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Old and new quasi-magic numbers

Nuclear structure physics

Eivind 1964

to Eivind at 70

Igal Talmi

The Weizmann Institute of Science

Rehovot, Israel

Welcome

- Let me first welcome the participants of this conference and thank the organizers for their initiative and for inviting us.
- It is clear that all of us are very interested in nuclear structure physics.
- It is also clear that all of us came here to pay tribute to Eivind Osnes who is 70 years young.
- Not only because of his accomplishments but largely for the person who he is.
- All of us wish him many more happy and active years.

Introduction

- I was asked to mention the early work

of Eivind.

- He started by doing simple shell model calculations about which I can report.
- Afterwards, he moved into the main subject of the conference, the nuclear many-body problem which will be discussed later in detail.

Plan

- I will start with pre-history, b.e. (before Eivind),in order to describe his early work, recycling some very old stuff…
- Then I will discuss some quasi-magic nuclei in the simple shell model. Much of it will be qualitative but will still display the simplicity of the shell model. The main message of my talk is to remind you of this simplicity.
- I hope that many-body calculations and in particular, ab initio ones, will eventually reach this kind of a simple description where it is applicable. We know that the nuclear many-body problem is very complicated. Still, nuclear data exhibit very simple and regular features.
- Many of us would like to see how this

SIMPLICITY ARISES FROM THE COMPLEXITY.

Magic numbers in the simple shell model

- Recently there is a “proliferation” of new magic or rather quasi-magic numbers. At the same time some magic numbers are “demoted” and seem to lose their magicity.
- In the simple shell model these are due to shell or sub-shell closures.

Two inaccurate statements are often made

- The first is that the realization “that magic numbers are actually not immutable occurred only during the past 10 years”.

This was realized almost 50 years ago.

- The other is that “the shell structure may change” ONLY “for nuclei away from stability”, “nuclear orbitals change their ordering in regions far away from stability”.

Such changes occur all over the nuclear landscape.

Attempts at proliferation occurred in the past as seen from Physical Review C

The quest for an effective interaction

- In the beginning, the ill-determined interaction between free nucleons was used (we were told to wait for better experiments…).
- The very singular short range behavior (hard core) dictated the necessity of a strongly renormalized effective interaction.
- This conference shows that the quest is still going on even if much progress has been actually achieved.

Calculating energies and wave functions in the shell model

- In the absence of reliable theoretical calculations, matrix elements of the effective interaction were determined from experimental data in a consistent way.
- Restriction to two-body interactions leads to matrix elements in n-nucleon states which are linear combinations of two-nucleon matrix elements.
- Nuclear energies may be calculated by using a smaller set of two-nucleon matrix elements determined consistently from experimental data.

The first successful attempt –low lying levels of 40K and 38Cl

- In the simplest shell model configurations of these nuclei, the 12 neutrons outside the 16O core, completely fill the 1d5/2, 2s1/2, 1d3/2 orbits while the proton 1d5/2 and 2s1/2 are also closed. The valence nucleons are

in 38Cl: one 1d3/2 proton and a 1f7/2 neutron

in 40K: (1d3/2)3Jp=3/2 proton configuration and

a 1f7/2neutron.

- In each there are states with J=2, 3, 4, 5
- Using levels of 38Cl, the 40K levels may be calculated and vice versa.

We were disappointed but not surprized. Why?

- The assumption that the states belong to rather pure jj-coupling configurations may have been far fetched. Also the restriction to two-body interactions could not be justified a priori.
- There was no evidence that values of matrix elements do not appreciably change when going from one nucleus to the next.
- Naturally, we did not publish our results

but then…

Conclusions from the relation between the spectra of 38Cl and 40K

- The restriction to two-body effective interaction

is in good agreement with some experimental data.

- Matrix elements (or differences) do not change appreciably when going from one nucleus to its neighbors (Nature has been kind to us).
- Some shell model configurations in nuclei are very simple. It may be stated that Z=16 is a proton magic number (as long as the neutron number is N=20).
- This was the first successful calculation in a series which culminated in more detailed ones with millions of shell model states included in the calculations (with only two-body forces) .

Could such simple relations arise from large scale calculations?

- Successful aproximations to the nuclear many body problem starting from a correct interaction between free nucleons will be very very important.
- They could be used in regions with little experimental information.
- The calculated wave functions would yield moments, transitions, reactions etc.
- Will simple relations, based on two-body effective interaction, emerge?

Which nuclei are magic?

In magic nuclei, energies of first excited

states are rather high as in 36S (Z=16, N=20)

where the first excited state is at 3.3 MeV,

considerably higher than in its neighbors.

Shell closure may be concluded if

valence nucleons occupy higher orbits

like 1d3/2 protons in the 38Cl, 40K case.

Shell closure may be demonstrated by

a large drop in separation energies (no

stronger binding of the closed shells!)

This is the case for 36S as seen inproton separation energies from N=20 isotones

General features of effective intreractions extracted from simple cases.

- The T=1 interaction is strong and attractive

in J=0 states.

The other T=1 interactions are weak and

repulsive on the average.

It leads to a seniority type spectrum.

- The T=0 interaction, between protons and

neutrons, is strong and attractive

on the average.

It breaks seniority in a major way.

These features are consistentwith the saturation properties of nuclear energies

- The potential well of the shell model is created

by the attractive proton-neutron interaction

which determines its depth and its shape.

- Hence, energies of proton orbits are determined

by the occupation numbers of neutrons and vice

versa. Thus, also shell closures depend on occupation numbers.

- These conclusions were published in 1960 addressing 11Be,

and in more detail in a review article in 1962.

The next example shows how such considerations may be applied.

Proton-neutron interactions- level order and spacings

- From a 1959 experiment it was “concluded that the assignment J= ½- for 11Be, as expected from the shell model is possible but cannot be established firmly on the basis of present evidence.”
- We did not think so.
- In 13C the first excited state, 3 MeV above the ½- g.s. has spin ½+ attributed to a 2s1/2 valence neutron. Which is the g.s.of 11Be obtained from 13C by removing

two 1p3/2 protons?

- Their interaction with a 1p1/2 neutron is expected to be stronger than their

interaction with a 2s1/2 neutron, hence, the

latter’s orbit may become lower in 11Be.

Experimental information on p3/2p1/2 and p3/2s1/2 interactions

How should this information be used?

- Removing two j-protons coupled to J=0 reduces the interaction with a j’-neutron by TWICE the averagejj’ interaction (the monopole part).
- The average interaction is determined by the position of the center-of-mass of the

jj’ levels,

V(jj’)=SUMJ(2J+1)<jj’J|V|jj’J>/SUMJ(2J+1)

Comments on the 11Be case

- Last figure is not an “extrapolation”. It is a graphic solution of an

exact shell model calculation in a rather limited space.

- The ground state is an “intruder” from a higher major shell. It can

be said that here the neutron number 8 is no longer a magic number.

- The calculated separation energy agreed fairly with a subsequent

measurement. It is rather small and yet, we used matrix elements

which were determined from stable nuclei.

- We failed to see that the sneutron wave function should be

appreciably extended and 11Be should be a “halo nucleus”.

It is amusing to see that similar arguments are presented as new

ones in 2001, 40 years later.

Nuclei in which N=20 is no longer a magic number

- A similar explanation applies to the disappearance of the

magic number N=20.

- In 36S and nuclei considered above, the 1f7/2 orbit is

sufficiently higher than s and d orbits and N=20 seems to

be magic.

- In nuclei with Z<16, the loss of proton-neutron interaction

affects more the s and d neutron orbits than the f orbit.

Thus, the neutron configuration is no longer the closed s,d shell

but has appreciable admixtures of 1f7/2 neutrons. In such nuclei,

N=20 is no longer a magic number.

Binding energies of Ca nuclei

- Valence neutrons in Ca nuclei with n=21 up to n=28

should occupy the 1f7/2 shell.

- Their spectra agree only roughly with this expectation,

indicating effects of interactions with higher configurations.

Ground states, far from perturbing states, have energies,

as well as neutron separation energies, which agree

very well with predictions of the seniority scheme for a

jn configuration

- BE(jn)-BE(jn-1)=(n-1)a+b{1+(-1)n}/2

States of odd (1f7/2)n configurations in Ca isotopes and N=28 isotones

Enter Osnes

- Deviations of Ca levels from (1f7/2)n predictions may be due to mixing of configurations with 2p3/2 neutrons whose orbit is rather low.
- This mixing was shown in 1966 by Osnes and Engeland to yield better

agreement with experimental data.

90Zr has been shown to have some magic properties

- It has a high 0+ first excited state lower than the first 2+ state.
- Interpreted by Ken Ford as excitation of the two 2p1/2 protons into the 1g9/2 orbit. States of nuclei between Zr and Sn were shown by several authors to belong to admixtures of (1g9/2)n and (1g9/2)n-2(2p1/2)2 configurations (good agreement for energies and e.m. transitions).
- In Zr isotopes beyond N=50, states seem to contain no proton excitations and to be due only to neutrons in the 2d5/2 orbit.

The quasi-magic nuclei 90Zr and 96Zr

- The neutron number N=50 is a magic number. The 40 protons occupy the closed shells of Z=28 and the practically closed 2p3/2, 1f5/2 orbits. In the ground state, the (2p1/2)2 state is mixed with the (1g9/2)2J=0 state but the state with the orthogonal wave function is rather high. In higher Zr isotopes the low lying levels are rather pure states of (2d5/2)n neutron configurations. Thus, 90Zr and 96Zr may be called quasi-magic nuclei.

Levels of heavier Zr isotopes

- Clearly exhibit an almost rotational spectrum indicating both protons and neutrons outside closed shells. Thus

the proton quasi-magic number Z=40 disappears for neutron numbers higher than N=98.

- Federman and Pittel in 1977attributed it to the strong attraction between 1g9/2 protons and 1g7/2 neutrons. With more 1g7/2 valence neutrons, protons are excited from the Z=40 shells into the 1g9/2 orbit.

On quasi-magic oxygen nuclei

- The lowest orbit In 17O is the 1d5/2 orbit. An expected situation like in Zr isotopes is roughly observed.
- In 18O and 20O some level spacings are not equal and the J=3/2 level in 19O is lower than calculated.

Still, the first excited state in 22O is at 3.27 MeV.

Is N=14 a quasi-magic number for Z=8?

- Single neutron and

two neutron separation

energies show a big

drop beyond 22O. It can

be said that

for Z=8, N=14 is a

quasi-magic number.

For N=82, proton number Z=64 is quasi-magic

- In 1978 Kleinheinz noticed that 146Gd has a 3- first excited state and the 2+ state lies higher than in neighboring nuclei. This fits with 6+8=14 valence protons occupying the closed 2d5/2 and 1g7/2orbits.
- Further experiments analyzed by theory (Blomqvist) show spectra of rather pure (1h11/2)n configurations. Thus,
- for N=82 proton number, Z=64 is quasi-magic

(unlike N=64 for Z=50). The drop in proton separation energies beyond Z=64 is rather small.

Conclusions

- There are very interesting phenomena when going towards nuclei near the stability limit. Some were observed and we expect more.
- Shell closures which appear and disappear due to changing nucleon occupation numbers have been known for many years and are not specific to new experiments.
- Such phenomena can be simply explained by the simple shell model. It is a challenge for nuclear many-body theories to derive the shell model

SIMPLICITY OUT OF THE COMPLEXITY

of their calculations.

Finally, “new” explanations

- After 40 years it is shown that “new magic numbers appear and some others disappear in moving from stable to exotic nuclei in a rather novel manner due to a particular part of the nucleon-nucleon interaction”.
- In the case of the s,d shell, “the dramatic change…is primarily due to the strongly attractive interaction between a proton in 0d5/2 and a neutron in 0d3/2”.
- Compare with a quotation from 1962 introducing the 11Be prediction: “The interaction energy” between a proton in the j-orbit and a neutron in the j’-orbit “is strong and attractive. Its strength depends on the specific j

and j’ orbits involved. As a result, when a certain proton shell is being filled, the relative positions of single neutron orbits may change considerably.”

(1f7/2)n configurations in Ca isotopes and N=28 isotones

- Level schemes in Ca isotopes and in N=28 isotones show deviations from

pure (7/2)n configurations.

- In the seniority scheme, spacings of corresponding levels are independent

of particle number n.

- Any two-body interaction is diagonal

in seniority for j=7/2.

The need of an effective interaction in the shell model

- In the Mayer-Jensen shell model,

wave functions of magic nuclei are well determined. So are states with one valence nucleon or hole in such nuclei.

- States with several valence nucleons are degenerate in the single nucleon Hamiltonian. Mutual interactions remove degeneracies and determine wave functions and energies of states.

On the magicity of N=28

- Protons between Z=21 and Z=28 (with N=28) occupy

the 1f7/2 orbit, not all states are pure but binding energies

follow the predictions of the seniority scheme. Also excited

states show features of a (7/2)n configurations.

Due to properties of the T=0 interactions, this would not be

so had the 28 neutrons not formed a closed shell.

It is amazing that even for Z=14, N=28 is a magic number

as demonstrated by the existence of a bound 42Si nucleus.

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