The mutable nature of particle-core excitations with spin in the one-valence-proton nucleus 133Sb

The gamma-ray decay of excited states of the one-valence-proton nucleus 133Sb has been studied using cold-neutron induced fission of 235U and 241Pu targets, during the EXILL campaign at the ILL reactor in Grenoble. By using a highly efficient HPGe array, coincidences between gamma-rays prompt with the fission event and those delayed up to several tens of microseconds were investigated, allowing to observe, for the first time, high-spin excited states above the 16.6 micros isomer. Lifetimes analysis, performed by fast-timing techniques with LaBr3(Ce) scintillators, reveals a difference of almost two orders of magnitude in B(M1) strength for transitions between positive-parity medium-spin yrast states. The data are interpreted by a newly developed microscopic model which takes into account couplings between core excitations (both collective and non-collective) of the doubly magic nucleus 132Sn and the valence proton, using the Skyrme effective interaction in a consistent way. The results point to a fast change in the nature of particle-core excitations with increasing spin.

The structure of atomic nuclei can be viewed from two general and complementary perspectives: a microscopic one, focusing on the motion of individual nucleons in a mean field potential created by all constituents, giving rise to the quantum shell structure, and a mesoscopic perspective that focuses on a highly organized complex system, exhibiting collective behavior. Ideal systems to investigate this duality should be nuclei composed of one valence particle and a doubly magic core in which the coupling between collective core excitations (phonons) and the valence nucleon strongly influences the structure of the system [1]. The understanding of this particle-phonon coupling is of primary importance, being responsible for the anharmonicities of vibrational spectra [1,2], the quenching of spectroscopic factors [3][4][5][6][7] and the reduction of β-decay half-lives in magic nuclei [8]; it is also the key process at the origin of the damping of giant resonances [9]. In general, the coupling between phonons and particles is at the basis of fermionic many-body interacting systems, both in nuclear physics and in condensed matter physics [10].
In reality, in nuclear physics a more complex scenario is realized: collective phonons are not the only excitations at low energy in doubly magic systems -usually states having real phonon character coexist here with excitations that are less collective or have no collective properties. In this respect, a benchmark region is around 132 Sn, which is one of the best doubly magic cores and exhibits low-lying both collective and non-collective excitations. In the present work, we had the goal of investigating the nature of particle-core excitations in the corresponding one-valence-proton nucleus 133 Sb, populated in cold-neutron induced fission. First, we aimed at identifying, experimentally, new high spin yrast states above the long-lived 16.6 µs isomer. This required a demanding technique which relies on measuring coincidences between γ rays prompt with the fission event and those delayed up to several tens of microseconds. Then, we studied transition probabilities through lifetime measurements of selected states. To interpret the data, a new microscopic and self consistent model has been developed, containing particle couplings to core excitations of various nature: in such a heavy mass region this cannot be treated with a shell model * Corresponding author: silvia.leoni@mi.infn.it (SM) approach as it would require full SM calculations in the configuration space that encompasses proton and neutron orbitals below and above 132 Sn [11,12]. We anticipate that experimental data on 133 Sb, in the light of model predictions, provide evidence for a fast change in the nature of particlecore excitations, from a collective character to a non-collective one with increasing spin.
So far, studies of particle-core excitations considered, almost exclusively, couplings with collective phonons of the core. The most known case is the multiplet in 209 Bi (one-proton nucleus with respect to the 208 Pb core) arising from the coupling of a h 9/2 proton with the 3 − phonon of 208 Pb (at 2615 keV), exhibiting one of the largest vibrational collectivity across the nuclear chart (34 W.u.) [1]. In other one-particle (1p) or one-hole (1h) nuclei around 208 Pb [13][14][15] and other magic nuclei [16][17][18][19][20][21][22][23], states originating from couplings of the 3 − phonon with single particle/hole have been located as well. In the past, the theoretical description of particle-phonon couplings relied on phenomenological models [1,2]. Now, microscopic approaches based on either Skyrme forces or Relativistic Mean Field (RMF) Lagrangians are feasible, but with applications limited to the description of single particle states [24][25][26][27] and giant resonances [28,29].
In the case of the 132 Sn core, the first three excitations, 2 + at 4041 keV, 3 − at 4352 keV and 4 + at 4416 keV show a less pronounced collectivity (of the order of 7 W.u.) with respect to the 3 − of 208 Pb, and the other states have 1p-1h character [30,31]. In consequence, the one-valence-proton nucleus 133 Sb, being bound up to 7.4 MeV (unlike 133 Sn with a neutron binding energy of only 2.4 MeV), is a perfect case to test, simultaneously, the coupling of a particle with core excitations of various nature.
On the contrary, no information exists on positiveparity levels above the long-lived 21/2 + isomer, which is known with x < 30 keV uncertainty in energy [33,36].
The γ-ray coincidence data on 133 Sb were obtained with a highly efficient HPGe array, installed in black, the decay below the long lived 21/2 + isomer, known prior to this work (being x < 30 keV the isomer energy uncertainty) [33]; in red, newly identified transitions, above the isomer. The half lives of the 13/2 + and 15/2 + statesdeduced from this work -are also given.
at the PF1B [37] cold-neutron facility at Istitut Laue Langevin (Grenoble, France). The ILL reactor is a continuous neutron source with an in-pile flux up to 1.5×10 15 neutrons cm −2 s −1 . After collimation to a halo-free pencil beam, the capture flux on target was 10 8 neutrons cm −2 s −1 . Two detector setups were used, the first consisting of 8 EX-OGAM clovers [38], 6 large coaxial detectors from GASP [39] and 2 ILL-Clover detectors, with a total photopeak efficiency of about 6% at 1.3 MeV. In the second setup, the GASP and ILL detectors were replaced by 16 LaBr 3 (Ce) detectors, named FATIMA array [40], for lifetime measurements by fast-timing techniques. This is the first time a large HPGe array has been installed around such a high intensity, highly collimated cold-neutron beam [41][42][43]. The campaign, named EXILL, lasted two reactor cycles (each ≈50 days long) and its main part consisted of two long runs of neutron induced fission on 235 U and 241 Pu targets. The use of a fully digital, triggerless acquisition system (with time stamp intervals of 10 ns) allowed event rates up to 0.84 MHz to be handled and to study coincidences among γ transitions separated in time by several tens of microseconds [44] -with analogue electronics, coincidences only across a few µs isomers could be studied with large Ge arrays. In 133 Sb, the 21/2 + isomeric state decays via a cascade of five transitions: an unknown isomeric transition with E γ < 30 keV followed by 62, 162, 1510 and 2792 keV γ rays that feed the 7/2 + ground state (see Fig. 1). Therefore, a search for high spin structures of 133 Sb was undertaken by considering coincidences between two classes of γ-rays: i) prompt γ-rays -coincident (within 200 ns) with a fission event (defined by γ-ray multiplicity equal or larger than 4, within 200 ns) and ii) delayed γ rays -emitted within 20 µs after the fission event and coincident (within 200 ns) with at least one of the four known transitions deexciting the 21/2 + isomer. First, we investigated a prompt-delayed matrix. Fig. 2 (a) and (b) show spectra of γ rays preceding the 16.6 µs isomer, obtained from the 241 Pu and 235 U targets, respectively. The γ rays observed in both data sets at 207.9(4), 318.0(4), and 561(1) keV are candidates for transitions occurring higher in the level scheme of 133 Sb. In addition, by exploiting the prompt-prompt coincidence histogram, constructed in coincidence with a delayed γ ray deexciting the isomer, a new weak 243-keV line was identified in coincidence with the 318-keV transition (see insets of Fig. 2). This line was then placed in cascade with the 318-keV γ ray, depopulating a level located at 561 keV above the isomer. This placement is strongly supported by the existence of a 561-keV transition that was found above the isomer. As the 318-keV transition has much higher intensity then the newly found 243-keV line, it can be placed as feeding the 21/2 + isomer. In this way, we have located three new levels, with energies 4734+x, 4844+x and 5087+x keV, as shown in Fig. 1.
In the second part of this work, fission data from 235 U and 241 Pu targets, taken with the setup including the LaBr 3 (Ce) scintillators, were used to extract the lifetimes of the 13/2 + and 15/2 + states of 133 Sb, by fast-timing techniques [45][46][47]. The analysis was based on triple coincidence events, within a time window of 200 ns, in which two γ-rays are detected in the LaBr 3 (Ce) scintillators and the third one in the Ge array. By setting the very selective gate on the 2792-keV line of 133 Sb recorded in the Ge, a (E γ1 ,E γ2 ,∆t) histogram was constructed, ∆t being the time difference between γ rays with E γ1 , E γ2 energies, measured by the LaBr 3 (Ce) detectors. Figure 3 shows the time distributions used in the analysis of the 13/2 + (a) and 15/2 + (b) states, respectively ( 235 U data set), and associated with the γ-ray cascades shown on the right [46]. All time spectra are background subtracted by considering a two-dimensional gate in the energy plane, around the corresponding coincidence peak (as shown in the insets). The time difference ∆C between the centroids of the time distributions provides the lifetime τ of the state (2τ = ∆C -PRD), after correction for the prompt response difference (PRD) [46]. From the 235 U data, half-lives T 1/2 = 32(10) ps and < 17 ps were deduced for the 13/2 + and 15/2 + states, respectively, in very good agreement with the 30(12) ps and < 21 ps values obtained from the 241 Pu data. This gives the average values T 1/2 = 31(8) ps and < 20 ps. Taking into account the decay branchings from the two levels [33], B(M1) values were extracted for the 15/2 + →13/2 + and 13/2 + →11/2 + transitions, yielding > 0.24 W.u. and 0.0042(15) W.u., respectively. This large difference, of almost two orders of magnitude, is clearly intriguing and brings a signature of some non-trivial change of configuration mixing in the 11/2 + , 13/2 + and 15/2 + states of 133 Sb, respectively.
In order to interpret the experimental findings, a new microscopic model has been developed with the aim of describing states with different degrees of collectivity. In our model we solve the Hamiltonian where we have written for simplicty j instead of nlj, and the phonons with angular momentum J are labelled by the index n. Our calculation has no free parameters and is self-consistent in the sense that both single-particle states and phonons come out of Hartree-Fock (HF) and Random Phase Approximation (RPA) calculations performed with the Skyrme SkX interaction [48]. This Hamiltonian can be diagonalized separately in different Hilbert subspaces with good angular momentum j and parity π. In each of these subspaces we have both one-particle states and so-called one particle-one phonon states. However, some of the so-called phonons turn out to be pure 1p-1h states, as it can be expected. We label all states as phonons just in keeping with the fact that they come out of the RPA diagonalization, but the word "excitation" would indeed be more appropriate. Table 1 shows the calculated excitations of the 132 Sn core, in comparison with known experimental data; we presently include those up to 5.5 MeV, together with the proton states of the 50-82 shell. We call our model "Hybrid Configuration Mixing" (HCM) and its details, as well as a detailed account of the results, will be published elsewhere.
The key point is that the correction for the nonorthonormality of the basis is taken into account [49] by solving the generalized eigenvalue problem (H − E λ N ) |λ = 0, where N is the overlap matrix between the basis states. Figure 4 shows, in the bottom panel, the calculated yrast and near yrast states of 133 Sb, arising from the coupling between the valence proton and core-excitations. A number of features is evident. The model does reproduce well the energy sequence of the high-spin states observed experimentally. Also, a fast evolution of the wave function composition is seen, from complex to non-collective character, with increasing spin. As shown in the top panels, the low spin states are dominated by the g 7/2 proton coupled to the 2 + phonon, while the highest spin excitations arise mostly from this valence proton coupled to the neutron h −1 11/2 f 7/2 noncollective core excitation. The states in between, at spin 13/2 + and 15/2 + , show instead a fragmented wave function involving the coupling of the valence proton to both the 4 + phonon and non-collective particle-hole excitations.  The states located in the present work above the 21/2 + isomer, at 4.844+x and 5.087+x MeV (being x<30 keV), clearly correspond to the excitations calculated at 4.83 and 5.11 MeV. As seen in Fig. 4, they arise from almost pure (>95%) configurations of πg 7/2 νf 7/2 h −1 11/2 character. One has to note that the existence of two yrast states above the 21/2 + isomer with spin-parity assignments of 23/2 + and 25/2 + were suggested on the basis of the shell-model calculations with adjusted empirical interactions by W. Urban et al. [32,33]. Similar sequences of states involving neutron particle-hole νf 7/2 h −1 11/2 excitations of the 132 Sn core have been identified above 4 MeV in the neighboring nuclei 134 Sb, 134 Te and 135 Te [50,51]. A characteristic feature of these multiplets is that their members are often connected by M1 and E2 competing transitions. The decay of a level located at 5.087 MeV, and assigned as the highest spin member 25/2 + of the πg 7/2 νf 7/2 h −1 11/2 configuration, has M1 and E2 branches which supports the suggested spin-parity assignments.
Concerning the two M1 transition probabilities that have been measured in the present work, theory provides the values of 0.021 W.u. and 0.001 W.u. in the case of 15/2 + → 13/2 + and 13/2 + → 11/2 + , respectively. The large ratio of ≈ 20 between the two values is in qualitative agreement with the experimental value of ≈ 60, and can be well understood from the point of view of our model, while it would not come out within the simple picture of Refs. [32,33]. The 15/2 + and 13/2 + states have very similar composition of the wave function, and the largest component is the 2p-1h configuration (πg 7/2 νh −1 11/2 f 7/2 ) which has an amplitude of the order of 0.4. The transition matrix element 13/2 + ||O(M 1)||15/2 + , that is 0.78 eµ N , would become 4.86 eµ N if a pure (πg 7/2 νh −1 11/2 f 7/2 ) component were assumed for both states, as in the simplified shell model of Refs. [32,33], leading to a B(M1) of 0.78 W.u. If we further assumed the same purity for the 11/2 + state, the transition matrix element would remain approximately the same with an associated B(M1) of 0.72 W.u. Instead, in our model, the compositions of the 13/2 + and 11/2 + states are very different (see Fig. 4), leading to a much more quenched value of the B(M1).
In summary, for the first time, excited states in 133 Sb were observed above the 21/2 + , 16.6 µs isomer, up to (25/2 + ), and lifetimes of the yrast excitations 13/2 + and 15/2 + were measured. To describe the structure of 133 Sb, a microscopic model named Hybrid Configuration Mixing Model" (HCM) was developed, which includes couplings with various types of core excitations. The model reproduces very well the energies of the observed excitations in 133 Sb and provides explanation for the large differences in M1 strength for transitions connecting neighbouring medium spin yrast states. The HCM model is, therefore, a very promising tool for describing low-lying spectra of odd nuclei made of a magic core and an unpaired nucleon. The present work calls for complementary studies with direct reactions to assess spectroscopic factors of single particle states of 133 Sb, in analogy with the case of the one-valence-neutron nucleus 133 Sn [7].  Figure 4: (Color online) Bottom panel: experimental and calculated energies (thick and thin lines, respectively) of the lowlying positive parity states of 133 Sb, in the spin range 9/2 + -25/2 + . In the calculations, other states below 5.5 MeV are shown at 19/2 + , 21/2 + and 23/2 + . Top panel: the components of each lowest state (with amplitude larger than 0.01), with configurations indicated in the legend. Lifetimes for the 13/2 + and 15/2 + states were measured in this work.