научная статья по теме MEAN FIELD AND BEYOND IN NUCLEI FAR FROM STABILITY LINES Физика

Текст научной статьи на тему «MEAN FIELD AND BEYOND IN NUCLEI FAR FROM STABILITY LINES»

HREPHAH 0H3HKA, 2004, moM 67, № 9, c. 1667-1671

MEAN FIELD AND BEYOND IN NUCLEI FAR FROM STABILITY LINES

©2004 P. F. Bortignon1)*, F. Barranco2), R. A. Broglia1),3), G.Colo1), E. Vigezzi4)

Received January 21, 2004

We calculate, for the first time, the state-dependent pairing gap of a finite nucleus (120Sn) diagonalizing the bare nucleon—nucleon potential (Argonne v14) in a Hartree—Fock basis. The resulting gap accounts for about half of the experimental gap. Going beyond mean field in the particle—particle (pp) channel, the combined effect of the bare nucleon—nucleon potential and of the induced pairing interaction arising from the exchange of low-lying surface vibrations between nucleons moving in time reversal states close to the Fermi energy accounts for the experimental gap. Examples for light, halo nuclei are also reported. The more studied effects of the particle-vibration coupling in the particle—hole (ph) channel, are discussed for the low-lying quadrupole vibration in 120Sn and the giant dipole resonance in the unstable oxygen isotopes and 132 Sn.

In the study of finite many-body systems such as the atomic nucleus with its rich variety of quantal size effects, structural properties, and fluctuations, the central problem has been to identify the appropriate degrees of freedom for describing the phenomena encountered. The complementary concepts referring to the independent motion of the individual nucleons and the collective behavior of the nucleus as a whole provide, in a mean field approach, the elementary modes of excitation needed to describe the system [1]. Beyond mean field, the unifying picture emerging from the interweaving of these degrees of freedom is well described in terms of nuclear field theory (NFT) [2—6] based on the particle-vibration coupling (for other related approaches, see, e.g., [7—10]). It has been applied to a number of schematic models and realistic situations [11 — 15] and its validity demonstrated. It thus provides a natural framework to assess the role different degrees of freedom play in the nuclear structure. In this contribution, recent examples are presented of the effects of the particle-vibration coupling in particle—particle (pp) and particle—hole (ph) channels, following in particular [ 16].

An important subject presently under intensive study concerns the characterization of an eventual long range component of the pairing interaction in nuclei [17—19]. In what follows, we use NFT to assess the importance the exchange of vibrations between

^Dipartimento di Fisica, Universita di Milano and INFN Sezione di Milano, Italy.

2)Departamento de Fisica Aplicada III, Universidad de Sevilla, Spain.

3)The Niels Bohr Insitute, University of Copenhagen, Denmark.

4)INFN Sezione di Milano, Italy.

E-mail: pierfrancesco.bortignon@mi.infn.it

pairs of nucleons moving in time reversal states have in building up pairing correlations in nuclei, taking also into account self-energy and vertex corrections, cf. [20] and references therein).

To this scope, we study the quasiparticle and vibrational spectrum of odd and even isotopes of single-closed-shell nuclei, where all the richness of the single-particle and collective degrees of freedom are fully expressed, avoiding the extracomplications of static deformations and associated rotations. The spectra of the A0 Sn isotopes, in particular those with mass number A = 119, 120, and 121, with their abundance of detailed experimental information, provide an excellent laboratory where to test the importance of the residual pairing interaction and its relation to self-energy processes.

In general one fixes the parameters of the effective interaction of nucleons in the nucleus, by requiring mean field theory, as a rule Hartree—Fock or Hartree— Fock—Bogoliubov theory if the system is superfluid, to reproduce the experimental findings: binding energies, mean-square radii, etc. This is equivalent to requiring that the solution of the Schrodinger equation describing the bound states of the electron— proton system, interacting through the Coulomb force, reproduces the energy levels of the hydrogen atom. We know that this is not possible, unless the renormalization effects arising from the electron— photon coupling are properly taken into account as prescribed by QED. Similarly, the parameters of the effective nuclear interaction should reproduce the experimental findings only when the particle-vibration coupling is allowed to renormalize, screen and dress the different modes of elementary excitation and the interaction among them. In fact, it will be

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concluded that a consistent description of the low-energy nuclear spectrum requires, aside from the bare nucleon—nucleon interaction, not only the dressing of single-particle motion through the coupling to the nuclear surface, to give the right density of levels close to the Fermi energy (and thus to an effective mass mm ^ m [21]), but also the renormalization of collective vibrational modes through vertex and self-energy processes, processes which are also found to play an essential role in the pairing channel, leading to a long range, state dependent component of the pairing interaction.

The formalism we shall use is based on the Dyson equation [18]. It can describe on equal footing the dressed one-particle state a of an odd nucleon renor-malized by the (collective) response of all the other nucleons (Figs. 1a— 1d of [16]), the renormalization of the energy frwv (Figs. 2a, 2b of [16]) and of the transition probability B(E\) (Figs. 2c, 2f of [16]) of the collective vibrations of the even system where the number of nucleons remains constant (correlated particle—hole excitations), and the induced interaction due to the exchange of collective vibrations between pairs of nucleons [17], moving in time reversal states close to the Fermi energy (Figs. 1e— 1g of [16]). We include both self-energy and vertex correction processes, thus satisfying Ward identities (cf., e.g., [20]). Within this framework, the self-consistency existing between the dynamical deformations of the density and of the potential sustained by "screened" particle-vibrations coupling vertices leads to renormalization effects which make finite (stabilize) the collectivity and the self-interaction of the elementary modes of nuclear excitation, in particular of the low-lying surface vibrational modes, providing an accurate description of many seemingly unrelated experimental findings, in terms of very few (theoretically calculable) parameters, namely: the fc-mass mk [21] and the particle vibration coupling vertex h(abv), associated to the process in which a quasiparticle changes its state of motion from the unperturbed quasiparticle state a to b, by absorbing or emitting a vibration v [1].

The Dyson equation describing the renormal-ization of a quasiparticle a, due to this variety of couplings is

Ea 0 0 -Ea

+

Ell(Ea) El2(Ea) Ï2l(Ea) E22(Ea) ,

(1)

= En

where En and Ej (i = j) are the normal and abnormal self-energies. The quantities Ea denote the quasi-particle energies obtained from a previous diagonal-ization of the bare nucleon—nucleon potential within the framework of the generalized Bogoliubov—Valatin transformation,

Sll(Ea ) = ]T

| F (abv )|2

bv

Ea - (Eb + hWv)

+

(2)

+

IW (abv )|2

Ea + (Eb + hWv) ) '

£l2(Ea) = V(abv)W(abv) x

bv

(3)

1

1

Ea - (Eb + hWv) Ea + (Eb + H^v)

With £22(Ea) = -En(-Ea).

Equation (1) is to be solved iteratively, and simultaneously for all the involved quasiparticle states. At each iteration step, the original quasiparticle states a with occupation numbers ua and va and quasiparticle energies Ea, becomes fragmented into the several eigenstates a of the energy-dependent eigenvalue problem of Eq. (1). For each positive eigenvalue Ea, there is a corresponding solution with eigenvalue -Ea. As in HFB theory, only positive solutions have to be included in the iterative procedure. In Eqs. ( 1)— (3), the phonon energies are denoted by Huv, and V(abv) or W(abv) label the particle-vibration vertex coupling the unperturbed quasiparticle a, with the configuration composed of the eigenstates b and of the phonon v. They are given by

V(abv) = h(abv)(uaVb - vaub), W (abv ) = h(abv )(ua Ub + vaVb ).

(4)

ya.

The basic particle-vibration vertex h(abv) is calculated as explained in [1]. The unperturbed quasipar-ticle energies and occupation factors, resulting from mean field calculations, are denoted by Ea, ua, and va while Ea, Ua, and va denote the corresponding renormalized quantities. The original quasiparticle strength become fragmented over the different eigen-states a with probability uUa + V^, while the renormal-ized occupation factors are obtained from the components of the eigenvectors, xa and ya, according to the relations Ua = XaUa + yaVa, Va = -yaUa + XaVa. The quantities ua and va are related to the spectroscopic factors measured in one-nucleon stripping and pickup reactions, respectively. One can also define [18, 20] a renormalized state-dependent pairing gap, through the relation Aa = 2EaUaVa/(ua + Vl), which in the

x

a

MEAN FIELD AND BEYOND

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limit of no fragmentation reduces to the usual BCS expression5).

We have also included vertex corrections which renormalize the particle-vibration vertices V (and W) according to

„ Xf V (acri

w = i^iTw 151

where X^ is the matrix element between the configurations c^ and bv.

In the calculations reported below, a Skyrme interaction (Sly4 parametrization, with mk ~ 0.7m [23]), was solely used to determine the properties of the bare single-particle states and the collective vibrations in the ph channel. On the other hand, in the pp (pairing) channel the interactions used were the bare nucleon— nucleon v14 Argonne potential and the exchange of collective vibrations.

As seen from Fig. 3 of [16] Hartree—Fock theory is not able to account for the experimental quasi-particle energies of the low-lying stat

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