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# Part III

## Appendices
# Chapter 7

# Selected publications

## 7.1 Shape Coexistence in Light Se Isotopes: Evidence for Oblate Shapes
# Shape Coexistence in Light Se Isotopes: Evidence for Oblate Shapes

J. Ljungvall,¹ A. Görgen,¹ M. Girod,² J.-P. Delaroche,² A. Dewald,³ C. Dossat,¹ E. Farnea,⁴ W. Korten,¹ B. Melon,³ R. Menegazzo,⁴ A. Obertelli,¹ R. Orlandi,⁵ P. Petkov,³,⁶ T. Pissulla,³ S. Siem,⁷ R. P. Singh,⁵,* J. Srebrny,⁸ Ch. Theisen,¹ C. A. Ur,⁴,† J. J. Valiente-Dobón,⁵ K. O. Zell,³ and M. Zielińska¹

¹CEA-Saclay, DAPNIA/SPhN, F-91191 Gif-sur-Yvette Cedex, France

²CEA-DIF, DPTA/SPN, Bruyères-le-Châtel, F-91297 Arpajon Cedex, France

³Institut für Kernphysik, Universität zu Köln, D-50937 Köln, Germany

⁴INFN Sezione di Padova, I-35131 Padova, Italy

⁵INFN-Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy

⁶INRNE, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria

⁷Department of Physics, University of Oslo, N-0316 Oslo, Norway

⁸Heavy Ion Laboratory, Warsaw University, Warsaw, PL-02097, Poland

(Received 17 December 2007; published 14 March 2008)

Lifetimes of states in the ground-state bands of ⁷⁰Se and ⁷²Se were measured using the recoil-distance Doppler shift method. The results deviate significantly from earlier measurements, requiring a revision of the conclusions drawn from a recent Coulomb excitation experiment concerning the shape of ⁷⁰Se. The new results lead to a coherent picture of shape coexistence in the neutron-deficient selenium and krypton isotopes. The coexistence and evolution of oblate and prolate shapes in this mass region is for the first time consistently described by new Hartree-Fock-Bogolyubov-based configuration-mixing calculations which were performed using the Gogny D1S interaction.

DOI: 10.1103/PhysRevLett.100.102502

PACS numbers: 21.10.Re, 21.10.Tg, 25.70.Hi, 27.50.+e

The nuclear shape is a very sensitive probe of the under-
lying nuclear structure and the interaction between the
nucleons. Observables such as the electric quadrupole mo-
ment or the transition strength between excited nuclear
states are hence important benchmarks for nuclear struc-
ture theory. Nuclei in the mass region A ≈ 70 close to the
N = Z line are known to exhibit a variety of nuclear
shapes, which can be attributed to large shell gaps at
both prolate and oblate deformation for proton and neutron
numbers 34 and 36. Coulomb excitation experiments have
yielded the quadrupole moments for several low-lying
states in ⁷⁴Kr and ⁷⁶Kr; their signs give firm evidence for
the prolate character of the ground-state band coexisting
with an excited oblate band built on a low-lying 0₂⁺ state
[1]. The analysis of the electromagnetic matrix elements
and the properties of the 0₂⁺ shape isomers [2] suggest an
inversion of the prolate and oblate configurations with an
oblate ground-state shape for ⁷²Kr. The relatively small
B(E2; 0₁⁺ → 2₁⁺) value measured via intermediate-energy
Coulomb excitation in ⁷²Kr [3] seems to support this
interpretation. While many theoretical models predict the
coexistence of prolate and oblate shapes [4–7], the inver-
sion of oblate and prolate configurations in the light Kr
isotopes is so far only correctly described by the so-called
excited VAMPIR variational approach [8] and the Hartree-
Fock-Bogoliubov (HFB) based configuration-mixing
method using the Gogny D1S interaction, as described in
Ref. [9].

The shape coexistence scenario in the neutron-deficient
Se isotopes is less clear. Several theoretical investigations
predict oblate ground-state configurations also for the Se
isotopes near N = Z [4,10,11]. The isotones ⁷²Se and ⁷⁴Kr

show a similar structure with an isomeric excited 0<sub>2</sub><sup>+</sup> state
just above the 2<sub>1</sub><sup>+</sup> state [12] and strongly decreasing B(E2)
strength for the transitions in the yrast cascade towards the
ground state [13], which can be attributed to a strong
mixing between prolate and oblate configurations at
low spin. In neighboring 70Se, on the other hand, a strong
increase of the B(E2) values was observed towards the
ground state [13]. This behavior is very unusual, and the
large B(E2; 2<sub>1</sub><sup>+</sup> → 0<sub>1</sub><sup>+</sup>) value for 70Se results in an unex-
plained discontinuity in the evolution of collectivity in the
chain of Se isotopes. The best candidate for shape coex-
istence appears to be 68Se, where two distinct rotational
bands have been observed. The ground-state band has a
lower moment of inertia and was interpreted as an oblate
rotational band, while the excited band was found consis-
tent with prolate shape [14]. The moments of inertia of the
rotational structures observed in 68Se and 70Se show strik-
ing similarities [15], even though the presumed oblate
rotation is not prevailing in 70Se. Consequently, 70Se was
thought to show a rapid transition from oblate shape near
the ground state to prolate shape at higher spins. A recent
low-energy Coulomb excitation experiment with a 70Se
beam from REX-ISOLDE, however, found a negative di-
agonal E2 matrix element for the 2<sub>1</sub><sup>+</sup> state, indicating
prolate shape for this state [16]. This result cast doubt on
the existence of low-lying oblate states in 70Se. As only the
integral cross section was measured in the Coulomb exci-
tation experiment, the result for the diagonal matrix ele-
ment relies on the independently measured lifetime of the
2<sub>1</sub><sup>+</sup> state. This lifetime is hence a crucial parameter for the
understanding of the shape coexistence in the light Se
nuclei. In this Letter we report on a new lifetime measure-
ment with improved precision, and we interpret the results in the light of new configuration-mixing calculations beyond the mean-field approach.

Lifetimes of low-lying states in $^{70}$Se and $^{72}$Se were measured at the Laboratori Nazionali di Legnaro using the recoil-distance Doppler shift method [17]. The states were populated in the reactions $^{40}$Ca($^{36}$Ar, $\alpha2p$)$^{70}$Se and $^{40}$Ca($^{36}$Ar, $4p$)$^{72}$Se at a beam energy of 136 MeV. Gamma rays were detected with the GASP detector array [18] comprising 38 escape-suppressed Ge detectors arranged in seven rings with respect to the beam axis. The isotopically enriched $^{40}$Ca target had a thickness of 0.5 mg/cm$^2$ and was evaporated onto a 2.0 mg/cm$^2$ gold foil which was facing the beam. The recoils had an average velocity of 10.73(3) $\mu$m/ps and were stopped in a gold stopper foil of 10 mg/cm$^2$ thickness. The Cologne plunger device [19] was used to control the distance between the target and the stopper foils. Data were collected for 12 distances between 8 and 400 $\mu$m for an average of 10 h per distance. Events with at least two prompt $\gamma$ rays detected in coincidence were recorded and sorted offline into $\gamma - \gamma$ matrices for further analysis.

To illustrate the quality of the data, spectra of the $2_1^+$ → $0_1^+$ transition in $^{70}$Se and the $4_1^+$ → $2_1^+$ transition in $^{72}$Se are shown in Fig. 1 for different distances between target and stopper foil. Similar spectra were obtained for all detector rings and plunger distances and for other transitions in both $^{70}$Se and $^{72}$Se. In all cases a coincidence gate was placed on the shifted component of the transition directly feeding the state of interest, so that effects of unknown side feeding were eliminated. The lifetimes of the states were extracted from the intensities of the stopped and Doppler-shifted components as a function of the plunger distance using the differential decay-curve method [17] following the procedure described in Ref. [20]. Since the lifetimes are relatively short, the contribution from $\gamma$ rays emitted during deceleration in the stopper foil had to be considered. An average stopping time of 1.45(18) ps was found in a

Monte Carlo simulation of the stopping process, which was also used to produce a set of line shape profiles for various lifetimes of the respective states [21]. The effect of the finite stopping time was included in the analysis by correcting for the fraction of $\gamma$ rays emitted during deceleration. The results are shown in Table I.

While the new lifetimes for $^{72}$Se are in reasonable agreement with the results of Heese et al. [13], the values for the $2_1^+$ and $6_1^+$ states in $^{70}$Se deviate considerably. The most likely reason for this disagreement are effects from unknown side feeding in a singles measurement. It is interesting to note that significant deviations from older singles measurements were also observed in neighboring Kr isotopes [20], so that one might speculate about unusual feeding patterns in this mass region, which may not be correctly described by the feeding models employed in the older singles measurements. In the case of $^{70}$Se, a delayed feeding component from the $13^-$ isomer with 2.3 ns lifetime [10] could have influenced the results of the previous measurement.

The experimental results in Table I are compared to HFB-based configuration-mixing calculations using the Gogny D1S interaction. The calculations contain no free parameters other than the globally derived parameters of the D1S nucleon-nucleon interaction [22,23]. The configuration mixing of the constrained HFB wave functions is calculated using the generator coordinate method with Gaussian overlap approximation. The five-dimensional generator coordinate comprises the collective quadrupole coordinates $q_0$ and $q_2$, which can be expressed in terms of the axial and triaxial deformation parameters $\beta$ and $\gamma$, and the three Euler angles. It has been shown recently that the inclusion of the triaxial degree of freedom is crucial for the correct description of shape coexistence in the light Kr isotopes [1,24]. The procedure to calculate the energy spectra, reduced transition probabilities, and spectroscopic quadrupole moments is described in detail in Ref. [9].

<table><caption>TABLE I. Comparison of the new lifetimes with literature values. The resulting $B(E2)$ values and the energies of the states are compared to the theoretical calculations. The theoretical spectroscopic quadrupole moments are also given.</caption><thead><tr><th rowspan="2">I<sup>π</sup></th><th colspan="2">τ (ps)</th><th colspan="2">E<sub>ex</sub> (keV)</th><th colspan="2">B(E2; l) (e<sup>2</sup> fm<sup>4</sup>)</th><th rowspan="2">Q<sub>S</sub> (e fm<sup>2</sup>)</th></tr><tr><th>New</th><th>[13]</th><th>Exp.</th><th>Theo.</th><th>Exp.</th><th>Theo.</th></tr></thead><tbody><tr><td colspan="8"><strong><i>70</i></strong>Se</td></tr><tr><td>2<sub>1</sub><sup>+</sup></td><td>3.2(2)</td><td>1.5(3)</td><td>945</td><td>818</td><td>342(19)</td><td>549</td><td>+16</td></tr><tr><td>4<sub>1</sub><sup>+</sup></td><td>1.4(1)</td><td>1.4(3)</td><td>2038</td><td>1800</td><td>370(24)</td><td>955</td><td>+12</td></tr><tr><td>6<sub>1</sub><sup>+</sup></td><td>1.9(3)</td><td>3.9(9)</td><td>3002</td><td>2703</td><td>530(96)</td><td>1404</td><td>-35</td></tr><tr><td colspan="8"><strong><i>72</i></strong>Se</td></tr><tr><td>2<sub>1</sub><sup>+</sup></td><td>4.2(3)</td><td>4.3(5)</td><td>862</td><td>742</td><td>405(25)</td><td>678</td><td>+6</td></tr><tr><td>4<sub>1</sub><sup>+</sup></td><td>3.3(2)</td><td>2.7(4)</td><td>1637</td><td>1660</td><td>882(50)</td><td>1277</td><td>-33</td></tr><tr><td>6<sub>1</sub><sup>+</sup></td><td>1.7(1)</td><td>2.3(2)</td><td>2467</td><td>2266</td><td>1220(76)</td><td>2123</td><td>-77</td></tr></tbody></table>

<figure><img src="image_33.png" /><figcaption>FIG. 1 (color online). Spectra showing the stopped and Doppler-shifted components observed in the forward detectors at 36° for the transitions and distances indicated. The simulated line shape of the stopped component (for a lifetime of 3.3 ps) and the Gaussian fit of the shifted component are shown for the transition in $^{72}$Se.</figcaption></figure>
The $B(E2)$ transition strength in $^{72}$Se strongly decreases from the $6_1^+$ to the ground state. The same behavior has been observed in the light Kr isotopes, where it is understood by an increased mixing of oblate and prolate configurations for the low-spin states [20]. The overall collectivity is higher in the Kr isotopes, which can be attributed to the deformation-driving effect of the $g_{9/2}$ proton orbital. The collectivity in $^{70}$Se is much smaller than in $^{72}$Se (due to less occupation of the $g_{9/2}$ neutron orbital), but a similar trend is observed: the $6_1^+ \rightarrow 4_1^+$ transition is more collective than the transitions below, which are similar in strength. The configuration-mixing calculations reproduce the energies of the states very well; the theoretical $B(E2)$ values, however, are systematically too large. On the other hand, the systematic trend, i.e., the lower collectivity of $^{70}$Se, and the reduction of the $B(E2)$ values towards the ground state is qualitatively reproduced by the calculations.

With the new values from the present measurement, the systematics of the $B(E2)$ values in the chain of light Se isotopes shows a smooth decreasing trend from the most collective isotopes $^{76}$Se and $^{74}$Se at midshell to a small value for $^{68}$Se [25], resolving the unexplained staggering caused by the very large $B(E2)$ value previously assigned to $^{70}$Se.

The revised lifetime of the $2_1^+$ state in $^{70}$Se has important consequences for the interpretation of the results obtained in the recent Coulomb excitation experiment [16]. This is illustrated in Fig. 2, which shows the $B(E2; 2_1^+ \rightarrow 0_1^+)$ value as a function of the spectroscopic quadrupole moment $Q_s$.

The full sloping lines represent the $1\sigma$ limits obtained from the measured Coulomb excitation cross section, which is sensitive to the quadrupole moment via the reorientation effect [26]. The dashed horizontal lines mark the $1\sigma$ limit of the $B(E2)$ value based on the old lifetime measurement by Heese *et al.* [13]. Only negative values of $Q_s$ are consistent with both the old lifetime and the Coulomb excitation measurements, and it was concluded that the shape associated with the $2_1^+$ state in $^{70}$Se is prolate [16]. The present lifetime measurement requires a revision of this conclusion. The $1\sigma$ limits for the $B(E2)$ value from the new measurement are shown by the full horizontal lines. The intersection of the areas of possible values clearly favors a positive value of the spectroscopic quadrupole moment, i.e., oblate shape. Even though the precision of the lifetime measurement is significantly improved, it does not yet permit a precise determination of the quadrupole moment due to the large uncertainty of the Coulomb excitation cross section and its relatively weak dependence on the quadrupole moment. A more stringent constraint would require a more precise Coulomb excitation experiment. While the theoretical $B(E2)$ value is too large, the calculation also favors a moderately oblate shape for the state, as indicated by the asterisk in Fig. 2.

The spectroscopic quadrupole moments for the low-lying states in $^{70}$Se and $^{72}$Se obtained in the calculations are given in Table I, and they are plotted in Fig. 3 as a function of spin. In addition, calculations were also performed for $^{68}$Se, and the resulting quadrupole moments are included in Fig. 3. The agreement between the calculated and experimental excitation energies of the states in $^{68}$Se is equally satisfying as for $^{70}$Se and $^{72}$Se. The $2_1^+$ states of all three Se isotopes can be associated with oblate shapes of moderate deformation. The ground-state band in $^{72}$Se evolves quickly into a highly collective prolate rotational

<figure><img src="image_34.png" /></figure>

FIG. 2 (color online). $B(E2)$ value as a function of the spectroscopic quadrupole moment $Q_s$ for the $2_1^+$ state in $^{70}$Se. The sloping lines indicate the $1\sigma$ range of values compatible with the Coulomb excitation measurement [16]. The horizontal solid (red) and dashed (blue) lines represent the $1\sigma$ ranges from the present and the earlier [13] lifetime measurements, respectively. The asterisk indicates the theoretical value.

<figure><img src="image_35.png" /></figure>

FIG. 3 (color online). Spectroscopic quadrupole moments obtained from the configuration-mixing calculations as a function of spin for the low-lying states in $^{68}$Se, $^{70}$Se, and $^{72}$Se.
band. This is consistent with the rapid increase of the $B(E2)$ with spin, found both experimentally and theoretically. The ground-state band in $^{70}$Se, on the other hand, stays oblate up to the $4_1^+$ state and turns prolate only with the $6_1^+$ state. This is consistent with the relatively small experimental $B(E2; 4_1^+ \rightarrow 2_1^+)$ and $B(E2; 2_1^+ \rightarrow 0_1^+)$ values, and the lower overall collectivity found in $^{70}$Se. Finally, in $^{68}$Se the ground-state band stays oblate, as already concluded by Fischer et al. from its low moment of inertia [14]. The quadrupole moments of the states in the excited band in $^{68}$Se are negative, confirming the previous interpretation of shape coexistence in this nucleus. It is interesting to note in this context that the calculations find the lowest excited $0_2^+$ state at 2.5 MeV excitation energy, which would explain why it has escaped observation so far, despite considerable experimental effort [14,15,25]. It also shows that the picture of two rotational bands built on oblate and prolate $0^+$ states is too simplistic to describe the shape coexistence scenario in this mass region. This has already been pointed out based on the experimental results in the light Kr isotopes, which show not only a complex mixing of prolate and oblate shapes, but also mixing between rotational and vibrational states [1]. It has also become evident that triaxial shapes play a crucial role in this mass region; a correct description of the shape coexistence is only achieved if the triaxial degree of freedom is included in the beyond-mean-field calculations [24].

From the totality of experimental and theoretical results found in the light Se and Kr nuclei, a consistent picture seems to emerge in which oblate shapes become more and more favored near the ground state as one approaches $N=Z$. In the chain of Se isotopes $^{68}$Se remains the best example for coexistence between oblate and prolate configurations. The conclusions drawn from the Coulomb excitation of $^{70}$Se, which contradicted this picture, have to be revised based on the new lifetime measurement. The shape coexistence in $^{70}$Se and $^{72}$Se is, however, more delicate than in $^{68}$Se or in the Kr isotopes, as the coexisting shapes do not prevail in collective bands to higher spin. Further experimental evidence for the shape coexistence scenario could come from multistep Coulomb excitation experiments.

In summary, a new RDDS lifetime measurement in $^{70}$Se and $^{72}$Se revealed considerable discrepancies with the literature values, requiring a revision of the conclusions drawn from the recent Coulomb excitation experiment on the shape of $^{70}$Se. HFB-based configuration-mixing calculations have been performed through solving a five-dimensional, microscopic collective Hamiltonian. The calculated energies are in rather good agreement with the

experimental results, while the $B(E2)$ values are systematically too large. The calculations support the interpretation of an oblate rotational ground-state band in $^{68}$Se. Both theoretical and experimental results indicate that the energy of the oblate configuration increases with neutron number, so that prolate shapes are prevailing more and more in the heavier isotopes.

This work was partly supported by the European Community FP6—Structuring the ERA—Integrated Infrastructure Initiative—Contract EURONS No. RII3-CT-2004-506065.

<footnote>*Permanent address: Inter University Accelerator Centre, New Delhi 110067, India.</footnote>

<footnote>†On leave from IFIN-HH Bucharest, R-077125, Romania.</footnote>

<footnote>[1] E. Clément et al., Phys. Rev. C **75**, 054313 (2007).</footnote>

<footnote>[2] E. Bouchez et al., Phys. Rev. Lett. **90**, 082502 (2003).</footnote>

<footnote>[3] A. Gade et al., Phys. Rev. Lett. **95**, 022502 (2005).</footnote>

<footnote>[4] W. Nazarewicz et al., Nucl. Phys. **A435**, 397 (1985).</footnote>

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<footnote>[7] M. Bender et al., Phys. Rev. C **74**, 024312 (2006).</footnote>

<footnote>[8] A. Petrovici et al., Nucl. Phys. **A665**, 333 (2000).</footnote>

<footnote>[9] J. Libert et al., Phys. Rev. C **60**, 054301 (1999).</footnote>

<footnote>[10] T. Mylaeus et al., J. Phys. G **15**, L135 (1989).</footnote>

<footnote>[11] A. Petrovici et al., Nucl. Phys. **A728**, 396 (2003).</footnote>

<footnote>[12] J. E. Draper et al., Phys. Rev. C **9**, 948 (1974).</footnote>

<footnote>[13] J. Heese et al., Z. Phys. A **325**, 45 (1986).</footnote>

<footnote>[14] S. M. Fischer et al., Phys. Rev. Lett. **84**, 4064 (2000).</footnote>

<footnote>[15] S. M. Fischer et al., Phys. Rev. C **67**, 064318 (2003).</footnote>

<footnote>[16] A. M. Hurst et al., Phys. Rev. Lett. **98**, 072501 (2007).</footnote>

<footnote>[17] A. Dewald et al., Z. Phys. A **334**, 163 (1989).</footnote>

<footnote>[18] D. Bazzacco, in *Proceedings of the International Conference on Nuclear Structure at High Angular Momentum*, Ottawa (AECL Report No. 10613, 1992), Vol. 2, p. 376.</footnote>

<footnote>[19] A. Dewald, in *Ancillary Detectors and Devices for Euroball*, edited by H. Grawe (GSI and the Euroball Ancillary Group, Darmstadt, 1998), p. 70.</footnote>

<footnote>[20] A. Görgen et al., Eur. Phys. J. A **26**, 153 (2005).</footnote>

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<footnote>[22] J. Dechargé and D. Gogny, Phys. Rev. C **21**, 1568 (1980).</footnote>

<footnote>[23] J.-F. Berger et al., Comput. Phys. Commun. **63**, 365 (1991).</footnote>

<footnote>[24] M. Girod et al. (to be published).</footnote>

<footnote>[25] E. Clément et al., Nucl. Instrum. Methods Phys. Res., Sect. A doi:10.1016/j.nima.2008.01.059 (2008).</footnote>

<footnote>[26] K. Alder and A. Winther, *Electromagnetic Excitation: Theory of Coulomb Excitation with Heavy Ions* (North-Holland, Amsterdam, 1975).</footnote>
7.2 Onset of collectivity in neutron-rich Fe isotopes:
Toward a new island of inversion?
Onset of collectivity in neutron-rich Fe isotopes: Toward a new island of inversion?

J. Ljungvall,¹,²,³ A. Görgen,¹ A. Obertelli,¹ W. Korten,¹ E. Clément,² G. de France,² A. Bürger,⁴ J.-P. Delaroche,⁵ A. Dewald,⁶

A. Gadea,⁷ L. Gaudefroy,⁵ M. Girod,⁵ M. Hackstein,⁶ J. Libert,⁸ D. Mengoni,⁹ F. Nowacki,¹⁰ T. Pissulla,⁶ A. Poves,¹¹

F. Recchia,¹² M. Rejmund,² W. Rother,⁶ E. Sahin,¹² C. Schmitt,² A. Shrivastava,² K. Sieja,¹⁰ J. J. Valiente-Dobón,¹²

K. O. Zell,⁶ and M. Zielińska¹³

¹CEA Saclay, IRFU, Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France

²GANIL, CEA/DSM-CNRS/IN2P3, Bd Henri Becquerel, BP 55027, F-14076 Caen, France

³CSNSM, CNRS/IN2P3, F-91405 Orsay, France

⁴Department of Physics, University of Oslo, PO Box 1048 Blindern, N-0316 Oslo, Norway

⁵CEA, DAM, DIF, F-91297 Arpajon, France

⁶Institut für Kernphysik, Universität zu Köln, D-50937 Köln, Germany

⁷Instituto de Física Corpuscular, CSIC-Universidad de Valencia, E-46071 Valencia, Spain

⁸Institut de Physique Nucléaire, CNRS/IN2P3-Université Paris-Sud, F-91406 Orsay, France

⁹Dipartimento di Fisica dell'Università and INFN Sezione di Padova, I-35131 Padova, Italy

¹⁰IPHC, CNRS/IN2P3 and Université Louis Pasteur, F-67037 Strasbourg, France

¹¹Departamento de Física Teórica, IFT-AM/CSIC, Universidad Autónoma, E-28049 Madrid, Spain

¹²INFN, Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy

¹³Heavy Ion Laboratory, Warsaw University, Warsaw, PL-02097, Poland

(Received 8 March 2010; published 15 June 2010)

The lifetimes of the first excited $2^+$ states in $^{62}$Fe and $^{64}$Fe have been measured for the first time using the recoil-distance Doppler shift method after multinucleon transfer reactions in inverse kinematics. A sudden increase of collectivity from $^{62}$Fe to $^{64}$Fe is observed. The experimental results are compared with new large-scale shell-model calculations and Hartree-Fock-Bogolyubov-based configuration-mixing calculations using the Gogny D1S interaction. The results give a deeper understanding of the mechanism leading to an onset of collectivity near $^{68}$Ni, which is compared with the situation in the so-called *island of inversion* around $^{32}$Mg.

DOI: 10.1103/PhysRevC.81.061301

PACS number(s): 21.10.Re, 21.10.Tg, 27.50.+e

It is a common feature of systems of interacting fermions to form a shell structure. In atomic nuclei the spin-orbit interaction lowers the energy of the orbitals with the highest angular momentum into the next lower oscillator shell with opposite parity, leading to the well-known sequence of magic numbers. While the shell structure and the resulting energy gaps between the orbitals explain many general properties of nuclei across the nuclear chart, it has become evident that the shell structure and magic numbers change for nuclei with large neutron excess. As protons and neutrons in such exotic nuclei occupy different orbitals compared to their stable counterparts, the effective single-particle energies are shifted. As a consequence, some shell gaps vanish and new ones emerge [1]. It is important to find experimental signatures for the changing shell structure in order to advance the theoretical description of exotic nuclei, e.g., by finding better shell-model interactions or better energy functionals for mean-field based models.

The region of neutron-rich nuclei between $Z = 20$ (Ca) and $Z = 28$ (Ni) is of particular interest for understanding the evolution of the shell structure for nuclei with large neutron excess. Adding protons to the $1f_{7/2}$ orbital changes the relative energies of the neutron $2p_{3/2}$, $2p_{1/2}$, and $1f_{5/2}$ orbitals due to the strongly attractive proton-neutron spin-flip interaction [2]. With the gradual filling of the neutron *fp* harmonic oscillator shell, excitations into the $1g_{9/2}$ intruder orbital become increasingly important. Subtle effects related to the energy gap between the *fp* shell and the $1g_{9/2}$ orbital

lead to a strong variation of collectivity for nuclei with $N \approx 40$. The nucleus $^{68}_{28}$Ni$_{40}$ has many features of a doubly magic nucleus with high excitation energy of the $2_1^+$ state [3] and a small $B(E2; 0_1^+ \rightarrow 2_1^+)$ value [4]. Mass measurements, on the other hand, show that the $N = 40$ gap is weak for $^{68}$Ni [5,6], and it has been argued that the small $B(E2)$ value does not necessarily indicate a shell gap at $N = 40$, but that it reflects the character of the $2_1^+$ state as a predominant neutron excitation [7]. The fragility of the $N = 40$ subshell gap is further evidenced by the developing collectivity in the neutron-rich Zn and Ge isotopes, for which the energy of the $2_1^+$ state drops sharply from $N = 38$ to $N = 42$ while the $B(E2)$ values increase [8,9]. The removal of protons from the $f_{7/2}$ orbital has a similar effect. The $2_1^+$ energies in the Fe isotopes drop sharply above $N = 36$. The removal of only two protons changes the energy of the $2_1^+$ state from 2033 keV in $^{68}$Ni to 573 keV in $^{66}$Fe [10]. The $2_1^+$ energy drops even further to 517 keV in $^{68}$Fe [11]. A similar behavior is observed in the Cr isotopes, where the $2_1^+$ energies decrease gradually beyond the $N = 32$ subshell closure [12]. Inelastic proton-scattering experiments [13] and shell-model studies [14] corroborate the development of substantial collectivity and deformation in the neutron-rich Cr isotopes.

The role of the $v_{g_{9/2}}$ orbital for the onset of collectivity in the region below $^{68}$Ni has been discussed controversially in the past. The excitation spectrum of $^{64}$Fe was established by Hoteling et al. and compared to shell-model calculations restricted to the *fp* model space [15]. Since the calculations
J. LJUNGVALL *et al.*

PHYSICAL REVIEW C 81, 061301(R) (2010)

reproduced the energy of the 2⁺ state reasonably well, it was concluded that the $\nu g_{9/2}$ orbital is not responsible for the low $2_1^+$ energy and that it becomes important only at higher energies, where the calculations could not reproduce the observed states. Lunardi *et al.* have studied neutron-rich Fe isotopes of both even and odd mass [16]. Comparing the results to shell-model calculations in the *fpg* model space it was concluded that the occupation of the $g_{9/2}$ orbital influences the structure of the neutron-rich Fe isotopes considerably. The increasing role of the $g_{9/2}$ orbital is also consistent with the decreasing excitation energy of 9/2⁺ isomers in odd-mass Fe nuclei [17]. In studies of the odd-mass Mn isotopes, Valiente-Dobón *et al.* have argued that including the effects of the $g_{9/2}$ orbital improves the shell-model description of <sup>61</sup>Mn<sub>36</sub> and <sup>63</sup>Mn<sub>38</sub> [18], while Crawford *et al.* have interpreted the level structure in <sup>61</sup>Mn with a low-level density below 1 MeV without the need to invoke neutron excitations into the $g_{9/2}$ orbital [19]. In the Cr isotopes there is clear evidence for the influence of the $g_{9/2}$ orbital already for <sup>59</sup>Cr<sub>35</sub> [20].

A deeper understanding of how collectivity develops in this region of the nuclear chart can come from experimental $B(E2)$ values. For the even-even nuclei $B(E2)$ values have been measured up to <sup>58</sup>Cr<sub>34</sub> [21], <sup>60</sup>Fe<sub>34</sub> [22], and <sup>70</sup>Ni<sub>40</sub> [23]. Here we report on a lifetime measurement for the $2_1^+$ states in <sup>62</sup>Fe and <sup>64</sup>Fe performed at the Grand Accélérateur National d'Ions Lourds (GANIL) using the recoil-distance Doppler shift (RDDS) method [24] in combination with multinucleon transfer reactions in inverse kinematics. The technique is similar to the one used in a recent lifetime measurement in <sup>50</sup>Ca and <sup>51</sup>Sc in direct kinematics at Legnaro [25]. Excited states in a wide range of neutron-rich nuclei were populated in multinucleon transfer reactions between a <sup>238</sup>U beam at 6.5A MeV and a <sup>64</sup>Ni target of 1.5 mg/cm² thickness. The targetlike reaction products were detected and identified event by event in the large-acceptance spectrometer VAMOS [26]. The optical axis of the spectrometer was positioned at 45° with respect to the beam axis, close to the grazing angle of the reaction. The focal plane detection system of the spectrometer allows the full reconstruction of the trajectories through the spectrometer and an unambiguous identification of the reaction products in mass, charge, and atomic number (see Fig. 1). The magnetic rigidity of the spectrometer was optimized for the transmission of <sup>64</sup>Fe.

The target was surrounded by nine large, segmented germanium clover detectors of the Exogam array [27], one at 180°, three at 135°, and five at 90° relative to the spectrometer axis. The typical average recoil velocity of the targetlike reaction products exiting the target foil was 37 µm/ps. In order to apply the RDDS technique the velocity of the recoils was slowed to approximately 32 µm/ps in a 4.7-mg/cm²-thick magnesium foil placed at micrometer distances behind the target. The different Doppler shifts of γ rays emitted before and after the degrader foil were measured in the germanium detectors under backward angles. The distance between target and degrader was adjusted and controlled by a compact *plunger* device [28]. Data was collected for six distances between 40 and 750 µm. An additional measurement was performed without degrader to determine the exact recoil velocities after the target.

The spectra of Fig. 2 show γ rays recorded in coincidence with ions identified as <sup>62</sup>Fe and <sup>64</sup>Fe for selected target-

<figure><img src="image_36.png" /></figure>

FIG. 1. (Color online) Typical identification plot showing the energy loss of the ions in the ionization chamber of VAMOS as a function of the mass-over-charge ratio, which is determined from the time of flight through the spectrometer and the magnetic rigidity. The lines indicate the loci of ions with $A = 64$ and $Z = 26$, respectively. The most intense peak corresponds to elastically scattered <sup>64</sup>Ni target nuclei. The data is shown for only one of 21 silicon detectors mounted in VAMOS and for one single charge state ($Q = 23$). The identification of the ions was performed in the same way for all detectors and all charge states reaching the focal plane.

degrader distances. A Doppler correction was performed using the recoil velocity measured in VAMOS after the degrader foil and the relative angle between the γ ray and the recoil vector. Two peaks are observed for the $2_1^+ \rightarrow 0_1^+$ transitions in both <sup>62</sup>Fe and <sup>64</sup>Fe, corresponding to the different recoil velocities before and after the degrader foil, respectively. The peaks are well separated for relative angles greater than 135°. Spectra were produced and analyzed separately for two angular ranges between 135° and 150° and between 160° and 180°, respectively. Decay curves $I_d/(I_d + I_f)$ were constructed from the intensities of the degraded ($I_d$) and fully Doppler shifted ($I_f$) components of the transitions as a function of target-degrader distance. The normalization factors ($I_d + I_f$) were found to be consistent with the number of ions identified in VAMOS for each distance. Lifetimes of the $2_1^+$ states in <sup>60</sup>Fe, <sup>62</sup>Fe, and <sup>64</sup>Fe were extracted separately for the two angular ranges from the decay curves of the $2_1^+$ and $4_1^+$ states using the differential decay curve (DDC) method [24] and then combined into a weighted average. In the DDC method the lifetime of a state is determined for each target-degrader distance from the slope of

<figure><img src="image_37.png" /></figure>

FIG. 2. Spectra showing the fully Doppler shifted and degraded components of the $2_1^+ \rightarrow 0_1^+$ transitions for three target-degrader distances in <sup>64</sup>Fe and for one distance in <sup>62</sup>Fe. The γ rays were observed at angles ranging from 160° to 180° with respect to the recoil direction.
<figure><img src="image_38.png" /><figcaption>FIG. 3. (Color online) Systematic of $B(E2; 2_1^+ \rightarrow 0_1^+)$ and $E(2_1^+)$ values for even-even nuclei around $^{68}$Ni. The data are taken from the compilation of Raman *et al.* [29] complemented by newer results for Cr [12,21], Fe [11], Ni [23], and Zn [8] isotopes. The panels on the right compare the experimental results for the chain of Fe isotopes with theoretical calculations (see text).</figcaption></figure>

the decay curves, i.e., from the intensity curves as a function of distance. Distances were converted into flight times by using the average velocities measured separately for the different nuclides without degrader foil. In this way the lifetime of the $2_1^+$ state in $^{60}$Fe was determined to be 11.4(12) ps, which is in good agreement with the literature value of 11.6(22) ps [22]. Lifetimes for the $2_1^+$ states in $^{62}$Fe and $^{64}$Fe were determined for the first time and found to be similar with 7.4(9) and 7.4(26) ps, respectively. The observed feeding times from the $4^+$ states were found to be slightly different in $^{62}$Fe and $^{64}$Fe. The decay curves are consistent with the assumption of equal feeding times from the observed $4^+ \rightarrow 2^+$ and other unobserved transitions. Systematic deviations of the lifetimes derived at individual distances would appear if the feeding times for observed and unobserved transitions were different. The measured lifetimes yield $B(E2; 2_1^+ \rightarrow 0_1^+)$ values of 214(26) and $470_{-110}^{+210} e^2 \text{ fm}^4$ for $^{62}$Fe and $^{64}$Fe, respectively.

The $B(E2; 2_1^+ \rightarrow 0_1^+)$ values and excitation energies of the $2_1^+$ states in this mass region are shown as a function of neutron number in the left part of Fig. 3. The large $2_1^+$ energy and small $B(E2)$ value for $^{68}$Ni could be interpreted as a sign for a subshell closure at $N = 40$. However, the chains of Zn and Fe isotopes above and below the proton shell closure, respectively, show a different behavior compared to the Ni isotopes. The drop in excitation energy of the $2_1^+$ states, which is observed for all isotopic chains except Ni, illustrates that the nuclei become more collective and deformed with increasing neutron number. The $B(E2)$ values show that collectivity develops more gradually in the Zn compared to the Fe isotopes, where the new data indicate a steep increase from $^{62}$Fe to $^{64}$Fe.

To understand the rapid onset of collectivity in $^{64}$Fe we have performed new shell-model calculations beyond the scope of

previous calculations in this mass region [30]. A large valence space was used comprising the *fp* shell for protons and the *fpgd* shell for neutrons outside a $^{48}$Ca core. The effective interaction was based on a realistic CD-Bonn potential adapted to the model space by many-body perturbation techniques [31] and modified in its monopole part. The interaction reproduces the global evolution of single-particle energies along the nickel and *N* = 40 isotopic and isotonic chains, respectively. Transition probabilities were obtained using a standard polarization effective charge of 0.5 *e*. The shell-model diagonalizations accommodate for up to 14 particle-hole excitations across the *Z* = 28 and *N* = 40 gaps in the case of $^{64}$Fe. The largest dimensionality tackled in the study was $7 \times 10^9$ for $^{64}$Fe. More details of the shell-model calculations and the new effective interaction will be discussed in a forthcoming article [32].

To illustrate the importance of including the $v_{g9/2}$ and $d_{5/2}$ orbitals, calculations restricted to the neutron *fpg* and *fp* valence spaces are also shown in Fig. 3. The comparison suggests that the *fpg* valence space is sufficient to describe the $2_1^+$ states in the Fe isotopes up to *N* = 36. The large $B(E2; 2_1^+ \rightarrow 0_1^+)$ value measured for $^{64}$Fe clearly demonstrates the influence of neutron excitations across the gap at *N* = 40. In fact, the $B(E2)$ value can only be explained if the $d_{5/2}$ orbital is also included in the model space. The effect becomes even more evident in $^{66}$Fe: Only the calculations in the full *fpgd* model space reproduce the low excitation energies of the $2_1^+$ state. It is well understood that the valence space should contain quasi-SU(3) partners, in this case the $g_{9/2}$ and $d_{5/2}$ orbitals, to allow the development of quadrupole collectivity [33].

For the ground state of $^{64}$Fe the *fpgd* shell-model calculations find average occupancies for the $v_{g9/2}$ and $d_{5/2}$ orbitals of 1.76 and 0.21, respectively, and this state is characterized by an intruder configuration with a strong two-particle two-hole component. Such intruder configurations become completely dominant in $^{66}$Fe$_{40}$ and also in $^{62}$Cr$_{38}$. In the latter case the occupancies of the $g_{9/2}$ and $d_{5/2}$ orbitals increase to 2.2 and 0.72, respectively, which drives the nucleus to an intrinsically deformed prolate shape (with $\beta \approx 0.35$) and causes a more rotational yrast sequence consistent with experimental observations [13]. The transition from spherical $^{68}$Ni to more proton deficient *N* ≈ 40 isotones with deformed intruder configurations seems to have some similarity to the situation in the so-called *island of inversion* of Na and Mg isotopes with *N* ≈ 20 (see, e.g., Ref. [1] for a review). In both cases the developing quadrupole collectivity can be related to the occupation of neutron intruder orbitals from the next oscillator shell, which are at the same time quasi-SU(3) partners: ($f_{7/2}$, $p_{3/2}$) for $^{32}$Mg and ($g_{9/2}$, $d_{5/2}$) in the present case. Active quadrupole partners are also available for the protons: ($d_{5/2}$, $s_{1/2}$) for Mg and ($f_{7/2}$, $p_{3/2}$) for Fe and Cr. The shell-model calculations furthermore find a reduction in the harmonic oscillator gap at *N* = 40 from $^{68}$Ni to $^{60}$Ca. The sudden increase of collectivity from $^{62}$Fe to $^{64}$Fe can thus be understood in a similar way as the increase of collectivity from $^{30}$Mg to $^{32}$Mg.

It is clear, on the other hand, that there is no shell closure at *N* = 40 comparable to the one at *N* = 20. At this point the analogy between the island of inversion, which is situated at
J. LJUNGVALL *et al.*

PHYSICAL REVIEW C 81, 061301(R) (2010)

the end of a long chain of spherical $N = 20$ isotones, and the $N = 40$ region breaks down. With the exception of $^{68}$Ni with closed proton shell, all $N = 40$ isotones develop collectivity and deformation (see left-hand part of Fig. 3). In a recent study within a microscopic collective model, the low-lying states in the $N = 40$ isotones (except $^{68}$Ni) were described as prolate deformed ground-state bands and quasi-γ bands with good overall agreement between the calculations and experimental data [34]. We have extended these studies along the chain of Fe isotopes. The $2_1^+$ energies and $B(E2)$ values were obtained in Hartree-Fock-Bogolyubov-based configuration mixing calculations using the the Gogny D1S force [35,36] and the generator coordinate method with Gaussian overlap approximation, treating axial and triaxial quadrupole deformations [37]. The calculations use a triaxial harmonic oscillator basis including 11 major shells. The collective masses are calculated as described in Ref. [38]. Albeit the $2_1^+$ energies are on average 30% too high, the systematic trend is reasonably well reproduced, considering that the model contains no free parameters (and no effective charges). The calculated $B(E2)$ values show only a small increase at $N = 40$, which can also be attributed to the occupancy of the $vG_{9/2}$ orbital. This occupancy is found to increase from approximately 0.6 in the ground state of $^{60}$Fe to 2.5 in $^{64}$Fe [39]. The occupancy of the $d_{5/2}$ orbital in the ground state of $^{64}$Fe is found to be 0.06, i.e., smaller than in the shell model. A more important difference, however, is the fact that the $N = 40$ gap remains more or less constant in the Gogny mean-field calculation, performed at deformation zero, with $\Delta E \approx 4$ MeV from $Z = 20$ to $Z = 40$ (see Fig. 1 in Ref. [34]), whereas in the shell model the gap is strongly reduced for $Z < 28$. Although it is not obvious that single-particle energies from shell model and

spherical mean-field calculations can be directly compared, since the latter do not contain any correlations beyond the mean field, the different development of the $N = 40$ shell gap could explain the more gradual increase of $B(E2)$ values in the Gogny calculation. On the other hand, proton excitations and the size of the $Z = 28$ gap could also contribute to the difference in collectivity. More theoretical work is needed to understand the origin of the differences between the shell-model and mean-field predictions. While experimental data on $^{60}$Ca remain out of reach with present-day techniques, the measurement of $B(E2)$ values in neutron-rich Cr isotopes will already shed more light on this question and allow making better predictions for a possible doubly magic nature of $^{60}$Ca.

In summary, the lifetimes of the $2_1^+$ states in $^{62}$Fe and $^{64}$Fe have been measured using the RDDS technique in combination with multinucleon transfer reactions in inverse kinematics. A steep increase of the $B(E2; 2_1^+ \rightarrow 0_1^+)$ value is observed from $^{62}$Fe to $^{64}$Fe. The comparison with both shell-model calculations and a microscopic collective model based on the Gogny force allows relating the increase in collectivity to the occupation of neutron intruder orbitals. The mechanism causing this onset of collectivity below $^{68}$Ni is found to be similar to the one responsible for the island of inversion around $^{32}$Mg. The two theoretical approaches predict a different structure of the more exotic nuclei with $N \approx 40$.

The authors thank the technical staff at GANIL for excellent support and A. Navin for fruitful discussions. This work was partly supported by DFG (Germany) under Contract No. DE1516/-1, MICINN (Spain) under project FPA2009-13377, and Comunidad de Madrid (Spain) under project HEPHACOS S2009/ESP-1473.

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7.3 The Orsay Universal Plunger System
<figure><img src="image_39.png" /></figure>

ELSEVIER

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Nuclear Instruments and Methods in
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<figure><img src="image_40.png" /></figure>

# The Orsay Universal Plunger System

J. Ljungvall<sup>a,*</sup>, G. Georgiev<sup>a</sup>, S. Cabaret<sup>a</sup>, N. Karkour<sup>a</sup>, D. Linget<sup>a</sup>, G. Sedes<sup>a</sup>, R. Chevrier<sup>b</sup>, I. Matea<sup>c</sup>, M. Niikura<sup>c</sup>, M.-D. Salsac<sup>d</sup>, B. Sulignano<sup>d</sup>

## ARTICLE INFO

Article history:

Received 8 February 2012

Received in revised form

26 March 2012

Accepted 27 March 2012

Available online 4 April 2012

Keywords:

In-beam γ-ray spectroscopy

Recoil-Distance Doppler Shift

Lifetimes

## ABSTRACT

A new plunger device for Recoil-Distance Doppler Shift and Time-Differential Recoil-In-Vacuum measurements has been developed at the CSNSM in collaboration with the IPN Orsay. It is conceived to be easily adapted to different experimental setups and facilities, in particular to the future radioactive beam facility Spiral2. An electronic feedback system keeping the distance between target and stopper foils constant, imperative for precision measurements of short lifetimes, was also developed.

The plunger device was commissioned with an experiment performed at the IPN Orsay Tandem accelerator laboratory using the Orgam germanium detector array. Excited states in $^{76}$Kr were populated using the reaction $^{45}$Sc($^{25}$Cl,2p2n)$^{76}$Kr. The lifetime $\tau$ of the $4_{1}^{1}$ state was determined to 4.3(6) ps, in accordance with previous measurement, confirming the functionality of the new plunger device. The commissioning experiment showed in particular that the electronic feedback system works as foreseen.

© 2012 Elsevier B.V. All rights reserved.

## 1. Introduction

Detailed spectroscopic information is, together with the nuclear mass, the experimentally accessible observables used to understand the many-body nuclear problem. The energies of excited states, although fundamental, give an incomplete picture of the nuclear-state wave function and are therefore by themselves insufficient for detailed scrutinizing of nuclear structure models. An additional sensitive probe of the nuclear wave function is the electromagnetic transition strength $B(E/ML : J_i \rightarrow J_f)$. Here $J_{if}$ is the spin of the initial (final) nuclear state, $L$ is the multipolarity of the emitted $\gamma$ ray, and $E/M$ indicates an electric- or a magnetic transition. The $B(E/ML : J_i \rightarrow J_f)$ is accessible via several different methods based either on the observation of the excitation, e.g. Coulomb excitation [1], or the decay of an excited state via $\gamma$-ray emission, e.g. methods based on the observed Doppler shift of the energy of the emitted $\gamma$ ray [1] or fast timing measurements [2]. The measurement of the lifetime of an excited state gives a completely model-independent result given that the transition de-exciting the state in question is not of mixed multipolarity.

In this paper a so-called "plunger device" [3] will be described. The "plunger" consists of a thin target foil for the production of recoiling exited nuclei and a "stopper" foil at a well defined distance from the target and is used with the Recoil Distance Doppler Shift (RDDS) method [4]. This method is based on the observation of the $\gamma$-rays emitted when the nuclear states de-excite. By stopping the recoiling nuclei in a "stopper" foil two distinctive velocity regimes are created, and hence the $\gamma$ rays emitted before or in the stopper foil can be separated using the observed Doppler shift of the $\gamma$-ray energies. $\gamma$-Rays emitted before the stopper foil will be observed with a "shifted" energy due to the Doppler shift. If the nuclei have been stopped the $\gamma$ rays will be observed without a Doppler shift of their energy, i.e. "unshifted". By comparing the relative intensities of the "shifted" and "unshifted" components of the $\gamma$-ray line, the lifetime $\tau$ can be extracted. Using the RDDS method together with a plunger device lifetimes of about one to a few hundreds of picoseconds can be measured with a precision comparable to that of the statistical uncertainty. Such experiments therefore deliver high quality data on electromagnetic transition strengths. The distance between the target and stopper foil used for RDDS experiments is approximately given by the product of the lifetime of the excited state and the recoil velocity of the nuclei. This corresponds in many cases to less than a few tens of micrometers. A major technical challenge for a plunger device is therefore to keep the

<footnote><sup>a</sup> CSNSM, Bâtiments 104 et 108, 91405 Orsay Cedex, France</footnote>

<footnote><sup>b</sup> CEA, DAM, DIF, F-91297 Arpajon, France</footnote>

<footnote><sup>c</sup> Institut de Physique Nucléaire, IN2P3-CNRS and Université Paris-Sud, F-91406 Orsay Cedex, France</footnote>

<footnote><sup>d</sup> CEA Saclay, IRFU, SPHN, F-91191 Gif-sur-Yvette, France</footnote>

<footnote>* Corresponding author.
E-mail address: joa.ljungvall@csnsm.in2p3.fr (J. Ljungvall).</footnote>
target- and stopper-foil parallel and at a fixed distance while they
are heated by the beam.

Because of the sensitivity of the B(E/ML : Jᵢ → Jf) value to the
wave function, and the model independent determination of
electromagnetic transition strengths given by plunger measure-
ments, this technique will play an important role in the physics
campaigns at future radioactive beam facilities, such as Spiral2 at
GANIL [5]. Traditionally used together with heavy-ion fusion
evaporation reactions recent developments have shown that the
RDDS technique is also suited to measure lifetimes of excited
states populated in multi-nucleon transfer reactions [6,7], inter-
mediate-energy Coulomb excitation studies [8], low-energy mul-
tistep Coulomb excitation studies [9], as well as together with
induced fission reactions to measure lifetimes of excited states in
fission fragments.

The design of the new plunger will be described in Section 2.
This will be followed in Section 3 by the description of a
successful commissioning experiment performed at the IPN Orsay
tandem. Future plans will be discussed in Section 4, to finally
conclude in Section 5.

## 2. Design

In designing a plunger device a set of characteristics/problems
have to be considered such as

* the possibility to vary the distances between target and stopper foil from a few micrometers to some tens of millimeters with a sub-micrometer precision,

* transparency of the device to minimize the absorption of the γ rays emitted from the de-exciting nuclei,

* a mechanism to stretch target and stopper foils,

* and an active feedback to compensate for changes in the target-stopper foil distance induced by the target-beam interaction.

The solutions used to address these concerns will be discussed in
the following paragraphs.

The heart of any plunger device is the motor that positions the
stopper foil relative to the target foil. There is a need to combine
high precision, i.e. the ability to position the stopper foil within
some tens of nanometers, with mechanics that are rigid enough to
allow for feedback movements. At the same time one has to
minimize the material around the target to keep the plunger
device transparent to γ rays. For traditional plunger devices,
designed to be used with fusion evaporation experiments, this
problem is usually solved by placing the mechanics far away from
the target using a coaxial geometry (e.g. see the plunger device in
Ref. [10]). Such a design has the disadvantage that the plunger
often becomes part of the beam line and hence difficult to adapt
to different experimental setups. A more severe problem with the
coaxial geometry is that it is best suited for experiments were the
reaction products leave the target with a relatively small angle
relative to the beam axis. An alternative is to use a linear piezo
motor assembly on which the mechanics for target and stopper
foils are mounted. It can be made very compact and allows for
large angles between incoming beam and outgoing reaction
products. This is the choice made for the Orsay Plunger. The base
of the plunger is the PI N-661 Miniature Linear Stage (see Fig. 1).
It is a piezo-electric linear motor with a movement range of
20 mm. The size of this industrially available assembly is
70 mm × 50 mm × 20 mm. Its linear motor can generate a force
of up to 10 N. The movement is performed by the upper half of
the PI N-661, which can be positioned with a precision of 40 nm.
The Orsay Plunger is literally built around the PI N-661 linear
motor. Its core is shown in Fig. 2. By the means of pre-threaded

<figure><img src="image_41.png" /><figcaption>Fig. 1. The PI N-661 Miniature Linear Stage.</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>Fig. 2. Main components of the Orsay Plunger.</figcaption></figure>

holes a stopper-foil support is mounted on the moving upper half
of the piezo motor. The PI N-661 itself is mounted on an
aluminum support plate, to which the target-holder support is
also attached. This structure is supposed to remain the same
independently of the type of experiment performed. Due to the
relative thickness of the PI linear motor assembly the amount of
material in the direction orthogonal to the beam and opposing the
target- and stopper-foil holders is quite large, but this is of lesser
importance since the observed Doppler shift of the energy of the
emitted γ rays usually is too small for RDDS measurements at
angles close to 90°.

The Orsay Plunger uses a target- and stopper-foil stretching
system based on pulling the foils over metallic cones that are
machined with high precision. This ensures that the foils can be
considered as planar surfaces allowing a precise definition of the
distance between them. In Fig. 3 the target stretcher assembly is
shown. The target foil is glued onto the target frame, that in turn
is positioned over the cone and attached with three screws.
Stretching the foil is done by tightening the screws forcing the
foil to slightly deform over the cone. The stretcher cone itself is
attached with three screws to a ring, which is firmly attached to
the plunger via the target-frame support. An identical assembly is
used to mount the stopper foil. This allows the adjustment of the
parallelism of the two foils by means of the three screws
mentioned above attaching the stretcher cone to the rings used
to fix the target- and stopper assemblies to the their respective
<figure><img src="image_43.png" /><figcaption>Fig. 3. Target frame and stretch cone shown without a target foil in place.</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>Fig. 4. Three dimensional view of the Ortec Plunger. The target and stopper foils are in place. The shaded part is a support structure specific for the commissioning experiment.</figcaption></figure>

supports. The chosen target size of 10 mm and, with it, the size of the target frames and cones, can easily be changed if needed, e.g. for a multi-nucleon transfer reaction experiment or a reaction with a secondary beam. In Fig. 4 the complete assembly as used during the commissioning experiment is shown. The target and stopper foil sides are indicated in the figure. The shaded structure in the figure are specific for the use at the IPN Orsay Tandem.

When measuring short lifetimes the variations of the target-
stopper distance, due to heating from the beam, might be
significant compared to the distance and they have to be cor-
rected for. An electronic feed-back system was developed for this
purpose, correlating the distance between the foils with the
capacitance measured between them: the two foils are used as
a capacitor whose value changes as function of the distance
between the two foils. A schematic of the electronics setup is
shown in Fig. 5. The feed-back system consists of a high-precision
pulser of type BNC Model PB-5, a spectroscopic amplifier Ortec
571, an impedance matcher circuit, a National Instruments BNC-
2110 BNC adapter, and a National Instruments PCIE-6361 X Series
DAQ—a PCI express card mounted in the control PC. The electro-
nics chain was tested off-line for stability during a period of 64 h,
showing a FWHM variation of 2.5‰, correlated with changes in
the ambient temperature. The PI N-661 linear stage was con-
trolled by the PI E-861 Controller, connected to the control PC via
a USB interface. The electronics were controlled by an application
developed under LabVIEW. The application provided the follow-
ing functionality:

* sampling of the output of the Ortec 571 amplifier via the PCIE-6361 DAQ board,

<figure><img src="image_45.png" /><figcaption>Fig. 5. Schematics of the electronics used to control the Orsay Plunger.</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>Fig. 6. A photo of the Orsay Plunger in the Orgam target chamber.</figcaption></figure>

* online processing to extract the pulse-height at the output of the Ortec 571 amplifier,

* a closed loop feedback controlling the PI N-661 to keep the capacitance as stable as possible,

* slow control of the plunger system, i.e. adjustment of target-stopper foil distance, starting and stopping of the feedback, loading and saving configurations, and security functions.

**3. Commissioning experiment**

A commissioning experiment was performed at the IPN Orsay
Tandem accelerator laboratory in order to verify that the Orsay
Plunger device performs as foreseen. In particular the electronic
feedback system, designed to keep the target-stopper foil dis-
tance constant, had to be proven reliable. The Orsay Plunger was
mounted in the Orgam target chamber, surrounded by the 24
n-type HPGe detectors of the Orgam detector array. The Germa-
nium detectors were equipped with BGO Compton suppression
shields. A photo of the plunger in the chamber is shown in Fig. 6.
The detectors were organized into three rings, according to their
angle relative to the beam axis. Rings 1–3 were located at 46.5°,
133.5°, and 157.6°, respectively.

The main criteria to choose the reaction for the commissioning
experiment were

* a high cross-section to allow for γγ coincidences,

* well known lifetimes of the lowest-laying yrast states to be able to cross-check the results,

* a high beam intensity from the IPN Orsay Tandem,

* and a target material suitable for a plunger experiment.

For the commissioning experiment the ⁴⁵Sc(³⁵Cl, 2p2n)⁷⁶Kr reaction, with a beam energy of 126 MeV, was found suitable. Targets
of 1.1 mg/cm²⁰⁵Sc, protected by thin layers of gold ( ≪ 1 mg/cm²) evaporated on each side, were used during the experiment. A 5 mg/cm² thick gold foil was used as a stopper foil.

Data were collected for a total of five days with beam intensities between 10 and 40 pA. Two different targets were used, one during the first two days, and a second target during the last three days. The first target was used for measuring distances longer than 50 µm. Before measuring shorter distances the target was changed as it was estimated that the second target was of higher quality. During the experiment it was (wrongly) estimated that the data from the first and second target could be use together by matching the normalized intensities for one overlapping distance. During the analysis, the data from the longer distances measured with the first target turned out to be incoherent, an effect most likely due to an insufficiently stretched target foil, which gave ill-defined target-stopper distances. Because of this only distances shorter than 50 µm were used in the lifetime analysis.

Problems with the spectroscopic amplifiers used during the experiment reduced the number of germanium detectors from 24 to 7, thus drastically reducing the statistics in γγ coincidence mode. The working detectors were partitioned as one in ring 1 and ring 3, respectively, and five in ring 2. Since the reaction had several open reaction channels of comparable strengths increasing the statistics by analyzing the data in singles mode was not possible due to too many overlapping γ-ray lines. A singles spectrum from ring 2 is shown in Fig. 7 where the γ rays from the transitions $2_1^+ \to 0_1^+$ (424 keV), $4_1^+ \to 2_1^+$ (611 keV), and $6_1^+ \to 4_1^+$ (824 keV) in ⁷⁶Kr are marked with arrows.

One of the main goals of the commissioning experiment was to verify the feed-back mechanism imperative for accurate measurements at short distances. All values read by the Orsay Plunger control computer were stored on disk to be used during the analysis of the experiment. A calibration was made by tracing the distance as a function of the pulse height at the output of the spectroscopic amplifier read by the DAQ in the control computer. This was done before the second target was exposed to beam, and redone after about 24 h of target irradiation with a beam intensity of 30 pA. A calibration function was then fitted to the combined calibration data set. Shown in Fig. 8 are the calibration points together with the fitted function. The difference between electrical contact and

<figure><img src="image_47.png" /><figcaption>**Fig. 7.** Zoom on the region on interest for ⁷⁶Kr, taken with ring number 2 in Orgam in singles. The first three transitions in the ground-state band, 424 keV, 611 keV, and 824 keV, respectively, are marked with arrows. In-flight components are marked with thicker dashed arrows. The lifetime of the $2_1^+$ state is too long to have an in-flight component.</figcaption></figure>

<figure><img src="image_48.png" /><figcaption>**Fig. 8.** Calibration curves for the capacitive measurement of the target-stopper distance for the plunger taken one day apart, showing the stability of the system. The distance scale on the y-axis has an arbitrary offset of about 5 µm. A fit to the measured points is also shown.</figcaption></figure>

<caption>**Table 1**</caption>

Distances as determined using the function are shown in Fig. 8. The quoted uncertainties are the RMS's of the distance distributions.

<table><thead><tr><td>Distance</td><td>Dist. (µm)</td></tr></thead><tbody><tr><td>1</td><td>11.24(8)</td></tr><tr><td>2</td><td>15.67(2)</td></tr><tr><td>3</td><td>21.03(2)</td></tr><tr><td>4</td><td>25.40(2)</td></tr><tr><td>5</td><td>34.23(3)</td></tr><tr><td>6</td><td>44.59(4)</td></tr></tbody></table>

so-called optical contact, i.e. when no light could be observed between the two foils, was 9 µm. During the analysis of the experiment the calibration function was used to find the relative distances with a high precision. These distances are shown in Table 1. Given in the table are the mean distances together with the RMS of the distance distributions. For the shortest distance, some minor variation can be seen with a RMS of 0.1 µm, corresponding to about 1%. This is because the distance was at the limit considering the quality of the target and stopper foils, and, relative to the distance, a large tolerance for the feed-back system had to be used. It is, however, clear that the feed-back is capable of keeping the target-stopper foil distance constant.

Data from the experiment were sorted into four γγ matrices corresponding to the four possible permutations of coincidences given the angle of the different Ge detectors relative to the beam direction. γ-Ray spectra were produced by gating in the γγ matrices on the γ-ray peak corresponding to the emission in flight of the γ ray feeding the state of interest. Spectra in which the γ ray de-exciting the state of interest was observed at the same energy were summed, giving three spectra to analyse for each distance. In each spectrum the intensity of the unshifted and shifted peak of the transition of interest was determined. The intensities for each distance were normalized using the total number of γ-rays observed from the Coulomb excitation of the gold stopper. Lifetimes were then extracted using the methods
<figure><img src="image_49.png" /><figcaption>**Fig. 9.** γ-Ray spectra for the $4_1^+ \to 2_1^+$ transition in ring 2 gated on the shifted component of the $6_1^+ \to 4_1^+$ transition. The broad peaks to the left are the shifted peaks of the γ rays from the $4_1^+ \to 2_1^+$ transition, whereas the unshifted component is at the energy corresponding to that of the γ-ray transition (611 keV).</figcaption></figure>

outlined in Ref. [3]. The expression

$$I(\gamma_1^{\text{sh}}, \gamma_2^{\text{us}}, x) = \tau v \frac{d}{dx} I(\gamma_1^{\text{sh}}, \gamma_2^{\text{sh}}, x) \quad (1)$$

is fitted to the experimental data consisting of the normalized γ-ray peak intensities $I(\cdots)$, using the $\chi^2$ method with $\tau$ as a parameter together with four parameters describing a piecewise differential function for $I(\gamma_1^{\text{sh}}, \gamma_2^{\text{sh}}, x)$, in this case a quadratic spline. In the expression $I(\cdots)$ the first argument $\gamma_1^{\text{sh}}$ refers to the gate used to extract the γ-ray intensity, i.e. the in-flight component of the $6_1^+ \to 4_1^+$, and the second argument refers to the transition de-exciting the state of interest. In this case the $4_1^+$ state whose lifetime was measured. The superscript sh(us) on the second argument refers to the shifted(unshifted) component of the γ-ray intensity. The velocity $v$ is determined from the observed Doppler shifts of the energies of the γ rays. In Fig. 9 example spectra used to fit the intensities are shown. All spectra were obtained by gating on the shifted component of the feeding $6_1^+ \to 4_1^+$ transition. By analyzing spectra gated on the in-flight component of the feeding transition systematic errors related to feeding times and the angular distributions of γ rays can be minimized [3]. Systematical errors arising from the so-called de-orientation effect can also be avoided [11]. The intensities of the shifted(in-flight) and unshifted(stopped) components were fitted by assuming fixed positions and widths for the γ-ray peaks, varying only the amplitude for each distance. The fits of Eq. (1) to the data are shown in Fig. 10 for the different rings in Orgam. The lifetime has been fitted for each ring separately, as well as using the data from all rings simultaneously giving a lifetime of 4.1(15) ps, 4.9(11) ps, 3.6(8) ps, and 4.3(6) ps, respectively. These values are consistent with the most recent and precise measurement of Görgen et al. [12], as well as other literature values as quoted in Table 2.

One should note that the large error in our measurement is not only due to low statistics, but also because the distances from which good data could be extracted were all quite short compared to the product of the lifetime of interest and the recoil velocity. Had the data taken with longer target-stopper distances been of the same quality as the data taken for the short distances the error would have been reduced.

## 4. Future development

The Orsay Plunger has been developed with future needs at GANIL and Spiral2 in mind. Apart from the classical use of RDDS

<figure><img src="image_50.png" /><figcaption>**Fig. 10.** Normalized intensity curves ($I(\gamma_1^{\text{sh}}, \gamma_2^{\text{sh}}, x)$ and $I(\gamma_1^{\text{sh}}, \gamma_2^{\text{sh}}, x)$) for the $4_1^+ \to 2_1^+$ transition gated on the in-flight component of $6_1^+ \to 4_1^+$ for rings 1–3. The lifetime deduced from all data is $\tau = 4.3(6)$ ps.</figcaption></figure>

<caption>**Table 2**</caption>

Comparison of literature values with the present work. Görgen et al. is the most recent value.

<table><thead><tr><td>Ref.</td><td>τ (ps)</td></tr></thead><tbody><tr><td>This work</td><td>4.3(6)</td></tr><tr><td>Görgen et al. [12]</td><td>3.67(9)</td></tr><tr><td>Wörrman et al. [13]</td><td>4.9(4)</td></tr><tr><td>Piercey et al. [14]</td><td>5(2)</td></tr><tr><td>Nolte et al. [15]</td><td>8.2(23)</td></tr></tbody></table>

together with fusion-evaporation reactions an extensive physics program using RDDS measurements together with high-intensity radioactive beams is also foreseen. Such experiments will often be performed using spectrometers such as VAMOS [16] and LISE [17] to identify the reaction products on an event-by-event basis. This will require the use of a degrader foil instead of a stopper foil and the modification of the target and stopper foil frames. As an example, a multi-nucleon transfer experiment using a light radioactive beam on a heavy neutron rich target will require that the beam hits the target at an angle of about 60°.

Another foreseen development is the use of the Orsay Plunger for high-precision gyromagnetic (g) factor measurements of excited states with lifetimes of the order of picoseconds. For states with such short lifetimes a very strong magnetic field is needed to achieve an observable modification of the angular distribution of the emitted γ rays within the lifetime of the excited state. One way to obtain such strong fields is to use the Transient Field technique. However, it has the shortcoming that the strength of the obtained magnetic field cannot be estimated using first principles and hence it relies on reference measurements to extract the g factor for the state of interest. Using a
plunger device together with the so-called “Time-Differential Recoil-In-Vacuum” (TDRIV) technique this problem can be overcome. This method uses the interaction of the nuclear magnetic moment with the magnetic field created by the electrons orbiting the recoiling ion as described by Goldring [18]. This interaction creates a time-dependent pattern in the angular distribution of the emitted γ-rays. The frequency of this pattern is directly proportional to the gyromagnetic factor of the nuclear state and to the magnetic field created at by the electrons orbiting the nucleus. For highly ionized light ions (e.g. with a single 1s electron) this magnetic field can be calculated from first principles with high precision. A plunger device, by which one can obtain a precise time-control and measurements on a picosecond scale, is an essential tool in a similar type of measurements. TDRIV measurements on short-lived states in light nuclei have provided a number of precise g-factor measurements. A development if this technique, to be applied with radioactive beams, has been proposed by Stuchbery et al. [19]. The essential part of it is that the nuclei of interest are not stopped in the second plunger foil but are to fly through it into vacuum. It is estimated that the integral de-orientation of the nuclear spins, due to the RIV effects, would not impede the measurement. Provided that this technique works in the “differential-plunger” regime it would allow for model-independent high-precision measurements of g factors in light exotic nuclei.

## 5. Conclusions

A new plunger device has been developed in collaboration between CSNSM and IPN Orsay. Named The Orsay Universal Plunger System (Oups) it is intended for, but not limited to, use at radioactive and stable-beam facilities like GANIL and Spiral2. The flexible design allows simple adaptation to most facilities with a γ-ray spectrometer.

The device has been commissioned at the Tandem facility at IPN Orsay. Excited states in the nucleus ⁷⁶Kr were populated using the heavy-ion fusion evaporation reaction ⁴⁵Sc(³⁵Cl, 2p2n)⁷⁶Kr. A lifetime

of 4.3(6) ps for the 4<sup>+</sup> state in ⁷⁶Kr, compatible with recent lifetime measurements, could be extracted. The electronic feed back system designed to keep the distant constant between the target and stopper foils was proven to function to full satisfaction. It is therefore concluded that the Oups is ready for performing RDDS measurements.

## Acknowledgment

The use of detectors from the Gammapool European Spectroscopy and France–UK (IN2P3/STFC) Loan-Pool Resources through the ORGAM (Orsay Gamma Array) project is gratefully acknowledged.

## References

[1] A. Görgen, Journal of Physics G: Nuclear and Particle Physics 37 (2010) 103101.

[2] H. Mach, et al., Journal of Physics G: Nuclear and Particle Physics 31 (2005) S1421.

[3] A. Dewald, S. Harissopulos, P. von Brentano, Zeitschrift Physik A 334 (1989) 163.

[4] T.K. Alexander, K.W. Allen, Canadian Journal of Physics 43 (1965) 1563.

[5] M. Lewitowicz, Nuclear Physics A 805 (1–4) (2008) 519c.

[6] J.J. Valiente-Dobón, et al., Physical Review Letters 102 (2009) 242502.

[7] J. Ljungvall, et al., Physical Review C 81 (2010) 061301(R).

[8] A. Dewald, et al., Physical Review C 78 (2008) 051302(R).

[9] W. Rother, et al., Nuclear Instruments and Methods in Physics Research Section A 654 (2011) 196.

[10] A. Dewald, in: H. Grawe (Ed.), Ancillary Detectors and Devices for Euroball, GSI and the Euroball Ancillary Group, Darmstadt, 1998, p. 70.

[11] P. Petkov, A. Dewald, P. von Brentano, Nuclear Instruments and Methods in Physics Research Section A 457 (2001) 527.

[12] A. Görgen, et al., The European Physical Journal A 26 (2) (2005) 153–157.

[13] B. Wörmann, et al., Nuclear Physics A 431 (1984) 170.

[14] R.B. Piercey, et al., Physical Review C 25 (1982) 1941.

[15] E. Nolte, et al., Zeitschrift Physik A 268 (1974) 267.

[16] S. Pullanhiotan, et al., Nuclear Instruments and Methods in Physics Research Section A 593 (2008) 343.

[17] R. Anne, et al., Nuclear Instruments and Methods in Physics Research Section A 257 (1987) 215.

[18] G. Goldring, in: R. Bock (Ed.), Heavy Ion Collisions, vol. 3, North-Holland, Amsterdam, 1982, p. 483.

[19] A. Stuchbery, et al., Physical Review C 71 (2005) 047302.
7.4 Investigation of high-K states in $^{252}$No
Investigation of high-*K* states in $^{252}$No

B. Sulignano,¹ Ch. Theisen,¹ J.-P. Delaroche,² M. Girod,² J. Ljungvall,³ D. Ackermann,⁴,⁵ S. Antalic,⁶ O. Dorvaux,⁷

A. Drouart,¹ B. Gall,⁷ A. Görgen,⁸ P. T. Greenlees,⁹ K. Hauschild,³ R.-D. Herzberg,¹⁰ F. P. Heßberger,⁴,⁵ U. Jakobsson,⁹

P. Jones,⁹ R. Julin,⁹ S. Juutinen,⁹ S. Ketelhut,⁹ W. Korten,¹ M. Leino,⁹ A. Lopez-Martens,³ M. Nyman,⁹ A. Obertelli,¹

J. Pakarinen,¹¹ P. Papadakis,¹⁰ E. Parr,¹⁰ P. Peura,⁹ J. Piot,⁷ P. Rahkila,⁹ D. Rostron,¹⁰ P. Ruotsalainen,⁹ J. Sarén,⁹ C. Scholey,⁹

J. Sorri,⁹ J. Uusitalo,⁹ M. Venhart,¹²,⁶ and M. Zielińska¹

¹CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France

²CEA, DAM, DIF, F-91297 Arpajon, France

³CSNSM, IN2P3-CNRS, F-91405 Orsay Campus, France

⁴GSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt, Germany

⁵Helmholtz Institut Mainz, D-55099 Mainz, Germany

⁶Department of Physics and Biophysics, Comenius University, Bratislava 842 48, Slovakia

⁷Institut Pluridisciplinaire Hubert Curien, F-67037 Strasbourg, France

⁸SAFE Kjemibygningen, Sem Særlandsvei 26, 0371 Oslo, Norway

⁹Department of Physics, University of Jyväskylä, FIN-40014 Jyväskylä, Finland

¹⁰Department of Physics, University of Liverpool, Liverpool L69 7ZE, United Kingdom

¹¹Isolde, CERN

¹²Institute of Physics, Slovak Academy of Sciences, Bratislava 845 11, Slovakia

(Received 3 April 2012; revised manuscript received 19 July 2012; published 16 October 2012)

In this paper we investigate the rotational band built upon a two-quasiparticle 8⁻ isomeric state of $^{252}$No up to spin $I^π = 22^−$. The excited states of the band were populated with the $^{206}$Pb($^{48}$Ca, 2n) fusion-evaporation reaction. An unambiguous assignment of the structure of the 8⁻ isomer as a 7/2⁺[624]ᵢ ⊗ 9/2⁻[734]ᵢ configuration has been made on the basis of purely experimental data. Comparisons with triaxial self-consistent Hartree-Fock-Bogoliubov calculations using the D1S force and breaking time-reversal as well as z-signature symmetries are performed. These predictions are in agreement with present measurements. Mean-field calculations extended to similar states in $^{250}$Fm support the interpretation of the same two-neutron quasiparticle structure as the bandhead in both *N* = 150 isotones.

DOI: 10.1103/PhysRevC.86.044318

PACS number(s): 21.10.–k, 23.20.Lv, 23.35.+g, 27.90.+b

# I. INTRODUCTION

In the past few decades exhaustive investigations of deformed nuclei in the transfermium region around *N* = 152 and *Z* = 100, where enhanced stability is observed, have been performed using in-beam and decay spectroscopy [1]. Nuclei in this region are produced with cross sections ranging from nanobarns to microbarns, high enough for detailed spectroscopic studies. Moreover, this region is characterized by the presence of *K* isomerism, which may enhance the stability of such nuclei against spontaneous fission [2], as in $^{270}$Ds [3], in $^{254}$No [4], and in $^{250}$No [5].

*K* isomers are due to the presence near the Fermi surface of either neutron or proton orbitals with a large angular momentum projection *K* along the symmetry axis. Excitation of particles into such an orbital leads to a *K* state for which the decay is strongly hindered according to the $\Delta K$ selection rule. Therefore these states with a strong signature represent a new prolific ground for experimental investigations. The study of *K* isomers and collective bands built on them provides a detailed probe of single-particle states and their coupling to the collective degrees of freedom. Famous examples are located around *Z* = 72, *N* = 106 and include the textbook example of the $K^π = 16^+$ isomer in $^{178}$Hf with a half-life of 31 yr (see for instance the review articles [6–8]). In the same way, the investigations of the nobelium isotopes ($Z$ = 102), thus, deliver data in a region close to the domain of superheavy

nuclei, where our knowledge of single-particle spectra and of pairing correlations is particularly limited. These studies can provide information relevant for the next shell closure, which is expected to be at *Z* = 114, 120, or 126 and *N* = 184 for spherical superheavy elements depending on the model (see [9–11] and references therein).

Ghiorso *et al.* were the first to discover *K* isomers in this region 39 years ago in $^{250}$Fm and $^{254}$No [12]. The decay of $^{254}$No was recently revisited by Herzberg *et al.* [13] and by Tandel *et al.* [14], where a second isomer was found. In the nucleus $^{252}$No, a new *K* isomer with a half-life of 110 ± 10 ms at an excitation energy of 1254 keV was discovered at SHIP (GSI, Darmstadt) [15] and confirmed at FMA (Argonne National Laboratory) [16], both experiments using decay spectroscopy techniques.

The isomer in $^{252}$No has been interpreted as a two-quasineutron state with the configuration 7/2⁺[624]ᵢ⊗9/2⁻[734]ᵢ with spin and parity $I^π = 8^−$. This assignment was made by analogy with other *N* = 150 isotones. In fact, $K^π = 8^−$ isomers have been found in $^{244}$Pu [16,17], $^{246}$Cm [16,18], $^{248}$Cf [19], and $^{250}$Fm [20]. Very similar level schemes are observed in $^{246}$Cm and $^{250}$Fm, where the $I^π = 8^−$ isomer decays into the ground-state rotational bands via an intermediate excited band.

According to previous calculations, using Hartree-Fock-Bogoliubov (HFB) mean-field calculations with the Gogny
D1S force [21], the two-quasineutron $K^\pi = 8^-$ excitation in $^{252}$No was predicted to lie at 1070 keV (without breaking time-reversal symmetry in the mean field [22]), in agreement with the experimental excitation energy $E_x = 1254$ keV. This result is in agreement with HFB calculations using the SLy4 interaction, which predict the two-quasineutron $K^\pi = 8^-$ excitation at $\sim 1300$ keV [23]. On the other hand, both HFB models predict the two-quasiproton states at higher excitation energies (i.e., $E_x \sim 2000$ keV). In contrast, the macroscopic-microscopic Woods-Saxon calculations predict $K^\pi = 8^-$ isomers at $\sim 1$ MeV for both two-quasineutron and two-quasiproton configurations [2, 16, 20]. Recent random phase approximation (RPA) calculations predict a two-neutron quasiparticle state at 1300 keV [24]. The differences between these predictions are related to different single-particle level schemes underlying the models.

The goal of this work is to deduce the structure of the $K^\pi = 8^-$ isomer in $^{252}$No with purely experimental data and to understand how this structure changes with the rotational frequency. Together with previous data on $^{250}$Fm [20], this will provide valuable reference points for theoretical models.

Strong constraints on the configuration assignment and models is provided (for $K^\pi \neq 0$ states) by the magnetic moment or gyromagnetic factor, which can be determined by measuring the *M1/E2* intensity ratio within the band. While the $K^\pi = 8^-$ isomer in $^{252}$No has been investigated using decay spectroscopy, this work extends the study to a rotational band built above this state via in-beam $\gamma$-ray spectroscopy, allowing for firm configuration assignments of the $8^-$ isomer.

## II. EXPERIMENT

The experiment was carried out at the Accelerator Laboratory of the University of Jyväskylä. A $^{48}$Ca$^{+10}$ beam, with an energy of 218 MeV and an average current of 30 pnA, impinged on 452 $\mu g/cm^2$ thick $^{206}$PbS (with an enrichment of 98.6%) targets mounted on a rotating wheel. The fusion-evaporation residues were selected by the gas-filled separator RITU [25] and implanted into the focal plane spectrometer GREAT [26]. GREAT consists of a multi-wire proportional counter (MWPC) to measure the time of flight and energy loss $\Delta E$ of the incoming ions and double-sided silicon strip detectors (DSSSD) to measure the energy and time of the ions and subsequent decays. The DSSSD are surrounded by germanium detectors (clover and planar) and PIN diodes for the detection of $\gamma$ rays and conversion electrons from the decay of the implanted recoil. Prompt $\gamma$ rays were detected in the 41 Compton-suppressed germanium detectors of the JUROGAM array. A $\gamma$-ray efficiency of 4.2% at 1332 keV was determined using calibration sources. The JUROGAM detector preamplifier signals were digitized with TNT2 digital pulse processor units [27]. This allowed a counting rate of up to 35 kHz for each germanium detector while retaining good energy and time resolution. Compared to standard analog shaping amplifiers and peak-sensing analog-to-digital converters (ADCs), the use of digital electronics allowed an increase of the average beam current by approximately a factor of 2. A time-stamped system total data readout (TDR) data

<figure><img src="image_51.png" /><figcaption>FIG. 1. Energy spectrum of $\alpha$ particles detected at the focal plane.</figcaption></figure>

acquisition system was used [28]. Data analysis was carried out using the GRAIN software package [29].

A total of 198 h of irradiation yielded $1.5 \times 10^4$ recoils detected at the focal plane correlated with the decay of $^{252}$No and/or its daughters. The identification of the recoils was performed by demanding a signal ($\Delta E$) in the MWPC detector in coincidence with an energy signal in the double-sided silicon strip implantation detector. Correlations between the time of flight and total energy were used to discriminate against scattered beam and target-like reaction products. The lifetime of the $^{252}$No ground state was determined by correlating evaporation residues (ERs) with the subsequent alpha decay ($\alpha$) or spontaneous fission (SF). Since the gain of the silicon detector electronics was optimized for the evaporation residues, events rendering overflow in the energy ADC and in anticoincidences with the MWPC were identified as SF events. The measured branching ratios for the different decay modes of the $^{252}$No ground state, namely, $b_\alpha = 65.3(5)\%$ and $b_{SF} = 33.9(3)\%$, are found to be in good agreement with the literature values [30]. The electron capture decay path could not be measured directly so the previously measured value of $b_\epsilon = 0.8\%$ [31] was assumed. The $\alpha$-particle energy spectrum in anticoincidence with MWPC events is shown in Fig. 1. From the time distribution a half-life of $2.43 \pm 0.13$ s was obtained for the ground state, in agreement with the evaluated data of $2.44 \pm 0.04$ s [32].

### A. Delayed spectroscopy

Isomeric states in heavy nuclei often decay to the ground-state band via strongly converted electromagnetic transitions. Correlating such electrons, detected in the DSSSD and identified via their energy, with the implanted ERs allows selection of the recoils which were implanted prior to the decay of the isomeric state. For details see, e.g., Jones [33]. The measured electron sum signals of the isomer decays detected in the same pixel as an ER event within a time window of 700 ms after the implantation are shown in Fig. 2. The insert in Fig. 2 shows the logarithm of the time difference between implantation of the recoil and its subsequent electron
<figure><img src="image_52.png" /><figcaption>FIG. 2. (Color online) Energy spectrum of conversion electrons observed within 700 ms after a fusion-evaporation residue was implanted in the same pixel of the DSSSD. The inset shows the logarithmic decay time distribution of conversion electrons correlated with ER recoils detected at the focal plane; no correlation with $\alpha$ decay or SF of $^{252}$No is required. The smaller contribution at 156 $\mu$s is ascribed to a short-lived isomer in $^{253}$No.</figcaption></figure>

decay. The red curve represents the time density distribution of electrons correlated to the evaporation residue of $^{252}$No, and the peak position determines the lifetime via the $\ln(\tau)$ relation. This method [34,35] allows the discrimination of contributions from sources having different lifetimes. The distinct cluster with a lifetime of 156 $\mu$s corresponds most likely to the de-excitation of an isomeric state in $^{253}$No. This is confirmed by the observation of the characteristic x rays of nobelium and few $\gamma$ rays with energies similar to that reported recently by Antalic et al. [36]. $^{253}$No nuclei have been produced via the $2n$ evaporation channel from reactions on the $^{207}$Pb contaminant present in the $^{206}$Pb target. The $\alpha$ decay of this channel is also visible in Fig. 1.

For an unambiguous identification and to remove the contribution from $^{253}$No, an additional requirement of $\alpha$ decay or SF of $^{252}$No observed subsequently in the same pixel was demanded. We should point out that among all possible open channels, $^{252}$No is the only nucleus having a SF branch; SF is therefore an unambiguous signature of this nucleus. A total of 3833 conversion electrons (ER-$e^-$ correlation) signals were detected in the DSSSD. A half-life of 109 $\pm$ 3 ms was found, in perfect agreement with 110 $\pm$ 10 ms quoted in [15].

Delayed $\gamma$ rays from the de-excitation of isomeric states were detected in the germanium detectors surrounding the focal plane. A coincidence with the conversion electrons from the decay of the isomeric state using the conditions 2 ms $<$ $\Delta t$(ER-$e^-$) $<$ 700 ms and $E_e^-$ $<$ 700 keV was demanded. See Fig. 3 for the $\gamma$-ray spectra. The deduced decay pattern from the isomeric state is found to be in perfect agreement with the previous measurements [15], as shown in Fig. 4.

Moreover, it was possible to confirm the transitions observed in earlier experiments. Indeed, the $6^-$ $\rightarrow$ $5^-$ (75 keV) M1 transition is clearly visible in Fig. 3, as are the $5^-$ $\rightarrow$ $3^-$ at 107 keV, $7^-$ $\rightarrow$ $5^-$ at 156 keV, $6^-$ $\rightarrow$ $4^-$ at 133 keV, and

<figure><img src="image_53.png" /><figcaption>FIG. 3. Gamma rays detected in prompt coincidence with conversion electrons in the clover detector (a) and in the planar germanium detector (b) of the GREAT spectrometer.</figcaption></figure>

$4^-$ $\rightarrow$ $2^-$ at 86 keV E2 transitions. Other peaks are also visible in the spectra, although at present status it is not possible to fit them into our decay scheme.

## B. Prompt spectroscopy at the target position

The de-excitation of states lying above the isomer was studied [see Fig. 5(a)] by correlating prompt $\gamma$ rays in the

<figure><img src="image_54.png" /><figcaption>FIG. 4. (Color online) $^{252}$No level scheme. Ground-state band measured up to spin $I^\pi = 20^+$ and comparison to HFB predictions to spin $24^+$ (gray box) are shown on the left-hand side. The rotational band measured on top of the $K^\pi = 8^-$ neutron isomer is shown (up to spin $I^\pi = 22^-$) in red and green colors for even and odd spin sequences, respectively. HFB level predictions for this band to spin $I^\pi = 23^-$ are shown inside the gray box. The $K^\pi = 2^-$ collective band observed at lower excitation energy is marked in blue.</figcaption></figure>
<figure><img src="image_55.png" /><figcaption>FIG. 5. (Color online) (a) Prompt γ-rays measured in JUROGAM in coincidence with an ER-e⁻ (Eₑ⁻ < 700 keV) pair observed at the same position in the DSSSSD within 700 ms. (b) Same spectrum with the additional requirement that an α particle or SF is correlated with the evaporation residue. (c) Projection of the γ-γ coincidences matrix using electrons and α/SF correlations. The region of nobelium x rays has been scaled down by a factor of 5 in all three spectra.</figcaption></figure>

JUROGAM array with fusion-evaporation residues observed in the focal plane of RITU. The additional requirement that an electron sum event was observed within 700 ms after a recoil at the same position in the DSSSSD was made. Higher selectivity was reached by detecting the characteristic α or SF decay following the evaporation residue within ΔT (ER-α/SF) < 12 s [see Fig. 5(b)]. Support for the assignment of the rotational band is given by the investigation of recoil-gated γ-γ coincidences. Figure 5(c) shows the projection of the recoil-gated γ-γ coincidence matrix using electrons and α/SF correlations. Although the statistics are rather weak, each (presumed) transition is in coincidence with at least one of the other transitions. The corresponding level scheme including also transitions from previous work [15,37] is shown in Fig. 4.

### C. Spin assignment

The spin *I* of the states in the rotational band built on top of the isomer can be assigned using a method extensively applied to superdeformed rotational bands in the mass 190 region [38,39]. The technique known as the Harris fit [40] consists in expanding the dynamic moment of inertia *J*⁽²⁾ in even powers of the rotational frequency *ħ*ω = *E*γ/2 averaged over two consecutive γ-ray transitions,

$$ \mathcal{J}^{(2)} = 4/\Delta E_{\gamma} = dI/d\omega = A + B\omega^2 + C\omega^4. \quad (1) $$

The resulting function is integrated with respect to ω to obtain the spin *I*, namely,

$$ I_{\gamma}(\omega) - I_{\gamma 0} = A\omega + (B/3)\omega^3 + (C/5)\omega^5 + 1/2. \quad (2) $$

The basic assumption of the technique is that of no alignment at rotational frequency ω ~ 0, i.e., *I*γ₀ = 0. This is indeed in perfect agreement with our HBF calculations, as discussed below in Sec. IV. The dynamic moment of inertia is displayed

in Fig. 6(b). Let us first make a comment on the procedure used to calculate the moment of inertia. Although the band is characterized by a sequence of Δ*I* = 1ħ transitions, this band is divided for convenience into two Δ*I* = 2ħ branches when calculating the moment of inertia. Looking at the moment of inertia, one notes the presence of a shallow minimum at a rotational frequency of ħω ~ 0.19 MeV, a feature that will be commented upon later. Obviously, the two points in the vicinity of this minimum, which strongly deviate from the otherwise smooth pattern, should be excluded from the Harris fit. The low statistics in the γ-ray spectra measurements (see Fig. 5) leads to uncertainties in the determination of the parameters entering the fitted expression [Eq. (2)]. The best fit leads to the estimate *I*₀ = 8.1 ± 1.9ħ for the bandhead level. In addition, the spin-fitting method introduced by Wu *et al.* [41] has been used as a cross-check of the above spin determination. The method gives for integer *I*₀ a minimum in the χ² function at *I*₀ = 8ħ. Finally, both methods are fully consistent with spin *I*₀ = 8ħ for the bandhead.

### D. Structure assignment

In the *N* = 150 ($^{246}$Cm and $^{248}$Cf) isotones the presence of the 8⁻ isomeric states is a common feature of this region, as was already pointed out in the introduction. The neutron 7/2⁺[624]ᵥ ⊗ 9/2⁻[734]ᵥ nature of this state has then been firmly established on the basis of different experimental means, e.g., transfer reactions and β decay. The similarity of the level schemes between $^{252}$No and those for the other isotones makes the assumption that also in this case the decay proceeds through the two-neutron components of the wave functions. A direct measurement of the structure of the isomeric states found in $^{252}$No produced via the fusion-evaporation reaction can be performed via the study of the γ rays emitted during the de-excitation of the band built on the isomeric state. The electromagnetic properties of the states were deduced and the magnetic moment, or the *g*<sub>K</sub> gyro-magnetic factor, was inferred. The *g*<sub>K</sub> factor depends on the two-quasiparticle configuration and depends in general on the scenario. It is therefore a crucial quantity for understanding the isomer structure and for a comparison with the models. As mentioned in the introduction, the *K*<sup>π</sup> = 8⁻ isomeric state can be interpreted either as a two-quasiproton (9/2⁺[624]<sub>π</sub> ⊗ 7/2⁻[514]<sub>π</sub>) or two-quasineutron (7/2⁺[624]ᵥ ⊗ 9/2⁻[734]ᵥ) configuration. Although new HFB calculations are presented in this article, the complex formalism for calculating rigorously the magnetic moment has not yet been implemented in the HFB code. Therefore the gyromagnetic factor inferred from Woods-Saxon-based calculations will be used. According to [42], the *g*<sub>K</sub> factor for these two 8⁻ configurations are 0.01 (7/2⁺[624]ᵥ ⊗ 9/2⁻[734]ᵥ) and 1.01 (9/2⁺[624]<sub>π</sub> ⊗ 7/2⁻[514]<sub>π</sub>), respectively. It is interesting to note that these values are very close to the asymptotic limit for singlet states (Σ = 0) which are energetically favored according to the Gallagher rule [43]. In this case the gyromagnetic factor of the spin cancels and one has only a contribution from the angular momentum *I* of the two protons (*g*<sub>K</sub> = *g*<sub>*l*</sub><sup>*p*</sup> = 1) or neutrons (*g*<sub>K</sub> = *g*<sub>*l*</sub><sup>*n*</sup> = 0). To determine the experimental *g*<sub>K</sub> factor, the well-known relationship between the observed
<figure><img src="image_56.png" /><figcaption>FIG. 6. (Color online) (a) and (b) Dynamic moment of inertia $J^{(2)}$ vs $\hbar\omega$ for the ground-state rotational band (dashed black lines) and for the isomeric $K^\pi = 8^-$ band, for the $^{250}$Fm and $^{252}$No (red triangles) isotones. (c) and (d) Same as (a) and (b) but for the kinematic moments of inertia. Triangular blue marks represent theoretical calculations using the D1S Gogny force. See text for details.</figcaption></figure>

γ-ray branching ratio, the $g_K$ factor, and decay probabilities was used, i.e.,

$$R^{\text{the}} = \frac{I(M1; J \rightarrow J-1)}{I(E2; J \rightarrow J-2)} = \frac{1.76[E_{\gamma}(M1)]^3 B(M1) \text{s}^{-1}}{1.22[E_{\gamma}(E2)]^5 B(E2) \text{s}^{-1}}, \quad (3)$$

where the reduced transition probabilities $B(E2)$ and $B(M1)$ are

$$B(E2, J \rightarrow J-2) = \frac{5}{16\pi} e^2 Q_0^2 \langle JK20|(J-2)K \rangle^2 \quad (4)$$

and

$$B(M1, J \rightarrow J-1) = \frac{3}{4\pi} \left( \frac{e\hbar}{2Mc} \right)^2 (g_K - g_R)^2 K^2 \langle JK10|(J-1)K \rangle^2. \quad (5)$$

The core gyromagnetic factor is taken to be $g_R = Z/A = 0.4$ in the hydrodynamic limit. While the $g_K$ factor changes strongly from one configuration to another, the electric quadrupole moment $Q_o$ does not exhibit large differences and is accurately predicted by the models. We have here taken that for the neutron two-quasiparticle HFB value, namely, $Q_o = 13.75$ e b. The experimental branching ratios are given in Table I for the three initial states for which both interband M1 and intraband E2 transitions were measured. We have assumed a small mixing ratio for the $\Delta I = 1\hbar$ transition according to [44]. Unfortunately, the γ-ray transitions are observed with low intensity, which translates into large error bars for the experimental ratio. According to Eq. (3), the

branching ratio is a parabolic function of $g_K$, as shown in Fig. 7 in the case of the $I^\pi = 14^-$ initial level. Decay probabilities lie on this parabola: $7/2^+[624]_\nu \otimes 9/2^-[734]_\nu$ with $g_K = 0.01$ (square symbol) and $9/2^+[624]_\pi \otimes 7/2^-[514]_\pi$ with $g_K = 1.01$ (circle symbol). The experimental intensity ratio is shown with the yellow area. The intersection of the experimental intensity ratio value with the parabola gives two possible $g_K$ values. It is clear that only the $K^\pi = 8^-$ neutron configuration agrees well with the measured ratio. The same conclusion can be drawn for the $15^-$ and $16^-$ initial states.

However, using single intensity ratios from low-statistic spectra can lead to large uncertainties. It is therefore more appropriate in this case to apply the analytical method described in detail in [45]. The technique is based on the idea that the transitions in a strongly coupled band will have easily visible stretched E2 transitions and interband $\Delta I = 1\hbar$ transitions clustered at low energies. As the band is built on a particular configuration the branching ratios in the band are easily modeled by using this configuration as input. This gives an estimate of the number of counts one would expect to find in this $\Delta I = 1\hbar$ region in the spectrum, even if the statistics are not large enough to analyze individual peaks. An “integral branching ratio” is therefore extracted, and this may often take

<table><caption>TABLE I. Experimental intensity ratios $I(M1)/I(E2)$.</caption><thead><tr><th>Initial level</th><th><i>I</i>(M1)/<i>I</i>(E2)</th><th><i>R</i><sub>exp</sub></th></tr></thead><tbody><tr><td>16<sup>-</sup></td><td><i>I</i><sub>γ</sub>(182)/<i>I</i><sub>γ</sub>(354)</td><td>0.49 ± 0.41</td></tr><tr><td>15<sup>-</sup></td><td><i>I</i><sub>γ</sub>(172)/<i>I</i><sub>γ</sub>(334)</td><td>0.37 ± 0.37</td></tr><tr><td>14<sup>-</sup></td><td><i>I</i><sub>γ</sub>(162)/<i>I</i><sub>γ</sub>(313)</td><td>0.65 ± 0.37</td></tr></tbody></table>
<figure><img src="image_57.png" /><figcaption>FIG. 7. (Color online) The relationship between intensity ratio $I(M1)/I(E2)$ and $g_K$, showing the values for the expected neutron (red square) and proton (blue circle) configurations compared with the experimental ratio (yellow area) for the decay of the $I^\pi = 14^-$ level. See text for details.</figcaption></figure>

the form of an upper limit to compare the experimental spectral shape to that predicted by the rotational model based on the different possible configurations. The details are explained in [45].

The number of experimentally deduced counts is denoted with $N_0$ and the number of expected counts in the same region for both candidate configurations with $N_n$ (two-quasineutron state $7/2^+[624]_v \otimes 9/2^-[734]_v$) or $N_p$ (two-quasiproton state $9/2^+[624]_\pi \otimes 7/2^-[514]_\pi$), using the easily observed intensity of the $E2$ transitions as normalization. This method was applied for the intraband transitions $I^\pi = 14^-$ to $I^\pi = 19^-$. The comparison between the deduced values $N_0$ and the Gaussian probability distribution found for the two different scenarios shown in Fig. 8 favors the neutron structure being about 3 times as probable as the proton structure (56% versus 14%) [45].

<figure><img src="image_58.png" /><figcaption>FIG. 8. (Color online) Expected total counts $N_n$ for the two-quasiparticle neutron configuration (green) and $N_p$ for the two-quasiparticle proton configuration (red) and observed counts $N_o$ (blue line) along with attached uncertainties (dashed blue lines).</figcaption></figure>

This technique was also applied to the rotational band built on top of the $K^\pi = 8^-$ isomer in $^{250}$Fm [45], the result being fully consistent with the interpretation of Greenlees et al. [20].

This method combined with the measurement of the $M1/E2$ ratio provides a strong experimental indication of the isomeric structure as a spin singlet two-quasineutron state. With $K^\pi = 8^-$ as the bandhead, the likely configuration is $7/2^+[624]_v \otimes 9/2^-[734]_v$.

### E. Hindrance factor

Long-lived $K$ isomers occur because the decay via transitions with a large $\Delta K$ is strongly hindered: there is no state with similar wave function available for the decay. These states with a high degree of $K$ purity can only be obtained for axially symmetric nuclei. The reduced hindrance factor $f_v$ gives an indication on the “goodness” of the quantum number $K$, or the $K$ purity of the state. It is defined as

$$f_v = (T_{1/2}^{\gamma}/T_{1/2}^{W})^{1/\nu}, \qquad (6)$$

where $T_{1/2}^\gamma$ is the partial $\gamma$-ray half-life and $T_{1/2}^W$ is the Weisskopf single-particle estimate. The exponent $1/\nu = 1/(\Delta K - \lambda)$ represents the degree of forbiddenness with $\lambda$ being the multipolarity of the decay radiation. A value of $f_v \sim 100$ is expected according to the systematics of Löbner [46]. In $^{252}$No, the reduced hindrance factor can be obtained from the decay to the $K^\pi = 0^+$ ground-state band ($8^- \to 8^+ E1$ transition at 710 keV) or from the decay to the $K^\pi = 2^-$ octupole band ($8^- \to 7^- M1$ transition at 25 keV). Using a branching ratio of 4% for the $E1$ transition at 710 keV [15], one obtains $f_v$ ($E1, 8^- \to 8^+$) = 178 and $f_v$ ($M1, 8^- \to 7^-$) = 218. The high value of the reduced hindrance factor testifies to the high purity of the two-quasiparticle $K^\pi = 8^-$ isomer and the small degree of mixing with both the ground-state band and the $K^\pi = 2^-$ excited band. This value favorably compares with the $K^\pi = 8^-$ isomer in the $N = 150$ isotones: $f_v$ ($^{250}$Fm, $E1, 8^- \to 8^+$) = 213, $f_v$ ($^{250}$Fm, $M1, 8^- \to 7^-$) = 192 [20], $f_v$ ($^{246}$Cm, $E1, 8^- \to 8^+$) = 212 [47], and $f_v$ ($^{244}$Pu, $E1, 8^- \to 8^+$) $\sim 200$ [17].

### III. THEORETICAL FRAMEWORK

Mean-field calculations were performed in triaxial oscillator bases including 14 shells using the D1S Gogny force [21,22]. The neutron and proton single-particle states obtained at equilibrium deformation are shown in Fig. 9, where single neutron gaps are predicted at $N = 150$ and $N = 152$ while the proton main gaps are predicted to be at $Z = 98$ and 104. The deformed shell gaps indicated by the experiments are $N = 152$ and $Z = 100$ [1]. The present neutron and proton level schemes at equilibrium deformation differ from those in previous publications [9,16,48,49].

To establish our notation we first define the quasiparticle vacuum HFB energy from minimization of the functional

$$\delta \langle \Phi | \hat{H} - \lambda_Z \hat{Z} - \lambda_N \hat{N} | \Phi \rangle = 0, \qquad (7)$$

where $\hat{H}$ is the nuclear Hamiltonian, and

$$\langle \Phi | \hat{N} | \Phi \rangle = N; \quad \langle \Phi | \hat{Z} | \Phi \rangle = Z, \qquad (8)$$