valence shell excitations in even-even spherical nuclei within microscopic model

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valence shell excitations in even-even spherical
nucleiwithin microscopic model

• Ch. Stoyanov
• Institute for Nuclear Research and Nuclear Energy
• Sofia, Bulgaria

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The model Hamiltonian
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Central forces
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Quasiparticle RPA(collective effects)
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Quasiparticle RPA (2)(quasiboson approximation)

• Jm denote a single-particle level of the average
  field for neutrons (or protons)
• The neutron ?µ means coupling to the total
  momentum ? with projection µ
• The quantity is
  Clebsch-Gordon coefficient
• Bogoliubov linear transformation

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Phonon properties

• Phonons are not only collective
• Collective ? many amplitudes
• Non-collective ? a few amplitudes
• Pure quasi-particle state ? only one amplitude
• Diverse Momentum and Parity Jp spin-multipole
  phonons
• The interaction could include any kind of
  correlations
• (particle-particle channel)
• LARGE PHONON SPACE

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Quasiparticle RPA (3)(collective effects)
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Harmonic vibrations
To avoid Pauli principle problem
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Microscopic description of mixed-symmetry states
in nearly spherical nuclei
Chavdar Stoyanov and N. Lo Iudice
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Introduction

• Low-lying isovector excitations are naturally
  predicted in the algebraic IBM-2 as mixed
  symmetry states. Their main signatures are
  relatively weak E2 and strong M1 transition to
  symmetric states.
• T. Otsuka , A.Arima, and Iachello, Nucl .Phys.
  A309, 1 (1978)
• P. van Isacker, K.Heyde, J.Jolie et al., Ann.
  Phys. 171, 253 (1986)

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Definitions

• The low-lying states of isovector nature were
  considered in a geometrical model as
  proton-neutron surface vibrations.
• is in-phase (isoscalar) vibration of
  protons and neutrons.
• is out-of-phase (isovector) vibration of
  protons and neutrons.
• A.Faessler, R. Nojarov, Phys. Lett., B166, 367
  (1986)
• R. Nojarov, A. Faessler, J. Phys. G, 13, 337
  (1987)

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Review paper

• N. Pietralla, P. von Brentano,
• and A. F. Lisetskiy,
• Prog. Part. Nucl. Phys. 60, 225 (2008).

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Microscopic calculations

• Within the nuclear shell modelA. F. Lisetskiy,
  N. Pietralla, C. Fransen, R. V. Jolos, P. von
  Brentano, Nucl. Phys. A677, 1000 (2000)
• Within the quasi-particle-phonon model (QPM)N.
  Lo Iudice and Ch. Stoyanov, Phys. Rev. C 62,
  047302 (2000)
• N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 65,
  064304 (2002)

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Definition

• In order to test the isospin nature of 2
  states the following ratio is computed

• This ratio probes
• The isoscalar ((2)lt1) and
• The isovector (B(2)gt1)properties of the 2
  state under consideration

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The dependence of M1 and E2 transitions on the
ratio G(2)/k0(2) in 136Ba.
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Structure of the first RPA phonons (only the
largest components are given) and corresponding
B(2) ratios for 136Ba
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The values of B(2) for 144Nd
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Explanation of the method used

• The quasi-particle Hamiltonian is diagonalized
  using the variational principle with a trial wave
  function of total spin JM

Where ?0 represents the phonon vacuum state and
R, P and T are unknown amplitudes ? labels the
specific excited state.
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Explanation of the method used (2)

• Taking into account the fermionic structure of
  the phonon operator, their commutation relations
  read

The second term is important in considering
multi-phonon states which may violate the Pauli
principle
The first term corresponds to the ideal boson
approximation
The second one takes into account the internal
fermionic structure of the phonons
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Explanation of the method used (3)

• The normalization condition for states reads

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Energies and structure of selected low-lying
excited states in 94Mo. Only the dominant
components are presented.
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94Mo level scheme./low-lying transitions/
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E2 transitions connecting some excite states in
94Mo calculated within QPM.
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M1 transitions connecting some excite states in
94Mo calculated within QPM.
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92 Zr
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(No Transcript)
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92 Zr Contribution of N and Z in the 2 QRPA
phonons
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Energy and phonon structure in 92 Zr.
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E2 and M1 transitions connecting excited st. in
92 Zr
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QPM, EXP and SM g-fact. of low-lying excited
st. in 92 Zr
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QRPA Results for N80 isotones
Two-quasi-particle proton states, entering into
the first 2 excitations
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Energy and structure of selected low-lying
excited states
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The N80 isotones
N. Pietralla et al., Phys. Rev. C
58, 796 (1998). G. Rainovski, N. Pietralla et
al., Phys. Rev. Lett. 96, 122501 (2006). T. Ahn,
N. Pietralla, G. Rainovski et al., Phys. Rev. C
75, 014313 (2007). K. Sieja et al., Phys. Rev.
C, v. 80 (2009) 054311.
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Experimental results
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Fermi energy as a function of the mass number
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Occupation probabilities
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Results on QRPA level
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QPM Results for N80 isotones
134Xe
134Xe
136Ba
138Ce
138Ce
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N84 Experimental results
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N84 theoretical description
N. Pietralla et al., Phys. Rev. C 58,
796 (1998). G. Rainovski, N. Pietralla et al.,
Phys. Rev. Lett. 96, 122501 (2006). T. Ahn, N.
Pietralla et al.,Phys. Rev. C 75, 014313 (2007).
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Two quasiparticle poles
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N84 theoretical description
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Comparison to the experiment
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Conclusions

• There are two modes in the low-lying quadrupole
  excitations isoscalar and isovector one.
• The properties of these two modes are close to
  IBM-2 symmetric and mixed-symmetry states.
• The coupling of the modes leads to variety of
  excited states. There are well pronounced
  regularities of E2 and M1 transitions connecting
  the states.
• The spin degree of freedom pays a dominant role
  in some states, such as second 1.

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Structure of the low-lying states in some N80
isotones
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Outlook

• Motivation
• A sketch of the mean-field model used.
• A look into the existing results.

• Results for the single particle-level schemes.
• Introducing the QRPA technique used in our work.
• Results for 138Ce.

• Conclusion

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Motivation

• Skyrme force - universal parameterization for all
  nucleons
• Consistency between mean field calculations and
  the QRP calculations

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Results on QRPA level
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Thank You for Your attention!!!