Spectroscopy of 74 Ge: from soft to rigid triaxiality

Spectroscopy of 7 Ge: from soft to rigid triaxiality J. J. Sun a, Z. Shi b, X. Q. Li a,, H. Hua a,, C. Xu a, Q. B. Chen a, S. Q. Zhang a, C. Y. Song b, J. Meng a, X. G. Wu c, S. P. Hu c, H. Q. Zhang c, W. Y. Liang a, F. R. Xu a, Z. H. Li a, G. S. Li c, C. Y. He c, Y. Zheng c, Y. L. Ye a, D. X. Jiang a, Y. Y. Cheng a, C. He a, R. Han a, Z. H. Li a, C. B. Li c, H. W. Li c, J. L. Wang c, J. J. Liu c, Y. H. Wu c, P. W. Luo c, S. H. Yao c, B. B. Yu c, X. P. Cao c, H. B. Sun d a School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 171, China b School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 1191, China c China Institute of Atomic Energy, Beijing 113, China d College of Physics Science and Technology, Shenzhen University, Shenzhen 51, China Abstract The first heavy ion fusion-evaporation reaction study for 7 Ge has been performed through the reaction channel 7 Zn( 7 Li,np) 7 Ge at beam energies of 3 and 35 MeV. Previously known yrast band is extended to higher spins and five new collective bands are established. Based on comparison with the neighboring 7,7 Ge isotopes, a distinct pattern of energy staggering S (I) is observed in the γ band of 7 Ge. The collective structure of 7 Ge, including the excitation energies and transition probabilities of ground-state band and γ band, is reproduced by the state-of-the-art five-dimensional collective Hamiltonian (5DCH) model constructed from the covariant density functional. By including the 7,7,7 Ge isotopes, systematical investigation of the structure evolution in Ge isotopes is performed. Based on the systematic comparisons and analysis, the triaxial evolution with spin in 7 Ge is revealed and 7 Ge is found to be the crucial nucleus marking the triaxial evolution from soft to rigid in Ge isotopes. Keywords: Spectroscopy, 7 Ge, Energy staggering, Triaxiality, Five-dimensional collective Hamiltonian, Covariant density functional 1. Introduction Being an extremely complicated quantum finite many-body system, the spatial distribution of an atomic nucleus exhibits a variety of intriguing geometric shapes. The possible shapes which the nucleus adopts result from the competition and delicate balance of the collective vibrational, rotational, and singleparticle degrees of freedom. Most nuclei have been found to have axially symmetric shapes. With the breaking of axial symmetry of the quadrupole deformation, some nuclei appear to have triaxial shapes. The triaxial degree of freedom of a nucleus has led to many interesting phenomena including the γ band [1], wobbling motion [1 3] as well as nuclear chirality [ ], and continues to be a hot issue in nuclear structure physics. One of nuclear regions with the triaxiality significantly involved is around neutron number N. In particular, for the even-even Ge isotopes where a gradual shape transition around N has been suggested experimentally [7 1], the lowlying second excited states (γ band-head) indicating the importance of the triaxiality have been observed in 7,7,7,7 Ge isotopes [9 1]. The experimental analyses and theoretical calculations suggested that the low-lying structures of 7,7,7,7 Ge isotopes have large γ values (γ 3 ) [9 11, 1, 5]. It Corresponding Author Email addresses: lixq@pku.edu.cn (X. Q. Li), hhua@pku.edu.cn (H. Hua) should be mentioned that whether the triaxiality in these nuclei is soft or rigid has not been discussed in these previous studies. To address this question, the energy staggering S (I) of adjacent levels within γ band [, 7] has been recently used to investigate the triaxial shape in 7 Ge []. For a γ-rigid rotor, the levels of γ band exhibit a (, ), (, ), (, ),... grouping, which is different from the γ-soft model pattern of, (, ), (, ),... grouping. Based on the analysis of S (I) in the γ band, 7 Ge was proposed to be the lone instance of a nucleus with a rigid triaxial shape in this mass region, while the neighboring Se and Kr isotopes were suggested to have softtriaxial shape []. It is therefore very important to explore how the triaxiality evolves from γ softness to γ rigidness with spin and isospin in this mass region. As a neighboring nucleus of 7 Ge, the spectroscopy of 7 Ge has been previously studied by many experiments [9]. S- ince few suitable stable target/heavy-ion beam combinations are available to populate 7 Ge via the fusion-evaporation reactions, the collective structure in 7 Ge has not been well developed. So far, only the yrast band of 7 Ge was established up to spin at 31 kev via 19 Os( Se,Xγ) deep-inelastic reaction [3, 31]. Thus, extending the spectroscopic study to 7 Ge would shed new light on the triaxial rigidness/softness and also the structure evolution in Ge isotopes. In this letter, the first heavy ion fusion-evaporation reaction study for 7 Ge through the reaction channel 7 Zn( 7 Li,np) 7 Ge is reported. The collective structure of 7 Ge is expanded sig- Preprint submitted to Physics Letter B May, 1

nificantly and the structure evolution in Ge isotopes is investigated in terms of the state-of-the-art five-dimensional collective Hamiltonian (5DCH) based on the covariant density functional theory (CDFT). Counts 1 33.7 5. 71.* 9.1 9. 999. & 999.7 11. 13. 95.3 319.* 35.1* 31.* 3.5* 39. 3.9 7. 7.* 5.1* 79.5 93. 511.5* 5.5* 59. & 53.5 53.1 553.* 51.1 3 5 7 9 111. 15.* 11.* 139.3* 15.* 115.5 193.7 1111. & 1113. 1131. 115. 119.* 117.1 133. 11. 1. 15.7 1331. 1 11 1 13 1 15 1 17.3 &..* 3.5* 3.5 5.7 (a) Gate: 595. kev 7.* 73.* 3.5 71.3 Energy (kev) 73.* 7. 737. 17. 79.* 77.7 151.3 151. 3.1*.7 9. (b) Gate: 595. kev 11.5 59.1*.3* 7.9 93. 99. 91.* Figure 1: Coincident γ-ray spectra with gating on 595.-keV transition. The peaks marked with stars are known contaminants from other nuclei, and the peaks marked with triangles are the deexcited transitions from individual levels of 7 Ge which don t belong to the established collective bands. 13. stretched E transitions. In general, stretched quadrupole transitions were adopted if DCO ratios were larger than 1., and stretched dipole transitions were assumed if DCO ratios were less than.. The DCO ratio is plotted as a function of γ-ray energy for most of the observed transitions in 7 Ge in Fig.. The partial level scheme of 7 Ge deduced from the present work is shown in Fig. 3. It was constructed from γ-γ coincident relationships, intensity balances, and DCO analyses. The only previous known yrast band (band 1) of 7 Ge is extended from spin at 3. kev to spin 1 at 591. kev. By requiring the coincidence with the known γ-ray transitions under spin of the yrast band in 7 Ge, seven new coincident γ-ray transitions of 53.5, 51.1,.7, 1331., 11.5, 151. and 13. kev are observed in the γ-ray spectra. Based on the D- CO ratio analyses and coincident relationships, band is firstly constructed in the present work. In the neighboring isotopes, Ge, the continuation of the ground-state band has been observed [33, 3] and was found to become the non-yrast band around spin. The similar γ-ray transition sequence and the spin at which band crossing occurs as, Ge indicate that the band of 7 Ge is the continuation of the ground-state band as well.. 1. 1.. Experiment and Results The present experiment was performed at the HI-13 tandem facility of the China Institute of Atomic Energy (CIAE). The high-spin states of 7 Ge were populated via the 7 Zn( 7 Li,np) 7 Ge fusion-evaporation reaction at the beam energies of 3 and 35 MeV. In the first run with 3 MeV beam energy, the target thickness was.15 mg/cm with.93 mg/cm Au backing. The deexcitation γ rays were detected by an γ detector array which consists of 1 high-purity germanium (HPGe) detectors with BGO anti-compton suppressors and t- wo planar HPGe detectors. In the second run with 35 MeV beam energy, the target thickness was 3. mg/cm with 15.75 mg/cm Au backing. The γ detector array consists of 11 highpurity germanium (HPGe) detectors with BGO anti-compton suppressors and two planar HPGe detectors. In both runs, the energy resolutions of these detectors were.-3. kev at 1.33 MeV. All detectors were calibrated using the standard 15 Eu and 133 Ba γ-ray sources. A total of. 1 coincident events were collected, from which a symmetric γ-γ matrix was built. The level scheme analysis was performed using the RADWARE program [3]. The typical γ-ray spectra gated on the 595.-keV transition in 7 Ge is shown in Fig. 1. In order to obtain the Directional Correlations of γ rays deexciting Oriented states (DCO) intensity ratios to determine the multipolarities of γ-ray transitions, the detectors around 9. with respect to the beam direction were sorted against the detectors around. to produce a twodimensional angular correlation matrix. To get clean DCO values for transitions in 7 Ge, gates were set on uncontaminated DCO Ratio 1. 1. 1...... 1 1 1 1 1 Energy (kev) Figure : The DCO ratio as a function of γ-ray energy for transitions in 7 Ge. The γ band (band 3) is newly established in the present work. The level at 1. kev, which has been interpreted as the band-head of γ band [11], is confirmed here. The levels at 197. and 15. kev in band 3 were reported in Refs. [35 39], and (3) + and (3, ) + were tentatively assigned to these t- wo levels, respectively [9]. The present DCO ratio analyses for the intraband and interband deexcited transitions from these two levels support the previous and assignments to the 197. and 15. kev levels, respectively. The levels at 97.1 and 3315. kev in band 3 have also been previously reported [3 ]. In Ref. [9], and were tentatively assigned to the 97.1 and 3315. kev levels, respectively, while Kosyak et al. [35] suggested assignment for the 97.1 kev level. According to the DCO values obtained in the current work, the two transitions of 999.7 and 115. kev have quadrupole transition characters, which support the spin and parity assignments for the 97.1 kev level by Kosyak et al. Therefore the 97.1 and 3315. kev levels are more likely to be and states, respectively. Here, the γ-ray decay of

Band Band (1 ) 3.1 b (9 ) 57. Band (9 ) 51.7 151. Band 1 (1 ) 591. 117.1 1. 1 531.9 11. (7 ) 1.5 a 11. 13. 93. (5 ) 3.7 3.9 59. 3. 5.7 9.15. 97. 39. (3 ) 9. 151.3 7 5.5 95.3 5 3579. 79.5 3.5 3 935.7 11. 77.7 17. 11.5 39..7 399.7 51.1 1331. 53.5 13. 1. 999. 3. 1111. 59. 115.5 13.7 7.9 Band 3 a 3315. 7. 115. 15. b (7 ) 31.3 1113. 5 97.1 53.1 999.7 7. 71.3 133. 3 197. 9. 33.7 93. 1. 111.. Band 5 ( ) 3.7 99. ( ) 33.9.3 737.. 1131. Figure 3: Partial level scheme of 7 Ge. Energies are in kev. The arrow widths are proportional to the relative transition probabilities. 595. 1. 595.. 1113. kev for which the DCO ratio could not be extracted is also assumed to be a stretched E transition. From a careful analysis of the coincident data, other three bands (bands, 5 and in Fig. 3), are newly established in 7 Ge. As compiled in Refs. [9], the 3 level at 935.7 kev (bandhead of band ) and the level at. kev (band-head of band 5) have been identified in a variety of studies. The present DCO analyses further support the spin and parity assignments of these two levels. For band, the level at 97. kev was previously reported with a tentative (3) assignment [9], while the level at 3. kev was reported to be [9]. The coupling between band a and b indicates that band a and b may be the signature partners. Thus, the is tentatively assigned to the 97. kev level. The orderings of transitions in bands, 5 and are well defined due to the presence of the band-crossing transitions. 3. Discussion The new structure information obtained presently for 7 Ge, together with what known previously for neighboring Ge isotopes, allows a systematic study of the triaxial evolution in Ge isotopes. As mentioned in Refs. [, 7], the energy staggering S (I) = [E(I) E(I 1)] [E(I 1) E(I )] E( 1 ) within the γ band has been utilized as an important signature to distinguish the soft or rigid triaxiality. In Fig., the experimental staggering S (I) of γ bands in 7,7,7 Ge is plotted as a function of spin. It is shown that, below the spin 9, both 7 Ge and 7 Ge exhibit pronounced oscillatory patterns but with the opposite phases. The staggering of 7 Ge with positive S (I) values at even spin and negative values at odd spin suggests 3 that 7 Ge has a rigid-triaxial shape [], while the staggering of 7 Ge follows the pattern expected for a γ-soft potential. The staggering of 7 Ge does not show a simple oscillatory pattern with the increase of spin. At low spin (I 5 ), its staggering resembles that of 7 Ge, then changes to the same phase as that of 7 Ge for higher spin states (I > 5 ). The distinct staggering pattern observed in 7 Ge implies that with the increase of spin, the triaxiality of 7 Ge undergoes an evolution from γ-soft as in 7 Ge to γ-rigid as in 7 Ge. S(I) 1-1 1-1 7 Ge Exp. 5DCH model 7 Ge 1 7 Ge 7 Ge SpinI( ) 1 Figure : The experimental staggering S (I) of 7,7,7,7 Ge isotopes in comparison with the 5DCH model calculations. The experimental data of 7 Ge is taken from Ref. [1] and the data of 7 Ge from Ref. []. To get a comprehensive understanding of the triaxiality and its rigidness/softness in Ge isotopes, the calculations of fivedimensional collective Hamiltonian (5DCH) model based on the covariant density functional theory (CDFT) have been performed []. The 5DCH model describes the nuclear collective

7 Ge 7 Ge 7 Ge 7 Ge............ Figure 5: (Color online) The potential energy surfaces in the -γ plane ( γ ) for 7,7,7,7 Ge calculated by constrained triaxial covariant density functional theory with the newly proposed PC-PK1. The energy separation between each contour line is 5 kev. excitations of quadrupole vibration and rotation with parameters determined by constrained self-consistent triaxial CDFT, and has given successful microscopic description for the nuclear shape transitions along with the isotopic and isotonic chains in different mass regions [3 7]. Starting from constrained triaxial CDFT calculations with the newly proposed point-coupling energy density functional PC-PK1 [], the calculated potential energy surfaces (PESs) in the -γ plane for 7 Ge and its neighboring isotopes 7,7,7 Ge are shown in Fig. 5. It can be seen that for all these Ge isotopes the γ degree of freedom plays an important role. With the increase of the neutron number, there exhibit two general trends in the PESs of Ge isotopes, i.e., from a triaxial to a prolate-like shape and from rather soft to rather rigid potential. In comparison with the 7,7 Ge, the PESs around the ground states in 7,7 Ge are less stiff in γ direction and show relatively γ-soft shapes. Fig. displays the calculated spectra by 5DCH model for the ground-state, γ and bands in 7,7,7,7 Ge isotopes, and the corresponding intraband and interband transition probabilities, in comparison with the available experimental data. As shown in Fig., the overall agreements between theoretical excitation energies and experimental data are good, especially for the low spin states. For example, the calculated E( )/E(+ 1 ) ratios, 1.9, 1.9,. and.5 for 7,7,7,7 Ge isotopes, are consistent with the experimental data 1.,.,. and 1.9, respectively. With the increase of the angular momentum, the deviations of the theoretical excitation energies from experimental data increase. In Fig. 7, the calculated and experimental angular moments as functions of rotational frequency for the ground-state and γ bands of 7,7,7,7 Ge isotopes are plotted. It can be seen that for the 7,7,7 Ge, the calculated results agree with the experimental data at low rotational frequency. At high rotational frequency, the deviations of the theoretical results from experimental data become large. Around spin, the quasiparticle alignment characters are clearly shown in the 7,7 Ge, where the ground-state band structures above band crossing have not been observed. For the 7 Ge which have softer potential, the deviations are even larger for both ground-state and γ bands. The major reason of the deviations could be due to the adiabatic approximation of the Hamiltonian in the 5DCH model, where the collective parameters such as the moments of inertia are calculated in the vicinity of zero rotational frequency. At high spin states of these Ge isotopes, especially the 7 Ge whose soft potential is susceptible to the increase of angular momentum, this approximation becomes rough. Comparing with the experimental data at high spin states, the 5DCH calculations underestimate the moments of inertia. Moreover, since the quasiparticle alignments occur around spin in 7,7,7 Ge, the deviations between the theoretical results and experimental data are enlarged. Using the variable moments of inertia with spin in the 5DCH model is expected to improve the agreements between theoretical results and experimental data. Such efforts are still on-going. As shown in Fig., the calculated transition probabilities of 7,7,7,7 Ge isotopes well reproduce the experimental values not only for intraband transitions within the ground-state band but also for transitions from the γ band to the groundstate band. For the 7 Ge, the calculated branching ratios B(E; + 1 )/B(E; + + 1 ) =.5 and B(E; + 1 1 )/B(E; + 1 + 1 ) = 1.5 are in good agreement with the available experimental data.1(7) and 1.33(), respectively. In Fig., the calculated bands for 7,7,7,7 Ge isotopes have also been presented to make prediction. Experimentally, the possible band-heads of bands in 7,7,7,7 Ge isotopes have been summarized in Ref. [5]. Recent systematic Coulomb excitation studies [1 1] suggested that the 1st excited + s- tates of 7,7,7 Ge are mainly the intruder states with almost spherical shapes. According to the present calculations, for the 7,7 Ge, the observed nd excited + states at.9 MeV ( 7 Ge), 1.75MeV ( 7 Ge) are close to the theoretical values.13 MeV ( 7 Ge), 1.71MeV ( 7 Ge), respectively. For the 7 Ge, the calculated band-head energy 1.59 MeV of band has 1. MeV difference from the known nd excited + state.9 MeV, while for the 7 Ge, the calculated band-head energy.37 MeV of band locate between the known 1st (1.57 MeV) and nd (3.35 MeV) excited + states. So far, the collective structures of bands have not been observed in these Ge isotopes. Thus, more studies on the -band collective structures in 7,7,7,7 Ge isotopes would further test the 5DCH model predictions and provide a comprehensive understanding the shape evolution along the Ge isotopic chain. To further inspect the staggering behavior of γ band in

.3 >11 Figure : (Color online) The calculated excitation energies and the intraband and interband B(E) values (in W.u.) of the ground-state bands, γ bands and bands in 7,7,7,7 Ge isotopes by 5DCH model compared with the experimental data. The experimental B(E) values are taken from Refs. [, 1 1, 9 51]. (1.75) (1.3) 9.5 (1.11) (.53) + 3.9 (.53).1.5 39..3 + 9(1).9() (.19) Excitation energy (MeV) 1. (9) +. (.19) (.55).3. 19. (.31) (.797) (3.95) 79.7..1 + (1.59) (1.1) 3(9) 5... 3..7. 59.1 (.35) 9. (3.53) 57. (3.193) 35.7 (.) (.5) 3.(1) (5.7) 99. 1.5 9. (7.1) 1 + (.9).1 59.9 1.5 (5.7) 1. (5.1) 59. (.13). (3.13) (.57) (.13) (3.533) (3.3) (.7) (.) (1.539) (1.1) 1. + 3().. (1.53) 19(11).53() (1.) + 1.(3) 3.7 (1.57) (1.1) 1. (.75) (.19) (3.9) 53..3. 1.3 17.5.7 3. (.91) (.7) (.37) 7.1 (.) 39.9 3. (3.51) +.1.3 1.5 53. (3.97) (3.71) (.737) 3. 1 + (5.).3. (7.) 17. 3. (.77) 7. 1 1 + (7.119) 1 +.1 (7.7) (.11) 1. 3.7 (.3) 5.9 1. (5.15) (5.19) 3.1 11 + (c) 7 Ge (d) 7 Ge + + (.3) 3. 1.1 7.7. (.3) 3.5().13( +1 - ) (.59) (.59) 3.7() Excitation energy (MeV) 1.1 (.11).9.3 (1.3) (1.79) 37(5) (.13). 31. (3.153) 5.9 (.53).5( +3-5 ) ( +9-11 ) +... (1.17) + 15( + -15 ) (3.5)..1.3 5.7.3 (.9) (.773) + 37( +1-37 ).5 9..7.1 1.5 (.9) 5.5.3 (.1) (.197).9 (3.7) ( +11-7 ) (5.71) (.3) (.) (.1) (3.3) (3.1) (.5) (.) (1.5) 1..3 3. (.) 91. 1.3 (.1) (5.1).5 (1.51) 5..1. (1.71) (1.) 3.(7) (.77) 59.9.1..7.3 (3.779) 1.. (.93).7 5. (.31) 5.3 (1.77) 7.7(3).() 3.1 (.35) + (.59).1 1. 3.3 (3.) (3.9) (.1) 7.9 (.371) 1.1 1.5 59. (.39) 3. (3.1) (3.315) (.97) (.15) (1.97) (1.) (5.77) 93.5 1 + 1.7..3 (.797) 9.3 (.3) 1.7 7.7 (5.95).3 (5.) (5.1) 1 + 7.9 (5.311) (7.799) 1 + (7.3) (7.379) 119. 1 + 1 3. (7.573) 1 + (7.191) 1 (a) 7 Ge 11 + (b) 7 Ge 5

7,7.7,7 Ge isotopes, in Fig., the resulting theoretical staggering values are compared with the experimental data. For the 7 Ge, the calculations reproduce not only the evolution pattern, but also the magnitude of experimental staggering S (I) very well. For spin I, the predicted staggering of 7 Ge begins to increase rapidly with spin and exhibits a pronounced oscillatory pattern. As demonstrated in the Bohr Hamiltonian and interacting boson approximation calculations [7], such pronounced oscillatory pattern at high spin region I is expected for the rigid-triaxial potential. The good agreement between the calculated staggering S (I) and experimental results is also obtained for the 7 Ge with spin up to I =. For the 7 Ge, the experimental oscillatory phases of the S (I) are reproduced by the 5DCH model, but at high spin region the 5DCH calculations overestimate the experimental S (I) values. Spin I ( ) 1 1 1 1 7 Ge 7 Ge Exp. ground band Cal. ground band Exp. band Cal. band.... 1. 7 Ge 7 Ge.... 1. (MeV) Figure 7: (Color online) Comparison of the 5DCH calculation with experimental data for 7,7,7,7 Ge for the relation between spin I and ω. With the 5DCH calculations, the expectation values of and γ ( and γ ) for each given collective state can be obtained with the definition =, γ = arccos( 3 cos 3γ / )/3, see Ref. [] for more details. Fig. displays the calculated deformation parameters and γ as functions of spin for ground-state bands of 7,7,7,7 Ge isotopes by the 5DCH model based on CDFT. The γ expectation values for the ground states of 7,7,7,7 Ge (γ = 7.7,.7,., and., respectively) are in good agreement with the available values (γ =.5 ( 7 Ge),.5 ( 7 Ge), 7.3 ( 7 Ge) deduced from the experimental Q 1 and B(E; 1 + 1 ) values in Coulomb excitation measurements [7, ], and γ =. ( 7 Ge),.3 ( 7 Ge),. ( 7 Ge) deduced from the coupled-channel analyses of the α scattering measurements with the asymmetric rotor model [1]). It can be seen that γ value of 7 Ge decreases rapidly with spin and γ value of 7 Ge remains nearly constant, while the trend of γ value of 7 Ge lies in between. For the 7 Ge, its γ value is also relatively stable. These features can be understood from the soft triaxiality in 7 Ge and rigid triaxiality in 7 Ge (The PES along γ direction in 7 Ge is also rigid). For 7 Ge with a γ soft potential, its γ deformation is susceptible to the increasing spin. For 7 Ge with a rigid triaxial potential, its γ deformation tends to be stable. For 7 Ge, a transitional character is shown. Moreover, the values of 7,7,7,7 Ge increase with spin in different slopes in accordance to the rigidness/softness of the nucleus. γ(deg).5..3. 3 7 Ge 7 Ge 7 Ge 1 1 1 1 1 SpinI( ) 7 Ge Figure : The calculated deformation parameters and γ as functions of spin for 7,7,7,7 Ge isotopes by the 5DCH model based on CDFT.. Summary The collective structure of 7 Ge has been expanded significantly via 7 Zn( 7 Li,np) 7 Ge fusion-evaporation reaction. The previously known yrast band is extended to higher spins and five new collective bands are established. The energy staggering S (I) of adjacent levels within the γ band, which is sensitive to structural changes, is used to investigate the triaxial rigidness/softness in 7 Ge. In comparison with the pronounced oscillatory staggering patterns in neighboring isotopes 7,7 Ge, the distinct staggering pattern observed in 7 Ge implies that with the increase of spin, the triaxiality of 7 Ge undergoes an evolution from γ-soft as in 7 Ge to γ-rigid as in 7 Ge. Starting from constrained triaxial CDFT calculations with the newly proposed point-coupling energy density functional PC-PK1, the collective structures of even-even 7,7,7,7 Ge isotopes are discussed within the framework of the 5DCH model. The calculated potential energy surfaces around the ground s- tates of 7,7,7,7 Ge isotopes indicate that comparison with the 7,7 Ge, 7,7 Ge are relatively γ-soft at low-lying states. The calculated 5DCH results based on CDFT reproduce the excitation spectra and transition probabilities for both the groundstate bands and γ bands in these Ge isotopes, and a good agreement with the experimental energy staggering of 7 Ge is also obtained. Based on the systematic comparisons and analysis, the triaxial evolution with spin in 7 Ge is revealed and 7 Ge is found to be the crucial nucleus marking the triaxial evolution from soft to rigid in Ge isotopes. ACKNOWLEDGMENTS This work is support by the Natural Science Foundation of China under Grant No. 111753, No. 11351, No. 11335, No. 11311, No. 1137517, No. 1137515, No. 1135, No. 115, and No. J113 and the

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