The Nuclear Shell Model –Past and Present

Igal Talmi

The Weizmann Institute of Science

Rehovot Israel

The main success of the nuclear shell model

• Nuclei with “magic numbers” of protons and neutrons exhibit extra stability.

• In the shell model these are interpreted as the numbers of protons and neutrons in closed shells, where nucleons are moving independently in closely lying orbits in a common central potential well.

• In 1949, Maria G.Mayer and independently Haxel, Jensen and Suess introduced a strong spin-orbit interaction. This gave rise to the observed magic numbers 28, 50, 82 and 126 as well as to 8 and 20 which were easier to accept.

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.

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

• In the early days, the rather mild potentials used for the interaction between free nucleons, were used in shell model calculations. The results were only qualitative at best.

Theoretical derivation of the effective interaction

A litle later, the bare interaction turned out to be too singular for use with shell model wave functions. It should be renormalized to obtain the effective interaction. More than 50 years ago, Brueckner introduced the G-matrix and was followed by many authors who refined the nuclear many-body theory for application to finite nuclei.

• Starting from the shell model the aim is to calculate from the bare interaction the effective interaction between valence nucleons. Also other operators like electromagnetic moments should be renormalized (e.g. to obtain the neutron effective charge!)

• Only recently this effort seems to yield some reliable results. This does not solve the major problem - how independent nucleon motion can be reconciled with the strong and short ranged bare interaction.

The simple 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 between 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 calculation –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 orbits 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 1f7/2 neutron.

• In each nucleus there are states with J=2, 3, 4, 5

• Using levels of 38Cl, the 40K levels may be calculated and vice versa.

Comparison with 1954 dataonly the spin 2- agreed with our prediction

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…

Comparison of our predictions with

experiment in 1955

Conclusions from the relation between the 38Cl and 40K levels

• 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=21).

• This was the first successful calculation and it was followed by more complicated ones carried out in the same way.

• The complete p-shell, p3/2 and p1/2 orbits (Cohen and Kurath), Zr isotopes (mostly the Argonne group), the complete d5/2,d3/2,s1/2 shell

(Wildenthal, Alex Brown et al) and others.• This series culminated in more detailed

calculations including millions of shell model states, with only two-body forces (Strasbourg and Tokyo). Not all matrix elements determined.

• Only simple examples will be shown.

Effective interactions, no longer restricted by the bare interactions, have been adopted

General features of the effective interaction extracted even from

simple cases• The T=1 interaction is strong and attractive

in J=0 states.• The T=1 interactions in other states are weak

and their average is repulsive.• It leads to a seniority type spectrum.

• The average T=0 interaction, between protons and neutrons, is strong and attractive.

• It breaks seniority in a major way.

Consequences of these features• 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.

• These conclusions were published in 1960 addressing 11Be, and in more detail in a review article in 1962. It will be considered later but first let us look at identical valence nucleons.

.

Seniority in jn Configurations

• The seniority scheme is the set of states which diagonalizes the pairing interaction

VJ=<j2JM|V|j2JM>=0 unless J=0, M=0• States of identical nucleons are characterized by v, the

seniority which in a certain sense, is the number of unpaired particles. In states in with seniority v in the

jv configuration, no two particles are coupled to states with J=0, i.e. zero pairing. From such a state, with given value of J, a seniority v state may be obtained by adding

(n-v)/2 pairs with J=0 and antisymmetrizing. The state, in the jn configuration thus obtained, has the same value

of J.

Seniority in NucleiSeniority was introduced by Racah in 1943 for classifying states of electrons in atoms.There is strong attraction in T=1, J=0 states of nucleons. Hence, seniority is much more useful in nuclei.The lower the seniority the higher the amount of pairing. Ground states of semi-magic nuclei are expected to have lowest seniorities, v=0 in even nuclei and v=1 in odd ones.

Seniority in jn configurations of identical nucleons

This is the theory of J=0 pairingS†=½Σ(−1)j-majm

†aj,−m†

Acting on the state with no nucleons creates a J=0 pair, S=(S†)† annihilates it.

HS†|0>=V0S†|0>Seniority v is defined by S|jvvJ>=0.

In j n configuration, seniority v states are (S†)(n−v)/2|jvvJ>

These states are eigenstates of the pairing interaction 2S†S

A single nucleon state is ajm†|0>. It has seniority v=1.

If H is a two-body interaction, Hajm

†|0>=0. This state with n=3 is S†ajm†|0>.

If H is diagonal in seniorityHS†ajm

†|0>=[H,S†]ajm†|0>+ S†Hajm

†|0>=[[H,S†],ajm

†]|0>+ajm†[H,S†|0>=

[[H,S†],ajm†]|0>+ajm

†HS†|0>=[[H,S†],ajm

†]|0>+ajm†V0S†|0>

If it is an eigenstate then[[H,S†],ajm

†]|0>=½WS†ajm†|0>

Actually, it is an operator equation

[[H,S†],ajm†]=½WS†ajm

†

From this follows[[H,S†],S†]=W(S†)2

States with seniority v=0. with maximum pairing are given by (S†)n|0>.

A useful lemmaA useful lemmaB(S†)N|1>=(S†)NB|1>+N(S†)N−1[B,S†]|1>+½N(N−1)(S†)N−2[[B,S†],S†]|1>+…

If B is a sum of one body and two-body operators, the series expansion stops. H(S†)(n−v)/2|jvvJ>=(S†)(n−v)/2H|jvvJ>+

½(n−v)(S†)(n−v)/ 2−1[H,S†]|jvvJ>+(1/8)(n−v)(n−v−2)(S†)(n−v−4)/2[[H,S†],S†]| jvvJ>

The energy of the jv state is denoted by

H|jvvJ>=E(jvvJ)|jvvJ>

For [H,S†]|jvvJ>, |jvvJ> is a linear combination of products of v creationoperators ajm

† with various m-values, denoted by C†, acting on |0>. Commuting [H,S†] through the ajm

† the result is

[H,S†]|jvvJ>= [H,S†]C†|0>=½WvC†S†|0>+

C† [H,S†]|0>={½Wv+V0}S†|jvvJ>

Eigenvalues of H diagonal in seniorityCollecting all terms, the result is

H(S†)(n−v)/2|jvvJ>=E(jvvJ)(S†)(n−v)/2|jvvJ>+

{n(n−1)/2−v(v−1)/2}(W/4)(S†)(n−v)/2|jvvJ>+

½(n−v)(V0−W/4)(S†)(n−v)/2|jvvJ>

Thus, the double commutator conditions make the states above, eigenstates of H.

Defining α=W/4 β= V0−W/4 the eigenvalues are equal to

E(jvvJ)+α{n(n−1)/2−v(v−1)/2}+ ½(n−v)β

Seniority binding energy formulaFor ground state energies of semi-magic nuclei,

v=0 states of even-even nuclei and v=1 states of odd-even ones, these expressions are simpler.

Using them, binding energies are expressed by

BE(jn)=BE(n=0)+nCj+αn(n−1)/2+½(n−v)β=

BE(n=0)+nCj+αn(n−1)/2+[n/2]β

A linear term, a quadratic term and a pairing term.

Properties of the seniority scheme ground states

• If a two-body interaction is diagonal in the seniority scheme, its eigenvalues in states with v=0 and v=1 are given by

αn(n-1)/2 + β(n-v)/2where α and β are linear combinations of two-body

energies VJ=<j2J|V|j2J>.This leads to a binding energy expression (mass

formula) for semi-magic nuclei

B.E.(jn)=B.E.(n=0)+nCj+αn(n-1)/2+ β(n-v)/2 It includes a linear, quadratic and pairing terms.

Neutron separation energies from calcium isotopes

A direct result of this behavior is that magic nuclei are not more

tightly bound than their preceding even-even neighbors.

Their magic properties (stability etc.) are due to the fact that nuclei beyond them

are less tightly bound.

Seniority: level spacingsindependent of n

From the expression of eigenvaluesE(jvvJ)+α{n(n−1)/2−v(v−1)/2}+ ½(n−v)β

follows an important feature of seniority.The derivation holds also for states with

seniority v’<v, it uses only the number v ofparticles. Hence, energies of |j nv’J> states with v’≤v are equal to energies of |jvv’J> states plus the same energy. Their differences are equal.

Properties of the seniority schemeexcited states

• If the two-body interaction is diagonal in the seniority scheme in jn configurations,

then level spacings are constant, independent of n.

Levels of even (1h11/2)n configurations

Levels of odd (1h11/2)n configurations

Properties of the seniority schememoments and transitions

• Odd tensor operators are diagonal and their matrix elements in jn configurations are independent of n.

• Matrix elements of even tensor operators are functions of n and seniorities. Between states with the same v, matrix elements are equal to those for n=v multiplied by

(2j+1-2n)/(2j+1-2v) Simple results follow for E2 transtions.

E2 matrix elements in (1h11/2)n configurations

B(E2) values in even(1h9/2)n configurations

• The rates of E2 transitions usually increase as more valence nucleons are added. Here, the opposite behavior is evident.

Generalized seniority

There are semi-magic nuclei where there is evidence that valence nucleons occupy several orbits. Still, they exhibit seniority features, in binding energies and constant 0 - 2 spacings. A generalization of seniority to mixed configurations is needed.

Generalized seniorityPair creation operators in different orbits

Sj’†, Sj”

†,… may be added to S†=Σ Sj† which

has, with its hermitean conjugate the same commutation relations as in a single j-orbit,

with 2j+1 replaced by 2Ω=Σ(2j+1). A more realistic pair creation operator is

S†=ΣαjSj†

If not all coefficients are equal the algebra is more difficult. Still,

Generalized seniority:ground statesThere are states with seniority-like features.

Assume that (S†)N|0> are exact eigenstates

H(S†)N|0>=EN(S†)N|0> HS†|0>=VS†|0>

In fact, using the lemma, it follows,

H(S†)N|0>=(S†)NH|0>+N(S†)N−1[H,S†]|0>+

½N(N−1)(S†)N−2[[H,S†],S†]|0>=

{NV+½N(N−1)W}(S†)N|0>

Generalized seniority mass formulaHere, V is the (lowest) eigenvalue of the two

nucleon J=0 states. It is obtained by diagonalization of the Hamiltonian matrix for N=2, J=0, including single nucleon energies and two-body interactions. The formula for binding energies of even-even nuclei is

BE(N)=BE(N=0)+NV+½N(N−1)W

Two-nucleon separation energies should lie on a straight line.

Generalized seniority in Sn isotopesBinding energies of tin isotopes were measured

in all isotopes from N=50 to N=82. Two neutron separation energies lie on a smooth curve and no breaks due to possible sub-shells are present. Still, the line has a slight curvature indicating a possible small cubic term in binding energies. Including a term proportional to N(N−1)(N−2)/6 led to very good agreement. Such a term could be due to weak 3-body effective interactions or due to polarization of the core by valence nucleons.

Generalized seniority: excited statesWhen not all αj coefficients are equal, it is no longer true that all

level spacings are constant. Still, there is one state for any given J>0 which may lie at a constant spacing above the J=0 ground state. Consider the operator

DJ†=(1+δjj’)−½Σ(jmj’m’|jj’JM)ajm

†aj’m’†

which creates an eigenstate of H in a two particle system, HDJ†|

0>=VJDJ†|0>.

If HS†DJ†|0>=[H,S†]DJ

†|0>+S†HDJ†|0>=

[[H,S†],DJ†]|0>+(V+VJ)S†DJ

†|0> is an eigenstate, the double commutator condition must be satisfied, [[H,S†],DJ

†]=WS†DJ†

from which follows H(S†)N−1DJ

†|0>={(N−1)V+VJ+½N(N−1)W(S†)N−1DJ†|0>+

{EN+V2−V}(S†)N−1DJ†|0>

This means that V2−V spacings are independent of N

In semi-magic nuclei, with only valence protons or neutrons

experiment shows features of generalized seniority

Constant 0-2 spacings in Sn isotopes

Yrast states of odd Sn isotopes

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. The g.s.of 11Be is 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 in 12B7

How should this information be used?

• Removing two j-protons coupled to J=0 reduces the interaction with a j’-neutron by TWICE the average jj’ 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)

Shell model prediction of the 11Be ground state and excited level

Comments on the 11Be shell model calculation

• 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 s neutron wave function should be appreciably extended and 11Be should be a “halo nucleus”.

Shell model wave functions do not include the short range correlations which are due to the strong bare interaction. Thus, they cannot be the real wave functions of the nucleus. Still, some states of actual nuclei demonstrate features of shell model states. The large radius of 11Be, implied by the shell model, is a real effect.

• The monopole part of the T=0 interaction affects positions of single nucleon energies.

• The quadrupole-quadrupole part in the T=0 interaction breaks seniority in a major way.

• In nuclei with valence protons and valence neutrons it leads to strong reduction of the 0-2 spacings – a clear signature of the transition to rotational spectra and nuclear deformation.

Levels of Ba (Z=56) isotopesDrastic reduction of 0-2 spacings

Ab initio calculations of nuclear many body problem

• Ab initio calculations of nuclear states and energies are of great importance. Their input is the bare interaction. If it is sufficiently correct and the approximations made are satisfactory, accurate energies of nuclear states, also of nuclei inaccessible experimentally, will be obtained.

• The knowledge of the real nuclear wave functions will enable the calculation of various moments and transitions, electromagnetic and weak ones, including double beta decay.

• There are still many difficulties to overcome.

Shell model wave functions are real to an appreciable extent

• We saw some evidence in the halo nucleus 11Be and in

• E2 transitions. This is also evident from pick-up and stripping reactions.

• Other evidence is offered by Coulomb energy differences.

Coulomb energy differences of 1d5/2 and 2s1/2 orbits

Will simplicity emergefrom complexity?

• Shell model wave functions, of individual nucleons, cannot be the real ones. They do not include short range and other correlations due to the strong bare interaction. Yet the simple shell model seems to have some reality and considerable predictive power.

• The bare interaction is the basis of ab initio calculations leading to admixtures of states with excitations to several major shells, the higher the better…

• Will the shell model emerge from these calculations? It has been so simple, useful and elegant and it would be illogical to give it up. Reliable many-body theory should explain why it works so well, which interactions lead to it and where it becomes useless.

Three-body interactions- where are they?

• There is no evidence of three-body interactions between valence nucleons. They could be weak, state independent or both. Still, three-body interactions with core nucleons

could contribute to effective two-body interactions between valence nucleons and to single nucleon energies.

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 ifvalence 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!)

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.

Various approaches to nuclear structure physics

The simple Ab initio

shell model Theoretical derivation calculations

of the

The original effective interaction Recent

approach developments

Ab initio calculationsnew developements

• A typical theory of this kind is the No Core Shell Model (NCSM). The input of all ab initio calculations is the bare interaction between free nucleons. After 60 years of intensive work, there are several prescriptions which reproduce rather well, results of scattering experiments but none has a solid theoretical foundation.

• No core is assumed and harmonic oscillator wave functions are used as a complete scheme.

• Some results of NCSM will be presented and discussed in the following, showing the present status and some problems.

Merits of ab initio Calculations

• Ab initio calculations of nuclear states and energies are of great importance. If the input, the bare interaction, is sufficiently correct and the approximations made are satisfactory, accurate energies of nuclear states, also of nuclei inaccessible experimentally, will be obtained.

• More important would be the knowledge of the real nuclear wave functions. This will enable the calculation of various moments and transitions, electromagnetic and weak ones, including double beta decay.

• There are still many difficulties to overcome.

Proton separation energies from N=28 isotones