The Random Phase Approximation in Nuclear Physics

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The Random Phase Approximation in Nuclear Physics

• Lay out of the presentation
• Linear response theory a brief reminder
• Non-relativistic RPA (Skyrme)
• Relativistic RPA (RMF)
• Extension to QRPA
• Beyond RPA .

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Linear Response Theory

• In the presence of a time-dependent external
  field, the response of the system reveals the
  characteristics of the eigenmodes.
• In the limit of a weak perturbing field, the
  linear response is simply related to the exact
  two-body Greens function.
• The RPA provides an approximation scheme to
  calculate the two-body Greens function. .

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• Adding a time-dependent external field

.
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First order response as a function of time
.
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Two-body Greens Function and density-density
correlation function
.
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Linear response function and Strength distribution
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Main results

• The knowledge of the retarded Greens function
  gives access to
• Excitation energies of eigenmodes (the poles)
• Transition probabilities (residues of the
  response function)
• Transition densities (or form factors),
  transition currents, etc of each excited state .
  
  

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TDHF and RPA (1)
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TDHF and RPA (2)
And by comparing with p.5
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Residual p-h interaction
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Analytic summation of single-particle continuum
1) u, w are regular and irregular solutions
satisfying appropriate asymptotic conditions
2) This analytic summation is not possible if
potential U is non-local .
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Approximate treatments of continuum (1)
T. Vertse, P. Curutchet, R.J. Liotta, Phys. Rev.
C 42, 2605 (1990) .

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Approximate treatments of continuum (2)

• Calculate positive-energy s.p. states with
  scattering asymptotic conditions, and sum over an
  energy grid along the positive axis, up to some
  cut-off
• Sum over discrete states of positive energy
  calculated with a box boundary condition .

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Transition densities and divergence of transition
currents
Solid GQR
Dotted empirical
Dashed low-lying 2
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Convection current distributions
GQR in 208Pb
Low-lying 2 in 208Pb
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Finite temperature
Applications evolution of escape widths and
Landau damping of IVGDR with temperature .
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RPA on a p-h basis
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A and B matrices
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Restoration of symmetries

• Many symmetries are broken by the HF mean-field
  approximation translational invariance, isospin
  symmetry, particle number in the case of HFB,
  etc
• If RPA is performed consistently, each broken
  symmetry gives an RPA (or QRPA) state at zero
  energy (the spurious state)
• The spurious state is thus automatically
  decoupled from the physical RPA excitations
• This is not the case in phenomenological RPA .

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Sum rules

• For odd k, RPA sum rules can be calculated from
  HF, without performing a detailed RPA
  calculation.
• k1 Thouless theorem
• k-1 Constrained HF
• k3 Scaling of HF .

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Phenomenological RPA

• The HF mean field is replaced by a parametrized
  mean field (harmonic oscillator, Woods-Saxon
  potential, )
• The residual p-h interaction is adjusted
  (Landau-Migdal form, meson exchange, )
• Useful in many situations (e.g., double-beta
  decay)
• Difficulty to relate properties of excitations to
  bulk properties (K, symmetry energy, effective
  mass, ) .

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Relativistic RPA on top of RMF
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Fermi states and Dirac states
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Single-particle spectrum
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The Hartree polarization operator
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Fermi and Dirac contributions
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The RRPA polarization operator

• Generalized meson propagator for
  density-dependent case (Z.Y. Ma et al., 1997) .

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Diagrammatic representation
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RRPA and TDRMF

• One can derive RRPA from the linearized version
  of the time-dependent RMF
• At each time, one assumes the no-sea
  approximation, i.e., ones keeps only the positive
  energy states
• These states are expanded on the complete set (at
  positive and negative energies) of states
  calculated at time t0
• This is how the Dirac states appear in RRPA. How
  important are they?
• From the linearized TDRMF one obtains the matrix
  form of RRPA, but the p-h configuration space is
  much larger than in RPA! .

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Effect of Dirac states on ISGMR
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Effect of Dirac states on ISGQR
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Effect of Dirac states on IVGDR
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Including continuum in RRPA
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QRPA (1)

• The scheme which relates RPA to linearized TDHF
  can be repeated to derive QRPA from linearized
  Time-Dependent Hartree-Fock-Bogoliubov (cf. E.
  Khan et al., Phys. Rev. C 66, 024309 (2002))
• Fully consistent QRPA calculations, except for
  2-body spin-orbit, can be performed (M. Yamagami,
  NVG, Phys. Rev. C 69, 034301 (2004)) .

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QRPA (2)

• If Vpp is zero-range, one needs a cut-off in qp
  space, or a renormalisation procedure a la
  Bulgac. Then, one cannot sum up analytically the
  qp continuum up to infinity
• If Vpp is finite range (like Gogny force) one
  cannot solve the Bethe-Salpeter equation in
  coordinate space
• It is possible to sum over an energy grid along
  the positive axis ( Khan - Sandulescu et al.,
  2002) .

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Pairing window method
K. Hagino, H. Sagawa, Nucl. Phys. A 695, 82
(2001) .
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2 states in 120Sn
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2 states in 120Sn, with smearing
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3- states in 120Sn, with smearing
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Beyond RPA (1)

• Large amplitude collective motion Generator
  Coordinate Method
• RPA can describe escape widths if continuum is
  treated, and it contains Landau damping, but
  spreading effects are not in the picture
• Spreading effects are contained in Second RPA
• Some applications called Second RPA are actually
  Second TDA consistent SRPA calculations of
  nuclei are still waited for.

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Beyond RPA (2)

• There exist models to approximate SRPA
• The quasiparticle-phonon model (QPM) of Soloviev
  et al. Recently, attempts to calculate with
  Skyrme forces (A. Severyukhin et al.)
• The ph-phonon model see G. Colo. Importance of
  correcting for Pauli principle violation
• Not much done so far in relativistic approaches .

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Beyond RPA (3)

• Particle-vibration coupling

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Effect of particle-vibration coupling
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Acknowledgments

• Thanks to Wenhui LONG for Powerpoint
  tutoring .