NEW PHENOMENON IN EXOTIC NEUTRON-RICH Sn ISOTOPES : ROLE OF 3-BODY FORCE

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NEW PHENOMENON IN EXOTIC NEUTRON-RICH Sn ISOTOPES
ROLEOF 3-BODY FORCE

• S. Sarkar, M. Saha Sarkar
• Bengal Engineering and Science University,
  Shibpur, Howrah - 711103, INDIA
• Saha Institute of Nuclear Physics, Kolkata
  700064, INDIA

E-mail ss_at_physics.becs.ac.in
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Experimental Chart of Nuclei

132Sn
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Introduction 132Sn region

• Nuclei with 50 ?Z ?56 and 82 ? N ? 88 in the
  ?(gdsh) ?(hfpi) valence space above the 132Sn
  core lie on or close to the path of astrophysical
  r-process flow.
• Their structure,particularly the binding energy
  (BE), low-lying excited states and beta decay
  rates at finite temperatures are important
  ingredients for nucleosynthesis calculations.
• Sn isotopes are of particular importance.
• Even Sn isotopes, say 136Sn, is known to be the
  classical "waiting point" nucleus in A130 solar
  system abundance peak under typical r-process
  condition.

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Experimental Status

• Spectroscopic information, such as BE and low
  lying spectrum, is known experimentally only for
  134Sn.
• Recently half-lives of 135-137Sn have been
  measured through ?-n decay process. No other
  information exists.Lifetimes of these nuclei are
  very small and production rates are also very low
  presenting challenges to spectroscopic studies.
• Reliable theoretical results are therefore
  necessary and useful.

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E(21) of Sn isotopes(A102-134)
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Theoretical Endeavours

• Shell Model Calculations
• Few valence particle nuclei above the doubly
  closed magic 132Sn core are generally described
  in the valence space consisting of
• Proton (1g7/2, 2d5/2, 2d3/2, 3s1/2 and 1h11/2)
  and
• Neutron (1h9/2, 2f7/2, 2f5/2, 3p3/2, 3p1/2 and
  1i13/2) orbitals.
• Remarkably good results for isotopes of Sn, Sb,
  Te, I, Xe, Cs with different interactions for
  134 ?A ?138 and 50? Z ? 56.

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Interactions used

• Primarily two types of interactions used
  realistic and empirical
• Empirical interactions the interaction derived
  from 208Pb region (Chou Warburton) which fails
  for N gt 84 specific matrix elements are tuned to
  reproduce known experimental levels (S. Sarkar
  and M. Saha Sarkar)
• Realistic interactions obtained starting with a G
  matrix derived from the CD-Bonn nucleon-nucleon
  interaction using the Q-box method (B.A. Brown,
  M. Hjorth-Jensen, T. T. S. Kuo, and E. Osnes,
  F. Andreozzi, L. Coraggio, A. Covello, A.
  Gargano)

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Structure of even-even A 138 isobars and the
yrast spectra of semi-magic Sn isotopes above the
132Sn core, S. Sarkar and M. Saha Sarkar,
PHYSICAL REVIEW C 78, 024308 (2008)
SMPN
CWG
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Casten-Sherill Systematics

• Casten and Sherill have pointed out that,
  although E(21 )Sn - E(21 )Te 400 keV for a
  given neutron number over most of the N 5082
  shell, the difference is only 119 keV for N 84

R. F. Casten and B. M. Sherill, Prog. Part. Nucl.
Phys. 45, S171 (2000).

• The difference for N 86 is 108 keV with SMPN.
  It is consistent with the trend discussed by
  Casten and Sherill. (Casten-Sherill Systematics)
• For CWG, this difference is 733 - 356 377 keV
  for N 86, which deviates from the trend.

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Present Work

• We have used
• extended the calculations for more neutron rich
  140Sn
• The shell model codes OXBASH and NUSHELL_at_MSU have
  been used
• The results are
• The E(21 ) for 140Sn is 1949 keV showing a
  sudden increase for N90, indicating a shell
  closure
• With CWG interaction, the 01 - 21 spacing
  remains nearly constant at around 750 keV for
  136-142Sn, except for a small increase at 140Sn
• To understand the implication of these completely
  different trends in the results using these two
  interactions, the theoretical results for these
  experimentally unobserved nuclei have been
  compared with the E(21) values of neutron rich
  nuclei in other mass regions for which
  experimental data are available.

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Comparison with neutron-rich isotopes
140Sn
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The shell closure at N90

• In order to put forward further evidence and to
  understand the shell closure at 140Sn more
  precisely, the effective single-particle energies
  (ESPE) for the neutron orbitals for the two
  Hamiltonians have been compared.
• The ESPE is defined as bare single particle
  energy (spe) added with the monopole part of the
  diagonal two body matrix elements (tbme).

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Neutron ESPEs with CWG and SMPN interactions for
increasing neutron numbers
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Origin of this new shell closure

• Spin-tensor decomposition of the two body matrix
  elements (tbmes). - central, antisymmetric
  spin-orbit (ALS), spin-orbit (LS) and tensor
  parts of tbmes identified
• For SMPN,
• the central and ALS part for 2f7/2- 2f7/2 tbmes
  account for majority of the downward shift of the
  ESPE of 2f7/2with increasing valence neutron
  number (n).
• tbmes involving 3p3/2 - dominant contribution
  from the central part.
• The central parts of 2f7/2 and 3p3/2 vary with
  similar slopes for increase in n.
• Variation in ALS part is primarily responsible
  for this observed shell gap at N90.

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Decomposition
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Decomposition of USD tbmes for oxygen isotopes
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The implication of ALS term?

• ALS component in the tbmes corresponds to those
• LS-coupled matrix elements which have S?S?, i.e.,
  terms non-diagonal in S (spin). Do not conserve
  total spin of the matrix elements.
• But the interactions which are parity conserving
  and isospin conserving must also conserve the
  total spin.
• Bare nucleon- nucleon force contains no ALS term.
• But effective interaction is not simply related
  to bare nucleon-nucleon force. Core polarisation
  corrections to the G-matrix give rise to non-zero
  but small ALS matrix elements.
• A characteristic feature common to many empirical
  effective interactions is strong ALS components
  in the tbmes.
• It usually arises from inadequate constraint by
  the data.
• It indicates the important contributions from
  higher order renormalisation or many body forces
  to the effective interactions.
• In empirical SMPN such many - body effects might
  have been included in some way through the
  modification of important tbmes.

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Features of Realistic Interaction

• Two-body realistic interactions derived from the
  free nucleon-nucleon force fail to reproduce some
  shell closures.
• increase of the 1d5/2 - 2s1/2 gap for Z8 and
  1f7/2 - 2p3/2 gap for Z20 (as a function of
  neutron number), required to explain empirical
  data are not obtained in the calculations with
  these interactions.
• It has been shown that the three-body forces have
  to be taken into account to reproduce these shell
  gaps.
• Otsuka et al. have proposed a three-body
  delta-hole mechanism to explain these shell gaps
  and they have shown that three-body forces are
  necessary to explain why the doubly-magic 24O
  nucleus is the heaviest oxygen isotope
• Zuker showed earlier that a very simple
  three-body monopole term can solve practically
  all the spectroscopic problems in the p, sd, and
  pf shells those were hitherto assumed to need
  drastic revisions of the realistic two-body
  potentials.

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Three body forces and CWG interaction

• A simple three-body monopole term in CWG as
  prescribed by Zuker
• Corrections in 2f7/2-2f7/2 and 2f7/2-3p3/2 tbmes
  similar to those in KB3 for 1f7/2-1f7/2 and
  1f7/2-2p3/2 tbmes.
• Included the effect of mass scaling. By
  (40/132)(1/3) factor.
• This factor reduces the effect of three - body
  correction on CWG compared to that in KB3.
• The correction terms included in the tbmes are
• V J,T1ffff (CWG3M) V J,T1ffff (CWG)-74 keV,
  for J0,4and 6
• V J2,T1 ffff (CWG3M) V J2,T1ffff (CWG) - 208
  keV and
• V J,T1 frfr (CWG3M) V J,T1 frfr (CWG)201 keV
  for J2, 3, 4 and 5.
• f stands for 2f7/2 and r stands for 3p3/2.
• The correction factor will be effective for
  nuclei for which the valence neutron number n 3.
  
• A shell gap for N90 now appears with CWG3M which
  is very close to that with SMPN.
• The E(21 ) energies of 136,138Sn are 0.639 and
  0.633 MeV, respectively.
• The E(21 ) energy of 140Sn predicted by CWG3M
  (1.889 MeV) is close to that predicted by SMPN
  (1.949MeV).

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Comparison with neutron-rich isotopes
21
Neutron ESPEs with CWG interactions for
increasing neutron numbers
22
Wavefunction structure for CWG
23
Wavefunction structure for SMPN
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Wavefunction structure for CWG3M

• For CWG3M, the wave function composition
• for the 0 g.s is (70.4) from the ?(2f7/2)8
  partition, similar to SMPN (75.8)
• But due to overestimation of the up-sloping trend
  of ?(3p3/2) ESPE and for non-inclusion of
  corrections for other spes,
• for 21 state, 29.0 originates from the
  ?(2f7/2)6(1h9/2)2 and 9.6 from
  ?(2f7/2)6(2f5/2)2.
• The effective energy gap between ?(2f7/2) and
  ?(2f5/2) (the lowest orbital which contributes to
  the composition of 2 state) single particle
  orbitals is 2.370 MeV.

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Conclusion

• A new shell closure at 140Sn has been predicted.
• ALS term in empirical interaction SMPN is found
  to be responsible for the gap observed in SMPN
  results.
• A simple three-body monopole term has been
  included in CWG to get CWG3M, which predicts a
  shell gap at N90 for Sn isotopes as well as
  decreasing 21 energies for 136,138Sn, similar to
  that from SMPN.
• This also indicates that three body effect plays
  an important role for shell evolution in neutron
  rich Sn isotopes above 132Sn, as also observed in
  sd and fp shells.
• The anomalously depressed 21 states in Sn
  isotopes having N84-88, and the new magic number
  for N90, might have interesting consequences for
  the r - process nucleosynthesis.

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Construction of the new Hamiltonian

• Modification of the CW5082 W.T. Chou and E.K.
  Warburton, Phys. Rev. C 45, 1720 (1992)
  Hamiltonian in the light of recently available
  information on binding energies, low-lying
  spectra of A134 Sn,Sb and Te isotopes.
• The spes of the single particle orbitals of the
  valence space above the 132Sn core have been
  replaced by the recently measured ones.
• The details of this modification procedure have
  been given in Sukhendusekhar Sarkar, M. Saha
  Sarkar, Eur. Phys. Jour. A 21 (2004) 61.
• The new Hamiltonians work remarkably well in
  predicting binding energies, low-lying spectra
  and electromagnetic transition probabilities for
  N82,83 and even for N gt 84 isotones of
  Sn,Sb,Te,I,Xe and Cs nuclei.

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References

• O. Sorlin, M. G. Porquet, Prog. Part. and Nucl.
  Phys. 61, 602 (2008).
• Takaharu Otsuka et al., Phys. Rev. Lett. 95,
  232502 (2005).
• http//www.nndc.bnl.gov.
• M. P. Kartamyshev, T. Engeland, M. Hjorth-Jensen,
  and E. Osnes, Phys. Rev. C 76, 024313 (2007) and
  references therein.
• S. Sarkar, M. Saha Sarkar, Phys. Rev. C 78,
  024308 (2008).
• Sukhendusekhar Sarkar, M. Saha Sarkar, Eur. Phys.
  Jour. A21, 61 (2004) and references therein S.
  Sarkar and M. Saha Sarkar, Phys. Rev. C 81,
  039803 (2010).
• B.A. Brown, N. J. Stone, J. R. Stone, I. S.
  Towner, and M. Hjorth-Jensen, Phys. Rev. C 71,
  044317 (2005).
• Sukhendusekhar Sarkar, M. Saha Sarkar, Phys. Rev.
  C 64, 014312 (2001) and references therein.
• L. Coraggio, A. Covello, A. Gargano, and N.
  Itaco, Phys. Rev. C 72, 057302 (2005) and
  references therein.
• R. F. Casten and B. M. Sherrill, Prog. Part.
  Nucl. Phys. 45, 171 (2000).
• B.A. Brown et al., Oxbash for Windows PC,
  MSU-NSCL Report No. 1289, (2004) Nushell_at_MSU, B.
  A. Brown and W. D. M. Rae, MSU-NSCL report
  (2007).
• M.W. Kirson, Phys. Lett. 47B, 110 (1973) Kenji
  Yoro, Nucl. Phys. A 333, 67 (1980) B.A. Brown,
  W A Richter and B H Wildenthal, J. Phys.G Nucl.
  Phys. 11, 1191 (1985) K. Yoshinada, Phys. Rev. C
  26, 1784 (1982).
• A. Poves and A. Zuker, Phys. Rep. 70, 235 (1981)
  26 A. P. Zuker, Phys. Rev. Lett. 90, 042502
  (2003).
• Takaharu Otsuka, Toshio Suzuki, Jason D. Holt,
  Achim Schwenk, and Yoshinori Akaishi,
  arXiv0908.2607v2, 25 Jan 2010.
• S. Sarkar, M. Saha Sarkar, arXiv0910.2119v1, v2
  nucl-th, 12 Oct 2009.

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THANK YOU
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Onset of deformation for nuclei (Z?54)

• Shell closure at N90 with SMPN for Sn isotopes
• Does it contradict the experimentally observed
  fact that N90 is suitable for onset of
  deformation for nuclei with Z ? 54, like Xe, Ba
  etc.) ?
• The neutron ESPEs for SMPN does not show much
  variation with increasing proton number at N90
  for Z gt 50.
• The proton ESPEs for SMPN favours the onset of
  collectivity at N90 for Z gt 50.
• Evidenced by the substantial reduction of the
  (1g7/2) and (2d5/2) energy gap with (1g7/2)8.
  This is very similar to the appearance of the new
  shell gaps for the oxygen isotopes which
  disappears at larger Z values

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Proton ESPEs with SMPN interactions for
increasing proton numbers for N90
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Proton ESPEs with SMPN interactions for
increasing proton numbers for N90
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Neutron ESPEs with SMPN interactions for
increasing proton numbers for N90
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132Sn region Shell Model calculations

• It has been pointed out that there should be
  many points of similarity between the
  spectroscopy of the doubly closed shell regions
  around 208Pb and 132Sn.
• The single particle orbits above and below the
  shell gap in the two cases are similarly ordered.
  Every single particle orbit in the 132Sn region
  has its counterpart in the 208Pb region, with
  same radial quantum numbers but one unit larger
  in angular momentum l and j values.
• As a consequence, effective interactions in the
  Sn region can be estimated from the corresponding
  well studied effective interactions constructed
  for nuclei in the 208Pb region.

• J. Blomqvist, in Proceedings of the 4th
  International Conference on Nuclei far from
  Stability, Denmark, 1981 (CERN, Geneva, 1981), p.
  536.
• W.T. Chou and E.K. Warburton, Phys. Rev. C 45,
  1720 (1992).
• Sukhendusekhar Sarkar, M. Saha Sarkar, Phys.
  Rev. C 64 (2001) 014312 and references therein.

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Procedure

• SINGLE PARTICLE ENERGIES (SPES)
• modified the CW5082 interaction. The valence
  space consists of five proton orbitals,
  1g7/2 , 2d5/2 , 2d3/2, 3s1/2 and 1h11/2 with
  energies 0.(-9.6629), 0.9624, 2.4396,
  2.6972, 2.7915 respectively, and
• 1h9/2, 2f7/2 , 2f5/2 , 3p3/2 , 3p1/2 and 1i
  13/2 for neutrons with energies in MeV,
  1.5609, 0.0 (-2.4553), 2.0046, 0.8537,
  1.6557, 2.6950, respectively with 132Sn as the
  inert core.
• CHANGE IN TWO BODY MATRIX ELEMENTS (TBMES)
• In SMN
• change the neutron-neutron and proton-neutron
  tbmes keeping the proton-proton tbmes the same
  as those in CW5082.
• In SMPN change the neutron-neutron,
  proton-neutron AND proton-proton tbmes .

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• Changes in two body matrix elements (tbmes)
• Neutron-neutron tbmes
• The six neutron-neutron diagonal tbmes with I
  0 were multiplied by a factor of 0.48. This
  factor is obtained by reproducing the
  binding energy of 134Sn (-6.365 MeV). All the
  binding energies in MeV are with respect to
  132Sn.
• Three excited states in 134Sn, predominantly
  from the neutron (2f7/2 )2, at energies 725.6,
  1073.4, and 1247.4 keV are used to modify the
• lt (2f7/2 )2 V (2f7/2 )2 gt 2,4,6 tbmes
  for neutrons.
• lt (1h9/2 2f7/2 ) V (1h9/2 2f7/2 ) gt 8
  changed to reproduce the energy of 8 level at
  2508.9keV.
• neutron proton tbmes
• Similarly, using binding energy (-12.952 MeV)
  and 1-, 2-, 3-, 4-, 7-, 8-, 10, 9, 10-,
  11- and 12- excited levels at energies
  13.0, 330.7, 383.5, 554.8, 283.0, 1073, 2434,
  2126, 4094, 4425 and 4517 keV respectively,
  of 134Sb, we have modified 12 dominant
  proton-neutron tbmes.

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137I
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137Te
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Evolution of collectivity in neutron -rich Sn
isotopes

• Level spectra of 135-138 Sn are unknown.
• The production rate very very low difficult to
  produce more neutron rich isotopes
• Astrophysical Scenario important - with respect
  to the r-process,
• 136Sn is a waiting-point nucleus for moderate
  neutron densities
• Theoretical suggestions and experimental hints
  that beyond the 132Sn core
• the Z50 shell gap quickly disappears and
• nuclear deformation shows up around N 87.
• Already around N84-85 a mild collectivity is
  recognised in the spectra of 137Te, 137I
• the spectra of 138Te and 139I show good
  vibrational characteristics.
• Thus it is of interest to see whether deformation
  develops in the close-to-dripline Sn isotopes.

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Both give similar agreement comparatively better
with SMPN R4E4/E2 indicate vibrational spectrum
Z52, N86
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136Te
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NEW FEATURE IN THESEMI-MAGIC neutron rich
isotopes
Depressed 2
Z50, N88
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The ESPE is defined as bare single particle
energy (spe) added with the monopole part of the
diagonal two body matrix elements (TBME). The
bare spe is originated from the interaction of a
valence nucleon with the doubly closed core. The
monopole interaction contribution is the (2J 1)
weighted average of the diagonal TBME, which
arises from the interaction of a valence nucleon
with the other valence nucleons.
Effective Single Particle Energy (ESPE)
Where stands
for the (diagonal) matrix element of a state
where two nucleons are coupled to an angular
momentum J and an isospin T.
If neutrons occupy j? and one looks into the
orbit j(? j?) as a proton orbit, the shift of the
single-particle energy of j is given by where nn
(j?) is (the expectation value of) the number of
neutrons in the orbit j?.