Continuum quasiparticle linear response theory using the Skyrme functional for multipole responses o

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Continuum quasiparticle linear response theory
using the Skyrme functional for multipole
responses of exotic nuclei

• Niigata University, Japan
• Kazuhito Mizuyama, Masayuki Matsuo, Yasuyoshi
  Serizawa

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Collective excitations in Unstable Nuclei and
Quasiparticle Random Phase Approximation
Neutron-rich unstable nuclei
Many weakly bound neutrons
Realistic description
Density functional theory Effective interaction
Continuum
Pairing
Continuum RPA S. Shlomo and G. Bertsch.
NPA243(1975). I. Hamamoto, H. Sagawa, and X. Z.
Zhang. PRC 57, R1064 (1998)
Self-consistent Skyrme QRPA J. Terasaki and J.
Engel. PRC 74, 044301, (2006).
Relativistic QRPA P. Ring, N. Paar, T. Niksic,
and D. Vretenar. NPA722, (2003).
Continuum QRPA M. Matsuo. NPA696, 371,
(2001). E. Khan, et al. PRC66, 024309, (2002).
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Continuum quasiparticle linear response
theoryand the collective excitations of unstable
nuclei
Continuum QRPA in coordinate space HFB formalism
M.Matsuo Nucl.Phys.A696(2001)371
M. Matsuo, K. Mizuyama, and Y. Serizawa. PRC 71,
064326, (2005).
Response function

• Shallow Fermi level
• Continuum
• Pairing correlation

HFB Greens function
Pairing and Exact boundary condition of out-going
wave for continuum states
correct asymptotic of out going wave
regular solution of q.p. wave function at r0.
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Continuum QRPA with the Skyrme functional
For the realistic description of unstable nuclei.

• Skyrme effective interaction
• Energy weighted sum rule (EWSR)

Self-consistency

• Skyrme HFB continuum QRPA
• Residual force
• (2nd derivative of the Skyrme functional)

-- Skyrme interaction --

• Velocity dependent terms (t1,t2) are important
  for
• The conservation of EWSR (including the
  enhancement factor for IV mode)
• Galilean invariance

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The purpose of this study

• The purpose of this study is to formulate the
  continuum QRPA with the Skyrme functional keeping
  the velocity dependent terms.

-- Skyrme-Hartree-Fock-Bogoliubov functional --
For the purpose of this study, the linear
response equation for various densities
fluctuations is needed
Spin dependent terms are dropped
Velocity dependent terms
Simple delta-type int.
Previous continuum QRPA
Important for the conservation of the EWSR

• Simple delta-type int.
• Landau-Migdal force

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From Previous continuum QRPA to the new
continuum QRPA
-- Previous continuum QRPA equation --
-- New continuum QRPA equation --
-- Response function --
-- Induced field --
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The treatment of singular terms in the response
function
-- Response function --
The treatment of delta-type singular terms in the
response function
(cf. K.F.LIU, N.V.GIAI, Phys.Lett.Vol.65,23(1976))
Regular terms
Singular terms
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Numerical calculation
-- Parameters --
Smoothing constant e1.0MeV
p-h channel
Skyrme interaction SkM for 20O, 54Ca
RMax15fm lcut7,8 Ecut60MeV
p-p interaction
V0 280MeV fm-3 for 20O , 285MeV fm-3 for
54Ca
?0 0.32 fm-3
Isovector dipole and Isoscalar quadrupole in
20O and 54Ca
-- Calculations --

• HFB ground state (Spherical)
• Continuum QRPA cal.

we compare the previous continuum QRPA with
Landau-Migdal force (LM) and the new continuum
QRPA (Full) in the neutron rich nuclei.
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Strength function
The structure is not changed between two
calculations in the IS E2 strength
IS
54Ca SkM IS IV 2
The structure is changed between two
calculations in the E1 strength
IV
Centroid energy
is different
about 2-3MeV in the IV mode.
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Energy weighted sum rule
EWSR (?0.32)
Running energy weighted sum
EWSR
Continuum QRPA with the Skyrme functional
satisfies the energy weighted sum rule about
95-99. On the other hand, Lndau-Migdal (LM)
approx. underestimated the sum rule about 15
in the E1 excitation, overestimated about 10 in
the IS quqdrupole excitation.
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Transition densities
54Ca SkM IV 1-
Low -lying

• The basic structures are same in the low-lying
  state and the GDR between two calculations.
• Low-lying state
• p-pair transition density is enhanced.
• IVGDR
• Transition densities are almost equivalent
  between two calculations, note that the peak
  energy is different.

Previous cQRPA
New cQRPA
IVGDR
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Transition densities
54Ca SkM IS 2
Low -lying

• The basic structures are same in the low-lying
  state and the GQR between two calculations.
• Low-lying state
• h-pair transition density is enhanced.
• ISGQR
• Transition densities are almost equivalent
  between two calculations, note that the peak
  energy is also same.

Previous cQRPA
New cQRPA
ISGQR
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Summary

• We formulated the continuum QRPA based on the
  Skyrme energy functional
• with keeping the velocity dependent terms.
• We applied the Skyrme continuum QRPA to the
  isovector dipole
• and the isoscalar quadrupole responces in
  neutron-rich O and Ca isotopes.
• The Skyrme continuum QRPA satisfies the EWSR.
• There are quantitative improvements,
• compared with the previous cQRPA (Landau-Migdal
  approx.)

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Translational symmetry and spurious mode
Improvement of the self-consistency
Violation of the self-consistency ? Spurious
mode at E?0
Renormalization factor fR
(residual force) ? fR (residual force)
The renormalization factor is one of the
indicators of the self-consistency
fR1Self-consistent fR?1Self-consistent is
violated
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Time-odd velocity-dependent terms effect in the
strength function
20O SkM IV 1-
Time-odd terms are most essential role to restore
the conservation of the EWSR.