Single Particle and Collective Modes in Nuclei R. F. Casten WNSL, Yale June, 2009 Wright Nuclear Structure Laboratory Surrey Mini-School Lecture Series

TINSTAASQ While I don’t mind hearing myself talk, these lectures are actually for YOU So, please ask questions if stuff isn’t clear.

A confluence of advances leading to a great opportunity for nuclear science Why we live in such cool times in nuclear physics (and are so lucky if we are at the beginnings of our careers) Breaching the technological wall

We can customize our system – fabricate “designer” nuclei to isolate and amplify specific physics or interactions The Four Frontiers 1.Proton Rich Nuclei 2.Neutron Rich Nuclei 3.Heaviest Nuclei 4.Evolution of structure within these boundaries Terra incognita — huge gene pool of new nuclei The scope of Nuclear Structure Physics

Remember, the nuclei are always right. It is us that have troubles and uncertainties about them. Moral: Never force an interpretation on a nucleus. The nucleus is talking to you trying to give you hints. Listen to it. Never do an experiment to prove that XXXX. Do an experiments to find out YYYY. Having said that, nuclei have spoken and given us some basic ideas about how they behave. Much the rest of these lectures will discuss a series of models that describe a lot of the data. However, they are exactly that, a series of models, not a single coherent unified framework Discovering that framework and developing a comprehensive understanding of nuclei will be your job.

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

Outline Introduction, survey of data – what nuclei do Independent particle model and Residual interactions – Particles in orbits in the nucleus – Residual interactions: results and simple physical interpretation – Multipole decomposition – Seniority – the best thing since buffalo mozzarella Collective models -- Geometrical – Vibrational models – Deformed rotors – Axially asymmetric rotors – Quantum phase transitions Linking the microscopic and macroscopic – p-n interactions The Interacting Boson Approximation (IBA) model

E (keV) JπJπ Simple Observables - Even-Even (cift-cift) Nuclei.. Masses

Empirical evolution of structure Magic numbers, shell gaps, and shell structure 2-particle spectra Emergence of collective features – Vibrations, deformation, and rotation

Energy required to remove two neutrons from nuclei (2-neutron binding energies = 2-neutron “separation” energies) N = 82 N = 84 N = 126

B(E2:  )    E2  

The empirical magic numbers near stability 2, 8, 20, 28, (40), 50, (64), 82, 126 These numbers, and a couple of R 4/2 values, are the only things I will ask you to memorize.

“Magic plus 2”: Characteristic spectra ~ 1.3 -ish

What happens with both valence neutrons and protons? Case of few valence nucleons: Lowering of energies, development of multiplets. R 4/2  ~2-2.4

Vibrator (H.O.) E(I) = n (   0 ) R 4/2 = 2.0 Spherical vibrational nuclei n = 0,1,2,3,4,5 !! n = phonon No.

Lots of valence nucleons of both types: emergence of deformation and therefore rotation (nuclei live in the world, not in their own solipsistic enclaves) R 4/2  ~3.33

Rotor E(I)  ( ħ 2 /2 I )I(I+1) R 4/2 = 3.33 Deformed nuclei – rotational spectra BTW, note value of paradigm in spotting physics (otherwise invisible) from deviations

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 ~ times per second! Instead of utter chaos, the result is very regular behavio u r, 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

Sudden changes in R 4/2 signify changes in structure, usually from spherical to deformed structure Onset of deformation Sph. Def. Observable Nucleon number, Z or N R 4/2 E2E2 1/E 2

Broad perspective on structural evolution: R 4/2 Note the characteristic, repeated patterns

B(E2; 2 +  0 + )

Ab initio calculations: One on-going success story

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

Independent particle model: magic numbers, shell structure, valence nucleons. Three key ingredients  V ij r UiUi r = |r i - r j | 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. First:

3 2 1 Energy ~ 1 / wave length n = 1,2,3 is principal quantum number E up with n because wave length is shorter Particles in a “box” or “potential” well Confinement is origin of quantized energies levels Second key ingredient: Quantum mechanics

= -

Nuclei are 3-dimensional What is new in 3 dimensions? – Angular momentum – Centrifugal effects

OK, I lied, I want you to memorize this notation also if you don’t know it already

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)

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 Third key ingredient

nlj: Pauli Prin. 2j + 1 nucleons

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

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

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 = 0. S o, 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 !!!

a) Hence J = 0

Let’s do Zr 51

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