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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?. nucleus. Sizes and forces.

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single particle and collective modes in nuclei

Single Particle and Collective Modes in Nuclei

Lecture Series

R. F. Casten

WNSL, Yale

Sept., 2008

sizes and forces
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
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 structurealsoSome preliminariesIndependent particle modeland clustering in simple potentialsConcept of collectivity(Note: many slides are VG images – and contain typos I can’t easily correct)




Simple Observables - Even-Even Nuclei








E (keV)

evolution of structure first the data
Evolution of structure – First, the data
  • Magic numbers, shell gaps, and shell structure
  • 2-particle spectra
  • Emergence of collective features, deformation and rotation



B(E2: 0+1  2+1)   2+1 E20+12

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
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.

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 number6870 72 74 76 78 80 82

Val. Neutr. number1412 10 8 6 4 2 0


Deformed nuclei – rotational spectra







E(I)  (ħ2/2I )I(I+1)

R4/2= 3.33

BTW, note value of paradigm in spotting physics (otherwise invisible) from deviations


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





Onset of deformation

Onset of deformation

as a phase transition

1 e 2 note similarity to r 4 2

Another, simpler observable

1/E2 – Note similarity to R4/2

E2, or 1/E2,

is among the first pieces of data obtainable in nuclei far from stability. Can we use just this quantity alone?




Nucleon number, Z or N

basic models
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
independent particle model magic numbers shell structure valence nucleons three key ingredients



r = |ri - rj|


Independent particle model: magic numbers, shell structure, valence nucleons.Three key ingredients


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.


Second key ingredient: Quantum mechanics

Particles in a “box” or “potential” well

Confinement is origin of quantized energies levels




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

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


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 spin-orbit force that lowers the energies of the

j = l + ½

orbits and raises those with

j = l – ½

pauli principle

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



Hence J = 0

applying the independent particle model to real nuclei
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

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).


So, we will have a Hamiltonian H = H0 + Hresid.where H0 is that of the Ind. Part. ModelThe eigenstates of H will therefore be mixtures of those of H0