Ⅱ-1. Nuclear Physics

RIKEN Accel. Prog. Rep. 51 (2018)

Triaxiality of neutron-rich 84,86,88 Ge from low-energy spectra
M. Lettmann,∗1 V. Werner,∗1 N. Pietralla,∗1 P. Doornenbal,∗2 A. Obertelli,∗3,∗2,∗1 T. R. Rodrı́guez,∗4
K. Sieja,∗5 and the SEASTAR 15 collaboration


∗1
∗2
∗3
∗4
∗5



Spectroscopic measurements of 84,86,88 Ge were performed within the SEASTAR campaign in 2015.1) The
spectroscopic results and theoretical predictions inspired intensive discussions of triaxial features for theGe isotopic chain. Two elementary models, which describe the nucleus as a rigid triaxial rotor or as softly
shaped, are competing in this region. Both models describe the breaking of the axial symmetry of the Bohr
Hamiltonian2) by introducing the triaxial deformation
parameter γ, which ranges from 0◦ (prolate shape) to60◦ (oblate shape), and the axial elongation β. The
maximum of triaxiality is reﬂected by γ = 30◦ . The
rigid triaxial rotor model by Davydov and co-workers3)
considers a well-deﬁned minimum for the potential energy surface while the model by Wilets and Jean4)
treats the potential independent of γ, introducing γsoftness. The diﬀerence between the soft and rigid
cases is manifested in the energy spacing between theodd and even members of the γ band. In the case ofa rigid triaxial rotor, the odd-spin levels are locatedcloser to the lower-lying even spin levels, whereas theodd spin levels are located closer to the higher-lying
even spin levels in the case of a γ-soft nucleus. This
energy diﬀerence is referred to as staggering.5,6)
At the RIBF, a 238 U beam with an energy of
345 MeV/u was impinged on a 3-mm-thick 9 Be target
at the entrance of BigRIPS.7) The isotopes of interest
were identiﬁed by BigRIPS and ZeroDegree spectrometer in two diﬀerent settings. The Ge isotopes wereproduced by knockout reactions inside the MINOS8)
LH2 target and the emitted γ radiation was detectedwith DALI2.9) The TPC of MINOS8) was used to improve the Doppler correction.
In total, 16 transitions in 84,86,88 Ge have been observed, ten of which were so far unknown. For 86 Geand 88 Ge, new level schemes are proposed, which areshown in Fig. 1 in red. The experimental results arecompared to a shell model calculation and a symmetryconserving conﬁguration mixing Gogny (SCCM) calculation in Fig. 1. The predicted sequences of states arein good agreement with the experimental results, although both theories overestimate the level energies inall cases. Nevertheless, the predicted R4/2 ≈ 2.5 agrees
with the data. Both calculations suggest a low-lying
γ band, which indicates some amount of triaxiality inboth isotopes. Furthermore, both theories predict a



Fig. 1. Systematics of the 86,88 Ge level energies from experiment compared to theoretical predictions from shellmodel (SM) and symmetry-conserving configuration
mixing Gogny (SCCM) calculations.

863+
Ge that is closer to the 2+
1 state of
2 state than to+
the 42 state of the γ band. A promising candidate forthis state is observed through a 380(8)-keV transitionof 86 Ge, because the strongest decay of the 3+
1 stateis expected to the 2+state.
Ashighlighted
before,2the staggering5,6) in the γ band should take a positive value for a rigid triaxial rotor. This would be thecase for a well-deformed rotor with E(J) ∼ J(J + 1) aswell, though in such a case, the γ-band head is at muchhigher values. So far only one nucleus in the mediumheavy mass region A < 100 is known with rigid triaxial features. This nucleus is 76 Ge, where a staggering
S(4) = 0.091(2) was found.10) With the level assignments presented in Fig. 1, a value of S(4) = 0.20(4)results for 86 Ge, pointing to an even larger degree oftriaxiality in the ground state than assigned to 76 Ge.
This results agrees with the predictions of both theories.

References
1) condensed from M. Lettmann et al., Phys. Rev. C 96,011301(R) (2017).
2) A. Bohr, B. Mottelson, Mat. Fys. Medd. Dan. Vid.
Selsk. 27 (1958).
3) A. S. Davydov, G. F. Filippov, Nucl. Phys. 8, 237(1958).
4) L. Wilets, M. Jean, Phys. Rev. 102, 788 (1956).
5) N. V. Zamfir, R. F. Casten, Phys. Lett. B 260, 265(1991).
6) E. A. McCutchan, D. Bonatsos, N. V. Zamfir,
R. F. Casten, Phys. Rev. C 76 024306(R) (2007).
7) T. Kubo et al., Prog. Theor. Exp. Phys. 2012 03C003(2012).
8) A. Obertelli et al., Eur. Phys. J. A 50 8 (2014).
9) S. Takeuchi et al., Nucl. Instrum. Methods Phys. Res.
A 763 8 (2014).
10) Y. Toh et al., Phys. Rev. C 87 041304(R) (2013).

Institut für Kernphysik, Technische Universität Darmstadt
RIKEN Nishina Center
CEA, Centre de Saclay
Departamento de Fı́sica Teórica, Universidad Autónoma deMadrid
IPHC, CNRS/IN2P3 et Université de Strasbourg- 47 -