1 Introduction

Transition probabilities are of special interest for the understanding of nuclear structure because of their sensitivity to the details of nuclear wave functions. Therefore, absolute transition probabilities or lifetime measurements of excited nuclear states are very important in γ-spectroscopy. Radiative lifetimes of excited nuclear levels in the picosecond region can be measured with the recoil distance Doppler shift (RDDS) method. The differential decay curve method (DDCM) [1-5], which has been proved to be a reliable analysis technique used for lifetime measurement data obtained from the RDDS experiments, such that mean-lifetimes of excited states in nuclei can be determined precisely. For standard DDCM analysis in γ-γ coincident mode, the mean lifetime of interest level can be determined at every target-to-stopper distance by gating on the Doppler-shifted component of higher lying transition. Thereafter, the normalized intensities of the stopped and Doppler-shifted γ rays for a transition depopulating this excited state were fitted with second-order polynomials, as described in Ref. [3], where one obtains two curves for the Doppler-shifted and unshifted components, which are used to extract the lifetime at each target-to-stopper distance. These curves are called coincidence decay curves.

The most significant errors associated with the measured mean-lifetime arise from the statistical uncertainties associated with the fitting of the decay curves and the range in the recoil velocities of the recoiling nuclei. In most of the work, there was a significant spread in recoil velocities due to the relatively low initial recoil velocity within the target. The low velocity of the recoiling compound nucleus results in a small energy separation between the stopped and Doppler-shifted peaks, which means that the shifted and unshifted peaks in the coincident spectra will overlap each other (see Fig. 1). In order to ensure that the gating region set on a flight peak does not include any contaminant γ rays from the tail of the corresponding stop peak, only partial selection of the shifted component could be used as the feeding component (see gate 1 in Fig. 1). Two major disadvantages will be caused by gating on the partial shifted peak. Firstly, the mean recoil velocity should be modified in the data analysis because only high speed flight nuclei in shifted components are selected for gating so it will bring some systematic deviations in the result. Secondly, the statistics of flight and stop peaks obtained from the gating partial shifted component in the spectra are usually poor and it is hard to fit the shifted and unshifted peaks.

Figure 1:Full-screen View

Sample for the gated region in the DDCM.

In this paper, the Doppler-shifted and unshifted peaks instead of the partial shifted component were selected as the gating region (see gate 2 in Fig. 1). This change will be beneficial in avoiding the disadvantages mentioned above. The mean-lifetimes of $2_1^{+}$ states in ¹³⁴Ce and ¹³⁸Nd determined by the new gating region were performed to confirm the reliability of this method.

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. A 0.62 mg/cm² thick ¹²³Sb foil, which was evaporated on a 2.2 mg/cm² thick Ta backing facing the beam, was used as the target. The flying residues nuclei were stopped by a 10 mg/cm² thick Ta stopper. The mean recoil velocity of the compound nucleus was ∼1% of the light speed, c. The lifetime measurement was performed by the RDDS method using the CIAE plunger device, which has been introduced in Ref. [6] 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 utilized to detect the deexcited γ-rays from the reaction residues. Three of these detectors were placed at 90, four at 153, and one at 42 with respect to the beam direction. The detectors were calibrated for γ-ray energies and efficiencies using the ¹³³Ba and ¹⁵²Eu standard sources. 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 coincidences data. For the data analysis of the lifetime of $2_1^{+}$ state in ¹³⁸Nd, only distance points shorter than 457 μM were used.

3 Data ANALYSIS METHOD

To simplify the calculation, the following discussion just contains the simple situation of γ-γ coincidence analysis in the DDCM. More complex detail can be found in Ref. [3,5]. According to the DDCM, the mean lifetime, τ, of the level i that is depopulated via transition A can be determined at every plunger distance x by gating either "indirectly" or "directly" (see Fig. 2). In the indirectly gating case, a gate is set on the Doppler-shifted component of a transition, say C, that depopulates a higher-lying state and feeds the level i, via intermediate transitions the last of which is the transition B, which directly feeds level i. Then the time evolution of the population, ni(t), of the state i is given by the well known differential equation:

Figure 2:Full-screen View

Level scheme sample for direct and indirect gating.

where $n_i^{CA}(t)$ is the number of nuclei in the state i at the time, t, which is populated and decays via the cascade C → B→ A. And λi denotes the decay constant of level i. Then the lifetime, τi, can be expressed as:

Here $N_A^{CA}(t)=N_i^{CA}(t)={\lambda }_i{\displaystyle {\int }_t^{\infty }}{n}_i^{CA}(t)\text{d}t$ and $N_B^{CA}(t)=N_k^{CA}(t)={\lambda }_k{\displaystyle {\int }_t^{\infty }}{n}_k^{CA}(t)\text{d}t$, because only the coincident cascade C → B → A in Fig. 2 was considered in the present discussion.

As can be seen in the discussion above, the time information of the coincident intensity must be contained in the lifetime measurement. To simplify the discussion, it is useful to define the notation $I_{(s\mathrm{,}u)}^{CA}$ as the coincident intensities of the unshifted (u) peak of transition A which is gated by the shifted (s) peak of transition C. If time, t, is the flight time of the recoil nuclei between creation in the target and arriving at the stopper, which means that the range of integration in Eq. (2) donates the intensities of unshifted component in gating spectra, then can be identified $N_i^{CA}$ with $I_{(u+s\mathrm{,}u)}^{CA}$, and $N_k^{CA}(t)$ with $I_{(u+s\mathrm{,}u)}^{CB}$. Eq. (2) can be rewritten as:

According to the time interval in the DDCM analysis, the coincident intensities in gating spectra can be split into several parts as follow:

Obviously, $I_{(u\mathrm{,}s)}^{CA}$ is equal to zero for the time order of transitions. Because $I_{(u+s\mathrm{,}u+s)}^{CA}$ is independent of time, t, an useful relation can be derived from Eq. (4) as:

Then Eq. (3) can be rewritten as:

or

The coincidence intensities discussed above denote absolute values which are usually not observed in an experiment. The measured coincidence intensities are proportional to absolute values, and the factors depend on the efficiencies for detecting the transition and the angular correlations of these transitions, respectively. Evidently, the factors between obtained intensities and absolute values are not the same for different transitions in coincident gating spectra, and the modifying factor for this difference is defined as α. In order to compare coincidence intensities, the following ratio is valid:

Using the observed coincidence intensity quantities one gets:

or

Using Eq. (8), the numerator of Eq. (10) can be changed to:

If υ is the mean recoil velocity of the nuclei, then the Eq. (9) and Eq. (10) can be rewritten as:

or

Moreover, a gate is often set on a direct feeding transition of the level of interest, e.g. a gate placed on the transition B for the example is shown in Fig. 2. This means that only the coincidence cascade B → A was considered. The level of interest is fed only via transition B and all other feeders are excluded, which means that the integral of $N_k^{CA}(t)$ in the numerator of Eq. (2) is changed into $N_B^{BA}(t)=N_k^{BA}(t)={\lambda }_k{{\displaystyle {\int }_t}}_{}^{\infty }{n}_k^{BA}(t)\text{d}t=I_{(u\mathrm{,}u+s)}^{BA}(t)$. The numerator of Eq. (7) is changed to:

So for the direct gating case, the lifetime, τi:

But in the present work, more attention was focused on the difference between Eq. (12) and Eq. (13). As can be seen from the discussion above, these two equations were both derived from the time differential equation (Eq. (1)). The difference occurred in Eq. (5). Denominators of Eq. (12) and Eq. (13) are the second term and third term of Eq. (5), respectively. However, for indirect gating case, only Eq. (13) was posed in Ref. [3], and then this equation was used widely in the latter indirect gating research. The main difference for these two equations is the gating region. In Eq. (12), coincident intensities were deduced by setting a cut on both Doppler-shifted and unshifted (u+s) components of a higher lying transition. But in Eq. (13), only a partial shifted (s) peak was selected as the gating region.

4 RESULTS

Two experimental data have been analysed to confirm that the new gating region selection can help to improve the statistics of flight and stop peaks in the gating spectra. The first comparison is that the lifetime measurement of 2⁺ state in ¹³⁴Ce, which has been presented in Ref. [7], was reanalysed with the new gating region, and the result reproduced the lifetime well within the experimental error.

The lifetime measurement of 2⁺ state in ¹³⁴Ce was performed by Husar et al. as 32.7(28) ps in 1976 [8]. Then in 2016, this lifetime was remeasured by Zhu et al. as 33.8(28) ps [7]. Differentiated form the direct gating case used in Ref. [7], in this paper, the experimental data presented by Zhu et al. was reanalysed with an indirect gate setting on both the Doppler-shifted and unshifted components of the 6⁺ → 4⁺ 814 keV γ-ray transition (see Fig. 3). Partial gating spectra of the 4⁺ → 2⁺ 639 keV and the 2⁺ → 0⁺ 409 keV transitions were posed in Fig. 4. Shifted and unshifted normalized intensities of these two transitions were fit by the NAPATAU program [11]. And the result 32.2(33) ps derived from Eq. (12) was consistent with the previous results in Ref. [8] as well as Ref. [7] within the experimental uncertainty, which means that Eq. (12) can also be used in the indirect gating case. The errors, Δτ, of the present results were calculated as $\Delta \tau =\sqrt{\Delta {\tau }_{\text{st}}^2+\Delta {\tau }_1^2+\Delta {\tau }_2^2}$ where Δτ_st are the statistical errors. The systematic errors, Δτ₁, are estimated uncertainties arising from contaminating lines that might be present but could not be observed directly in the present measurement. These values were estimated on the basis of a careful inspection of the total τ curves. Outside the sensitive range, the effects of contaminating lines show up in a pronounced way. In addition, it was taken into account how many different gates and different detectors were used in the data analysis because this is strongly correlated to the possibility of unobserved contaminations. The error Δ_τ2=2%·τ gives an estimate of systematic errors arising from several small effects for which no corrections were performed (Fig. 5).

Figure 3:Full-screen View

Partial level scheme of ¹³⁴Ce and ¹³⁸Nd. Information is taken from ¹³⁴Ce [9], ¹³⁸Nd [10].

Figure 4:Full-screen View

Left: Projected stopped and backward-shifted components for the 409 keV, 2⁺ → 0⁺ transition in ¹³⁴Ce from gating on the backward-shifted and unshifted components of the 814 keV transition. Right: Corresponding backward-shifted and unshifted components of the 639 keV 4⁺ → 2⁺ transition.

Figure 5:Full-screen View

Decay curves and lifetime determination of the 409 keV, 2⁺ → 0⁺ transition in ¹³⁴Ce. The middle and lower panel show the Doppler-shifted and stopped intensities vs distance. The mean-lifetime at each distance is determined (using Eq. (12) and is shown in the upper panel as function of distance. The weighted average of these values yields the overall mean-lifetime.

The second comparison is peak shapes in gating spectra of the $2_1^{+}\to 0_1^{+}$ transition in ¹³⁸Nd. Due to the neutron damage and poor statistics of forward (42°) detectors, only backward (153°) detectors were used to analyse the lifetime of the $2_1^{+}$ state in ¹³⁸Nd. However, a 723 keV contaminate γ-ray was found in the left side of 729 keV $4_1^{+}\to 2_1^{+}$ transition in the 90° spectrum (see Fig. 6). So the indirect gating case is suitable for the data analysis of the lifetime of the $2_1^{+}$ state in ¹³⁸Nd. In the present work, both gating regions described in Eq. (12) and Eq. (13) were tried. As can be seen in Fig. 8, the wider the gating region selected, the better peak shapes of the Doppler-shifted and unshifted component can be taken in the gating spectra, especially for the spectra of long target-stopper distance. Finally, Eq. (12) was used to fit curves of the shifted and unshifted normalized intensities by the NAPATAU program [11], too. And the resultant coincidence decay curves and τ plot are shown in Fig. 7. From the resulting decay curves, the deduced mean-lifetime of the $2_1^{+}$ state in ¹³⁸Nd is 10.9(12) ps. The result will be used to identify the shape coexistence and evolution in ¹³⁸Nd in the future.

Figure 6:Full-screen View

(a) Total project of 90° in the lifetime measurement experiment of ¹³⁸Nd. The 723 keV γ-ray is a contaminate from ¹³⁵Ce; (b)Background-corrected coincidence spectrum obtained by gating the 723 keV γ-ray at 90°, the 521 keV γ-ray peak can be seen in the gating spectrum clearly.

Figure 8:Full-screen View

Left: Projected stopped and backward-shifted components for the 521 keV, $2_1^{+}$ → $0_1^{+}$ transition and the 729 keV, $4_1^{+}$ → $2_1^{+}$ transition in ¹³⁸Nd from gating on the backward-shifted components of the 884 keV $6_1^{+}$ → $4_1^{+}$ transition. Right: the same as left, but from gating on the backward-shifted and unshifted components of the 884 keV transition.

Figure 7:Full-screen View

The same as Fig. 4, but for the 521 keV, $2_1^{+}$ → $0_1^{+}$ transition in ¹³⁸Nd.

5 SUMMARY

A new development of indirect gating case in the differential decay curve method has been introduced in the present work. The gate region was extended from partial shifted peak to both shifted and unshifted components. This change has been confirmed by reanalysing the lifetime of the 2⁺ state in ¹³⁴Ce in ref. [7]. The result of 32.2(33) ps was fit well with the previous published values [7,8] within the experimental uncertainty. This development was also used to analyse the lifetime of $2_1^{+}$ state in ¹³⁸Nd. The statistics of flight and stop peaks in the gating spectra was improved obviously. The lifetime 10.9(12) ps will be used in further discussion.