Interpreting and predicting structure

Useful interpretative models;p-n interaction

Second Lecture

We will now discuss some simple collective models to interpret structure

Two classes: Macroscopic or phenomenological and microscopic (femtoscopic, really)

We will mostly discuss macroscopic models. Such models, generally, are NOT predictive. They need to be “fed” some

data in a given nucleus. However, once fed, they can predict (correlate )huge quantities of data.

They also give an intuitive grasp of the kinds of structure observed, whereas complex microscopic models, while more

(but not completely) predictive are often less transparent physically (note: this is my opinion, others might disagree)

Macroscopic Models

Examples: Bohr-Mottelson model, Interacting Boson Approximation (IBA) Model, Geometric Collective Model (GCM), and some simplified versions of these. We have

already encountered some examples of these models when we briefly discussed the vibrator and rotor models

Probably the most successful, phenomenalogically, of the collective models has been the IBA, for the wide range of

structures it can accommodate and for its extreme economy of parameters. This will be the subject of three full morning

lectures in mid-June. For now, some very simple approaches.

Development of collective behavior in nuclei

• Results primarily from correlations among valence nucleons.

• Instead of pure “shell model” configurations, the wave functions are mixed – linear combinations of many components.

• Leads to a lowering of the collective states and to enhanced transition rates as characteristic signatures.

• How does this happen? Consider mixing of states. The following will be a little mathematical and uses some quantum mechanics. If you don’t follow it, don’t worry.

A illustrative special case of fundamental importance

T

Lowering of one state. Note that

the components of its wave function are all equal and

in phase

Consequences of this: Lower energies for collective states, and enhanced transition rates. Lets look at the latter in a

simple model.

W

The more configurations that mix, the stronger the

B(E2) value and the lower the energy of the

collective state. Fundamental property of

collective states.

2+

0+

Transition rates (half lives of excited levels) also tell us a lot about structure

B(E2: 0+1 2+

1) 2+1 E20+

12

Collective enahancement

Two very useful techniques:

Anharmonic Vibrator model (AHV)(works even for nuclei very far from vibrator)

E-GOS Plots(Paddy Regan, inventor)

ANharmonic Vibrator – Very general tool

2+ level is 1-phonon state, 4+ is two phonon, 6+ is 3-phonon … BUT R4/2 not exactly 2.0 in vibrational nuclei. Typ. ~2.2-2.4

Write: E(4) = 2 E(2) + ewheree is a phonon - phonon interaction.

Can fit any even-even nucleus by fixing e from E(4) . Allows one to predict higher lying states.

How many interactions in the 3 – phonon 6+ state?

Answer: 3: E(6) = 3 E(2) + 3e

How many in the n – phonon J = 2n state? Answer: n(n-1)/2

AHVHence we can write, in general, for any “band”:

E(J) = n E(2) + [n(n – 1)/2] en = J/2

Very general approach, even far from vibrator structures. E.g., pure rotor, E(4) = 3.33E(2), hence e= 1.33 E(2)

e /E(2) varies from 0 for harmonic vibrator to 1.33 for rotor (and 0.5 for gamma-soft rotor) !!!

What to do with very sparse data???

Example: 190 W. How do we figure out its structure?

P. Regan et al, private communication

W-Isotopes

N = 116

E-GOS PlotsEGamma Over Spin

Plot gamma ray energies going up a “band” divided by the spin of the initial state

E-Gos plots are, for spin, what AHV plots are for N or Z

Evolution of structure with J, N, or Z

E-GOS plots in limiting cases

Vibrator: E(I) = hw(I/2)

Hence Eg(I) = hw

Rotor: E(I) = h2/2J [I (I + 1)]

Hence Eg(I) = h2/2J [4I - 2]

AHV (R4/2 = 3.0): E(4) = 2E(2)+e=2E(2)+E(2)

E(I)=(I/2)E(2)+[(I/2)(I/2-1)/2]E(2)=(I/2)2E(2) Hence Eg(I) = 2(I + 2)

R = 2 = constant

R4/2 = 3.0 (AHV)

Comparisons of AHV and E-GOS with dataJ = 2 4 6 8 10 12 14

98-Ru 652 1397 2222 3126 4001 4912 5819

Egamma 652 745 825 904 875 911 907

AHV 652 1397 2235 3166 4190 5307 6517 (e=93)e /E(2) = 0.14

E-GOS 326 189 137 113 87.5 76 64.8

----------------------------------------------------------------------------------------------------------------166-Dy 77 253 527 892 1341 1868 2467

Egamma 77 176 274 365 449 527 599

AHV 77 253 528 902 1375 1947 2618(e=99)e /E(2) = 1.29

E-GOS 38 44 46 46 45 44 42.8

102 Ru

R4/2 = 3.0 (AHV)

Vibr rotor

Estimating the properties of nuclei

We know that 134Te (52, 82) is spherical and non-collective.

We know that 170Dy (66, 104) is doubly mid-shell and very collective.

What about:

156Te (52, 104) 156Gd (64, 92) 184Pt (78, 106) ???

All have 24 valence nucleons. What are their relative structures ???

Thus far, we have mostly discussed WHAT nuclei are doing. Now a key

question is WHY does structure vary the way it does and ways in which

we can exploit such understanding.

There are many aspects of this but a dominant driver of structural evolution is the competition

between two interactions – pairing and the proton-neutron interaction

Valence Proton-Neutron Interaction

Development of configuration mixing, collectivity and deformation – competition

with pairing

Changes in single particle energies and magic numbers

Partial history: Goldhaber and de Shalit (1953); Talmi (1962); Federman and Pittel ( late 1970’s); Casten et al (1981); Heyde et al (1980’s); Nazarewicz, Dobacewski et al (1980’s); Otsuka et al( 2000’s); Cakirli et al (2000’s); and many others.

Sn – Magic: no valence p-n interactions

Both valence protons and

neutrons

The idea of “both” types of nucleons – the p-n interaction

If p-n interactions drive configuration mixing, collectivity and deformation, perhaps they can be exploited to

understand the evolution of structure.

Lets assume, just to play with an idea, that all p-n interactions have the same strength. This is not realistic,

as we shall see later, since the interaction strength depends on the orbits the particles occupy, but, maybe,

on average, it might be OK.

How many valence p-n interactions are there? Np x Nn

If all are equal then the integrated p-n strength should scale with NpNn

The NpNn Scheme

NpNn scheme

Valence Proton-Neutron Interactions Correlations, collectivity, deformation. Sensitive to magic numbers.

NpNn

Scheme

Highlight deviant nuclei

P = NpNn/ (Np+Nn)p-n interactions perpairing interaction

The idea that p-n interactions produce collectivity and a lowering of collective states is seen everywhere, as the Sn-Xe figure shows.

It has, in particular, one spectacular phenomenon –

Intruder states

Consider the following 2p-2h

excitation across a magic number

Clearly, this requires considerable energy. However, if the upper orbit, j’, has high

overlap with the neutrons, then, as the number of valence neutrons increases, this state

can decrease sharply in energy. Consider what

happens when the shell gap is a) large and b) small

Consider the large Z = 50 gap. Towards mid-neutron shell this excited state drops in energy

and becomes a low lying excitation. Since it involves orbits from another major shell which come into the spectra of levels from the main

shell, it is called an “intruder state”

Consider the small, subshell, gaps at Z = 40 and 64. Here the “intruder state” can drop enough that is drops BELOW

the “ground state” and becomes the new, deformed, ground state. That, in fact, is the mechanism and reason

for the sudden onset of deformation that we have seen in the Sm region.

56 58 60 62 64 66 68 701.61.82.02.22.42.62.83.03.23.4

N=84 N=86 N=88 N=90 N=92 N=94 N=96R

4/2

Z

Disappearance of Z = 64 proton shell gap as a function of neutron number

The NpNn scheme: Interpolation vs Extrapolation

Predicting new nuclei with the NpNn Scheme

All the nuclei marked with x’s can be predicted by INTERpolation