Single Particle and Collective Modes in Nuclei Lecture Series R. F. Casten WNSL, Yale Sept., 2008
TINSTAASQ
You disagree? nucleus So, an example of a really really stupid question that leads to a useful discussion: Are nuclei blue?
Sizes and forces Uncertainty Principle: ΔE Δt > h Δm Δx/c > h Nuclear force mediated by pion exchange: m ~ 140 MeV Range of nuclear force / nuclear sizes ~ fermis --------------------------------------------------------------------------------- Uncertainty Principle: Δx Δ p > h Characteristic nuclear energies are 10 5 times atomic energies: 10 ev 1 MeV
Probes and probees E = h / λ 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 structure also Some preliminaries Independent particle model and clustering in simple potentials Concept of collectivity (Note: many slides are VG images and contain typos I can t easily correct)
Simple Observables - Even-Even Nuclei.. R E (4 + = 1 4 / 2 E (2 + 1 Masses 1000 ) ) 400 4 + B( E 24 ; + 2+ ) 2 + 1 1 + 1 + 1 B( E22 ; 0 ) 0 0 + E (kev) J π 1 B( E2; J J ) Ψ E2 Ψ 2J + 1 i f i f i 2
Evolution of structure First, the data Magic numbers, shell gaps, and shell structure 2-particle spectra Emergence of collective features, deformation and rotation
he magic numbers: special benchmark numbers of nucleons
B(E2: 0 + 1 2+ 1 ) 2+ 1 E2 0+ 1 2 2 + 0 + Be astonished by this: Nuclei with 100 s of nucleons orbiting 10 21 times/s, not colliding, and acting in concert!!!
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.
Magic plus 2 : Characteristic spectra E (4 = E (2 + 1 4 / 2 + 1 ) ) R < 2.0
What happens with both valence neutrons and protons? Case of few valence nucleons: Lowering of energies, development of multiplets. R 4/2 ~2
Spherical vibrational nuclei Vibrator (H.O.) E(I) = n (h ω 0 ) R 4/2 = 2.0 n = 0,1,2,3,4,5!! n = phonon No.
(Z = 52) Neutron number 68 70 72 74 76 78 80 82 Val. Neutr. number 14 12 10 8 6 4 2 0
Lots of valence nucleons of both types R 4/2 ~3.33
Deformed nuclei rotational spectra Rotor E(I) ( ħ 2 /2I )I(I+1) R 4/2 = 3.33 BTW, note value of paradigm in spotting physics (otherwise invisible) from deviations 8 + 6 + 4 + 2 + 0 +
Broad perspective on structural evolution: R 4/2 Note the characteristic, repeated patterns
Sudden changes in R 4/2 signify changes in structure, usually from spherical to deformed structure R 4/2 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 Def. Sph. 84 86 88 90 92 94 96 N Onset of deformation Ba Ce Nd Sm Gd Dy Er Yb R 4/2 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 Def. Sph. 56 58 60 62 64 66 68 70 Z Onset of deformation as a phase transition N=84 N=86 N=88 N=90 N=92 N=94 N=96
Another, simpler observable Observable 1/E 2 Note similarity to R 4/2 1/E(2 1 + ) 0,014 Z=56 Z=58 0,012 Z=60 Z=62 0,010 Z=64 Z=66 Z=68 0,008 0,006 0,004 0,002 Nucleon number, Z or N 0,000 84 86 88 90 92 94 96 Neutron Number R 4/2 E 2 1/E(2 1 + ) 0,014 N=84 N=86 0,012 N=88 N=90 0,010 N=92 N=94 N=96 0,008 0,006 0,004 0,002 E 2, or 1/E 2, is among the first pieces of data obtainable in nuclei far from stability. Can we use just this quantity alone? 0,000 56 58 60 62 64 66 68 Proton Number
B(E2; 2 + 0 + )
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
One on-going success story
Independent particle model: magic numbers, shell structure, valence nucleons. Three key ingredients First: V ij r = r i -r j U i r 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 3 1 Confinement is origin of quantized energies levels 2 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 2 2 d R 2 nl r + 2 nl m 2 dr r h () h l( l 1) + E U () r R () 0 2 2 nl r = m 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 = ħω (2n+l) E (n,l) = E (n-1, l+2) e.g., E (2s) = E (1d) Add spin-orbit force
nlj: Pauli Prin. 2j + 1 nucleons
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 spinorbit force that lowers the energies of the j = l + ½ orbits and raises those with j = l ½
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
a) Hence J = 0
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
Independent Particle Model Uh oh!!! Trouble shows up
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).
Backups
So, we will have a Hamiltonian H = H 0 + H resid. where H 0 is that of the Ind. Part. Model The eigenstates of H will therefore be mixtures of those of H 0
Wave fcts: