Etat de lieux de la QRPA = state of the art of the QRPA calculations

G. Colò / E. Khan

Espace de Structure Nucléaire Théorique SPhN, Saclay, January 11-12, 2005

QRPA = Quasi-particle RPA

• Method known since 40 years in nuclear physics• Strong peak of activity since ~ year 2000. Why ?

E*, S(E*) inelastic cross sections

N,Z

Pairing vibrations, 2n transfer cross sections

N+2,Z

half-lives, GT strengths, charge exchange cross sections

N+1,Z-1

Study nuclear transitions on the whole chart !(isotopic chains, open shells, drip-line nuclei, …)

1) Single quasiparticle (mean field) wave functions and energies : WS, HF+BCS, HFB ?

2) Interaction : separable, self-consistency, pairing ?

Necessity to have a microscopic approach

QRPA inputs

Single quasiparticle spectrum

HF

Single particle states

0

2

HFB

Single quasiparticle states

0

22)(E

-50 50

As already said, the QRPA equations were derived by M. Baranger many years ago. However, most of the calculations done in earlier times were NOT self-consistent. The single quasiparticle states and the interaction were often the result of an empirical choice.

On the other hand, there is recent interest in trying to define a microscopic, universal “Energy Functional” for nuclei and nuclear matter. Within this framework, the HF (or HFB) define the minimum of the functional, while the self-consistent RPA (or QRPA) equations describe the small oscillations around this minimum.

Non-relativistic framework : this talk

RMF : cf. D. Vretenar

Meaning of self-consistency within RPA:

the restoring force which governs the nuclear oscillations is derived from the energy functional.

Veff(1,2) → E[ρ] = < H > = ∫ ρ(1) Veff(1,2) ρ(2) , building the density ρ as a combination of independent particles wavefunctions

h[ρ] = δE / δρ = 0 defines the mean field; the restoring force is: δ2E / δρ2

If pairing is introduced, the energy functional depends on both the usual density ρ=<ψ+(r)ψ(r)> and the abnormal density κ=<ψ(r)ψ(r)> (κ=<ψ+(r)ψ+(r)>).

-

TDHFB :

External field :

Generalised density R0

**1

with

How to derive the QRPA equations ?

U= (UV)

Matrix elements which enter the QRPA equations

When we evaluate the quantities that we have called V, and which are the second derivatives of the energy functional, matrix elements between quasi-particle configurations are obtained. They are written in terms of particle-hole (p-h) and particle-particle (p-p) matrix elements.

There is no compulsory reason for the effective interaction to be the same in the p-h and p-p channel.

hh,hhpp,hhph,hh

hh,pppp,ppph,pp

hh,phpp,phph,ph

VVVVVVVVV

V

The Gogny force is designed so that it allows treating the p-h and p-p channels on the same footing.

When the Skyrme force is used in the p-h channel, a popular choice is to complement it with a zero-range density dependent pairing force (which is usually fitted on the ground-state pairing in the nucleus, or in the mass region, of interest).

Effective interactions

Pairing, divergence & cutoff

•Pairing :

•Zero range UV divergence in the pp channel •Prescription (V0 , Ecutoff) : stable

Advantages and disadvantages of self-consistency:

it allows testing a functional on known cases, and making extrapolations for unknown situations (e.g., predictions for exotic nuclei) without introducing free parameters.

… disadvantages : heavy treatment more difficult (not impossible) to carry out large-scale calculations, including deformed systems, and to go beyond QRPA

Examples of recent non self-consistent QRPA

•Macro/Micro (Möller) : WS for the single particle spectrum, constant G pairing, and separable GT residual interactionInterest : large-scale rates calculations

•FR-QRPA (Faessler) : takes into account the Pauli principle and allows the Ikeda sum rule to be verified ( decay) Interest : go beyond the quasi-boson approximation

Advantages of QRPA compared to other methods (e.g., shell model):

• simplicity, also from the computational point of view;

• there is no “core” (that is, no need of effective charges);

• it is possibile to study highly excited states.

• provides densities and transition densities

Disadvantage:

• not all the many-body correlations are taken into account.

By definition, states which include components like four quasi-particles cannot be described within the framework of standard QRPA.

Discrete vs. continuum :

•Continuum is important close to, or at, the drip lines. Continuum calculations are by definitions more complete than discrete calculations, and require a coordinate space formalism for the QRPA equations

•Discrete calculations have the advantage that they more directly provide the information about the wavefunctions on a quasi-particle basis (important for, e.g., the particle decay).

Conclusion: there is some degree of complementarity !

Microscopic calculations on the market…

•Skyrme:

HFBCS (WS) continuum K.Hagino,H.Sagawa, NPA695 (2001) 82

HFB (WS) continuum M.Matsuo, NPA696 (2001) 371

HFB continuum E.Khan et al., PRC66 (2002) 024309

HFBCS discrete G.Colò et al., NPA722 (2003) 111c

HFB discrete M.Yamagami,N.Van Giai, PRC69 (2004)

HFB discrete J.Terasaki et al. (unpublished, see nucl-th)

•Gogny:

HFB discrete G.Giambrone et al., NPA726 (2003) 6

HFB discrete S. Peru et al.

Most of the calculations drop the residual Coulomb and spin-orbit force

Charge-exchange calculations

•Skyrme:

BCS discrete P.Sarriguren et al., PRC67 (2003) 044313

HFB discrete (canonical basis) J.Engel et al., PRC60 (1999) 014302

M.Bender et al., PRC65 (2002) 054322

HFBCS discrete S.Fracasso,G.Colò

•DF-Fayans:

HFB continuum I.Borzov et al., PRC67 (2003) 025802

Some results…

QRPA gives, as a rule, reasonable results as far as the energy location of Giant Resonances is concerned (within 1 MeV or so). In fact, pairing is relatively unimportant for high-lying states and QRPA is almost equivalent to RPA.

S. Kamerdzhiev et al.

Width !!

(exp. ~4.9 MeV)

Need to go beyond QRPA

S. Goriely et al.

Pigmy dipole appears in microscopic HFB+QRPA calculations

Necessity for truly deformed QRPA

Results for the neutron-rich oxygen isotopes

•difficulties for 18O : deformed ?

•pairing decreases in the heavier 22,24O

•sensivity of the low-lying states on the pairing interaction

•Matsuo et al. :

E2 = 2.5 2.7 2.9 3.3 [MeV]

B(E2) = 17 18 20 18 e2.fm4

E. Khan et al.

Gogny QRPA

G. Giambrone et al.

30Ne

32Mg

36S

34Si38Ar

M.Yamagami,N.Van Giai

In the N=20 isotones self-consistent QRPA performs well

In the sulfur isotopes QRPA input in a folding model calculation reproduces the inelastic (p,p’) cross sections

D.T.Khoa et al.

D. Sarchi et al.

K.Hagino and H.Sagawa

Sn isotopes

120Sn

Astrophysics : (n,) rates

Deviation up to a factor 10

QRPA/Hybrid

QRPA/QRPA

T=1.5 109 K

Discrepancy pheno/micro

Agreement HF+BCS QRPA / HFB QRPA

Charge-exchange and ν-induced reactions

C. Volpe et al.

Σk r2k Y2(Ωk) t+

full = QRPA, dashed = SM

12C

used for calculation of (νe,e)(νμ,μ)

GT

Important to extend these calculations to exotic nuclei

It is difficult to assess in general the reliability of QRPA for the low-lying states. These are sensitive to the details of the shell-structure around the Fermi energy.

• The neglected terms in the calculations (e.g., the two-body spin-orbit residual force) may play a role;

• the pairing force used is not universal, is simply fitted to the g.s. We need to systematically study its influence;

• anharmonicities ?

Continuum treatment can affect GR in drip-line nuclei.

Short conclusions