Intermediate-energy Coulomb excitation of 104Sn:
Moderate strength decrease
The reduced transition probability of the first excited 2+ state in the nucleus 104Sn was measured via Coulomb excitation in inverse kinematics at intermediate energies. A value of 0.163(26) b was extracted from the absolute cross-section on a Pb target, while the method itself was verified with the stable 112Sn isotope. Our result deviates significantly from the earlier reported value of 0.10(4) b and corresponds to a moderate decrease of excitation strength relative to the almost constant values observed in the proton-rich, even- 106–114Sn isotopes. Present state-of-the-art shell-model predictions, which include proton and neutron excitations across the shell closures as well as standard polarization charges, underestimate the experimental findings.
Across the Segré chart of nuclei the tin isotopes take an eminent position. Besides containing the longest chain of isotopes in-between two doubly-magic nuclei, in this case 100Sn and 132Sn, accessible to nuclear structure research, the valley of stability against -decay crosses this chain at mid-shell. This allows for systematic studies of basic nuclear properties from very proton-rich to very neutron-rich nuclei. Of high interest in this context is the robustness of the proton shell closure when the magic numbers are approached. Experimentally, the correlated gap size can be inferred from mass measurements when data from neighboring isotones is available. The magnitude of the proton gap is well known for neutron-rich nuclei beyond the end of the major shell and shows a maximum for 132Sn Wang et al. (2012). On the proton-rich side, however, experimental information is more scarce and only indirect evidence for a good shell closure exists, e.g., the large Gamov-Teller strength observed in the decay of 100Sn Hinke et al. (2012).
Complementary shell evolution probes can be obtained from the transition energies, , and their reduced transition probabilities , in short . While the values of the tin isotopes between 100Sn and 132Sn are well established and exhibit only very little variation – the highest value is 1.472 MeV for 102Sn, the lowest is 1.132 MeV for 124Sn Lipoglavsek et al. (1996); Raman et al. (2001) – the transition strengths follow a different pattern. The a priori expectation is a curve showing maximum collectivity at mid-shell and smoothly decreasing towards the shell closures, reflecting the number of particles times the number of holes available within the major shell. This perception is put on a formal base for a single -shell by the seniority scheme (see, e.g., Ref. Casten (2001)), which predicts constant excitation energies and a parabolic pattern for the transition strengths. It has been shown that these key characteristics remain valid in the generalized seniority scheme as long as the orbits within the major shell are filled with the same rate, while for different level occupancies a shallow minimum for the values can be obtained at mid-shell Morales et al. (2011).
In recent years, several experimental findings generated the large interest regarding the strengths pattern in the tin isotopes. While the neutron-rich isotopes with follow the anticipated trend of smoothly decreasing values towards the major shell closure Radford et al. (2004); Allmond et al. (2011) well described by large-scale shell-model (LSSM) calculations Banu et al. (2005); Ekström et al. (2008), the proton-rich nuclei take a different path. Commencing with the stable isotope a steadily growing deviation from the shell-model expectations was observed with almost constant values for the isotopes Banu et al. (2005); Cederkäll et al. (2007); Vaman et al. (2007); Ekström et al. (2008); Doornenbal et al. (2008); Kumar et al. (2010). This triggered a revisit of the stable tin isotopes via direct lifetime measurements, yielding generally lower transition strengths than the adopted values given in Ref. Raman et al. (2001) and even a local minimum for the mid-shell nucleus Jungclaus et al. (2011).
While the value of 102Sn remains missing for a complete pattern of the tin isotopes within the major shell, a first attempt for 104Sn with limited statistics has recently been made Guastalla et al. (2013). The result of 0.10(4) b indicates a steep decrease of excitation strength in agreement with LSSM calculations. In order to ameliorate the experimental situation, a new measurement of the transition strength in 104Sn is desirable. Here, we report on the first extraction in the unstable, proton-rich tin nuclei from absolute Coulomb excitation cross-sections. Previously deduced values relied on target excitation at “safe” Cederkäll et al. (2007); Ekström et al. (2008) and intermediate Vaman et al. (2007) energies or used a stable tin isotope with known excitation strength as normalization Banu et al. (2005); Guastalla et al. (2013). In fact, all reported values from intermediate-energy Coulomb excitation measurements above 100 MeV/nucleon rely on the latter method Banu et al. (2005); Bürger et al. (2005); Saito et al. (2008) and so far no attempt has been made to determine absolute cross-sections at these high energies. Therefore, in the present work the stable 112Sn isotope, which has a known value, was Coulomb excited as well in order to validate the method.
The experiment was performed at the Radioactive Isotope Beam Factory (RIBF), operated by the RIKEN Nishina Center and the Center for Nuclear Study of the University of Tokyo. A 124Xe primary beam was accelerated up to an energy of 345 MeV/nucleon and impinged on a 3 mm thick Be production target at the F0 focus of the BigRIPS fragment separator Kubo et al. (2012). The method was applied to select and purify secondary beams of 104Sn and 112Sn in two subsequent measurements. The beam cocktail compositions were identified event-by-event. An ionization chamber located at the focal point F7 measured the energy loss , yielding the fragments’ element number . The combination of position and angle measurements at the achromatic focal point F3 and the dispersive focal point F5 with parallel plate avalanche counters (PPAC) Kumagai et al. (2001) and a time-of-flight (TOF) measurement with two plastic scintillators placed at the focal points F3 and F7 enabled the deduction of the mass-to-charge ratio . For the 104,112Sn secondary beams, momentum acceptances were 2.2% and 0.9%, respectively.
The secondary beams were transported to the focal point F8, where a 557 mg/cm2 thick Pb target was inserted to induce Coulomb excitation reactions. At mid-target, the secondary beam energies were 131 and 154 MeV/nucleon for the 104,112Sn fragments. In order to enhance the number of tin fragments in the fully stripped charge state, a 6 mg/cm2 thick aluminum foil was placed behind the reaction target. Scattering angles were determined with two PPACs located 1430 and 930 mm upstream and one PPAC located 890 mm downstream the secondary target. The PPACs’ position resolution in X and Y was 0.5 mm (), allowing for a scattering angle reconstruction resolution of about 5 mrad, while an angular straggling of 6–8 mrad was calculated with the ATIMA code ati (). Grazing angles, calculated using the formulas given in Ref. Wollersheim et al. (2005), were 28 and 23 mrad for 104,112Sn and their respective energies in front of the reaction target. Due to the scattering angle resolution and the angular straggling, a cut on “safe” angles would have led to a loss of a large fraction of the -ray yield. Therefore, no angular cut was applied and contributions from nuclear excitations were determined from inelastic scattering on a 370 mg/cm2 thick carbon target.
To detect -rays from the transitions, the reaction target was surrounded by the DALI2 array Takeuchi et al. (2003). It consisted of 186 NaI(Tl) detectors, covering center-of-crystal angles from 19 to 150 degrees. The efficiency of the DALI2 spectrometer was measured with 60Co and 88Y stationary sources and agreed within 5 % to simulations using GEANT4 Agostinelli et al. (2003). For the 1.33 MeV -ray emitted by the stationary 60Co source, a full energy peak (FEP) detection efficiency of 14 % and an energy resolution of 6 % (FWHM) were measured for the full array. Radiation arising from secondary bremsstrahlung produced from the ions’ deceleration in the reaction target was the anticipated main source of background. Therefore, the beam pipe at the F8 focus was enclosed by 1 mm of lead and 1 mm of tin shields. In addition, only forward angle detectors in the rest-frame were analyzed. After Doppler shift correction for a 1.26 MeV -ray emitted in-flight, values of 10 % and 8 % (FWHM) were expected for the FEP efficiency and energy resolution, respectively.
Reaction products behind the reaction target were identified by the ZeroDegree spectrometer Kubo et al. (2012), using the previously described TOF method from focus F8 to focus F11. Angular acceptances were mrad vertically and mrad horizontally for particles passing ZeroDegree with the central momentum. Including efficiencies of 83 and 76 % for scattering angle determination, 180 and 920 particles per second of 104,112Sn ejectiles were detected in the ZeroDegree spectrometer in their fully-stripped charge state. Figure 1 displays the -ray spectra measured in coincidence with fully-stripped 104,112Sn ions detected in BigRIPS and ZeroDegree after applying the Doppler shift correction. The two transitions were observed at 1258(6) and 1253(6) keV, close to the literature values of 1260 and 1257 keV Raman et al. (2001). The intensities were determined by fitting the experimentally observed spectra with simulated response functions on top of exponential background.
The measured inelastic cross-sections are composed of contributions from nuclear excitation, Coulomb excitation to the state, and Coulomb feeding from higher lying states (). In addition, the ZeroDegree angular acceptance depends on the momentum distribution of the secondary beam and has to be corrected for. Thus, a measured cross-section on the Pb target can be converted to a value only if and are quantified.
Inelastic scattering cross sections to the state of 40(4) and 48(4) mbarn were measured for the 104,112Sn isotopes on the carbon target. These values were reproduced with the DWEIKO code Bertulani et al. (2003) by selecting nuclear vibrational excitations and deformation lengths of and 0.45(2) fm. Optical potentials were derived for the calculations as described in Ref. Furumoto et al. (2012) using the microscopic folding model with the complex G-matrix interaction CEG07 Furumoto et al. (2008, 2009) and the density presented in Ref. Chamon et al. (2002).
For inelastic scattering of 112Sn on the Pb target, the cross-section to the state was mbarn, while a cross-section of mbarn was calculated with DWEIKO using the value of 0.242(8) b Raman et al. (2001); Kumar et al. (2010), the derived deformation length, and the angular transmission shown in Fig. 2 (a). Thus, a Coulomb feeding contribution of mbarn was determined. Figure 2 (c) displays the measured differential cross-section as function of scattering angle. It is compared to the calculated nuclear cross-sections and to the Coulomb excitation cross-section. Coulomb excitations are dominant for all scattering angles, as nuclear contributions are sizable only around the grazing angle. A larger nuclear contribution would have resulted in a maximum at the grazing angle. All calculations were convoluted with the detector resolution, the angular straggling, and the observed ZeroDegree scattering angle transmission.
The Coulomb feeding can be attributed to single-step Coulomb excitations to higher lying states and subsequent decay via the state, while multi-step excitations play only a minor role at intermediate energies. For example, the value to the state at 2355 keV in 112Sn has a strength of 0.087(12) b Jonsson et al. (1981) and decays through the state. This translates to a feeding contribution of 10(2) mbarn. For excitations, the total strength measured in the heavier 116–124Sn isotopes between two and four MeV corresponds to about 10 % of that to the first excited state Bryssinck et al. (2000), while no experimental information is available for states at higher energies.
The observed Coulomb feeding contributions in 112Sn can be used to evaluate the Coulomb feeding for 104Sn. In a simple picture, it originates from the excitation and fragmentation of the excitation strength to many states between 2 MeV and the proton separation energy (7.554(5) MeV for 112Sn and 4.286(11) MeV for 104Sn Wang et al. (2012)). Assuming an uniform excitation strength distribution in this region, the same excitation strength and correcting for the angular transmission shown in Fig. 2 (b) results in a Coulomb feeding of mbarn for 104Sn, mainly due to the lower value. This estimation is corroborated by a higher peak-to-background ratio for 104Sn despite a lower total cross-section, showing that fewer high lying excited states are populated.
For inelastic scattering of 104Sn on the Pb target, a cross section of mbarn was measured. Taking the previously determined nuclear contributions and the Coulomb feeding into account, a of 0.163(26) b was deduced. Note that due to the lower beam energy and the reduced scattering angle acceptance, nuclear contributions were significantly suppressed ( mbarn), as can be seen in the differential cross-section in Fig. 2 (d). The new value is displayed in Fig. 3 together with known data in-between the two doubly-magic tin nuclei. Our result deviates significantly from the value of 0.10(4) b obtained in Ref. Guastalla et al. (2013). It corresponds to only about 30 % excitation strength decrease compared to the even- 106–114Sn isotopes and shows that the reduction is much more shallow than previously suggested.
Correlated with the recent experimental progress approaching 100Sn, various calculations have been presented Banu et al. (2005); Ekström et al. (2008); Morales et al. (2011); Jiang et al. (2012); Bäck et al. (2013); Guastalla et al. (2013). Very instructive are the LSSM calculations presented with the first intermediate-energy Coulomb excitation experiment on proton-rich tin isotopes Banu et al. (2005). Within that work, two sets of calculations were performed, using 100Sn and 90Zr as inert cores, respectively, and an effective interaction derived from the CD-Bonn potential Machleidt (2001). The former, denoted LSSMa, used a neutron model space with the , , , , and orbitals, the latter, denoted LSSMb, contained the proton , , , , and () orbitals as well. Neutron effective charges of were used for the 100Sn core calculations to compensate the neglect of proton excitations across the shell while the 90Zr core calculations allowed up to four-particle-four-hole proton excitations and used “standard” neutron and proton effective charges of and . The results are added in Fig. 3 to the experimental values between the two doubly magic nuclei and yield inverted parabola in agreement with the neutron rich nuclei but fail to reproduce the enhancement for proton-rich nuclei.
In the most recent shell model prediction, denoted LSSMc in Fig. 3, the calculations were expanded to a 80Zr core and a model space, thereby allowing neutron as well as proton excitations across the gap Guastalla et al. (2013). The standard effective charges were used and truncation was applied depending on the nuclei’s neutron number due to computational limits. However, the inclusion of neutron excitations across the gap augmented the values only slightly. For 104Sn, a value of about 0.1 b is predicted, well below our experimental finding, and also the experimental values for 106Sn are underestimated.
Different suggestions to break the symmetry in the theoretical pattern have been made ranging from refined tuning of the proton-neutron monopoles Banu et al. (2005); Cederkäll et al. (2007), inclusion of excitation across the shell Banu et al. (2005), a shell gap reduction Ekström et al. (2008), to simply using two different sets of single particle levels and effective charges for the lower and upper half of the shell Jiang et al. (2012). In an alternative approach that included the neutron and orbitals as model space and single-particle energies fitted to experimental data Qi and Xu (2012), isospin-dependent effective charges as proposed by Bohr and Mottelson Bohr and B.Mottelson (1975) were introduced into the calculations Bäck et al. (2013). The neglect of proton excitations was compensated by normalizing the effective charges to in the middle of the shell for 116Sn resulting in () in the lower (upper) half of the shell. Indeed, a good overall agreement is observed for very neutron and proton rich nuclei, as shown in Fig. 3 by LSSMd. However, the collectivity increase on the proton-rich side commences later than observed in experiments Doornenbal et al. (2008); Kumar et al. (2010); Jungclaus et al. (2011) and the large effective charges are also coincident with the proton gap minimum around 108Sn (see, e.g., Fig. 4 of Ref. Doornenbal et al. (2008)).
In combination with the neglect of proton excitations, this effective charge adjustment can therefore only be regarded as an interim solution until sufficient computing power becomes available. The importance of proton excitations across the shell for a correct description of the pattern can be easily inferred from the difference in “matter” deformation lengths obtained from the carbon target and the charge deformation lengths obtained from the values. With and fm, the latter are (9) and 0.71(2) fm for 104,112Sn and hence about 50 % larger than the values of and 0.45(2) fm obtained with the carbon target.
In summary, a value of 0.163(26) b was measured for 104Sn in intermediate-energy Coulomb excitation. The drop in excitation strength is much smoother than obtained in Guastalla et al. (2013) and cannot be reproduced by present LSSM calculations using standard effective charges as well as proton and neutron excitation across the shell. Moreover, it was demonstrated that given the significant scattering angle resolution and angular straggling at energies well above 100 MeV/nucleon, nuclear excitation should be explicitly taken into account in determinations rather than suppressed in an inaccurate angular cut. Coulomb feeding from higher lying states cannot be neglected but can easily be determined from known values. A simple scaling of measured cross-sections for the 104,112Sn pair would have led to a 10 % lower assignment for 104Sn. Therefore, we suggest that future absolute cross-section measurements at energies well above 100 MeV/nucleon include calibration runs of nuclei with known values on a high target and nuclear excitation on a low target. Such an approach allows for the use of very thick reaction targets and thus gives access to more exotic nuclei.
Acknowledgements.We would like to thank the RIKEN Nishina Center Accelerator Division for providing the high 124Xe primary beam intensity and the BigRIPS team for preparing secondary beams with high purities for the isotopes of interest. We acknowledge financial support from the Spanish Ministerio de Ciencia e Innovación under contracts FPA2009-13377-C02-02 and FPA2011-29854-C04-01, the OTKA under contract number K100835, and the European Research Council through the ERC Starting Grant MINOS-258567.
- Wang et al. (2012) M. Wang, G. Audi, A. Wapstra, F. Kondev, M. MacCormick, X. Xu, and B. Pfeiffer, Chin. Phys. C 36, 1603 (2012).
- Hinke et al. (2012) C. Hinke et al., Nature 486, 341 (2012).
- Lipoglavsek et al. (1996) M. Lipoglavsek et al., Z. Phys. A 356, 239 (1996).
- Raman et al. (2001) S. Raman, C. W. N. Jr., and P. Tikkanen, Atom. Data and Nucl. Data Tab. 78, 1 (2001).
- Casten (2001) R. Casten, Nuclear Structure from a Simple Perspective (Oxford University Press, 2001).
- Morales et al. (2011) I. Morales, P. V. Isacker, and I. Talmi, Phys. Lett. B 703, 606 (2011).
- Radford et al. (2004) D. Radford et al., Nucl. Phys. A 746, 83c (2004).
- Allmond et al. (2011) J. Allmond et al., Phys. Rev. C 84, 061303(R) (2011).
- Banu et al. (2005) A. Banu et al., Phys. Rev. C 72, 061305 (2005).
- Ekström et al. (2008) A. Ekström et al., Phys. Rev. Lett. 101, 012502 (2008).
- Cederkäll et al. (2007) J. Cederkäll et al., Phys. Rev. Lett. 98, 172501 (2007).
- Vaman et al. (2007) C. Vaman et al., Phys. Rev. Lett. 99, 162501 (2007).
- Doornenbal et al. (2008) P. Doornenbal et al., Phys. Rev. C 78, 031303 (2008).
- Kumar et al. (2010) R. Kumar et al., Phsys. Rev. C 81, 024306 (2010).
- Jungclaus et al. (2011) A. Jungclaus et al., Phys. Lett. B 110, 695 (2011).
- Guastalla et al. (2013) G. Guastalla et al., Phys. Rev. Lett. 110, 172501 (2013).
- Bürger et al. (2005) A. Bürger et al., Phys. Lett. B 622, 29 (2005).
- Saito et al. (2008) T. Saito et al., Phys. Lett. B 669, 19 (2008).
- Kubo et al. (2012) T. Kubo et al., Prog. Theor. Exp. Phys. 2012, 03C003 (2012).
- Kumagai et al. (2001) H. Kumagai, A. Ozawa, N. Fukuda, K. Sümmerer, and I. Tanihata, Nucl. Instrum. and Meth. A 470, 562 (2001).
- (21) Atima, http://http://web-docs.gsi.de/~weick/atima/.
- Wollersheim et al. (2005) H. Wollersheim et al., Nucl. Instrum. and Meth. A 537, 637 (2005).
- Takeuchi et al. (2003) S. Takeuchi et al., in RIKEN Accelerator Progress Report (RIKEN, 2003), vol. 36, p. 148.
- Agostinelli et al. (2003) S. Agostinelli et al., Nucl. Instr. Meth. A 506, 250 (2003).
- Bertulani et al. (2003) C. A. Bertulani, C. M. Campbell, and T. Glasmacher, Comp. Phys. Com. 152, 317 (2003).
- Furumoto et al. (2012) T. Furumoto, W.Horiuchi, M.Takashina, Y.Yamamoto, and Y.Sakuragi, Phys. Rev. C 85, 044607 (2012).
- Furumoto et al. (2008) T. Furumoto, Y. Sakuragi, and Y. Yamamoto, Phys. Rev. C 78, 044610 (2008).
- Furumoto et al. (2009) T. Furumoto, Y. Sakuragi, and Y. Yamamoto, Phys. Rev. C 80, 044614 (2009).
- Chamon et al. (2002) L. Chamon et al., Phys. Rev. C 66, 014610 (2002).
- Jonsson et al. (1981) N. Jonsson, A. Backlin, J. Kantele, R. Julin, M. Luontama, and A. Passoja, Nuclear Physics A 371, 333 (1981).
- Bryssinck et al. (2000) A. Bryssinck et al., Phys. Rev. C 61, 024309 (2000).
- Jiang et al. (2012) H. Jiang, Y. Lei, G. Fu, Y. Zhao, and A. Arima, Phys. Rev. C 86, 054304 (2012).
- Bäck et al. (2013) T. Bäck, C. Qi, B. Cederwall, R. Liotta, F. G. Moradi, A. Johnson, R. Wyss, and R. Wadsworth, Phys. Rev. C 87, 031306(R) (2013).
- Machleidt (2001) R. Machleidt, Phys. Rev. C 63, 024001 (2001).
- Qi and Xu (2012) C. Qi and Z. Xu, Phys. Rev. C 86, 044323 (2012).
- Bohr and B.Mottelson (1975) A. Bohr and B.Mottelson, Nuclear Structure, Vol. II (Benjamin, New York, 1975).