Signatures of shape transition in odd-A neutron-rich Rubidium isotopes

Signatures of shape transition in odd-A neutron-rich Rubidium isotopes


The isotopic evolution of the ground-state nuclear shapes and the systematics of one-quasiproton configurations are studied in odd- Rubidium isotopes. We use a selfconsistent Hartree-Fock-Bogoliubov formalism based on the Gogny energy density functional with two parametrizations, D1S and D1M, and implemented with the equal filling approximation. We find clear signatures of a sharp shape transition at in both charge radii and spin-parity of the ground states, which are robust, consistent to each other, and in agreement with experiment. We point out that the combined analysis of these two observables could be used to predict unambiguously new regions where shape transitions might develop.

21.60.Jz, 21.10.Pc, 27.60.+j

The study of the nuclear shape evolution as the number of nucleons changes is nowadays a highly topical issue in nuclear physics from both theoretical and experimental points of view (see for example Refs. (1); (2); (3); (4) and references therein). Specially interesting are those situations where the nuclear structure suffers drastic changes between neighbor nuclides. These structural variations lead often to sudden changes of particular nuclear properties that can be used as signatures of phase/shape transitions (5); (6). This is the case of the neutron-rich isotopes with masses . Intense experimental (7); (8); (9) and theoretical (10); (11); (5); (6) efforts are being done to better characterize the structural evolution of the ground and excited states in this region of the nuclear chart. In particular, we have recently studied in Refs. (5); (6) the structural evolution in even and odd neutron-rich Sr, Zr, and Mo isotopes. In this work we concentrate on the study of neutron-rich odd-mass Rb isotopes that have received considerable attention recently (12); (13); (14); (15) and that exemplify a general pattern where structural fluctuations lead to observable effects.

Being an odd- nucleus, the spin and parity of odd- Rb isotopes are determined by the state occupied by the unpaired proton. The spectroscopic properties of the various isotopes will be determined by the one-quasiproton configurations that, in principle, are expected to be rather stable against variations in the number of even neutrons. However, as it is known in neighboring nuclei, approaching , the isotopes become well deformed (1); (5) and the abrupt change in deformation induces signatures in nuclear bulk properties like the two-neutron separation energies and the nuclear charge radii, as well as in spectroscopic properties. In particular, the spin and parity of the nuclear ground state might change suddenly from one isotope to another, reflecting the structural change.

Figure 1: (Color online) Single particle energies for protons and neutrons in Sr as a function of the axial quadrupole moment . The Fermi level is depicted as a thick dashed red line. The results have been obtained with the Gogny-D1S EDF.

We analyze in this work the bulk and spectroscopic properties of neutron-rich Rb odd isotopes within the selfconsistent Hartree-Fock-Bogoliubov (HFB) approximation based on the finite range and density dependent Gogny energy density functional (EDF) (16). Previous studies in this region including triaxiality (5) suggest that the -degree of freedom would not play a significant role in these isotopes and therefore, axial symmetry is assumed as a selfconsistent symmetry in this study.

In addition to the well known D1S parametrization (17) of the Gogny-EDF, we also consider the most recent parametrization D1M (18). From this comparison we evaluate not only the robustness of our results, but we also explore the capability of D1M to account for the phenomenology of odd- nuclei, not so well studied yet. The description of the odd- nuclei involves additional difficulties because the blocking procedure requires the breaking of time-reversal invariance, making the calculations arduous (19); (20). In the present study we use the equal filling approximation (EFA), a prescription widely used in mean-field calculations to preserve the advantages of time-reversal invariance. In this approximation the odd nucleon sits half into a given orbital and half into its time-reversed partner. The microscopic justification of the EFA is based on standard ideas of quantum statistical mechanics (21). The predictions arising from various treatments of the blocking have been studied in Ref. (22), concluding that the EFA is sufficiently precise for most practical applications. More details of our procedure can be found in Ref. (6).

Figure 2: (Color online) Experimental (a) excitation energies and spin-parity assignments (12); (13); (14); (15); (24) compared with Gogny-D1S HFB-EFA results (b) for one-quasiproton states in odd- Rb isotopes. Prolate configurations are shown by black lines, oblate ones by red lines, and spherical ones by blue lines.

The proton and neutron single-particle energies (SPEs) are shown in Fig. 1 as a function of the axial quadrupole moment for the even-even Sr isotope. Fermi levels are plotted with thick dashed (red) lines. Asymptotic (Nilsson) quantum numbers are also shown in the proton case for the values where the energy minima are located (vertical arrows) in both oblate and prolate sectors. As one can see, the valence protons occupy the shell and, if deformed, they start to fill the orbitals coming down from the shell. Neutrons occupy states belonging to the shell and approaching they start to populate the intruder orbitals coming down from the oscillator shell at large deformations ( b). These neutrons polarize the protons and drive them to populate the shell at similar deformations. Thus, the underlying nuclear structure in this mass region is very sensitive to the occupancy of these single-particle orbitals and the result is a rapid change in the nuclear deformation and in the spectroscopic properties as a function of both neutron and proton numbers.

In particular, Rb isotopes approaching will jump abruptly from slightly deformed oblate or prolate shapes to strongly prolate shapes. As a consequence, the unpaired proton that finally determines the spin and parity of the whole nucleus in Rb isotopes, will jump from the at low deformation to the orbital at large deformation. This is fully consistent with the Jahn-Teller picture, which states that nuclei avoid regions with high single-particle level densities around the Fermi level, thus favoring deformation where the energy gaps are higher. One sees large energy gaps at large prolate deformations for the orbital. This is also consistent with the Federman-Pittel mechanism (23), which points to the neutron-proton interaction as the driven force toward quadrupole deformation. This force is particularly intense between spin-orbit partners (neutron and proton in our case), as well as between orbitals with the same radial quantum numbers and large orbital angular momenta differing by one unit and (neutron and proton in our case).

Figure 3: (Color online) Gogny-D1S excitation energies of single-quasiproton prolate (a) and oblate (b) states in Rb isotopes. Hole states are plotted below zero energy and particle states are plotted above. The absolute ground states are indicated with a circle.
Figure 4: (Color online) Same as in Fig. 3, but for Gogny D1M.
Figure 5: (Color online) Isotopic evolution of the quadrupole deformation parameter of the energy minima obtained from Gogny D1S and D1M calculations.

In Fig. 2 we can see the experimental excitation energies and spin-parity assignments (a) in odd- neutron-rich Rb isotopes. They are compared to the one-quasiproton states predicted by the Gogny-D1S HFB-EFA calculation (b). The excited states in a given isotope are referred to the corresponding ground state, regardless its shape. Prolate configurations in our calculations are shown by black lines, oblate ones by red lines, and spherical ones by blue lines. The quasiparticle states are labeled by their quantum numbers. The most important configurations are joined by dashed lines following the isotopic evolution. In addition, the ground states are labeled by their asymptotic quantum numbers.

Experimentally (12); (13); (14); (15); (24) one observes ground states in Rb, then states in Rb, and finally states in the heavier isotopes. The states appear all the way up to Rb as excited states. The most striking feature observed is a jump at from to ground states.

The theoretical interpretation of these features can be understood from the analysis of our results in the lower panel (b). Rb isotopes evolve from spherical shapes in Rb around with the spherical , , and shells involved, to slightly deformed shapes (oblate and prolate almost degenerate) in Rb, and finally to well deformed prolate shapes in Rb. In the lighter isotopes the three spherical shells mentioned above are very close in energy and thus, we can see all the split levels coming from them at very close low-lying excitation energies. In the case of Rb isotopes we obtain oblate ground states with both oblate and prolate low-lying excited states very close in energy. For instance, we see that the experimental ground states in Rb appear in the calculation as excited states at less than 0.1 MeV. In the case of Rb the experimental ground states appear as prolate excited states, while the calculations produce oblate ground states. The ground states of the heavier isotopes are predicted to be in agreement with experiment. The calculations also predict excited low-lying states with an oblate nature, which are observed experimentally.

In the next figures we compare the spectroscopic properties of the Gogny D1S and D1M parametrizations. Contrary to Fig. 2, in Fig. 3 for D1S and in Fig. 4 for D1M, we have separated the prolate (a) and oblate (b) states and have plotted the most relevant hole states below zero energy and the particle states above. The absolute ground states, either oblate or prolate, are indicated with a circle. Specifically, in the figures corresponding to the prolate shape, we have plotted the evolution of the states, which are ground states for Rb. In the upper region we find the , while in the lower region we have the , , , and coming up from the shell. In the oblate case (b) we can see the state, which is ground state in Rb, the as particle state and the and from as hole states.

Very similar results are obtained from both parametrizations, which answer our original questions about robustness of the calculations and reliability of D1M. The only difference worth mentioning is that D1M produces slightly lower excited states. This feature can be understood from its larger effective mass that makes the single particle spectrum somewhat more dense with D1M.

To further illustrate the role of deformation and spin-parity assignments in the isotopic evolution, we display in Fig. 5 the axial quadrupole deformation of the energy minima as a function of with both D1S and D1M parametrizations. The deformation of the ground state for each isotope is encircled. We can see that starting at the semi-magic isotope with we get two minima in the prolate and oblate sectors. These are in the prolate case and (or very close in energy) in the oblate case. The calculations with the two forces predict oblate ground states, but with prolate solutions which are very close in energy and that could perfectly become ground states for slightly different type of calculations. Thus, in Rb the and are practically degenerate (see Fig. 2). In Rb (Rb) they are separated by less than 0.2 (0.5) MeV. The experimental information seems to favor prolate solutions since ground states are observed. The ground states of heavier isotopes starting at Rb are in agreement with experiment and strongly prolate. The oblate configurations appear all the way toward heavier isotopes.

Figure 6: Gogny-D1S HFB results for (a) and (b) in odd- Rb isotopes compared to experimental data from Ref. (25); (26) for masses and from Ref. (27) for radii. Results for prolate, oblate, and spherical minima are displayed with different symbols (see legend). Open circles correspond to ground-state results.
Figure 7: Same as in Fig. 6, but for Gogny D1M.

Finally, we show in Fig. 6 for D1S and in Fig. 7 for D1M, the charge radii differences (a) defined as , calculated with the same corrections as in Ref. (5), and the two-neutron separation energies (b). We compare our results for with isotope shifts from laser spectroscopy experiments (27) and our results for with masses from Refs. (25); (26). The results for agree in general with the measurements and are very similar between D1S and D1M. A somewhat better agreement is found with D1M at the beginning of the shell (). In any case, it would be interesting to see whether the incipient experimental tendency to raise the values in the heavier isotopes and not predicted by the calculations, still persists in more exotic nuclei. In this respect, we should remark that for the heavier Rb isotopes the mass data are not completely reliable because they are based on -endpoint measurements. It has been shown that this type of indirect measurements systematically tend to underestimate the -values leading to very strong binding. This problem has been discussed in Ref. (28) for neutron-rich Y and Nb isotopes. Thus, it would be very helpful to extend and improve mass measurements reducing the still large uncertainties in neutron-rich Rb isotopes.

The nuclear charge radii plotted in the upper figures show a jump at , where the radius suddenly increases. This experimental observation is well reproduced in our calculations, where the encircled ground states show that the jump occurs between and . This jump correlates well with the one observed in the spin-parity of the ground states displayed in Fig. 2 from to . Theoretically, the jump in could be understood as the result of the change from an oblate to a prolate shape. But it could also be interpreted as a jump from a slightly prolate shape to a well deformed prolate nucleus, because the oblate and prolate shapes in the transitional nuclei have very similar absolute values of deformation (see Fig. 5). This similarity gives raise to practically the same charge radii, as it can be seen by comparing the prolate (upward red triangles) and oblate (downward blue triangles) radii in this region. The only difference worth mentioning between D1S and D1M is the oblate radii at which corresponds to different configurations ( in D1S and in D1M) and different deformations (see Fig. 5).

Then, we see that contrary to , which does not signal any clear signature in Rb isotopes, clearly indicates the existence of a shape transition in agreement with experiment and well correlated to the spin-parity jump. It would be also very interesting to extend the isotope shifts measurements, which are rather old, to more exotic isotopes to confirm also the prolate stabilization.

In summary, we have studied the shape evolution in odd- Rb isotopes from microscopic selfconsistent Gogny-EDF HFB-EFA calculations. We have analyzed various sensitive nuclear observables, such as two-neutron separation energies, charge radii, and the spin-parity of ground states in a search for signatures of shape transitions. We have found that, although the masses are not very sensitive to these shape changes in Rb isotopes, the charge radii and the spin-parity are. In addition, the signatures found are all consistent to each other and point unambiguously to the existence of a shape transition. The correlations between all of these signatures are useful to establish a protocol to look for shape transitions in the future in other regions of the nuclear chart and to predict and identify them. We have also analyzed the isotopic evolution of the one-quasiproton configurations and have compared the predictions of two different Gogny-EDF parametrizations, demonstrating the robustness of our calculations.

The experimental information available in neutron-rich Rb isotopes and in general in this mass region is still very limited. It would be highly desirable to extend the experimental programs for mass, charge radii, and spectroscopic measurements to these exotic regions at existing facilities like ISOLTRAP (29) at ISOLDE/CERN and IGISOL (30) at the University of Jyväskylä, or at future ones like MATS and LaSpec (3) at the FAIR facility, where we can learn much about structural evolution in nuclear systems.

The theoretical efforts should be also pushed forward by improving the formalism including triaxial degrees of freedom in those regions where this can be an issue or dealing with the odd systems with exact blocking treatments. The quality of our mean field description could be improved from configuration mixing calculations in the spirit of the Generator Coordinate Method with the quadrupole moment as generator coordinate. Nevertheless, such an approach for odd nuclei is out of the scope of the present study. It is already known (31) that such a configuration mixing reduces the jump in predicted in pure mean field approximations when crossing shell closures, improving the agreement with experiment. This could be particularly relevant for the light isotopes considered in the present study, where the spherical minima are rather shallow. For heavier isotopes, the two minima, oblate and prolate, are separated by spherical barriers of about 3 MeV and appear about 1 MeV apart. The effect here is not expected to be significant because a single shape would be enough to account for the properties studied. A simple two-state mixing model could be used to mix the intrinsic deformed configurations into the physical states (1). Assuming that the wave functions are localized at largely different values of the deformation coordinate, one can neglect the cross terms. The final effect would be to place the value of the radius in between the oblate and prolate values according to the mixing amplitude of those intrinsic configurations into the physical wave function. The consequence would be a slightly smoother transition.


This work was supported by MICINN (Spain) under research grants FIS2008–01301, FPA2009-08958, and FIS2009-07277, as well as by Consolider-Ingenio 2010 Programs CPAN CSD2007-00042 and MULTIDARK CSD2009-00064. We thank Prof. J. Äysto and Prof. P.M. Walker for valuable suggestions and discussions.


  1. J. L. Wood et al., Phys. Rep. 215, 101 (1992).
  2. M. Bender, P.-H. Heenen, and P.-H. Reinhard, Rev. Mod. Phys. 75, 121 (2003).
  3. D. Rodríguez et al., Eur. Phys. J. Special Topics 183, 1 (2010).
  4. R. Rodríguez-Guzmán and P. Sarriguren, Phys. Rev. C 76, 064303 (2007); P. Sarriguren, R. Rodríguez-Guzmán and L.M. Robledo, Phys. Rev. C 77, 064322 (2008); L.M. Robledo, R.R. Rodríguez-Guzmán and P. Sarriguren, Phys. Rev. C 78, 034314 (2008); L.M. Robledo, R. Rodríguez-Guzmán and P. Sarriguren, J. Phys. G: Nucl. Part. Phys. 36, 115104 (2009); R. Rodríguez-Guzmán, P. Sarriguren, L.M. Robledo and J.E. García-Ramos, Phys. Rev. C 81, 024310 (2010).
  5. R. Rodríguez-Guzmán, P. Sarriguren, L.M. Robledo and S. Perez-Martin, Phys. Lett. B691, 202 (2010).
  6. R. Rodríguez-Guzmán, P. Sarriguren, and L.M. Robledo, Phys. Rev. C 82, 044318 (2010).
  7. W. Urban, et al., Nucl. Phys. A689, 605 (2001).
  8. P. Campbell, et al., Phys. Rev. Lett. 89, 082501 (2002).
  9. F.C. Charlwood et al., Phys. Lett. B674, 23 (2009).
  10. J. Skalski, S. Mizutori, and W. Nazarewicz, Nucl. Phys. A617, 281 (1997).
  11. F. R. Xu, P. M. Walker, and R. Wyss, Phys. Rev. C 65, 021303(R) (2002).
  12. G. Lhersonneau, B. Pfeiffer, H. Gabelmann, and K.-L. Kratz, Phys. Rev. C 63, 054302 (2001).
  13. D. Bucurescu et al., Phys. Rev. C 76, 064301 (2007).
  14. J. K. Hwang et al., Phys. Rev. C 80, 037304 (2009).
  15. G. S. Simpson et al., Phys. Rev. C 82, 024302 (2010).
  16. J. Dechargé and D. Gogny, Phys. Rev. C 21, 1568 (1980).
  17. J. F. Berger, M. Girod, and D. Gogny, Nucl. Phys. A428, 23c (1984).
  18. S. Goriely, S. Hilaire, M. Girod and S. Péru, Phys. Rev. Lett. 102, 242501 (2009).
  19. T. Duguet, P. Bonche, P.-H. Heenen, and J. Meyer, Phys. Rev. C 65, 014310 (2001).
  20. L. Bonneau, P. Quentin, and P. Möller, Phys. Rev. C 76, 024320 (2007).
  21. S. Perez-Martin and L.M. Robledo, Phys. Rev. C 78, 014304 (2008).
  22. N. Schunck et al., Phys. Rev. C 81, 024316 (2010).
  23. P. Federman and S. Pittel, Phys. Lett. B69, 385 (1977); Phys. Rev. C 20, 820 (1979).
  24. Evaluated Nuclear Structure Data Files (ENSDF):
  25. G. Audi, A. H. Wapstar, and C. Thibault, Nucl. Phys. A729, 337 (2003).
  26. S. Rahaman et al., Eur. Phys. J. A 32, 87 (2007).
  27. C. Thibault et al., Phys. Rev. C 23, 2720 (1981).
  28. U. Hager et al., Nucl. Phys. A 793, 20 (2007).
  29. M. Mukherjee et al., Eur. Phys. J. 35, 1 (2008).
  30. A. Jokinen et al., Int. J. Mass Spectrom. 251, 204 (2006).
  31. M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 78, 054312 (2008).
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