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.
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
E(I) = n (0 )
n = 0,1,2,3,4,5 !!
n = phonon No.
Neutron number6870 72 74 76 78 80 82
Val. Neutr. number1412 10 8 6 4 2 0
E(I) (ħ2/2I )I(I+1)
BTW, note value of paradigm in spotting physics (otherwise invisible) from deviations
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
and its extensions to weakly bound nuclei
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.
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 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?
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
Then, add in a spin-orbit force that lowers the energies of the
j = l + ½
orbits and raises those with
j = l – ½
This, plus the clustering of levels in simple potentials, gives nuclear SHELL STRUCTURE
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
Shell model too crude. Need to add in extra interactions among valence nucleons outside closed shells. These dominate the evolution of Structure
Trouble shows up
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