Core-coupled states and split proton-neutron quasi-particle multiplets in Ag
Neutron-rich silver isotopes were populated in the fragmentation of a Xe beam and the relativistic fission of U. The fragments were mass analyzed with the GSI Fragment separator and subsequently implanted into a passive stopper. Isomeric transitions were detected by 105 HPGe detectors. Eight isomeric states were observed in Ag nuclei. The level schemes of Ag were revised and extended with isomeric transitions being observed for the first time. The excited states in the odd-mass silver isotopes are interpreted as core-coupled states. The isomeric states in the even-mass silver isotopes are discussed in the framework of the proton-neutron split multiplets. The results of shell-model calculations, performed for the most neutron-rich silver nuclei are compared to the experimental data.
pacs:21.10.-k, 21.10.Hw, 21.10.Re, 21.10.Tg, 23.20.Lv, 23.35.+g, 27.60.+j
The Nuclear Shell model MGM48 () was introduced in the mid-20 century and its major success was the description of the magic numbers. Recently, it was suggested that for extremely neutron-rich nuclei the ordering of the orbits, and hence the magic numbers as we know them from nuclei close to the line of stability, may change due to diffuseness of the nuclear surface Do94 (); SF02 (). Further, it was pointed out that an experimental fingerprint of the changing structure may be found in the ratio systematics for the neutron-rich even-even nuclei, because in the case of weakening of the term in the shell model potential, the ratio would strongly depend on the occupation of the single-particle orbits Ch95 (). The shell quenching was also suggested to be the origin of the poor theoretical description of the -process abundance in the region below the peak Pf97 (); Di03 ().
However, recent studies performed at GSI show no need of shell quenching to explain the structure of Cd Ju07 (). It was pointed out that if a neutron shell erosion is present in the isotonic chain, it should take place deep below Sn in the proton shell. Among the candidates for strongly pronounced shell quenching effects are the neutron-rich zirconium nuclei SF02 (). Of particular interest is Zr which would be a magic nucleus in terms of the “classic” magic numbers. This nucleus is far from being accessible experimentally, but given that the degree of collectivity depends on the number of valence particles, a change in the neutron magic numbers would also affect the structure of the neutron mid-shell zirconium nuclei. To search for deviations from the mid-shell behavior, IBM-1 calculations were performed La09 () assuming the persistence of the =50 and 82 magic numbers. The spectroscopic observables, level energies and transition strengths in Zr were predicted using a Hamiltonian parametrized with respect to nuclei lying close to the line of -stability. Further, the excited states in Zr were experimentally observed from -decay studies performed at RIKEN Su11 (). The experimental level energies are in good agreement with the model predictions suggesting that Zr is indeed a mid-shell nucleus. In Zr Su11 (), an isomeric decay was also experimentally observed, revealing the energy of the first excited state. It was shown, that in the zirconium isotopic chain the state gradually increases in energy from the mid-shell nucleus towards the next neutron magic number. Even though the evolution of the states in the neutron-rich zirconium isotopes hints at the preservation of the “classical” magic numbers when departing from the neutron mid-shell Zr nucleus, more experimental data are needed to draw firm conclusions.
A different approach to the problem is the systematic analysis of the states in the odd-mass and odd-odd nuclei and in particular the positioning of the unique-parity states relative to the normal-parity states. These high- intruder states appear at the upper part of the shells, where low- normal parity states are present, and are often responsible for the islands of isomerism emerging in the vicinity of the magic numbers. Also, in the odd-odd nuclei at the top of the shells, long-lived isomers emerge from the structure of the split proton-neutron multiplets Pa79 (). Thus, the islands of isomerism in specific mass regions are directly related to the existence of the magic numbers and the identification of the opposite parity states in the odd-A and odd-odd nuclei can give a direct measure of whether the shells are quenched or not.
In the region below Sn, the unique-parity orbits are and arising from the fourth proton and fifth neutron oscillator shell, respectively, leading to the appearance of positive-parity states in the odd-A (Z=47) silver isotopes and states of opposite parities in the even-mass Ag nuclei. The interplay between these unique-parity and the normal-parity orbits often gives rise to isomers close to the ground state. A well-known example is Ag, which has a -decaying isomer with a half-life of 160 ms and a 46-ms ground state Kr00 (), leading to a Ag stellar half-life of the order of 80 ms. Such long-lived isomerism can shed light on the observed -process overabundance in the mass region. Therefore, of particular interest is the search for isomeric states in the most neutron-rich odd-mass and odd-odd nuclei, just below the doubly magic Sn nucleus.
Prior to this study, the shell structure in nuclei around Sn was studied in a number of experiments on isomeric decays. The Sn nuclei were studied at LOHENGRIN Pi00 () and during the -RISING campaign Lo08 (). Isomeric decay studies were performed for Sn Pi11 (), Cd Ju07 (); Ca09 (); Na10 () and In Go09 () nuclei in the RISING Stopped beam campaign Re08 () and for In at LOHENGRIN Sc04 (). The present work extends these studies towards the most neutron-rich silver nuclei.
Ii Experimental Set Up
The neutron-rich Ag nuclei were produced during two experiments performed at GSI, Darmstadt, using the fragmentation of a Xe beam Ju07 (); Ca09 (); Na10 (); Go09 () and the fission of a U beam Br10 () at relativistic energies. In these experiments, the beams were accelerated to 750 MeV/A by the SIS-18 synchrotron and impinged on Be targets of 1 and 4 g/cm thickness, respectively. The cocktail of fragments, produced in the fragmentation or fission, was analyzed with the GSI FRagment Separator (FRS) Ge92 (). The ions were separated by means of their magnetic rigidities, times of flight, energy losses and their positions in the middle and final focal plane of the separator. The nuclei were slowed by an aluminum wedge shaped degrader and implanted into a copper or plastic stopper, placed at the final focal plane. Delayed -rays were detected by the RISING multidetector array Pi07 (), comprising 105 HPGe detectors, mounted as 15 Cluster detectors. The signals were digitized by Digital Gamma Finder (DGF) modules providing energy and time information.
Iii Experimental Results and Data Analysis
iii.1 Odd- nuclei
Prior to the current study, isomer-delayed transitions were observed in neutron-rich Ag St09 (). Although the low-energy transitions directly depopulating the isomeric state were not seen in that work, half-lives of ns and ns were deduced from the distributions of the high-energy transitions de-exciting states below the isomers in Ag and Ag, respectively. An overall deviation of 3 keV in the -ray energies reported in St09 () with respect to the present values is observed. Alternative level schemes from isomeric decays, were also presented in a PhD thesis To06 (). The present work gives revised and extended level schemes of Ag, available due to the superior efficiency of the RISING multidetector array which enables -ray coincidences to be clearly established. Preliminary results on the isomeric decays in Ag were reported in La11 (), but a manuscript was not sent for the conference proceedings. During the completion of the present manuscript, a revised level scheme of Ag was published in Ka12 ().
The nucleus Ag was produced in both experiments and observed in the FRS setting centered on the transmission of fully stripped Rh (fission) and Cd (fragmentation) ions, respectively.
Fig. 1 shows a -ray spectrum observed in coincidence with Ag ions, within a 3.75 s wide coincidence window opened 125 ns after the implantation. The most intense transitions, presented in Fig. 1, were previously reported and placed in level schemes St09 (); To06 (). The inset of Fig. 1 shows a summed time spectrum for the strongest transitions in Ag. The half-life of the isomeric state (16) ns was deduced from the fit to the slope. This value is in good agreement with 396(37) ns measured in St09 () and within with 0.32(3) s To06 ().
Sample coincidence spectra are shown in Fig. 2. Fig. 2(a) shows that the 713-keV transition is in strong coincidence with the 685-keV transition, and in weaker coincidence with the 732-keV, 335-keV and 349-keV transitions. In Fig. 1, the 349-keV peak is strong, while the 335-keV peak is weak, which suggests that the 349-keV transition feeds a level, which subsequently decays via a branch of transitions, some of which are parallel to the 713-keV transition. This interpretation is supported by Fig. 2(b) which shows the 349-keV line in coincidence with the 1049-keV and 1076-keV transitions. The 1049-keV and 1076-keV transitions are observed to be in mutual anti-coincidence. The energy spectrum, presented in Fig. 2(c) shows a peak with an energy of 629 keV, which is in coincidence with the 769-keV -ray. The 769-keV and 629-keV transitions are not observed to be in coincidence with the 713-keV and 685-keV transitions.
Transition not placed in the level scheme.
Transition not observed experimentally but deduced from the intensity balance at the 1425-keV level. The relative intensity is calculated assuming an multipolarity for the 48-keV transition and multipolarity for the de-exciting transitions.
In the present work, the weak 732-keV transition is observed in anti-coincidence with the 685-keV transition, which suggests an ordering of the 685-keV and 713-keV transitions opposite to that suggested in St09 (). Also, no experimental evidence was found for the 714-717-keV doublet reported in Ref. St09 (). A level scheme based on the coincidence studies performed in the present work, is shown in Fig. 3. Table 1 lists the -ray energies and intensities, as observed in the present study.
The delayed coincidence method An82 () was used to estimate the half-lives of the 656-keV and 740-keV levels. It shows that the half-life of the two excited states is shorter than the DGF time binning, which is 25 ns per channel. Such a short half-life is consistent with a dipole or quadrupole nature for the 629-keV and 713-keV transitions.
Coincidence spectra for the weak 382-keV, 390-keV and 593-keV transitions are shown in Fig. 4. The three -rays are in mutual coincidence. A half-life of (20) ns is obtained from the fit to the summed time distributions for the 382-keV, 390-keV and 593-keV transitions. In the level scheme, this sequence is placed in parallel to the -rays de-exciting the 393-ns isomer, given that no experimental evidence was found for coincidences between the two branches.
Two weak transitions with energies of 741 keV and 1008 keV are also observed in coincidence with the Ag ions, as shown in Fig. 1. Due to the poor statistics no evidence for coincidences with the strongest gamma transitions was found. However, the two unplaced transitions may be related to a weak and fragmented -decay branch de-exciting the 1076-keV level. Such a scenario is supported by the observed intensity dis balance for the 1076-keV level.
The spin and parity assignments to the states in Ag are based on the -ray decay pattern, observed in the present work, and on the systematics.
: The systematics in Fig. 5 show that is the ground state in the silver nuclei, while is the first excited state and is the second excited state. becomes the ground state in Ag Fr09 () and is the lowest lying positive parity state until Ag St09 (). The ground state for the silver nuclei is . The level energy in the heavier silver isotopes is unknown. Thus, based on the systematics, , and are the most probable candidates for the ground state in Ag. However, because of the strong -decay feeding to the 264-keV level in Cd, the and assignments to the ground state have been ruled out Oh04 (). Given also that the fission process populates mainly yrast states, is assigned to the lowest lying state of the strongly populated sequence, de-exciting the 393-ns isomer. Again, based on the systematics, is adopted for the 27-keV level.
: Fig. 5 shows that the 11/2 state appears at approximately 600 keV above the state and that it is correlated to the level energies in the even-even cadmium nuclei. Also, in all silver isotopes with more than two valence neutrons outside the shell closure the 11/2 state is lower in energy than the state. Therefore, based on the systematics, and were assigned to the 656-keV and 740-keV levels, respectively. These assignments are also consistent with the dipole or quadrupole nature of the 629-keV and 713-keV transitions, discussed above.
: The 1473-keV level decays via a weak 732-keV transition to the () state, but not to the levels with lower spin. Therefore, can be expected for the 1473-keV state. Similarly, the 1425-keV level does not decay directly to the level. Therefore, a tentative assignment can be made for this level. Furthermore, in order for the 1473-keV level to be an isomeric state, a low-energy transition is expected to compete with the 732-keV transition. Such a transition would be a 48-keV transition linking the 1473-keV level with the 1425-keV state. For a 48-keV transition of or nature the conversion coefficients calculated with BrIcc Bricc () are and and the hindrance factors would be and W.u., respectively. Both values are consistent with the hindrance factor systematics in ref. La11 (), but for higher multipolarities the 48-keV transition would be enhanced from three to nine orders of magnitude and hence very unlikely. Also, the intensity balance, performed for the 1425-keV level, suggests that the 48-keV -ray intensity will be 69% of the intensity of the 713-keV transition in case of multipolarity, and 7.5% if it is an transition. Because, the 48-keV transition is not observed in the present experiment, multipolarity was adopted. Hence, and assignments are made to the 1473-keV and 1425-keV states, respectively. Thus, assuming pure and nature of the isomeric transitions in Ag, W.u. and W.u. A similar isomeric decay has been observed in In with =3.3(5) W.u. Sc04 ().
: Given that the 1076-keV level decays to the ground state and is fed by the 349-keV transition from the state, a assignment to the level was made.
In the present work, Ag was observed in the Rh FRS setting from U fission and in the Cd FRS setting from the fragmentation of the Xe beam. Fig. 6 shows -rays observed in coincidence with the Ag ions within a 3.85 s wide time window opened 125 ns after implantation. A group of four intense transitions at energies of approximately 700 keV and a strong 103-keV -line is observed. Coincidence spectra are shown in Fig. 7. Fig. 7(a) presents coincidences between the 103-keV and the 728-keV transitions.
The 765-keV -rays are only in coincidence with the 714-keV -rays, as shown in Fig. 7(b).
Fig. 7(c) shows coincidences between the 683-keV and 714-keV transitions and Fig. 7(d) – the coincidences between 670-keV and 728-keV transitions. Fig. 7(e) shows that the 103-keV -rays are in coincidence with the most intense -rays but not in coincidence with the 765-keV and 787-keV transitions.
The level scheme, presented in Fig. 8, is based on the coincidences observed in the present work. The energy difference between the 1501-keV and 1478-keV levels is 23 keV, and hence impossible to detect with the set up used. Also, because of the poor statistics, no half-life information was obtained from the time distribution of the 765-keV transition, which makes it difficult to determine whether the 1478-keV level is fed by the 1501-keV isomer or it is a different isomeric state.
The four transitions with energies of 670 keV, 683 keV, 714 keV and 728 keV were previously observed and placed in a level scheme St09 (). Based on the coincidences between the 787-keV and 714-keV -rays, the ordering of the 714-keV and 683-keV sequence is inverted with respect to St09 ().
The inset of Fig. 6 shows a summed time spectrum for the isomeric state in Ag and the fit to the slope of the distribution gives a half-life of 474(35) ns for the 1501-keV isomeric state. The half-life, obtained in the present work, overlaps with 473(111) ns given in St09 () and with 0.44(9) s from To06 () and is consistent with s in Ka12 ().
The delayed coincidence method An82 () was used to estimate the half-life of the 714-keV and 670-keV levels. It showed that the half-life of the two excited states is shorter than the DGF time binning. This observation is consistent with a dipole or a quadrupole nature of the 714-keV and 670-keV transitions.
Several weak lines with energies of 74 keV, 129 keV, 155 keV, 187 keV, 215 keV, and 311 keV were observed in coincidence with the Ag ions. The 311-keV, 215-keV and 187-keV lines were observed in mutual coincidence and in a tentative coincidence with a weaker 147-keV transition as shown in Fig. 9(a), (b) and (c). The half-life of 80(17) ns was deduced from the slope of the time spectrum shown in Fig. 9(d). Coincidences between this group of transitions and the transitions de-exciting the 474-ns isomer were not observed, which enables the placement of this group of transitions in parallel to the main branch.
Not placed in the level scheme.
The -ray energies and intensities are listed in Table 2. An intensity imbalance between the 714-keV and 683-keV transitions was reported in St09 (), with the 683-keV transition being the stronger. The analysis, performed in the present work, shows indeed an imbalance of 17%, but in the opposite direction. However, the 765-keV and 787-keV transitions were also observed in coincidence with the 714-keV transition, helping to balance the intensity for the 714-keV level. The intensity balance for the 1398-keV level suggests that the 103-keV transition is converted with , which is close to the theoretical value of 1.3 for an transition calculated with BrIcc Bricc ().
In Ag, the spin and parity assignments are based on the observed -decay pattern, on analogy with Ag, and on the systematics presented in Fig. 5. In contrast to the level scheme proposed in St09 (); Ka12 (), was assigned to the lower-lying excited state in Ag. For the 1501-keV level, the =1.18(11) W.u. and W.u. values, calculated with and Bricc (), are consistent with the hindrance factor systematics La11 ().
iii.2 Even-A Ag nuclei
Prior to our study two isomeric states with half-lives of 0.20 s and 0.55 s and and were observed in Ag Kr00 (); Ta07 (), but no isomeric decay transitions had been seen. The present work reports on a new shorter-lived isomer, observed in the relativistic fission of U and the fragmentation of Xe. In these experiments the FRS was tuned to transmit Rh and Cd respectively.
Fig. 11 shows a -ray spectrum observed in delayed coincidence with the Ag ions within a 8.75-s wide time window opened 2.75 s after implantation. A single -ray with an energy of 91 keV is observed. The inset of Fig. 11 shows the time distribution of the 91-keV transition. A half-life of (10) s was obtained from the slope, which suggests or multipolarity, given that only W.u. and W.u. calculated with and Bricc () are consistent with the hindrance factor systematics La11 ().
Similar low-lying s isomeric states were observed in the neutron-rich In Sc04 () and related to the decay of the and states. Also, an excited state is present in In Sc04 (). It is 1692 keV above the level in In and decreases in energy to 688-keV above the ground state in In. In Ag, however, is assigned to the ground state Ta07 () on the basis of the logft and 5.9 to the daughter 2+ and (4+) levels, respectively, and is assigned to the 0.55-s isomer in Kr00 (). Therefore, based on the analogy with In, was tentatively assigned to the isomeric level observed in the present study. Given that the multipolarity of the 91 keV transition is or , the final state can be the excited state or the ground state.
The level energy keV of the 0.55-s isomer is estimated from its half-life and the transition strength in In Sc04 (). This enables the placement of the state in between the isomeric state and the ground state of Ag.
Prior to our study, two -rays with energies of 155 keV and 1132 keV were observed in mutual coincidence and associated with Ag To06 ().
In the present work, Ag was observed in the Rh FRS settings from U fission and in the Cd and Cd FRS settings from fragmentation of the Xe beam. Fig. 12 shows -ray spectra, observed in delayed coincidence with Ag nuclei within a 5.75 s time window opened 450 ns after implantation. Two transitions with energies 75 keV and 155 keV were observed to be in coincidence. The two -rays are observed also in Ka12 (). The 1132-keV line, reported previously in To06 (), is not confirmed by the present study. The inset of Fig. 12(a) shows the summed time spectrum for the 75-keV and 155-keV transitions and the half-life, obtained from the fit to the slope is 1.46(20) s, which is consistent with the 1.62 s in Ka12 ().
The 155-keV -ray is delayed with respect to the 75-keV -ray, which enables the ordering made in the level scheme in Fig. 10. However, the two transitions are of low energy and therefore a time-walk correction has to be made in order to properly determine the half-life of the intermediate state. The time-walk of the prompt distribution as a function of the energy was determined by using data for Cd from Ho07 (); Ca09 () and for Ag from the present experiment. Time distributions, calculated as the difference between the detection time of the feeding and de-exciting transitions, were analyzed. The centroids of the prompt distribution were obtained for several transitions in Cd and Ag where the level of interest is fed by a low-energy transition and de-excited by a high-energy transition. The centroid of the time distribution is plotted on Fig. 13 as a function of the low-energy feeding transition. The energy-time dependence was fitted with a function of the type , where is the centroid of the time distribution, is the energy of the feeding transition in keV and nskeV, ns are the parameters obtained from the fit. A mirror symmetric function JM10 () for the position of the prompt distribution can be obtained from a set of states fed by a high-energy transition and decaying via a low-energy transition. In the Ag case, the two transitions have low energies. Therefore, the Ag data point was plotted on Fig. 13 after time-walk correction was applied for the 75-keV feeding transition. Thus, a half-life of 0.14(5) s was obtained from the difference between the Ag data point and the value for =155 keV. The W.u. and W.u., values, calculated for the 155-keV transition with and Bricc (), are consistent with the systematics La11 (). An multipolarity assignment for the 155-keV transition is also possible for .
Table 3 shows that the 75-keV transition has an intensity of 74(38)% of that of the 155-keV transition. The intensity balance for the 155-keV level leads to in case of pure 155-keV transition and to if 155-keV transition is a pure transition. These conversion coefficients are consistent with an nature of the 75-keV transition. Thus, W.u. was obtained with .
Ag is the most neutron-rich nucleus in the silver isotopic chain studied in the present experiment. The ions were transmitted through the FRS in the Cd setting during the Xe beam fragmentation experiment. The energy spectrum, shown on Fig. 14, was incremented for delayed transitions in coincidence with the Ag ions within a 48-s wide time window placed 0.58 s after the prompt- flash. A single ray at energy of 254 keV was detected by RISING. The inset of the figure presents the time distribution of the 254-keV line and a half-life of 27(6) s was obtained from the slope. The isomeric transition in Ag is also reported in Ka12 () and a lower limit of the half-life is given to be 20 s.
An multipolarity assignment for the isomeric transition is based on comparison between the measured half-life with the Weisskopf estimates for the partial half-life of a 254-keV transition, and the hindrance factor systematics for low-lying transitions in the mass region La11 (). W.u. is calculated with Bricc (). The nature of the isomeric transition is also consistent with the isomeric decay in the In isotone as shown in Fig. 10, where the , =23 s isomer decays via a 248-keV transition to the ground state with W.u. Therefore, and are assigned to the 254-keV and the ground state levels in Ag.
The systematics, presented in Fig. 5, show that the level energies in the odd-mass silver nuclei evolve smoothly with the neutron number following the trend of the first level of the Cd core. Often, such states are interpreted as core coupled states.
Based on the relative positioning of the level with respect to the level, two groups of nuclei can be distinguished. For the level energy is higher than the level energy. For the relative positioning of the two levels inverts. Extrapolating the trend towards , it can be expected that becomes the ground state again. While the group of nuclei with less then six valence particles or holes can be understood in the single-particle framework the anomalous position of the state hints at a more complex structure.
The origin of this anomaly was initially related to the three-proton cluster configuration Ki66 (). Having 47 protons, the only available proton sub-shell for the silver valence proton holes is . Also, given that this is a unique-parity orbit, the low-lying positive parity states can arise only from the configuration. By using the expansion coefficients of the three-particle matrix elements of a two-body interaction in terms of two-particle matrix elements for the configuration of identical nucleons He94 (), energy spectra were calculated for the Ag isotopes. Fig. 15 shows the energy spectra, obtained from
where are the two-body matrix elements, parametrized with respect to the neighboring even-even cadmium nuclei, and are the coefficients of fractional parentage listed in BK71 (). With the exception of Ag and Ag, this simple approach gives a good overall description of the experimental data and in particular it reproduces correctly the sequence of the levels, hinting at the importance of the cluster configuration in understanding the structure of the silver nuclei. However, the systematics in Fig. 15 show also that the theoretical and level energies are systematically overestimated and the description of the level energy in Ag and in Ag is poor. Moreover, the negative-parity and levels in Ag, are outside the model space. A more realistic description of the excited states in the odd-mass silver isotopes, and in Ag nuclei in particular, should include a larger space and a certain degree of collectivity. A small quadrupole deformation was suggested already at Ag to account for the isomeric transition strength between the and states CKN75 (). Further away from the magic number rotational bands based on the Nilsson orbit appear Hw02 (). As the magic number is approached, a reduction of collectivity should take place and the excited states are expected to have purer wave functions.
Given that the above approach does not take into account the proton-neutron and neutron-neutron interactions, which are important in the mid-shell and transitional regions, shell model calculations were performed for the Ag odd-even nuclei using the NuShell code Br04 (). The full jj45pn space, involving , , , proton and , , , , neutron orbitals was used. The theoretical spectra, shown in Fig. 16 are obtained with the jj45pna interaction, parametrized with respect to the nuclei lying close to the doubly magic Sn. The theoretical level energies evolve smoothly from Ag to Ag, predicting the state to be the lowest-lying positive-parity state. In Ag, the wave function of the ground state and the yrast states consist of almost pure configuration with an amplitude of 68% for the ground state. In the lighter Ag the respective wave functions are more fragmented, because of the larger valence space. Even though, the positive parity states observed in Ag are present in the calculated level schemes, their sequence is incorrect. In particular, the level energy in Ag appears 635 keV above the state and the ordering of the doublet is opposite to the experimental level scheme. Also, the energies of the isomeric and levels in Ag are overestimated by NuShell by 300 to 500 keV. Similar deviations from the experimental data were observed also in the neutron-rich indium isotopes Sc04 (). The Shell model with the jj45pna interaction and and effective charges gives an overestimated =18.5 W.u. value for the isomeric 48-keV transition in Ag with respect to the experimentally observed 6.8(8) W.u. given in sect.III.A.1.
A different theoretical approach was given many years ago by a model where a cluster of three valence protons was coupled to quadrupole vibrations Pa73 (). In these calculations, the particles are allowed to move in the , and sub-shells and are coupled to a Sn vibrational core. Even though, there are no specific calculations for the neutron-rich silver isotopes, a qualitative analysis can be made given that the even-even medium-mass cadmium nuclei have a smooth behavior with their first phonon being at approximately 500-600 keV. A good overall description of the medium-mass silver nuclei, where the anomaly takes place, can be obtained with a coupling parameter strength . The energy levels were calculated in Pa73 () for the negative parity states up to and for the positive-parity states up to . The level has a seniority zero-phonon and one-phonon component, as well as a one-phonon contribution, each with an amplitude of approximately 20%. The level has a dominant contribution of the zero-phonon component with an amplitude of 30% and a one-phonon component with a contribution of 17%. The state is more fragmented with zero-phonon, one-phonon and two-phonon components with amplitudes of 16%, 21% and 12%, respectively. For the state, the components with amplitudes higher than 10% are the zero-phonon and the one-phonon.
The model calculations give W.u. or 16.0 W.u. values for the 48-keV isomeric transition in Ag, obtained with two sets of effective charges and and 2.5. Both values overestimate the experimental one. Nevertheless, the model gives a good qualitative description of the level sequence and the multiplet structure in particular, hinting at the importance of zero- and one- phonon excitations in understanding the structure of the lowest-lying positive-parity yrast states.
In the low spin regime, the cluster-vibration and the shell model predict different transition strengths. The shell model calculations give W.u. for the 27-keV transition in Ag, while the cluster-vibration model predicts a more enhanced transition with W.u. Even though, the present experiment does not allow the measurement of the half-life and the mixing ratio of the 27-keV transition in Ag, the models have clear predictions which can be tested in future experiments.
The low-lying excited states in the odd-odd neutron-rich silver nuclei can be described by using the cluster-vibration model, where the proton-neutron residual interaction is a result of quadrupole and spin vibration phonon exchange between the odd particles and the nuclear core Pa79 (). As a result, split multiplets with level energies as a function of the nuclear spin arise. The level energies obey the parabolic rule
where the quadrupole and spin-vibrational contributions to the splitting of the multiplet are given by
Here, and denote the quasi-proton and the quasi-neutron energies, and are deduced from the experimental data. and are the proton and neutron total angular momenta, and is the nuclear total angular momentum. If and are both particle-like or both hole-like the parameter , otherwise . The parameter is defined as if the Nordheim number and if . Otherwise, for and for , where the symbols and distinguish the cases and . The quadrupole and spin vibration coupling strengths are defined as
where is the occupation probability for the level and .
This approach was used extensively in the mid-shell indium isotopes Ki88 (), where a good overall agreement with the experimental data was achieved. The level energies of Ag were calculated by using the single-particle neutron energies taken from the Sn level scheme Ka11 (), which is presented in Fig. 17. The neutron occupation probabilities, listed in Table 4, were calculated from the spectroscopic factors , obtained in the Sn(d,p)Sn reaction BH73 (). The coupling strengths MeV and MeV were deduced from the and the energy of the first phonon excitation in Cd. The proton occupation probability is deduced from Va76 ().
Fig. 17 shows the calculated parabolas for the , and multiplets in Ag. Within each of the multiplets, the excited states decay via fast transitions, and hence the observed isomeric states can arise only from transitions connecting states of different multiplets. The lowest-lying states, belonging to each of the multiplets, are the , and states, which is consistent with the experimental data.
Shell model calculations were also performed for Ag with the full jj45pn space. Results are presented in Fig. 18. In Ag, the main component of the wave function (w.f.) is with an amplitude of 30%, followed by with 10%. The w.f. of the state has a leading component with an amplitude of 39%. Weaker components are and with 13% and 11%, respectively. The state consists of , contributing 34% of the w.f., and which has an amplitude of 24% of the w.f. All other components contribute with smaller amplitudes. In the lighter nuclei, these configurations become diluted and the wave functions are much more fragmented. The relative position of the theoretical and is the same as the ordering of the experimental levels. However, in contrast to the spin/parity assignments made in the present work, the theoretical is pushed below the states in Ag. The behavior of the level in the neutron-rich even-even Ag nuclei is not surprising, since in the neutron-rich even-A indium nuclei a strong variation of the state with respect to the state is also observed and attributed to a weakening of the effects of the interaction when moving from In to In Sc04 (). It was suggested also that the underestimation of the interaction in the region of nuclei with may limit the predictive power of the Shell model.
This work presents results on two previously known sub-microsecond isomers and on two new isomeric states in Ag. Because of the high efficiency and granularity of RISING, coincidences were established enabling the construction of the level schemes. A new microsecond isomer was observed in Ag. The isomeric state in Ag is observed to decay via a 75-keV transition and a half-life of the intermediate state was deduced. An isomeric state with a half-life of 27(6) s is observed in Ag. The spin/parity assignments are based on the systematics and the observed decay pattern.
The positive-parity states of the odd-mass nuclei were analyzed within a coupling scheme, based on a simple angular momentum re-coupling algebra, and the results compared to shell model calculations, performed in a larger space. A reasonable description of the isomeric and the level energies is achieved. The biggest discrepancy between the theoretical calculations and the Ag data is in the level energy, which appears 635 keV above the ground state. In Ag, the theoretical level energy for the is also overestimated by approximately 300 keV. Further qualitative analysis was made in the framework of the cluster-vibration model.
The excited states in the even-mass silver nuclei were analyzed with the shell model and within a phenomenological approach based on a quadrupole and spin-vibrational proton-neutron interaction.
Even though the phenomenological approaches used in the present work give a satisfactory description of particular levels, more realistic calculations using a larger valence space fail in reproducing the level energy sequence in several cases. This may be explained by an inapplicability of the interaction parametrized with respect to the nuclei placed close to Sn and by a certain degree of collectivity presented in the nuclei with few valence holes to the and magic numbers. Nevertheless, the intruder and orbits seem to play an important role in understanding the structure of the neutron-rich silver nuclei and the isomerism in Ag in particular.
Acknowledgements.This work is supported by the Bulgarian National Science Fund under contract No: DMU02/1, UK STFC, Royal Society, the Spanish Ministerio de Ciencia e Innovación under contracts FPA2009-13377-C02-02 and FPA2011-29854-C04-01 and the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), the Swedish Science Council and OTKA contract number K100835.
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