Single Particle and Collective Modes in Nuclei

Lecture SeriesR. F. CastenWNSL, YaleSept., 2008

TINSTAASQ

So, an example of a really really stupid question that leads to a useful discussion:

Are nuclei blue?

nucleus

You disagree?

Sizes and forces

Uncertainty Principle: E t > h m x/c > h

Nuclear force mediated by pion exchange: m ~ 140 MeV

Range of nuclear force / nuclear sizes ~ fermis

---------------------------------------------------------------------------------

Uncertainty Principle: x p > h

Characteristic nuclear energies are 105 times atomic energies: 10 ev 1 MeV

Probes and “probees”

E = h /

Energy of probe correlated with sizes of probee and production devicesAtoms – lasers – table top

Nuclei – tandems, cyclotrons, etc – room sizeQuarks, gluons – LHC – city size

Overview of nuclear structure

also

Some preliminaries

Independent particle modeland clustering in simple potentials

Concept of collectivity

(Note: many slides are VG images – and contain typos I can’t easily correct)

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

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

Evolution of structure – First, the data

• Magic numbers, shell gaps, and shell structure

• 2-particle spectra

• Emergence of collective features, deformation and rotation

The magic numbers:” special benchmark numbers of nucleons

B(E2: 0+1 2+

1) 2+1 E20+

122+

0+

Be astonished by this: Nuclei with 100’s of nucleons orbiting 1021 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

)2(

)4(

1

12/4

E

ER < 2.0

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

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

Vibrator (H.O.)

E(I) = n ( 0 )

R4/2= 2.0

Spherical vibrational

nuclei

n = 0,1,2,3,4,5 !!n = phonon No.

Neutron number 68 70 72 74 76 78 80 82

Val. Neutr. number 14 12 10 8 6 4 2 0

(Z = 52)

Lots of valence nucleons of both types

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

Broad perspective on structural evolution: R4/2

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

Sph.

Def.

Sph.

Def.

1/E2 – Note

similarity to R4/2

Observable

Nucleon number, Z or N

R4/2

E2

E2, or 1/E2,

is among the first pieces of data

obtainable in nuclei far from stability. Can we use just this quantity

alone?

Another, simpler observable

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 Modeland 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

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

=

-

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

But nuclei are 3- dimensional. What’s new in 3-dimensions?Angular momentum, hence centrifugal effects.

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

nlj: Pauli Prin. 2j + 1 nucleons

Too low by 10

Too low by 12

Too low by 14

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

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

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.

• 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

Shell model too crude. Need to add in extra interactions among valence nucleons outside closed

shells.

These dominate the evolution of Structure

Independent Particle Model – Uh –oh !!!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).

Shell Structure

Backups

So, we will have a Hamiltonian

H = H0 + Hresid.

where H0 is that of the Ind. Part. Model

The eigenstates of H will therefore be mixtures of those of H0

Wave fcts: