Role of dynamical particle-vibration coupling in reconciliation of the d3/2 puzzle for spherical proton emitters

It has been observed that decay rate for proton emission from d3/2 single particle state is systematically quenched compared with the prediction of a one dimensional potential model although the same model successfully accounts for measured decay rates from s1/2 and h11/2 states. We reconcile this discrepancy by solving coupled-channels equations, taking into account couplings between the proton motion and vibrational excitations of a daughter nucleus. We apply the formalism to proton emitting nuclei 160,161Re to show that there is a certain range of parameter set of the excitation energy and the dynamical deformation parameter for the quadrupole phonon excitation which reproduces simultaneously the experimental decay rates from the 2d3/2, 3s1/2 and 1h11/2 states in these nuclei. Typeset using REVTEX 1 Physics of nuclei close to the neutron and proton drip lines is one of the most active and exciting research areas of the current nuclear physics. Nuclei beyond the proton drip line are unstable against proton emission, but, since a proton has to penetrate the Coulomb barrier, their lifetime is sufficiently long to study their spectroscopic properties. Thanks to the recent experimental developments of production and detection methods, a number of ground-state as well as isomeric proton emitters have recently been discovered, which has stimulated many experimental and theoretical works [1–19]. For proton emitters in the A ∼ 150 region, proton emissions from the 1h11/2, 3s1/2, and 2d3/2 orbitals have been observed. It has been pointed out that a spherical calculation based upon a one dimensional optical potential with spectroscopic factor estimated in the BCS approximation systematically underestimates the measured decay half-lives for proton emissions from the 2d3/2 state, while the same model works well for emissions from the 1h11/2 and 3s1/2 states in both odd-Z even-N nuclei and odd-Z odd-N nuclei [5,9]. For the Lu nucleus, this discrepancy was attributed to the effects of oblate deformation of the core nucleus Yb, which alter both the decay dynamics and the BCS occupation probability of the valence proton [9,15,20]. The coupled-channels calculations with β2 ∼ −0.15 have successfully explained the measured decay half-lives for this nucleus [15]. However, for proton emitters such as Ta, Re, and Re, the static deformation parameter of the core nuclei Hf, W, and W is estimated to be β2 = −0.053, 0.080, and 0.089, respectively, based on the macroscopic-microscopic mass formula [21], and thus the deformation effects will be much smaller. Those nuclei are nearly spherical, and vibrational excitations should be considered instead of the deformation effects and the associated rotational excitations [22]. For vibrational nuclei, the excitation energy of collective excitations is in general significantly larger than that for rotational nuclei. This makes the channel coupling effects weaker in spherical nuclei. We notice, however, that the penetration probability is extremely sensitive to other degrees of freedom especially at deep barrier energies, regardless of the value of their excitation energies [23,24]. It is thus of interest and important to explore the role of collective core excitations during proton emission decays of spherical nuclei. The aim of this paper is to solve the coupled-channels equations for spherical proton emitters in order to investigate whether the effects of vibrational excitations of the daughter nucleus consistently account for the measured decay half-life from the 1h11/2, 3s1/2, and 2d3/2 states. We particularly study proton emissions from the 1h11/2 and 3s1/2 states of Re [4] and from the 2d3/2 state of Re [3], assuming that the vibrational properties are identical between the core nuclei W and W. We shall exclude in our study the proton emitter Ta, since the experimental data is somewhat ambiguous due to the competition between the proton emission and the β/EC decay. Since the properties of excited states of these proton-rich nuclei are not known, we shall study the dependence of the decay rate on the excitation energy as well as on the dynamical deformation parameter of the vibrational phonon excitation of the daughter nucleus. We will then extract a possible combination of these two from the experimental data. We will also discuss the dependence on the multipolarity of the phonon state. We consider the following Hamiltonian for the spherical proton emitting systems: H = − h̄ 2μ ∇ + Vcoup(r, α) +Hvib, (1) where r is the coordinate for the relative motion between the valence proton and the daughter 2 nucleus, and μ the reduced mass for this motion. α is the coordinate for the vibrational phonon of the daughter nucleus. It is related to the dynamical deformation parameter as αλμ = βμ √ 2λ+1 (aλμ + (−) aλμ), where λ is the multipolarity of the vibrational mode and aλμ(aλμ) is the creation (annihilation) operator of the phonon. Hvib = h̄ω ∑ μ a † λμaλμ is the Hamiltonian for the vibrational phonon. In this paper, for simplicity, we do not consider the possibility of multi-phonon excitations, but include only excitations to the single phonon state. The coupling Hamiltonian Vcoup(r, α) consists of three terms, i.e., Vcoup(r, α) = V (N) coup(r, α) + V (ls) coup(r, α) + V (C) coup(r, α). The nuclear term reads V (N) coup(r, α) = − V0 1 + exp ( r−R0−R0 αλ·Yλ(r̂) a ) , (2) where the dot denotes a scalar product. We have assumed that the nuclear potential is given by the Woods-Saxon form. Notice that we do not expand the nuclear potential but include the couplings to the all orders with respect to the phonon operator α [25,26]. The matrix elements of this coupling Hamiltonian are evaluated using a matrix algebra, as in Ref. [25]. As for the Coulomb V (C) coup as well as the spin-orbit V (ls) coup terms, the effects of higher order couplings are expected to be small [26], and we retain only the linear term. The Coulomb term thus reads V (C) coup(r, α) = ZDe 2 r + 3ZDe 2 Rc 1 2λ+ 1 ( Rc r )λ αλ · Yλ(r̂) (for r > Rc) (3)