Electromagnetic transition rates in the nucleus ¹³⁶₅₈Ce

The experiment was performed at the ALTO Tandem Accelerator Facility in Orsay, France. The states of interest in ¹³⁶Ce were populated via the ¹²⁸Te(¹²C,4nγ) fusion-evaporation reaction at a beam energy of 60 MeV and 5 pnA current on target. The target was a 95% isotopically enriched ¹²⁸Te foil of 2.6 mg cm⁻² thickness on a 20 mg cm⁻² Pb backing. The estimated production cross section of 550 mb for ¹³⁶Ce was calculated using the PACE4 code.

Our experiment was part of the first ν-ball campaign, for which the ν-ball Detector System [17] was constructed. The ν-ball spectrometer is a hybrid array combining high-purity co-axial Germanium detectors (HPGe), clover detectors and lanthanum bromide LaBr₃(Ce) scintillator detectors. This coupling allows us to combine the excellent energy resolution of the Germanium detectors together with the excellent time resolution of the LaBr₃(Ce) detectors. The achieved total photopeak efficiency at 1.3 MeV was 6.7%, and the peak-to-total ratio for the Germanium part of the array was 50%. The clover Germanium detectors were placed in two rings of twelve detectors at 75.50^o and 104.50^o with respect to the beam axis. The LaBr₃(Ce) detectors were placed in two rings of ten, at 46.5^o and 34^o with respect to beam axis and at 11 cm and 15 cm away from the target, respectively. These detectors were previously used in the FATIMA spectrometer [15] and were operated with PMTs (photomultiplier tubes) set to an anode output of 1 V/MeV, which corresponds to approximately 1400 V (PMT supply voltage). As shown in figure 3 of [16], the energy response of the FATIMA LaBr3(Ce) detectors has been found to be linear for energies below 3 MeV. For higher energies the unfolding non-linearity is relatively weak.

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Using the digital acquisition system FASTER [17], the achieved time resolution was about 250 ps for LaBr₃(Ce). That system makes the ν-ball a fully digital large fast-timing spectrometer with time resolution similar to analogue electronics.

The aim of the experiment was to measure the unknown lifetime of the ${6}_{1}^{+}$ state in ¹³⁶Ce. The expected value for the reduced transition probability between the first excited 6⁺ and 4⁺ states in ¹³⁶Ce is $B(E2;{6}_{1}^{+}\to {4}_{1}^{+})\,\gt \,$ 0.0046 W.u., which corresponds to T_1/2 < 5 ns. Assuming that the $B(E2;{6}_{1}^{+}\to {4}_{1}^{+})$ in ¹³⁶Ce is 1 W.u., the value will be between the known from the literature values for ¹³⁸Ce and ¹³⁴Ce [1, 6]. This corresponds to a lifetime of 27 ps for the ${6}_{1}^{+}$ state of ¹³⁶Ce. Lifetimes in the range between 5 ns and 27 ps are well within reach of the fast-timing technique.

Lifetimes of interest were extracted using the Generalized Centroid Difference (GCD) method [18]. This method uses two distributions (delayed and antidelayed) of the time difference ΔT between two successive γ rays, the feeding and decay transitions of the level of interest. The delayed time spectra distribution labeled as (D) is obtained when a transition which feeds the state provides the start signal and a decay transition from this state provides the stop signal for the ΔT. In the reverse case, the resulting ΔT distribution is labeled as antidelayed (AD). Swapping the start and stop detectors is equivalent to swapping the sequence of the feeder and decay transitions of the cascade and gives mirror-symmetric distributions. The mean lifetime of the state of interest can be determined from the difference between the centroids of these time spectra C_D and C_AD :

$$\begin{eqnarray}&&{\rm{\Delta }}C\equiv {C}_{D}-{C}_{{AD}}\equiv 2\tau +{PRD}.\end{eqnarray} \tag{ 1 }$$

In this equation, PRD (Prompt Response Difference) describes the mean time walk characteristics of the experimental setup [19]. A ¹⁵²Eu calibration source, which covers the energy range starting from 121keV up to 1408keV, was used to obtain the PRD curve. Both time spectra—delayed and antidelayed were produced by selecting several feeder-decay combinations for states with well-known lifetimes from the calibration source. By measuring the centroid differences ΔC and using equation (1), the PRD is obtained. The data points are then fitted using the function [16, 20]:

$$\begin{eqnarray}&&{PRD}({E}_{\gamma })=\displaystyle \frac{a}{\sqrt{{E}_{\gamma }^{2}+b}}+{{cE}}_{\gamma }+d\end{eqnarray} \tag{ 2 }$$

as it is shown in figure 3. The precision of the PRD fit is defined as two times the standard root-mean-squared deviation.

The lifetimes of interest were extracted by analyzing triple-γ coincidences. The coincidences were formed by using γ-γ data in LaBr₃(Ce) detectors which are in an additional coincidence with a chosen energy registered in the HPGe detectors. The latter coincidence condition improves the peak-to-background ratio in the LaBr₃(Ce) detectors and reduces the influence of possible contaminants. A schematic representation of the levels of interest in ¹³⁶Ce is shown in figure 4.

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To determine the lifetime of the ${6}_{1}^{+}$ state we set a cleaning gate on the γ-ray transitions directly populating the ${8}_{1}^{+}$ state (see figure 4) and measured the centroid difference between the delayed and antidelayed time distributions obtained with the 775 keV–900 keV coincidence in LaBr₃(Ce) detectors. The time-difference spectra obtained are shown in figure 5 together with the corresponding double gated LaBr₃(Ce) and HPGe projections. On panel (a) are shown projections of the lanthanum bromide scintillator and high purity germanium detectors energy spectra, generated after applying double HPGe-LaBr₃(Ce) gates on the triple gamma HPGe-LaBr₃(Ce)-LaBr₃(Ce) and HPGe-HPGe-LaBr₃(Ce) matrices. The gate on LaBr₃(Ce) is set on the transition populating the level of interest. The scaling factor between spectra from germanium detectors and the generated LaBr₃(Ce) spectra is 3.3. The blue lines show the gate width used for the depopulating transition, set on the LaBr₃(Ce) to produce the time difference spectra. The peak to background ratio for the depopulating transition is equal to 2.7(2). The green lines indicate the regions where the gates used to extract the centroid difference of the background were placed which is a necessary correction step [22] before using equation (1). The red line represents the linear fit of the background centroid difference below the full energy peak of the decay transition (ΔC${}_{d}^{{BG}}$). The PRD curve of the experimental setup is shown in blue. The green points correspond to the experimentally obtained centroid difference. On panel (b) is shown the time difference distributions (delayed and antidelayed) for the 775–900keV pair transitions used to extract the lifetime of the ${6}_{1}^{+}$ state. The panel (c) is the same as panel (a) with the LaBr₃(Ce) gate set on the depopulating transition, with the red line used to extract ΔC${}_{f}^{{BG}}$ for the feeder determined in analogues way as ΔC${}_{d}^{{BG}}$. The scaling factor for the generated LaBr₃(Ce) spectra is 3.1 and the peak to background ratio value is 3.9(3).

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The determined centroid difference between the delayed and antidelayed distributions is equal to ΔC_exp =164(31) ps. The centroid difference ΔC_FEP corrected for background is defined by [23, 24]:

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{{FEP}}\equiv {\rm{\Delta }}{C}_{{\exp }}+{\tilde{t}}_{{cor}},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{\tilde{t}}_{{cor}}\equiv \tfrac{{t}_{{cor}}({E}_{d}){\left(p/b\right)}_{f}+{t}_{{cor}}({E}_{f}){\left(p/b\right)}_{d}}{{\left(p/b\right)}_{f}+{\left(p/b\right)}_{d}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{t}_{{cor}}({E}_{f,d})\equiv \tfrac{{\rm{\Delta }}{C}_{{\exp }}-{\rm{\Delta }}{C}_{f,d}^{{BG}}}{{\left(p/b\right)}_{f,d}}.\end{eqnarray} \tag{ 5 }$$

The calculated centroid difference ΔC_FEP is 110(32) ps and equation (1) becomes [25]

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{{FEP}}=2\tau +{PRD}.\end{eqnarray} \tag{ 6 }$$

The final value for the lifetime of the ${6}_{1}^{+}$ state in ¹³⁶Ce is determined to be $\tau ({6}_{1}^{+})$ = 60(17) ps (T_1/2 = 42(12) ps), with the corresponding $B(E2;{6}_{1}^{+}\to {4}_{1}^{+})=0.23(9)\times {10}^{-2}{e}^{2}{b}^{2}$ (0.55(21) W.u.).

Additionally, we measured the well-known lifetimes for the yrast ${5}_{1}^{-}$ and ${7}_{1}^{-}$ states in ¹³⁶Ce [26, 27] (see table 1). In the case of ${7}_{1}^{-}$ the experimental half-life of T_1/2 = 306(20) ps was determined by taking the time difference between the 971 keV (9⁻ → 7⁻) and 329 keV (7⁻ → 5⁻) transitions in the LaBr₃(Ce) detectors with a cleaning gate on the 806 keV (11⁻ → 9⁻) transition detected in a HPGe detector (see figure 6). In figure 7 is shown the time distribution spectrum of the I^π = 5⁻ yrast state. The resulting half-life value of T_1/2=493(18) ps was determined by using the time difference spectra between the 446 keV (6⁻ → 5⁻) and 664 keV (5⁻ → 4⁺) transitions in the LaBr3(Ce) detectors with a cleaning gate on the 552 keV (4⁺ → 2⁺) transition and partial exponetial fit (see figure 7). The agreement with literature of the lifetime values of these two low-lying ${5}_{1}^{-}$ and ${7}_{1}^{-}$ states serves as validation for the obtained value of $\tau ({6}_{1}^{+})$ = 60(17) ps in ¹³⁶Ce.

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Table 1. The T_1/2 values of the excited states in ¹³⁶Ce measured in the current study.

Elev	Iπ	Eγ	T1/2	T1/2(lit.) a	λ L	B(λ L)exp	B(λ L)IBM1
[keV]		[keV]	[ps]	[ps]		[W. u]	[W. u]
1978	${5}_{1}^{-}$	644	493(18)	496(23)	E1	0.193(1)×10−5
2214	${6}_{1}^{+}$	900	42(12)		E2	0.55(21)	54.72
2307	${7}_{1}^{-}$	329	306(20)	270(24)	E2	11.1(7)
2366	${6}_{2}^{+}$	1052	27(8)		E2	0.39(12)

^a Taken from [27].

In the present experiment the lifetime of the ${6}_{2}^{+}$ state in ¹³⁶Ce was also measured (figure 8). The half-life of T_1/2 = 27(8) ps was determined by taking the time difference between the 1052 keV $({6}_{2}^{+}\to {4}_{1}^{+})$ and 623 keV $({8}_{1}^{+}\to {6}_{2}^{+})$ transitions in the LaBr₃(Ce) detectors with a cleaning gate on the 106 keV $({10}_{1}^{+}\to {8}_{1}^{+})$ transition. The T_1/2 values for all measured states in ¹³⁶Ce in the current study are summarized in table 1.

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The values for reduced transition probabilities from the theoretical calculations using the IBM1 model for Cerium isotopes are taken from [31]. The calculated values for the energy levels of the excited states generally match the experimental data below the 10⁺ state well, with a larger deviation observed for the heavier Cerium isotopes with mass numbers A = 136 and A = 138. However, the model only reasonably well describes the reduced transition probabilities for the lighter Cerium isotopes (figure 9). Beyond ¹³⁰Ce, the theoretical values significantly surpass the experimentally measured values. For ¹³⁶Ce and ¹³⁸Ce, the discrepancy reaches 2-3 orders of magnitude.

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Similar calculations using the IBM1 model have been carried out for the isotonic Barium chain [32]. The energy levels of the excited states from the yrast band for the lighter isotopes with mass number less than A = 132 are well reproduced. The experimental values for the reduced transition probability $B(E2;{6}_{1}^{+}\to {4}_{1}^{+})$ are also well described. Unfortunately, experimental information is lacking for the isotopes ¹³²Ba and ¹³⁴Ba, which does not allow us to compare the behavior of this isotonic chain with that for Cerium. However, experimental information on the reduced transition probabilities for the heavier ¹³⁶Ba and ¹³⁸Ba is available and we observe a discrepancy with the model predictions of 2-3 orders of magnitude, as in the case of the heavy Cerium isotopes.

From the comparisons made, it can be observed that for the light isotopes in the Barium and Cerium isotopic chains the IBM1 model reproduces the reduced transition probabilities and the energy levels reasonably well. However, it completely fails in describing the behavior of the heavier isotopes.

The successful application of the Shell model strongly depends on the total size of the configuration space, which for nuclei with mass numbers around 130 exceeds 10¹¹, far surpassing the number of configurations that can be handled with the available computing power.

Calculations for Cerium isotopes are notably challenging due to their distance from the closed proton and neutron shells, and theoretical data are only available for the isotopes ¹³⁸Ce and ¹⁴⁰Ce [1] for which the NUSHELLX@MSU code [33] was used. Similarly to the IBM1 model, the Shell model reproduces the energies of the low lying levels of the yrast band below the 10⁺ state. However, the discrepancies between calculated and experimental values for the reduced transition probabilities between the excited ${6}_{1}^{+}$ state and the ${4}_{1}^{+}$ state still persist, although they are an order of magnitude better than those obtained with the IBM1 model calculations.

Shell model calculation for the isotope of interest ¹³⁶Ce could not be carried out due to lack of available computing power.

Barium isotopes contain two protons fewer than Cerium isotopes, which significantly reduces the degrees of freedom and allows calculations using the Shell model. Calculations carried out for Barium isotopes with mass numbers between 132 and 138 [30] show good agreement for the values of the energies of the excited levels from the yrast band below the 8⁺ states. In figure 10, it is clearly seen that for ^136,138Ba the calculated values for the reduced transition probabilities follow the behavior of the experimental ones but are strongly overestimated. Unfortunately, there is a lack of experimental data that would allow for comparison of the behavior of the reduced transition probabilities at ¹³²Ba and ¹³⁴Ba.

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Based on the results from the two models, new experimental data for ¹³²Ba and ¹³⁴Ba would definitely allow for the models to fully explore the evolution of the reduced transition probabilities for the entire Barium and Cerium isotonic chains.