Energy 1 / wave length n 1,2,3 is
principal quantum number E up with n
because wave length is shorter
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-

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Nuclei are 3-dimensional

• What is new in 3 dimensions?
• Angular momentum
• Centrifugal effects

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OK, I lied, I want you to memorize this notation
also if you dont know it already
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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 h? (2nl) E
(n,l) E (n-1, l2) e.g., E (2s) E
(1d)
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Pauli Principle
Third key ingredient

• 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
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nlj Pauli Prin. 2j 1 nucleons
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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?
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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 !!!
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Compared to SHO, will mostly affect orbits at
large radii higher angular momentum states
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So, modify Harm. Osc. by squaring off the outer
edge. Then, add in a spin-orbit force that
lowers the energies of the j l ½ orbits
and raises those with j l ½
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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
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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 ms
• 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 !!!

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a)
Hence J 0
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Lets do 91 40Zr51
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Homework