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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R. F. Casten
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
B(E2: 0+1 2+1) 2+1 E20+12
Be astonished by this: Nuclei with 100’s of nucleons orbiting 1021 times/s, not colliding, and acting in concert !!!
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 of few valence nucleons:
E(I) = n (0 )
n = 0,1,2,3,4,5 !!
n = phonon No.
(Z = 52) of few valence nucleons:
Neutron number6870 72 74 76 78 80 82
Val. Neutr. number1412 10 8 6 4 2 0
Deformed nuclei – rotational spectra of few valence nucleons:
E(I) (ħ2/2I )I(I+1)
BTW, note value of paradigm in spotting physics (otherwise invisible) from deviations
Broad perspective on structural evolution: of few valence nucleons:R4/2
Note the characteristic, repeated patterns
Sudden changes in R of few valence nucleons:4/2 signify changes in structure, usually from spherical to deformed structure
Onset of deformation
Onset of deformation
as a phase transition
Another, simpler observable of few valence nucleons: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
B(E2; 2 of few valence nucleons:+ 0+ )
and its extensions to weakly bound nuclei
One on-going success story of few valence nucleons:
U of few valence nucleons:i
r = |ri - rj|
rIndependent 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: of few valence nucleons: 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
- of few valence nucleons:
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
nlj: 3-dimensions?Pauli Prin. 2j + 1 nucleons
Too low by 14 3-dimensions?
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. nuclear Binding Energies.
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 !!!
~ nuclear Binding Energies.
Compared to SHO, will mostly affect orbits
at large radii – higher angular momentum states
So, modify Harm. Osc. By squaring off the outer edge. nuclear Binding Energies.
Then, add in a spin-orbit force that lowers the energies of the
j = l + ½
orbits and raises those with
j = l – ½
Third key ingredient nuclear Binding Energies.Pauli Principle
This, plus the clustering of levels in simple potentials, gives nuclear SHELL STRUCTURE
Clusters of levels nuclear Binding Energies.+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
a) nuclear Binding Energies.
Hence J = 0
Shell model too crude. Need to add in extra interactions among valence nucleons outside closed shells. These dominate the evolution of Structure
Independent Particle Model – Uh –oh !!! among valence nucleons outside closed shells.
Trouble shows up
Shell Structure among valence nucleons outside closed shells.
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 among valence nucleons outside closed shells. H = H0 + Hresid.where H0 is that of the Ind. Part. ModelThe eigenstates of H will therefore be mixtures of those of H0
Wave fcts: among valence nucleons outside closed shells.