Pairing Vibrational States Research Articles

Collective treatment of the isovector pair correlations: Boson representation

The theoretical approach to consideration of the Hamiltonian with pairing forces using a technique of the finite boson representation is developed. It is shown that a simultaneous description of the pairing vibrational state in $^{56}$Ni and the pairing rotational states with $T$=0 in the neighboring $N=Z$ nuclei is possible if the pairing Hamiltonian takes into account only isovector monopole pairing. However, the calculated energies of the pairing rotational states of $N=Z$ nuclei removed from $^{56}$Ni by 12 and more nucleons exceed significantly the experimental values. The possible reason of this discrepancy is discussed.

Elastic response of the atomic nucleus in gauge space: Giant Pairing Vibrations

Due to quantal fluctuations, the ground state of a closed shell system $ A_{0}$ can become virtually excited in a state made out of the ground state of the neighbour nucleus $ \vert gs(A_0+2) \rangle $ ( $ \vert gs(A_0-2) \rangle $ ) and of two uncorrelated holes (particles) below (above) the Fermi surface. These $ J^{\pi} = 0^{+}$ pairing vibrational states have been extensively studied with two-nucleon transfer reactions. Away from closed shells, these modes eventually condense, leading to nuclear superfluidity and thus to pairing rotational bands with excitation energies much smaller than $ \hbar\omega_{0}$ , the energy separation between major shells. Pairing vibrations are the plastic response of the nucleus in gauge space, in a similar way in which low-lying quadrupole vibrations, i.e. surface vibrations with energies much smaller than $ \hbar\omega_{0}$ whose eventual condensation leads to quadrupole deformed nuclei, provide an example of the plastic nuclear response in 3D space. While much is known, in particular concerning its damping, regarding the counterpart of quadrupole plastic modes, i.e. regarding the giant quadrupole resonances (GQR), $ J^{\pi} = 2^{+}$ elastic response of the nucleus with energies of the order of $ \hbar\omega_{0}$ , little is known regarding this subject concerning pairing modes (giant pairing vibrations, GPV). Consequently, the recently reported observation of L = 0 resonances, populated in the reactions 12C(18O,16O)14C and 13C(18O,16O)15C and lying at an excitation energy of the order of $ \hbar\omega_{0}$ , likely constitutes the starting point of a new field of research, that of the study of the elastic response of nuclei in gauge space. Not only that, but also the fact that the GPV have likely been serendipitously observed in these light nuclei when it has failed to show up in more propitious nuclei like Pb, provides unexpected and fundamental insight into the relation existing between basic mechanisms --Landau, doorway, compound damping-- through which giant resonances acquire a finite lifetime, let alone the radical difference regarding these phenomena displayed by correlated (ph) and (pp) modes.

Proton pairing vibrational states around the doubly magic nucleusPb208

Proton pairing vibrational states in $^{208}\mathrm{Pb}$ isotones with $76\ensuremath{\le}Z\ensuremath{\le}88$ protons are described like neutron pairing vibrational states in lead isotopes with $120\ensuremath{\le}N\ensuremath{\le}132$ neutrons within the model of pair addition and pair removal phonons. The anharmonicity calculated with three parameters reproduces the known energies within less than 5%. The interaction between the pairing phonons for protons and neutrons is similar in a striking manner.

Search for pairing-vibration states of even Ca isotopes in 40Ca + 208Pb transfer reactions

Multi-nucleon transfer reactions in $^{40}$Ca+$^{208}$Pb have been studied at several bombarding energies close to the Coulomb barrier. Light reaction products have been identified in mass and charge with a time-of-flight spectrometer. The energy spectra of the inclusive two neutron pick-up channel show a population in a narrow region of excitation energies which corresponds to the predicted energy of pairing vibration states in $^{42}$Ca.

Search for pairing-vibration states in multinuclon transfer reactions

Multinucleon transfer reactions in 40Ca+ 208Pb have been studied at several bombarding energies close to the Coulomb barrier. Light reaction products have been identified in mass and charge with a time-of-flight spectrometer. The energy spectra of the inclusive two neutron pick-up channel show a population in a narrow region of excitation energies which corresponds to the predicted energy of pairing-vibration state in 42Ca.

Multipair transfer in collisions between heavy nuclei.

On the basis of recent progress in our ability to predict cross sections for pair transfer, the interesting possibility of observing multipair transfer in collisions between heavy nuclei is reexamined. The results indicate that 3--4 pair transfer between pairing vibrational and pairing rotational states should be observable at low bombarding energies.

A coupled-channel formalism in gauge space

In the framework of the intrinsic coordinate-dependent phase-shifts formalism a simple method to evaluate cross sections for multiple transfer of nucleon pairs in heavy-ion collisions is presented. The nucleus-nucleus potential is parametrized with an appropriate intrinsic coordinate which describes deformations in an abstract gauge space. The different pairing rotational and vibrational states correspond to the initial or final states in the transfer process. Angular distributions for transfer of one or several pairs of nucleons are obtained. The formalism is applied to some examples and effective gauge deformation parameters are determined.

E0 decays of 0+ states in 146Gd: Search for two-phonon octupole excitations.

Conversion-electron measurements following the $^{144}(\mathit{\ensuremath{\alpha}}$,2n) and $^{144}\mathrm{He}$,n) reactions reveal high-energy E0 transitions at 3020, 3485, and 3639 keV. The lowest of these deexcites the known two-neutron pairing vibrational state in $^{146}\mathrm{Gd}$. It is not clear whether either of the others can be associated with the ${0}^{+}$ member of the ${3}^{\mathrm{\ensuremath{-}}}$\ifmmode\times\else\texttimes\fi{}${3}^{\mathrm{\ensuremath{-}}}$ two-phonon octupole multiplet.

Investigation of pairing vibration states in 88Sr and 90Zr via the (d, 6Li) reaction and search for α-vibration states

With a 45 MeV deuteron beam the 94Mo(d, 6Li) 90Zr and 92Zr(d, 6Li) 88Sr reactions have been studied to search for α-vibration states in 90Zr and in 88Sr. No evidence is found for such a state in both nuclei with a strength > 5 % than that of the ground state. Strongly excited 0 + states in 88Sr and 90Zr are those also seen in two-nucleon stripping and pickup reactions.

Multi-step shell-model treatment of six-particle systems

A method is presented of solving the six-particle systems in several steps. In the first step, the two-particle spectrum is calculated within the standard shell model. In the second step, the four-particle system is solved within a (correlated) basis consisting of the vector-coupled two-particle states previously evaluated. Finally, in the third step the six-particle system is calculated using a basis formed by vector coupling the two- and four-particle spectra evaluated in the first two steps. The method is applied to analyse the spectra and (p, t) reactions leading to 204Pb and 202Pb and good agreement with experimental data is obtained. A liaison between this calculation and the pairing model is found and the importance of the Pauli principle upon the pairing vibrational states is attested.

136,138Ba(t, p) and the systematics of neutron pairing vibrations at N = 82

The 136,138Ba(t, p) reaction has been studied at E t = 17 MeV. The neutron monopole and quadrupole pairing vibration states were located in 138Ba at 3.612 MeV and 4.043 MeV, respectively. These results, when combined with previous observations in the Ce and Nd isotopes at the N = 82 closed shell, lead to a linear relationship between the two-neutron binding energies and the proton number. Predictions of locations of other pairing vibration states at the N = 82 shell are possible from these relations.

Alpha-cluster pickup from Te and Sn isotopes with the (d, 6Li) reaction

The reactions 122, 124, 126, 128, 130Te(d, 6Li) 118, 120, 122, 124, 126Sn and 112, 116, 118, 120, 122, 124, 126Sn(d, 6Li) 108, 112, 114, 116, 118, 120Cd have been investigated at E d = 33 MeV for states up to E x = 3 MeV or higher using magnetic analysis. Alpha-particle spectroscopic factors and reduced α-widths have been extracted with zero- and finite-range distorted wave Born approximation (DWBA) theory. Spectroscopic amplitudes were calculated from shell-model configurations including core excitations and a semi-microscopic analysis was performed for ground state transitions, proton pairing vibration states, as well as certain states with J π ≠ 0 +. Good agreement is obtained for the transitions to the Cd ground states provided DWBA is normalized to α-decay. The ground states and proton pairing vibration states in the heavy Sn isotopes appear to be strongly mixed. Increased collectivity, predicted by the interacting boson model, affects the excitation energies and transition strengths of the 0 + proton pairing vibration states in Sn in the middle of the neutron shell. Selectivity and coherence phenomena are also prevalent for states with J π ≠ 0 + and are interpreted microscopically. In particular, coherent contributions from proton pair and neutron pair excitations lead to enhancements not observed in two-nucleon transfer. The mass excess of 120Cd was measured as −83975 ± 25 keV. About 25 previously unknown states were observed in the Sn and Cd isotopes and about 30 spin-parity assignments were made.

Pairing vibrational and isospin rotational states in a particle number and isospin projected generator coordinate method

Pairing vibrational and isospin rotational states are described in different approximations based on particle number and isospin projected, proton-proton, neutron-neutron and proton-neutron pairing wave functions and on the generator coordinate method (GCM). The investigations are performed in models for which an exact group theoretical solution exists. It turns out that a particle number and isospin projection is essential to yield a good approximation to the ground state or isospin yrast state energies. For strong pairing correlations (pairing force constant equal to the single-particle level distance) isospin cranking (- ωT x ) yields with particle number projected pairing wave function also good agreement with the exact energies. GCM wave functions generated by particle number and isospin projected BCS functions with different amounts of pairing correlations yield for the lowest T = 0 and T = 2 states energies which are practically indistinguishable from the exact solutions. But even the second and third lowest energies of charge-symmetric states are still very reliable. Thus we conclude that also in realistic cases isospin rotational and pairing vibrational states may be described in the framework of the GCM method with isospin and particle number projected generating wave functions.

O + states observed in Cd and Sn nuclei with the ( 3He, n) reaction

The ( 3He, n) reaction has been studied using targets of 104, 106, 108, 110Pd and 106, 110, 112, 114, 116Cd at bombarding energy of 25.4 MeV. Angular distributions were obtained from 0° to 25° and the results compared with the predictions of the single-step zero-range DWBA theory. In addition to the ground-state transition, numerous L = 0 transitions were observed up to 5 MeV excitation in the residual nuclei. The pairing vibration state was found to be fragmented in all final nuclei and, except for 106, 108Cd, a strong L = 0 transition was observed to a state at an excitation energy much below that predicted for the pairing vibration. If this state is assumed to be a fragment of the pairing vibrational state, then the total strength to excited 0 + states is in reasonable agreement with the strength observed in the corresponding Sn( 3He, n)Te ground-state transition. However, the centroid energy of the pairing vibrational state is then much lower than expected. No transitions with L > 0 were excited with measurable strength.

Two-nucleon transfer processes in the lead region

In the lead region a theoretical study of two-nucleon transfer processes is made with the help of the generator coordinate method. Both two-proton and two-neutron transfer reactions are described. The two-phonon pairing vibrational states in 208Pb are especially considered. Energies, relative cross sections and angular distributions are presented and compared, when possible, with experiments.

Study ofpf-shell nuclei with a symmetry conserving generator coordinate method

Bands based on the 0+ ground state and the first excited 0+ pairing vibrational state of48Ti,52Cr and56Fe are studied with the generator coordinate method. The generating wave functions for each value of the angular momentumJ are angular momentum and particle number projected selfconsistent Hartree-Fock-Bogoliubov states where the constrained amount of pairing correlations serves as the generator coordinate. The interaction is given by reaction matrix elements derived from the Hamada-Johnston force. The basis includes the four lowest oscillator shells.

Microscopic, nonharmonic description of rotations and pairing vibrations in deformedpf-shell nuclei

Angular-momentum- and particle-number-projected self-consistent Hartree-Bogoliubov wave functions are used within the generator-coordinate method for the description of excited states in $^{48}\mathrm{Ti}$ and $^{56}\mathrm{Fe}$. Bands based on the ground state and first pairing-vibrational state respectively are studied. In the ground band of $^{56}\mathrm{Fe}$ an antipairing effect occurs parallel with an increase of correlations in the excited band, so that for higher $J$ the ground band becomes less paired than the excited one.[NUCLEAR STRUCTURE $^{48}\mathrm{Ti}$, $^{56}\mathrm{Fe}$; calculated rotational spectra. Hartree-Bogoliubov, generator-coordinate method, $J$ projection.]

Pairing vibrational states and the generator coordinate method

The pairing vibrational states and the two-neutron transfer cross sections between these states are calculated in Ni, Sn and Pb isotopes by the generator coordinate method (GCM). The particle number fluctuation of the BCS functions is handled by projecting in a good approximation on sharp particle numbers. The results agree quite well with the experimental data.

Influence of the quadrupole-pairing interaction on the Jπ=0+ vibrations in the Pt region

The dependence of the energy of the lowest pairing-vibrational state on the quadrupole-pairing strengthG2 is investigated in the Pt isotopes. The calculation are made using the Random Phase Approximation for the collective states with the pairing forces treated in the BCS approximation.

Theory of Pairing Vibration Mode

A simple and exactly solvable model is proposed which exhibits all the characteristic features of the pairing vibration model of A. Bohr; the near-harmonic level sequences and the strong enhancements of the (L=O) two-nucleon transfer reaction. This model consists of degenerate many-j single particle levels and degenerate hole levels with an energy-gap between them. The pairing interactions are taken into account, except for those between the particle and hole states. Using the empirical values of the single particle and hole energies together with the strength of the pairing interaction, the model is compared with the experi­ mental data in Ni, Sn and Ph isotopes to get quantitative agreement. This model is further used as the basis states to diagonalize the more realistic case of the non-degenerate model with pairing interactions among all levels. It is found that our pair­ ing vibration mode remains predominantly in the ground state of the non-degenerate model. 1l has been proved to be a convenient way of the classification of o+ states of spherical even nuclei. The essential idea is to view the quanta associating with the particle pair correlation as a boson with energy hw, while the quanta arising from pair correlation of holes is regarded as an another kind of boson with the same energy. Taking these two kinds of boson as building stones, a o+ state can be considered as consisting of w bosons of' the particle pair and w bosons of the hole pair, its energy being given by E= (w+w)hw measured from the (w=w=O) o+ state. This pairing vibrational state will be denoted as ( w, w). Bohr1l and Nathan2l made a systematic survey of this scheme in the regions around Ni66, Sn114 and Pb 208• Since the bosons in this model are related to the pairing correlation, the pairing vibration states will be characterized by the strong enhancement of (L= 0) two-nucleon transfer reaction. After this proposal, many high resolution experiments of the two-nucleon transfer reaction have been performed to test the pairing vibration scheme. Microscopic treatment of the pairing vibration model has been made first by Bes and Broglia8l and later by several authors.4l All these works are based on the formal analogy of the pairing correlation to the conventional random-phase­ approximation treatment of nuclear surface vibration, so that the microscopic foundation of the pairing vibration model seems to be not necessarily clear. In view of the above situation, we shall propose an exactly solvable, but nevertheless, realistic model which exhibits all the characteristic features of the pairing vibration model--near-harmonic level sequences and the strong enhance