The Nuclear Shell Model

1
The Nuclear Shell Model Past and Present

• Igal Talmi
• The Weizmann Institute of Science
• Rehovot Israel

2
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.

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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.

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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.

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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.

6
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)3Jp3/2 proton
  configuration and
• a 1f7/2 neutron.
• In each nucleus there are states with J2, 3, 4,
  5
• Using levels of 38Cl, the 40K levels may be
  calculated and vice versa.

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Comparison with 1954 dataonly the spin 2- agreed
with our prediction
8
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

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Comparison of our predictions with experiment in
1955
10
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 Z16 is a
  proton magic number (as long as the neutron
  number is N21).

11

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

• 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.

12
General features of the effective interaction
extracted even from simple cases

• The T1 interaction is strong and attractive
• in J0 states.
• The T1 interactions in other states are weak and
  their average is repulsive.
• It leads to a seniority type spectrum.
• The average T0 interaction, between protons and
  neutrons, is strong and attractive.
• It breaks seniority in a major way.

13
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.
• .

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Seniority in jn Configurations

• The seniority scheme is the set of states which
  diagonalizes the pairing interaction
• VJltj2JMVj2JMgt0 unless J0,
  M0
• 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 J0, 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 J0 and antisymmetrizing.
  The state, in the jn configuration thus obtained,
  has the same value
• of J.

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Seniority in Nuclei

• Seniority was introduced by Racah in 1943 for
  classifying states of electrons in atoms.
• There is strong attraction in T1, J0 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, v0 in even nuclei
  and v1 in odd ones.

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A useful lemma
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Properties of the seniority scheme ground states

• If a two-body interaction is diagonal in the
  seniority scheme, its eigenvalues in states with
  v0 and v1 are given by
• an(n-1)/2 ß(n-v)/2
• where a and ß are linear combinations of
  two-body energies VJltj2JVj2Jgt.
• This leads to a binding energy expression (mass
  formula) for semi-magic nuclei
• B.E.(jn)B.E.(n0)nCjan(n-1)/2 ß(n-v)/2
• It includes a linear, quadratic and pairing
  terms.

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Neutron separation energies from calcium isotopes
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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.

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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.

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Levels of even (1h11/2)n configurations
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Levels of odd (1h11/2)n configurations
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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 nv multiplied by
• (2j1-2n)/(2j1-2v)
• Simple results follow for E2 transtions.

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E2 matrix elements in (1h11/2)n configurations
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B(E2) values in even(1h9/2)n configurations
33

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

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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.

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Generalized seniority

• Pair creation operators in different orbits
• Sj, Sj, may be added to SS Sj which has,
  with its hermitean conjugate the same commutation
  relations as in a single j-orbit,
• with 2j1 replaced by 2OS(2j1). A more
  realistic pair creation operator is
• SSajSj
• If not all coefficients are equal the algebra is
  more difficult. Still,

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Generalized seniorityground states

• There are states with seniority-like features.
• Assume that (S)N0gt are exact eigenstates
• H(S)N0gtEN(S)N0gt HS0gtVS0gt
• In fact, using the lemma, it follows,
• H(S)N0gt(S)NH0gtN(S)N-1H,S0gt
• ½N(N-1)(S)N-2H,S,S0gt
• NV½N(N-1)W(S)N0gt

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Generalized seniority mass formula

• Here, V is the (lowest) eigenvalue of the two
  nucleon J0 states. It is obtained by
  diagonalization of the Hamiltonian matrix for
  N2, J0, including single nucleon energies and
  two-body interactions. The formula for binding
  energies of even-even nuclei is
• BE(N)BE(N0)NV½N(N-1)W
• Two-nucleon separation energies should lie on a
  straight line.

38
Generalized seniority in Sn isotopes

• Binding energies of tin isotopes were measured in
  all isotopes from N50 to N82. 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.

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Generalized seniority excited states

• When not all aj coefficients are equal, it is no
  longer true that all level spacings are constant.
  Still, there is one state for any given Jgt0 which
  may lie at a constant spacing above the J0
  ground state. Consider the operator
• DJ(1djj)-½S(jmjmjjJM)ajmajm
• which creates an eigenstate of H in a two
  particle system, HDJ0gtVJDJ0gt.
• If HSDJ0gtH,SDJ0gtSHDJ0gt
• H,S,DJ0gt(VVJ)SDJ0gt is an eigenstate,
  the
• double commutator condition must be satisfied,
• H,S,DJWSDJ
• from which follows
• H(S)N-1DJ0gt(N-1)VVJ½N(N-1)W(S)N-1DJ0
  gt
• ENV2-V(S)N-1DJ
  0gt
• This means that V2-V spacings are independent of N

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In semi-magic nuclei, with only valence protons
or neutrons experiment shows features of
generalized seniority

• Constant 0-2 spacings in Sn isotopes

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• Yrast states of odd Sn isotopes

42
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
• latters orbit may become lower in 11Be.

43
Experimental information on p3/2p1/2 and
p3/2s1/2 interactions in 12B7
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How should this information be used?

• Removing two j-protons coupled to J0 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(2J1)ltjjJVjjJgt/SUMJ(2J1)

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

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• The monopole part of the T0 interaction affects
  positions of single nucleon energies.
• The quadrupole-quadrupole part in the T0
  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.

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Levels of Ba (Z56) isotopesDrastic reduction of
0-2 spacings
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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.

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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.

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

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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.

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Which nuclei are magic?

• In magic nuclei, energies of first excited
• states are rather high as in 36S (Z16, N20)
• 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!)

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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.

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Various approaches to nuclear structure physics

• The simple
  Ab initio
• shell model Theoretical derivation
  calculations
• of the
• The original effective interaction
  Recent
• approach
  developments

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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.

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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.

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Proton separation energies from N28 isotones