Lifetime measurements in ¹³⁸Nd

Lifetime measurements in 138 Nd 1 first-author 2 corresp wxg@ciae.ac.cn wxg@ciae.ac.cn wxg@ciae.ac.cn 1 3 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China China Institute of Atomic Energy, Beijing 102413, China China Institute of Atomic Energy, Beijing 102413, China College of Physics, Jilin University, Changchun 621010, China College of Physics, Jilin University, Changchun 621010, China Lifetimes of the 2 1 + , 4 1 + , 7 2 - , 10 2 + , 12 2 + , and 14 1 + states in 138 Nd populated via the 123 Sb( 19 F, 4n) 138 Nd fusion-evaporation reaction were measured with the recoil distance Doppler shift technique in combination with the differential decay curve method. The <math id="M1"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> value fit well with the systematic trend in the Nd isotope chain and Grodzins rule, which proved that 138 Nd is a transitional nucleus. Lifetime measurement Recoil distance Doppler shift technique Differential decay curve method 1　Introduction 1 For the N = 78 isotonic chain, the shapes of the even–even rare earth nuclei ranging from Z = 60 to 70 were predicted to have a shape change from prolate to oblate. [ 1 1 , 2 2 ]. The observations of the γ -vibrational bands in 132–138 Nd [ 3 3 - 6 6 3-6 ] in this region proved the theoretical trend in some sense. Moreover, electromagnetic transition probabilities provide a sensitive test for understanding the nuclear structure. For the transitional nucleus 138 Nd, experimental data on electromagnetic transition probabilities are still scarce, and further studies need to be conducted. In the present work, lifetimes of the 2 1 + , 4 1 + , 7 2 - , 10 2 + , 12 2 + , and 14 1 + states were measured with the recoil distance Doppler shift (RDDS) technique. The data were analyzed by using the differential decay curve method (DDCM). The lifetime of the 2 1 + state has been published in Ref. [ 7 7 ] previously, and the other lifetimes will be presented in this study. It should be noted that another study also using the RDDS technique and the DDCM to measure lifetimes of several states in 138 Nd was published by Bello Garrote et al. [ 8 8 ]. Most of the results in the present work are close to the values in Ref. [ 8 8 ], except for the 7 2 - state. In their work, the experimental results, especially for the two different 10 + states, associated with proton and neutron configurations, respectively, have been explained in detail by various theories. Therefore, in the present work, the focus is concentrated on the systemic trend of the low-lying states in the 138 Nd isotopes and N = 78 isotonic chains. 2　EXPERIMENTAL DETAILS 1 The present work was performed at the HI-13 tandem accelerator of the China Institute of Atomic Energy (CIAE) in Beijing. Excited states in 138 Nd were populated using the 123 Sb( 19 F, 4n) 138 Nd fusion-evaporation reaction at a beam energy of 87 MeV. The reaction was chosen based on a cross-section calculation performed using the code PACE4. Lifetimes were measured using the CIAE plunger device, which has been introduced in Ref. [ 9 9 ], and was utilized to set and keep the distance between the target and stopper with a relative precision of 0.3 μ m. Eight Compton-suppressed high-purity Ge (HPGe) detectors were used to detect γ rays from the residues. Three of these detectors were placed at 90°, four at 153°, and one at 42° with respect to the beam direction. Thirteen different target-to-stopper distances (5, 9, 15, 25, 41, 70, 100, 166, 275, 457, 758, 1259, and 2000 μ m) were used to record the γ – γ coincidence data. The first five distances were measured for 8 h. The last three distances were measured for 4 h, and the rest were measured for 6 h. In addition to those distances mentioned above, another 3- μ m distance, with a measuring time of 2 h, was used in the data analysis of the 4 1 + state, because the lifetime of this state is quite short. The NAPATAU code [ 10 10 ] was used in the data analysis. The mean recoil velocity of the compound nucleus was ~ 1% of the light speed c . Owing to the neutron damage and poor statistics of the forward (42°) detector, only backward (153°) detectors were used for data analysis. The backward total projection at a distance of 100 μ m is shown in Fig. 1 Fig. 1 . 3　DATA ANALYSIS AND RESULTS 1 The experimental data were analyzed by using the DDCM, which has been proven to be a precise method for determining the lifetime of excited states [ 11 11 - 13 13 11-13 ]. For the DDCM, the mean lifetime τ i at level i and distance x can be determined as follows (see Fig. 2 Fig. 2 ): <math display="block" id="M2"><mrow><msub><mi>τ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>υ</mi><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where subscripts s and u are the Doppler-shifted and unshifted components, respectively, <math id="M3"><mrow><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup></mrow></math> refers to the normalized intensity I of the unshifted component of the depopulating transition A obtained by gating on the shifted peak of transition B , and υ denotes the recoil velocity. In the indirect gating case, the gate is set on the shifted peak of transition C . It will transform into <math display="block" id="M4"><mrow><msub><mi>τ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>A</mi></mrow></msubsup><mo>−</mo><mi>α</mi><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>B</mi></mrow></msubsup></mrow><mrow><mi>υ</mi><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>A</mi></mrow></msubsup></mrow></mfrac></mrow></math> and <math display="block" id="M5"><mrow><mi>α</mi><mo>=</mo><mfrac><mrow><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>A</mi></mrow></msubsup><mo>+</mo><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>A</mi></mrow></msubsup></mrow><mrow><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>B</mi></mrow></msubsup><mo>+</mo><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>B</mi></mrow></msubsup></mrow></mfrac><mn>.</mn></mrow></math> Before being used to determine the lifetime, the intensities of the shifted and unshifted components were normalized for differences in beam intensity and measuring time at different target-to-stopper distances. However, if the energies of transitions A and B are the same, gating on the shifted component of transition B also gates on the shifted component of transition A . The shifted peak of “transition” in the gating spectrum comprises two parts (see Fig. 3 Fig. 3 ). The first part is the shifted component of transition A obtained by gating on the shifted peak of transition B , and the second part is the shifted component of transition B , which is obtained from gating on the shifted component of transition A . However, the unshifted component of transition A can be obtained by gating only on the shifted peak of transition B , because the unshifted component of transition B cannot be seen from the gating on the shifted component of transition A . In this special case, the following relationship can be obtained: <math display="block" id="M6"><mrow><msubsup><mi>I</mi><mi>s</mi><mi>A</mi></msubsup><mo>=</mo><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msubsup><mo>+</mo><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup><mn>.</mn></mrow></math> Unfortunately, <math id="M7"><mrow><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msubsup></mrow></math> and <math id="M8"><mrow><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup></mrow></math> cannot be determined from the gating spectra. However, based on their logical relationship, the following equation holds: <math display="block" id="M9"><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Then Eq. 4 can be replaced with <math display="block" id="M10"><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mi>s</mi><mi>A</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mi>A</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> This means that, in the present case, Eq. 1 can be rewritten as <math display="block" id="M11"><mrow><msub><mi>τ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>I</mi><mi>u</mi><mi>A</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>υ</mi><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mi>s</mi><mi>A</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><msubsup><mi>I</mi><mi>u</mi><mi>A</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>υ</mi><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>I</mi><mi>s</mi><mi>A</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> 3.1　 Lifetime of the 7 2 - state 2 The lifetime the of 7 2 - state cannot be determined easily using Eq. 1, because energies of the strongest populated and depopulated transitions for this level are all close to 557 keV, and other populated and depopulated transitions are much weaker than the strongest one (see Fig. 4 Fig. 4 ) [ 14 14 ]. In other words, the 680 keV 9 3 - → 7 2 - and 470 keV 7 2 - → 5 2 - transitions are too weak to use for lifetime determination in our data. When we gated on the shifted component of the 557 keV γ peak, the shifted and unshifted components of the 470 keV 7 2 - → 5 2 - transition can barely be seen in the gating spectra. Therefore, in the present work, the lifetime of the 7 2 - state could not be extracted as done in Ref. [ 8 8 ] by fitting the shifted and unshifted components of the 470 keV 7 2 - → 5 2 - transition in the gating spectra, and the 557 keV 7 2 - → 6 1 + transition is the only choice to fit the differential decay curve. However, when we set a cut on the shifted component of the <math id="M12"><mrow><msubsup><mn>9</mn><mn>2</mn><mo>−</mo></msubsup><mo>→</mo><msubsup><mn>7</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> 556 keV transition, we will also set a cut on the shifted part of the <math id="M13"><mrow><msubsup><mn>7</mn><mn>2</mn><mo>−</mo></msubsup><mo>→</mo><msubsup><mn>6</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> 557 keV transition at the same time. Therefore. in the lifetime analysis of the 7 2 - state, Eq. 7 was used. According to the analysis above, the <math id="M14"><mrow><msubsup><mi>I</mi><mi>u</mi><mi>A</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> and <math id="M15"><mrow><msubsup><mi>I</mi><mi>s</mi><mi>A</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> trends fitted by the NAPATAU code are shown in the right panels of Fig. 5 Fig. 5 . Partly backward shifted and unshifted gating spectra are shown in the left panels of Fig. 6 Fig. 6 for the γ peak at 557 keV. The mean value of 9.5(10) ps fits well with the result in Ref. [ 8 8 ] within the experimental uncertainty. The mean lifetime of 19.0(20) ps, which is twice the mean value of 9.5(10) ps, is set as the final result for the 7 2 - state. 3.2　 Lifetimes of other states 2 In addition to the lifetimes of the 7 2 - state mentioned above and the lifetime of the 2 1 + state determined in Ref. [ 7 7 ], the lifetimes of the 4 1 + , 10 2 + , 12 2 + , and 14 1 + states were measured in the present work. For the 4 1 + , 10 2 + , and 14 1 + states, a direct gating case was used. However, for the 12 2 + state, an indirect gating case was used because the shifted peak of the 792 keV 14 1 + → 12 2 + transition will overlap with the unshifted peak of the 787 keV 13 2 + → 12 2 + transition. Moreover, the shifted and unshifted components of the 453 keV 10 2 + → 9 2 - transition were fit in the 10 2 + state lifetime determination, because the energy of the 329 keV 10 2 + → 9 3 - transition is close to the energy of the 331 keV 7 1 - → 5 1 - transition. Partly backward shifted and unshifted gating spectra of the 503 keV 12 2 + → 10 2 + transitions are shown in the right panels of Fig. 6 Fig. 6 . The decay curves and τ plots of 4 1 + , 10 2 + , 12 2 + , and 14 1 + states are shown in the left panels of Figs. 5 Figs. 5 and 7 7 . From the resulting decay curves, the deduced mean lifetimes of these states were determined to be 2.0(4), 50.5(41), 11.5(11), and 2.3(3) ps, respectively, and these states are all labeled in Fig. 4 Fig. 4 . A comparison of the results from the current work to those of Ref. [ 8 8 ] is given in Table 1 Table 1 . 1 1 E x ［ keV ］ 1 1 <math id="M16"><mrow><msubsup><mi>J</mi><mi>n</mi><mi>π</mi></msubsup></mrow></math> 1 1 Present work ［ ps ］ 1 1 Ref. [ 8 8 ] ［ ps ］ 1 1 521 1 1 2 1 + 1 1 10.9(12) 1 1 13.9(2) 1 1 1250 1 1 4 1 + 1 1 2.0(4) 1 1 1.9(2) 1 1 2691 1 1 7 2 - 1 1 19.0(20) 1 1 9.5(8) 1 1 3700 1 1 10 2 + 1 1 50.5(41) 1 1 47.8(11) 1 1 4203 1 1 12 2 + 1 1 11.5(11) 1 1 14.0(4) 1 1 4995 1 1 14 1 + 1 1 2.3(3) 1 1 4　DISCUSSION 1 The lifetime of the 2 1 + state in the present work yields a reduced transition probability of <math id="M17"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> = 0.19(2) e 2 b 2 , or 45(5) W.u. This value lies in the middle of the value of 39(4) W.u. in 136 Ce [ 17 17 ] and the value 51(4) W.u. in 140 Sm [ 18 18 ]. To identify the variation in the collectivity in the A = 130 mass region, the value of B ( E 2) of the 2 1 + state in the present work was compared to those values in the neighboring even–even isotopes (see Fig. 8 Fig. 8 ). As can be observed in Fig. 8 Fig. 8 , when the neutron number increases, the B ( E 2) values of the 2 1 + states decrease, which means that collectivity decreases. The B ( E 2) value usually correlates with the energy of the 2 1 + state. When the collectivity increases, the B ( E 2) value increases and the energies of the 2 1 + state decrease. This conclusion was recognized by Grodzins [ 19 19 ] and is known as Grodzins rule. Within an isobaric chain, Grodzins rule can be described as follows [ 20 20 ]: <math display="block" id="M18"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><mn>2</mn><mo stretchy="false">)</mo><mo>↑</mo><mo>=</mo><mo stretchy="false">(</mo><mi>H</mi><mn>1</mn><mo>−</mo><mi>H</mi><mn>2</mn><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msup><mi>E</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>Z</mi><mn>2</mn></msup><msup><mi>A</mi><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mn>,</mn></mrow></math> where <math display="block" id="M19"><mrow><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mi>A</mi><mn>2</mn></mfrac><mfrac><mrow><mn>1.0070</mn><mo>+</mo><mn>0.0128</mn><msup><mi>A</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><mn>1</mn><mo>+</mo><mn>0.0064</mn><msup><mi>A</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac><mn>.</mn></mrow></math> In Eq. 8, <math id="M20"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><mn>2</mn><mo stretchy="false">)</mo><mo>↑</mo></mrow></math> is given by e 2 b 2 and E keV, which is the level energy of the 2 1 + state; Z , N , and A are the proton number, neutron number, and mass number, respectively; and H 1 and H 2 are the Habs fit values, which can be found in Ref. [ 20 20 ]. For the Nd isotopic chain, H 1 = 3.38 and H 2 = 0.35. Values obtained from the modified Grodzins formula were compared to the experimental results, as shown in Fig. 8 Fig. 8 . Except for 132 Nd [ 21 21 ], the experimental B ( E 2) values of the 2 1 + states for the chain of Nd isotopes fit well with the systematic trend of the modified Grodzins rule. This means that the closer the neutron number gets to the N = 82 subshell, the weaker is the collectivity. 138 Nd has been discussed as a transitional nucleus [ 25 25 ]. To identify the shape information of the low-lying states in 138 Nd and the neighboring even–even nuclei, the excitation energy ratio R 4/2 , defined as E (4 1 + )/ E (2 1 + ), and the ratio B 4/2 , defined as <math id="M21"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>4</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> / <math id="M22"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> , are proposed (see Fig. 9 Fig. 9 ). For the shell configuration, R 4/2 <math id="M23"><mrow/></math> 2 and B 4/2 ~ 1; for a geometric vibrator, R 4/2 = 2–2.2 and B 4/2 ~ 2; for the γ -unstable nuclei, R 4/2 = 2.2–2.4 and R 4/2 ~3.3, B 4/2 ~ 1.4 for an ideal rotor, respectively. As can be observed in Fig. 9 Fig. 9 , the R 4/2 values of both Nd isotopes and N = 78 isotone chains show that 138 Nd should be a γ -soft or transitional nucleus, as discussed in Ref. [ 6 6 ]. The potential energy surface calculation in Ref. [ 8 8 ] also supports this conclusion. However, the B 4/2 value in the present work is close to 1.0. Owing to the lack of <math id="M24"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>4</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> values in 136 Nd and 140 Nd, the trend followed by the B 4/2 value is not clear in the Nd isotope chain. However, the B 4/2 value in 138 Nd is smaller than the neighboring results for 136 Ce and 140 Sm in the isotone chain. This phenomenon may be caused by the lifetime of the 4 1 + state in 138 Nd, because the lifetime of the 4 1 + state is close to the lower limit of the RDDS used in the present work and in Ref. [ 8 8 ]. The <math id="M25"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>4</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> value is 47(9) W.u. in the current experiment, which is close to the result of 49(7) W.u. in Ref. [ 8 8 ] but smaller than the values of 56(10) W.u. in 136 Ce and 69(5) W.u. in 140 Sm, although the <math id="M26"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> values in the present work and Ref. [ 8 8 ] fit smoothly into the trends of 136 Ce and 140 Sm. Moreover, in Ref. [ 8 8 ], the experimental <math id="M27"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>4</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> value is smaller than the theoretical prediction, which means that the lifetime of the 4 1 + state in 138 Nd may be shorter than the present result. 5　SUMMARY 1 In addition to the lifetime of 10.9(11) ps of the 2 1 + state reported in Ref. [ 7 7 ], lifetimes of the 4 1 + , 7 2 - , 10 2 + , 12 2 + , and 14 1 + states in 138 Nd were measured using the RDDS technique and the DDCM in the present work. Most of the results in this experiment were similar to those in Ref. [ 8 8 ], except for the lifetime of the 7 2 - state, which is twice that found in Ref. [ 8 8 ]. 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Görgen et al. , Structure of low-lying states in 140 Sm studied by Coulomb excitation . Phys. Rev. C 93 , 054303 ( 2016 ). doi: 10.1103/PhysRevC.93.054303 http://doi.org/10.1103/PhysRevC.93.054303 Structure of low-lying states in 140Sm studied by Coulomb excitation All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Jian Zhong, Xiao-Guang Wu, Shi-Peng Hu, Ying-Jun Ma, Yun Zheng, Cong-Bo Li, Guang-Sheng Li, Bao-Ji Zhu, Tian-Xiao Li, Yan-Jun Jin, Yan-Xiang Gao, and Qi-Wen Fan. The first draft of the manuscript was written by Jian Zhong and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. The authors would like to thank the staff of the HI-13 tandem accelerator at the China Institute of Atomic Energy for the steady operation of the accelerator. 10.1007/s41365-021-00953-4 This work was supported by the National Natural Science Foundation of China (Nos. U1932209, 11975315, U1867210, and 11905134) , the Leading Innovation Project (Nos. LC192209000701 and LC202309000201) , the Continuous Basic Scientific Research Project (Nos. BJ20002501 and WDJC-2019-13) , and the China National Nuclear Corporation (No. FA18000201). 2021-03-11 2021-05-14 2021-05-26 2021-10-04 2021-10-01 2021-11-29 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00953-4 http://dx.doi.org/10.1007/s41365-021-00953-4 . 2021 Nuclear Science and Techniques 10 32