Evidence of octupole-phonons at high spin in 207 Pb

A lifetime measurement of the 19 / 2 − state in 207 Pb has been performed using the Recoil Distance Doppler-Shift (RDDS) method. The nuclei of interest were produced in multi-nucleon transfer reactions induced by a 208 Pb beam impinging on a 100 Mo enriched target. The beam-like nuclei were detected


Editor: D.F. Geesaman
Keywords: AGATA spectrometer γ -Ray tracking VAMOS++ spectrometer Plunger device Nuclear deformation Octupole phonon and identified in terms of their atomic mass number in the VAMOS++ spectrometer while the prompt γ rays were detected by the AGATA tracking array. The measured large reduced transition probability B(E3, 19/2 − → 13/2 + ) = 40(8) W.u. is the first indication of the octupole phonon at high spin in 207 Pb.
An analysis in terms of a particle-octupole-vibration coupling model indicates that the measured B(E3) value in 207 Pb is compatible with the contributions from single-phonon and single particle E3 as well as E3 strength arising from the double-octupole-phonon 6 + state, all adding coherently. A crucial aspect of the coupling model, namely the strong mixing between single-hole and the phonon-hole states, is confirmed in a realistic shell-model calculation. The occurrence of collective vibrations, when a lattice of atoms or molecules oscillates uniformly at a single frequency forming a quantum-mechanical phonon, is a well-known phenomenon. Such vibrations correspond, in classical mechanics, to wave-like normal modes. Quantum-mechanical phonons, however, exhibit particlelike properties, too. The excitation spectra of several different many-body systems can be described as a superposition of elementary excitation modes that are (approximately independent) fluctuations about equilibrium. There is a close relation between the internal structure of the system and the nature of these fluctuations, which may lead to density vibrations or shape oscillations. In nuclei the character of collective vibrations follows from the observation that some are spherical, like doubly-magic nuclei, while others are deformed, like most rare-earth nuclei. In an intermediate situation the shape can undergo large fluctuations about one of the equilibrium shapes. In contrast to molecules, the nuclear energy scales related to vibrational and single-particle excitations are of the same order, and thus their interweaving has profound consequences.
Doubly-magic nuclei have a spherical equilibrium shape. Among them, the 208 Pb isotope, with Z = 82 protons and N = 126 neutrons, is the heaviest known doubly-magic nucleus. Its first-excited state has been established to be of natural-parity octupole type, J π = 3 − c , at an excitation energy of E x (3 − c ) = 2615 keV, about 800 keV lower than the neutron shell-gap energy at N = 126, the index c stands for collective. The highly enhanced and collective transition connecting the 3 − c level to the 0 + ground state has been measured to have a reduced transition probability of B(E3, 3 − c → 0 + ) = 34.0(5) W.u. [1], that is, it exceeds by 34 times the Weisskopf unit or single-particle estimate. The 3 − c state is interpreted as a one-phonon excitation corresponding to a nuclear surface vibration of octupole character while its microscopic structure is understood as the coherent and collective superposition of one-particle-one-hole (1p-1h) excitations across the neutron and proton shell gaps.
Provided that this 3 − c state represents the first phonon of the octupole vibration, it is expected that the double-octupole quar- [2]. In the case of a fully harmonic vibration, all members of this quartet, and in particular the 6 + c level, decay to the one-phonon state with the characteristic reduced transition probability have been undertaken to identify the members of the two-phonon octupole quartet [3][4][5][6][7][8][9][10][11]. Candidates for the lower-spin members have been proposed [10,11] but the 6 + c member has not been identified as yet. On the basis of a large-scale shell-model calculation, including up to 2p-2h excitations, Brown [12] concluded that the 6 + c member of the double-octupole quartet is fragmented. Furthermore, he found that there are 0 + , 2 + , and 4 + states with a concentrated double-octupole strength but decaying via weak E1 and E2 transitions, which in themselves are not strong evidence for the special double-octupole nature of a state.
In the nuclei neighboring 208 Pb, with one valence particle or hole, the particle-octupole-phonon model favors strong coupling between the orbitals j 1 = l 1 ± 1/2 and j 2 = l 2 ± 1/2 if | j 1 − j 2 | = |l 1 − l 2 | = 3, preserving the relative orientation of the spin and orbital angular momenta [13,(Vol. II,p. 419)]. In addition to the particle or hole states, several excitations have been found and interpreted as a collective octupole phonon |3 − c coupled to a particle or hole. Because of the strong coupling mentioned above, such states are expected to mix i.e. | j , the latter being a particle or hole coupled to a double-octupole phonon. Given this mixing, it has been suggested in Ref. [14], in analogy to the case of 147 Gd [15], that the characteristic enhancement of the 208 Pb, should be reflected in an enhanced B(E3, J 2 → J 1 ) value in the odd-mass nucleus.
For 207 Pb, among the available neutron orbitals, p 1/2 , p 3/2 , f 5/2 , f 7/2 , h 9/2 and i 13/2 , forming a major shell 82 ≤ N ≤ 126, only the j 1 = νi 13/2 and j 2 = ν f 7/2 satisfy the strong coupling rule, described above. The corresponding states, 13/2 + and 7/2 − , dominantly of single-hole character, are well studied [18]. The 19/2 − state and the corresponding 2485 keV transition to the 13/2 + state were assigned to 207 Pb by Schramm et al. [6], and the E3 character of the transition was recently determined by Shand et al. [19]. The 13/2 + , 7/2 − , and 19/2 − states were analyzed in terms of particle-octupole-vibration coupling in Ref. [14] using the experimentally known level energies and assuming the dominance of the above-mentioned orbitals. This coupling scheme is depicted in which can be calculated in a variety of ways. Its absolute value can be deduced from the excitation energies of the 13/2 + , 7/2 − , and 19/2 − levels in 207 Pb, leading to |h| = 0.725 MeV [14]. Alternatively, it is obtained in the context of the particle-vibration coupling model [13, (Vol. II, p. 418)], where it depends on the radial overlap of the particles and the oscillating potential at the surface of the nucleus and the zero point amplitude of the nuclear vibration. Hamamoto [20] for the case of 207 Pb obtained the value of h = 0.710 MeV. Finally, it can also be calculated with the shellmodel expression, where the particle-hole matrix elements can be obtained from particle-particle matrix elements using the Pandya transformation [21] which gives the separate contributions of the neutron-neutron (νν) and neutron-proton (νπ ) interactions. The sums are over the neutron and proton particle-hole excitations that constitute the octupole phonon. The amplitudes a ν kk and a π ll are obtained microscopically in a shell-model calculation for 208 Pb with 24 single-particle energies taken from Ref. [22] and with the realistic nucleon-nucleon interaction as given in Ref. [23]. Although the off-diagonal matrix elements in the expression for h generally are small and of varying sign, multiplied with amplitudes they act coherently, giving rise to a large mixing matrix element with the value of h = 0.655 MeV. This is the hallmark of collective behavior, which therefore is found to be present in a realistic shell-model description. The consistency of the values for the mixing matrix element derived with three totally different approaches lends support to the hole-octupole-phonon interpretation of states in 207 Pb.
In the following the experimental value of h = 0.725 MeV is used.
The experimental value of h = 0.725 MeV was determined assuming the contribution of the collective vibrational-phonons to the 13/2 + and 19/2 − states. Due to the presence of the specific orbits, the f 7/2 and i 13/2 for 207 Pb, a strong particle-octupolevibration coupling is expected to attract an admixture of the double octupole state to the low-lying yrast 19/2 − state, that can decay by the characteristic enhanced E3 transition. The main part of the double octupole state remains however in the higher lying 19/2 − , which could be fragmented and have different decay The measurement of sub-nanosecond lifetimes of high spin states in nuclei near 208 Pb is very challenging. These high spin states can be efficiently populated in multi-nucleon transfer reactions of heavy ions at the energies near the Coulomb barrier [6,14,23]. The excited products of interest are distributed near the grazing angle, far away from the beam axis, in contrast to fusion reactions. Multi-nucleon transfer reactions produce hundreds of nuclei at the same time, therefore some selection of the reaction products is required. It can be obtained using γ − γ coincidences or using a mass analyzer or magnetic spectrometer to determine the mass number. The direct measurement of the atomic number at Z ∼ 82 for low energy ions is not possible today. Mass analyzers have typically low acceptance and are restricted to operation near 0 • . Further, the use of the plunger technique, for measurement of sub-nanosecond lifetimes of states populated in multi-nucleon transfer reactions, requires an event-by-event measurement of the recoil velocity vector. In this work the VAMOS++ spectrometer was used to identify, for the first time, the atomic mass number of the lead-like ejectiles at energies ranging from 1 to 2 MeV/u. The required mass resolution was reached only for a part of the focal plane setup (∼ 15%), which resulted in reduced statistics. In this letter we present the results of the first lifetime measurement of the J π = 19/2 − level in 207 Pb, proving the oneoctupole phonon nature of this state and suggesting the existence of a double-octupole 6 + c state in 208 Pb. The experiment was performed at the Grand Accélérateur National d'Ions Lourds, Caen, France using the RDDS method [24], in combination with a multi-nucleon transfer reaction in inverse kinematics. A 208 Pb beam at 6.25 A MeV impinged on an enriched 1.9 mg/cm 2 -thick 100 Mo target followed by a 2 mg/cm 2 -thick Ni degrader. Beam-like reaction products were detected and identified on an event-by-event basis in the large-acceptance VAMOS++ spectrometer [25,26]. The optical axis of the spectrometer was positioned at 26 o with respect to the beam axis, at the grazing angle of the beam-like products. The VAMOS++ spectrometer allowed the identification of the reaction products in mass-over-charge ( A/Q ) and atomic charge ( Q ), and provided the velocity vector ( V ) necessary for the Doppler correction. Fig. 2 shows a typical two-dimensional identification matrix obtained in the present experiment. The X -axis represents the mass-  over-charge ratio as a function of the atomic-charge state. Mass resolution of ∼ 0.9/208 (FWHM) was obtained. The analysis procedure is further detailed in Ref. [27]. Excited-state half-lifes (T 1/2 ) were measured using the RDDS technique with the plunger device of the University of Cologne [28]. Doppler-corrected prompt γ rays, emitted before and after the Ni degrader foil, were measured by the HPGe AGATA tracking array [29,30] placed at backward angles in a compact geometry (target-to-detector distance of 148.5 mm).
The γ -ray energy Doppler correction was performed using the recoil velocity ( V ), obtained from the VAMOS++ spectrometer, after the Ni degrader, and the position of the first γ -ray interaction obtained from the Orsay Forward Tracking algorithms using standard parameters [31]. it is a contaminant from the random coincidence resulting from the inelastic scattering of the beam. The transitions marked with a circle correspond to the 100 Mo decay following Coulomb excitation, Doppler corrected using the velocity vector of the beam-like ion, measured after the degrader. Fig. 4 shows Doppler corrected γ -ray spectra measured in the AGATA spectrometer, selected on mass A = 207 in the VAMOS++ spectrometer, for five target-to-degrader distances (75 μm, 200 μm, 625 μm, 1000 μm, and 2000 μm) for the relevant transitions used for the lifetime measurement. Since the Doppler correction used the velocity measured after the degrader, the unshifted (U) component corresponds to the events where the γ -ray was emitted after the degrader and shifted (S) to the events where gamma-ray was emitted before the degrader. The velocity of ions detected in VA-MOS++ ranged from 14 to 22 μm/ps, and the decrease of the velocity in the degrader was typically about 13%. Events with a relative angle greater than 138 • , between the γ -ray and the outgoingparticle velocity vector, were selected to enhance the clear separation between the shifted (S) and unshifted (U) components of the γ -ray transitions. The parameters required for the Doppler correction using the AGATA and VAMOS++ spectrometers were obtained using the inelastic scattering of the 208 Pb in a data set without the thick Ni degrader. On the left panel of Fig. 4 Fig. 4). The γ -ray transition intensities were determined assuming for all distances the same width and centroid for the peaks. The normalization using the sum of the shifted (S 19/2 − ) and unshifted (U 19/2 − ) components of the 2485 keV transition is in agreement, within the statistical uncertainties, with the normalization using the 7/2 − 1 → 5/2 − 1 transition. The former has a higher statistical error due to the weak intensity of the shifted (S 19/2 − ) component. In the following, all quoted errors are statistical. In agreement with the level scheme of 207 Pb [19], γ -γ -coincidence analysis showed two transitions above the 13/2 + state populating the 19/2 − state: the 21/2 − → 19/2 − and 23/2 − → 19/2 − transitions with the respective energies of 592 keV and 749 keV and feeding of 20(6)% and 37(5)%, respectively. These states, having a very long effective lifetime, are taken into account in the analysis, following the method described in Ref. [24]. The lifetime was extracted from the first three distances where the RDDS analysis showed maximum sensitivity. The lifetime analysis procedure was verified using the known decay of the 2 + feedings from the 3 + and 4 + states, in reasonable agreement with the published value.
The result of the lifetime analysis for the 19/2 − state decaying by the 2485 keV transition in 207 Pb is presented in Fig. 5. The  Fig. 1(d) and with the two-to-one-phonon strength   The coefficients α and β can be taken from the analysis in [14].
With Woods-Saxon radial wave functions and an effective charge in 208 Pb. All contributions, including the two-phonon state, add coherently to reach maximum collectivity. The measured value is compatible, within the error bar, with the predicted value. However the experimental uncertainty remains too large to disentangle the presence of the strong particle-octupole coupling and the twophonon state. To achieve this goal, the experimental uncertainty for the case of 207 Pb has to reach at least the level of 2%. Further, a more precise determination of the effective charge, which is a main source of uncertainties in the calculations, would be required.
In summary, a large B(E3, 19/2 − → 13/2 + ) = 40(8) W.u. reduced transition probability has been measured in 207 Pb based on the lifetime measurement of the 19/2 − state using the RDDS technique. Such collective character indicates that the dominant component of this state is a single-hole excitation coupled to the octupole phonon of the 208 Pb core. The energy lowering of the 2485 keV transition in 207 Pb, as compared to the 2615 keV transition in 208 Pb, is consistent with a mixing with a state containing the double-octupole-phonon excitation. The measured reduced transition probability is compatible with a contribution from the twoto-one-octupole-phonon E3 transition. Further information on the double-octupole-phonon state can be obtained by a more precise lifetime measurement of the 19/2 − state in 207 Pb or of the corresponding 21/2 + state in 209 Pb, where the B(E3) was predicted to be 50 W.u. [14]. In addition, a more accurate measurement of the lifetime of the 15/2 − state in 209 Pb is mandatory to improve the precision of the E3 effective charge.