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**Shell Model and Collective Models in Nuclei Experimental and theoretical perspectives**

R. F. Casten WNSL, Yale Univ. RIKEN, January, 2010

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**Lecture 1 Introduction to Nuclear Structure and the Independent Particle Model (IPM)**

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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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**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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**Simple Observables - Even-Even (cift-cift) Nuclei**

. . Simple Observables - Even-Even (cift-cift) Nuclei 1000 4+ 400 2+ Masses 0+ Jπ E (keV)

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**Survey: 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) Sn Ba Sm Hf Pb 5 7 9 11 13 15 17 19 21 23 25 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 Neutron Number S(2n) MeV N = 82 N = 126 N = 84

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2+ 0+ 8

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2+ 0+ B(E2: 0+1 2+1) 2+1 E20+12 9

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**The empirical magic numbers near stability**

2, 8, 20, 28, (40), 50, (64), 82, 126 These are the famous magic numbers that have been benchmarks of structure for 60 years. Recently, with studies of exotic nuclei we are now beginning to realize that they are not as robust as we have thought. Studies in this area are one of the major thrusts of nuclear structure research today.

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Shell Structure Mottelson (Nobel Prize for the Unified Model, 1975) – ANL, Sept 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).

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**“Magic plus 2”: Characteristic spectra**

~ 1.3 -ish 12

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**Lowering of energies, development of multiplets. R4/2 ~2-2.4**

What happens with both valence neutrons and protons? Case of few valence nucleons: Lowering of energies, development of multiplets. R4/2 ~2-2.4 13

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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 types: emergence of deformation and therefore rotation (nuclei live in the lab frame) R4/2 ~3.33

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**Deformed nuclei – rotational spectra**

Rotor E(I) ( ħ2/2I )I(I+1) R4/2= 3.33 0+ 2+ 4+ 6+ 8+

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**Broad perspective on structural evolution: R4/2**

Note the characteristic, repeated patterns

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**Sudden changes in R4/2 signify changes in structure, usually from spherical to deformed structure**

Onset of deformation

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**Think about the striking regularities in these data**

Think about the striking regularities in these data. Take a nucleus with A ~ 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 behavior, 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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B(E2; 2+ 0+ )

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**Ab initio calculations: One on-going success story**

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But we won’t go that way – too complicated for any but the lightest nuclei. We will make some simple models – microscopic and macroscopic Let’s 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: Vij r Ui r = |ri - rj| 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 3 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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- = 28

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**Nuclei are 3-dimensional**

What is new in 3 dimensions? Angular momentum Centrifugal effects

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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 = ħ (2n+l) E (n,l) = E (n-1, l+2) e.g., E (2s) = E (1d)

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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

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**nlj: Pauli Prin. 2j + 1 nucleons**

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**The plot gives B.E.s PER nucleon.**

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? 36

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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 !!! 37

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**~ Compared to SHO, will mostly affect orbits**

at large radii – higher angular momentum states 38

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**orbits and raises those with**

The nuclear potential: a rounded square well (Wood-Saxon shape) works quite well in reproducing the magic numbers provided we add in a spin-orbit force* that lowers the energies of the j = l + ½ orbits and raises those with j = l – ½ * Maria Goeppert Mayer, Haxel, Jensen, and Suess, 1948, Nobel Prize 1963

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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 m’s M = j + (j – 1) + (j – 2) + …+ 1/2 + (-1/2) + … + [ - (j – 2)] + [ - (j – 1)] + (-j) = 0 M = 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 44 44

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**Ignore protons (magic), consider 51 neutrons**

Let’s do 91 40Zr51 Ignore protons (magic), consider 51 neutrons 46 46

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**Independent Particle Model**

Some great successes (for nuclei that are “doubly magic plus 1”). Clearly inapplicable for nuclei with more than one particle outside a doubly magic “core”. In fact, in such nuclei, it is not even defined. Thus, as is, it is applicable to only a couple % of nuclei. Residual interactions and angular momentum coupling to the rescue.

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Homework Note that only those circled can be done before considering residual interactions 57 57

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**Independent Particle Model – Uh –oh !!!**

Trouble shows up

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