1　Introduction

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, 2]. The observations of the γ-vibrational bands in ^132–138Nd [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 ¹³⁸Nd, experimental data on electromagnetic transition probabilities are still scarce, and further studies need to be conducted.

In the present work, lifetimes of the 2₁⁺, 4₁⁺, 7₂^-, 10₂⁺, 12₂⁺, and 14₁⁺ 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 2₁⁺ state has been published in Ref. [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 ¹³⁸Nd was published by Bello Garrote et al. [8]. Most of the results in the present work are close to the values in Ref. [8], except for the 7₂^- 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 ¹³⁸Nd isotopes and N = 78 isotonic chains.

2　EXPERIMENTAL DETAILS

The present work was performed at the HI-13 tandem accelerator of the China Institute of Atomic Energy (CIAE) in Beijing. Excited states in ¹³⁸Nd were populated using the ¹²³Sb(¹⁹F, 4n)¹³⁸Nd 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], and was utilized to set and keep the distance between the target and stopper with a relative precision of 0.3 μm. 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 μm) 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-μm distance, with a measuring time of 2 h, was used in the data analysis of the 4₁⁺ state, because the lifetime of this state is quite short. The NAPATAU code [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 μm is shown in Fig. 1.

Fig. 1Full-screen View

Backward (153°) total projection at a target–stopper distance of 100 μm. The main γ-ray transitions in ¹³⁸Nd are marked in the spectrum.

3　DATA ANALYSIS AND RESULTS

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 [11-13]. For the DDCM, the mean lifetime τi at level i and distance x can be determined as follows (see Fig. 2):

Fig. 2Full-screen View

Schematic decay scheme [7].

where subscripts s and u are the Doppler-shifted and unshifted components, respectively, $I_{(s\mathrm{,}u)}^{BA}$ 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

and

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). 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:

Fig. 3Full-screen View

(Color online) Shifted and unshifted components of transitions A and B in the gating spectrum.

Unfortunately, $I_{(s\mathrm{,}s)}^{AB}$ and $I_{(s\mathrm{,}s)}^{BA}$ cannot be determined from the gating spectra. However, based on their logical relationship, the following equation holds:

Then Eq. 4 can be replaced with

This means that, in the present case, Eq. 1 can be rewritten as

3.1　 Lifetime of the 7₂^- state

The lifetime the of 7₂^- 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) [14]. In other words, the 680 keV 9₃^- → 7₂^- and 470 keV 7₂^- → 5₂^- 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 7₂^- → 5₂^- transition can barely be seen in the gating spectra. Therefore, in the present work, the lifetime of the 7₂^- state could not be extracted as done in Ref. [8] by fitting the shifted and unshifted components of the 470 keV 7₂^- → 5₂^- transition in the gating spectra, and the 557 keV 7₂^- → 6₁⁺ transition is the only choice to fit the differential decay curve. However, when we set a cut on the shifted component of the $9_2^{-}\to 7_2^{-}$ 556 keV transition, we will also set a cut on the shifted part of the $7_2^{-}\to 6_1^{+}$ 557 keV transition at the same time. Therefore. in the lifetime analysis of the 7₂^- state, Eq. 7 was used.

Fig. 4Full-screen View

Partial level scheme for ¹³⁸Nd. Lifetimes measured in the present work are also labeled in the level scheme. The width of the arrows represents the relative intensity of the transitions. Relative intensity information was taken from Ref. [14]. The lifetime of the 3175 keV 10₁⁺ state with an asterisk was taken from Refs. [15, 16].

According to the analysis above, the $I_u^A(x)$ and $I_s^A(x)$ trends fitted by the NAPATAU code are shown in the right panels of Fig. 5. Partly backward shifted and unshifted gating spectra are shown in the left panels of 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] 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 7₂^- state.

Fig. 5Full-screen View

Decay curves and lifetime determination of the 729 keV 4₁⁺ → 2₁⁺ transition (left) and 557 keV γ transitions (right) in ¹³⁸Nd. The middle and lower panels show the Doppler-shifted and stopped intensities versus distance. The mean lifetime at each distance is determined and is shown in the upper panel as a function of distance. The weighted average of these values yields the overall mean lifetime.

Fig. 6Full-screen View

(Color online) Left: Partial stopped and backward-shifted components for the 557 keV γ transitions in ¹³⁸Nd from gating on the backward-shifted components of the 557 keV transition. Right: Partly backward-shifted and unshifted components of the 503 keV 12₂⁺ → 10₂⁺ transition in ¹³⁸Nd from gating on the backward-shifted components of the 847 keV 16₂⁺ → 14₁⁺ transition.

3.2　 Lifetimes of other states

In addition to the lifetimes of the 7₂^- state mentioned above and the lifetime of the 2₁⁺ state determined in Ref. [7], the lifetimes of the 4₁⁺, 10₂⁺, 12₂⁺, and 14₁⁺ states were measured in the present work. For the 4₁⁺, 10₂⁺, and 14₁⁺ states, a direct gating case was used. However, for the 12₂⁺ state, an indirect gating case was used because the shifted peak of the 792 keV 14₁⁺ → 12₂⁺ transition will overlap with the unshifted peak of the 787 keV 13₂⁺ → 12₂⁺ transition. Moreover, the shifted and unshifted components of the 453 keV 10₂⁺ → 9₂^- transition were fit in the 10₂⁺ state lifetime determination, because the energy of the 329 keV 10₂⁺ → 9₃^- transition is close to the energy of the 331 keV 7₁^- → 5₁^- transition. Partly backward shifted and unshifted gating spectra of the 503 keV 12₂⁺ → 10₂⁺ transitions are shown in the right panels of Fig. 6. The decay curves and τ plots of 4₁⁺, 10₂⁺, 12₂⁺, and 14₁⁺ states are shown in the left panels of Figs. 5 and 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. A comparison of the results from the current work to those of Ref. [8] is given in Table 1 .

Fig. 7Full-screen View

Decay curves and lifetime determination of the 792 keV 14₁⁺ → 12₂⁺ (left), 503 keV 12₂⁺ → 10₂⁺ (middle), and 453 keV 10₂⁺ → 9₂^- transitions (right) in ¹³⁸Nd. The middle and lower panels show the Doppler-shifted and stopped intensities versus distance. The mean lifetime at each distance is determined and is shown in the upper panel as a function of distance. The weighted average of these values yields the overall mean lifetime.

Table 1Full-screen View

Comparison of results between the present work and those of Ref. [8].

Ex ［ keV ］	$J_n^{\pi }$	Present work ［ ps ］	Ref. [8] ［ ps ］
521	21+	10.9(12)	13.9(2)
1250	41+	2.0(4)	1.9(2)
2691	72-	19.0(20)	9.5(8)
3700	102+	50.5(41)	47.8(11)
4203	122+	11.5(11)	14.0(4)
4995	141+	2.3(3)

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4　DISCUSSION

The lifetime of the 2₁⁺ state in the present work yields a reduced transition probability of $B(E{2;2}_1^{+}\to 0_1^{+})$ = 0.19(2) e²b², or 45(5) W.u. This value lies in the middle of the value of 39(4) W.u. in ¹³⁶Ce [17] and the value 51(4) W.u. in ¹⁴⁰Sm [18]. To identify the variation in the collectivity in the A = 130 mass region, the value of B(E2) of the 2₁⁺ state in the present work was compared to those values in the neighboring even–even isotopes (see Fig. 8). As can be observed in Fig. 8, when the neutron number increases, the B(E2) values of the 2₁⁺ states decrease, which means that collectivity decreases. The B(E2) value usually correlates with the energy of the 2₁⁺ state. When the collectivity increases, the B(E2) value increases and the energies of the 2₁⁺ state decrease. This conclusion was recognized by Grodzins [19] and is known as Grodzins rule. Within an isobaric chain, Grodzins rule can be described as follows [20]:

Fig. 8Full-screen View

(Color online) Systematics of B(E2) values for the even–even Nd isotopic chain. The experimental values for ¹³²Nd, ¹³⁴Nd, ¹³⁶Nd, ¹³⁸Nd, and ¹⁴⁰Nd are taken from Ref. [21], Ref. [22], Ref. [23], this work, and Ref. [24], respectively.

where

In Eq. 8, $B(E2)\uparrow$ is given by e²b² and E keV, which is the level energy of the 2₁⁺ state; Z, N, and A are the proton number, neutron number, and mass number, respectively; and H1 and H2 are the Habs fit values, which can be found in Ref. [20]. For the Nd isotopic chain, H1 = 3.38 and H2 = 0.35. Values obtained from the modified Grodzins formula were compared to the experimental results, as shown in Fig. 8. Except for ¹³²Nd [21], the experimental B(E2) values of the 2₁⁺ 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.

¹³⁸Nd has been discussed as a transitional nucleus [25]. To identify the shape information of the low-lying states in ¹³⁸Nd and the neighboring even–even nuclei, the excitation energy ratio R_4/2, defined as E(4₁⁺)/E(2₁⁺), and the ratio B_4/2, defined as $B(E{2;4}_1^{+}\to 2_1^{+})$/$B(E{2;2}_1^{+}\to 0_1^{+})$, are proposed (see Fig. 9). For the shell configuration, R_4/2 $$ 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, the R_4/2 values of both Nd isotopes and N = 78 isotone chains show that ¹³⁸Nd should be a γ-soft or transitional nucleus, as discussed in Ref. [6]. The potential energy surface calculation in Ref. [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 $B(E{2;4}_1^{+}\to 2_1^{+})$ values in ¹³⁶Nd and ¹⁴⁰Nd, the trend followed by the B_4/2 value is not clear in the Nd isotope chain. However, the B_4/2 value in ¹³⁸Nd is smaller than the neighboring results for ¹³⁶Ce and ¹⁴⁰Sm in the isotone chain. This phenomenon may be caused by the lifetime of the 4₁⁺ state in ¹³⁸Nd, because the lifetime of the 4₁⁺ state is close to the lower limit of the RDDS used in the present work and in Ref. [8]. The $B(E{2;4}_1^{+}\to 2_1^{+})$ value is 47(9) W.u. in the current experiment, which is close to the result of 49(7) W.u. in Ref. [8] but smaller than the values of 56(10) W.u. in ¹³⁶Ce and 69(5) W.u. in ¹⁴⁰Sm, although the $B(E{2;2}_1^{+}\to 0_1^{+})$ values in the present work and Ref. [8] fit smoothly into the trends of ¹³⁶Ce and ¹⁴⁰Sm. Moreover, in Ref. [8], the experimental $B(E{2;4}_1^{+}\to 2_1^{+})$ value is smaller than the theoretical prediction, which means that the lifetime of the 4₁⁺ state in ¹³⁸Nd may be shorter than the present result.

Fig. 9Full-screen View

Excitation energy ratio R_4/2 (upper panels) and the ratio of B(E2) values B_4/2 (lower panels) for the even–even Nd isotopic chain (left) and N = 78 isotones (right). Information for ¹³²Nd, ¹³⁴Nd, ¹³⁴Ba, ¹³⁶Ce, ¹³⁸Nd, and ¹⁴⁰Sm was obtained from Refs. [21, 26], Refs. [22, 27], Ref. [27], Ref. [17], this work, and Refs. [24, 28], respectively.

5　SUMMARY

In addition to the lifetime of 10.9(11) ps of the 2₁⁺ state reported in Ref. [7], lifetimes of the 4₁⁺, 7₂^-, 10₂⁺, 12₂⁺, and 14₁⁺ states in ¹³⁸Nd 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], except for the lifetime of the 7₂^- state, which is twice that found in Ref. [8]. The $B(E{2;2}_1^{+}\to 0_1^{+})$ value in the present work fits smoothly into the systematics of the even–even Nd isotope chain and the prediction of Grodzins rule. R_4/2 exhibits a smooth trend in the A = 130 mass region, which indicates that ¹³⁸Nd should be a γ-soft nucleus, but the B_4/2 value deviates from it. This may be due to the lifetime of the 4₁⁺ state being too short to be measured using the RDDS method.