Single Particle and Collective Modes in Nuclei

R. F. CastenWNSL, Yale

June, 2009

Wright Nuclear Structure Laboratory

Surrey Mini-School Lecture Series

TINSTAASQWhile I don’t mind hearing myself talk, these lectures are actually for

YOUSo, please ask questions if stuff isn’t clear.

A confluence of advances leading to a great opportunity for nuclear A confluence of advances leading to a great opportunity for nuclear sciencescience

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

1 12 4 2( ; )B E

1000 4+

2+

0

400

0+

E (keV) Jπ

Sim

ple

Ob

serv

able

s -

Eve

n-E

ven

(ci

ft-c

ift)

Nu

clei

1 12 2 0( ; )B E

212 2

2 1( ; )i f i f

i

B E J J EJ

. .

)2(

)4(

1

12/4

E

ER

Masses

Empirical evolution of structure

• Magic numbers, shell gaps, and shell structure

• 2-particle spectra

• Emergence of collective features –Vibrations, deformation, and rotation

Sn

Ba

SmHf

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(2

n)

Me

VEnergy required to remove two neutrons from nuclei

(2-neutron binding energies = 2-neutron “separation” energies)

N = 82

N = 84

N = 126

2+

0+

B(E2: 0+1 2+

1) 2+1 E20+

122+

0+

The empirical magic numbers near stability

• 2, 8, 20, 28, (40), 50, (64), 82, 126

• These numbers, and a couple of R4/2 values, are the only things I will ask you to memorize.

“Magic plus 2”: Characteristic spectra

)2(

)4(

1

12/4

E

ER ~ 1.3 -ish

What happens with both valence neutrons and protons? Case of few valence nucleons:

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

Vibrator (H.O.)

E(I) = n ( 0 )

R4/2= 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)

R4/2 ~3.33

0+2+4+

6+

8+

Rotor

E(I) ( ħ2/2I )I(I+1)

R4/2= 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 ~100-200. 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 behaviour, 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 R4/2

signify changes in structure, usually from spherical to deformed

structure

Onset of deformation

Sph.

Def.

Observable

Nucleon number, Z or N

R4/2

E2

1/E2

Broad perspective on structural evolution: R4/2

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

Vij

r

Uir = |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.

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

22 2

2 2( ) ( 1)( ) ( ) 0

2 2nl

nl nld R rh h l lE U r R r

m mdr r

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. 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 !!!

a)

Hence J = 0

Let’s do 91 40Zr51

Homework