On Systematics of Double-Beta Decay Half Lives

. Recommended 2 β − (2 ν ) half-lives and their systematics [1, 2] are examined in the framework of a semi-empirical approach. Impacts of nuclear deformation, transition energy, shape coexistence, and forbidden transitions on half-live values were observed. These ﬁndings were used to predict T 1 / 2 for 36 isotopes of interest. Current results are compared with available data.


Introduction
Double-beta decay was originally introduced by M. Goeppert-Mayer in 1935 [3] as a nuclear disintegration with simultaneous emission of two electrons and two neutrinos: There are several possible double-beta decay processes: 2β − , 2β + , β + , 2 and two major decay modes: two-neutrino (2ν) and neutrinoless (0ν). 2ν-mode is not prohibited by any conservation law and occurs as a second-order process compared to the regular β-decay. 0νmode differs from the 2ν-mode by the fact that no neutrinos are emitted during the decay. This normally requires that the lepton number is not conserved and neutrino should contain a small fraction of massive particles that equals its anti-particle (Majorana neutrino).
Since 1935, thousands of double-beta decay works were published in the literature. Analysis of the Nuclear Science References database [4] contents shows that ≈ 35% of papers include experimental results, and the rest are theoretical. Published theoretical results provide a very extensive range of possible scenarios and observables. As of today, 2β(2ν)-decay is experimentally observed in many nuclei [1,5,6], and it represents an interest to analyze these findings, deduce trends, and constrain theoretical pursuits.

Analysis of Recommended Values
Double-beta decay observations and limits have been compiled by several groups [1,[5][6][7]. These compilations were used to produce adopted or recommended values [1,6]. NNDC recommended numbers represent the best available values, further measurements will result in the addition of * e-mail: pritychenko@bnl.gov new and improvements to existing values. T 2ν 1/2 values are often described as follows [8]: where the function G 2ν (E, Z) results from lepton phase space integration and contains all relevant constants, and From the Eq. 2 one may conclude that decay half-lives depend on transition energy, nuclear charge, and deformation.
It will be useful to analyze the recommended halflives using Grodzins' approach [9]. In this analysis, we will consider only 2β − -decay parameters for 0 + → 0 + ground-state transitions, i.e. transitions without γ-rays and adopt deformation parameters (β 2 ) from Pritychenko et al. [10]. Previously, Primakoff and Rosen [11,12] predicted that for ββ(2ν) decay, the phase space available to the (four) emitted leptons is roughly proportional to the eighth through 11 th power of energy release, and it is known half-lives depend on dimensionless Coulomb energy parameter ξ ≈ ZA −1/3 [13][14][15]. Furthermore, several authors showed the anti-correlation between nuclear deformation [16] and the two-neutrino mode nuclear transition matrix element [17][18][19]. This will modify our result for 2β-transition as follows: where T 2ν 1/2 in years, E in MeV and a1, a2 and a3 are fitting parameters.
Next, we would use the method of least squares and the evaluated half-lives [1] to deduce fitting parameters. The fitting procedure parameters are given in Table 1. The least-squares fit data confirm theoretical findings that halflives are inversely proportional to transition energy and quadrupole deformation parameter values in eight and fifth degrees [12,17], respectively. In addition, the observed transition energy dependence upholds our previous result on 128,130 Te half-life systematic [1,2].

Semi-Empirical Approach Predictions
Eq. 3 allows us to deduce half-life times for all nuclei of interest as shown in Table 2. The comparison of the presently calculated and previously evaluated half-lives is shown in Fig. 1. Further analysis of the calculated to evaluated values ratio reveals the three cases of the current fit under prediction: 76 Ge, 82 Se, and 150 Nd, one case of overprediction: 96 Zr, and small deviations for 128 Te. The analysis of nuclear properties of even-even atomic nuclei reveals that in 76 Ge → 76 Se, 82 S e → 82 Kr, and 150 Nd → 150 Sm transitions the intermediate nuclei ( 76 As, 82 Br, and 150 Pm) ground-state parity is negative while in all other cases it is positive. This implies that these transitions are more complex than the Fermi/Gamow-Teller allowed and superallowed transitions where the selection rules for beta decay involve no parity change of the nuclear state (The spin of the parent nucleus can either remain unchanged or change by ±1), and first forbidden transitions could be present. Empirical evidence shows that adoption of a reduced deformation parameter β 2 (reduced) ≈ 2 3 β 2 resolves the predicted values deviations.
The small 96 Zr deformation parameter of 0.0615(33) [10] is responsible very large predicted half-life. It is much lower than the semi-magic 90 Zr parameter of 0.0907 (24). Both values have been deduced from the quadrupole collectivities of the first 2 + 1 states in zirconium nuclei. The recent survey of the first excited states spin and parity assignments in even-even nuclei shows that 2 + 1 is observed in 96% of cases [21], and 0 + 2 , 1 − 1 , and 3 − 1 assignments are also present. The 0 + 2 first excited state is observed in 96 Zr, and it provides a clear indication of a shape coexistence when the deformed 0 + 2 and spherical 2 + 1 excited states coexist in the same nucleus. The current model requires an increased deformation parameter β 2 (increased) ≈ 5 2 β 2 for spherical 2 + 1 states to reproduce the recommended half-life. The 0 + 1 first excited states are observed in 3% of even-even nuclei, and the impact of this correction 2ν-mode of 2β-decay is limited to 96 Zr and 98 Mo.
The small deviation for 128 Te is since that the tellurium values were not directly measured, they have rather deduced from the 130 Te half-lives and the observed ratios of daughter products. Thus, the earlier low-precision measurements impacted the recommended 128 Te half live. The recommended value could be corrected with adoption of the ratio from experiments with high statistics and recalculation of individual 128 Te half-lives. Recently, the application of the latest nuclear data values helped to resolve discrepancies in older fission yields and nuclear reaction cross sections [23,24], and it should help to clarify the tellurium data.
The discussed above examples explain the issues with the predicted half-life least-squares fit and show the impact of nuclear structure effects on their values. A similar task was recently attempted by a Nanjing-Lanzhou group using the Geiger-Nuttall law and the Viola-Seaborg formula arguments [25]. These authors assumed that the logarithm of half-life is inversely proportional to transition energy as a starting point and deduced fitting parameters to reproduce the Institute of Theoretical and Experimental Physics (ITEP), Moscow values [26]. Heavy reliance on αand βdecay fitting procedures by this group and the addition of a S = 2 parameter for neutron number magic nuclei to explain nuclear structure effects are not sufficient to explain the large variety of nuclear structure effects in double-beta decay nuclei and provide reliable predictions.

Conclusion
The two-neutrino mode of 2β − -decay has been observed by various groups in 11 even-even nuclei [22], and several sets of recommended half-lives were deduced [1,6]. 128,130 Te data analysis led to observation of 1 E 8 energy trend for T 2ν 1/2 recommended values [1]. The inclusion of quadrupole deformation parameters, for simulation of the nuclear structure effects in predicted half-lives, revealed the impact of first-forbidden transitions and shape coexistence in several nuclei. The current work energy and quadrupole deformation parameter trends were used to explain the nuclear systematics of recommended values and calculate 2β − (2ν)-decay half-lives for all nuclei of interest.

Acknowledgments
The author is indebted to Vladimir Tretyak (Kyiv Institute for Nuclear Research), Balraj Singh (McMaster University) for useful discussions, and the NSDD network members for their tireless work on the ENSDF library evaluations. Work at Brookhaven was funded by the Office of Nuclear Physics, Office of Science of the U.S. Department of Energy, under Contract No. DE-SC0012704 with Brookhaven Science Associates, LLC. Table 2. Empirical Rule Predictions: 2β − (2ν)-decay predicted, recommended and experimental values for 0 + → 0 + transitions. Input data were taken from [1,10,20,22]. Nuclei and half-lives with probable contributions of intermediate nucleus negative parity states, shape coexistence, and 0ν decay mode are marked with †, ‡ and * characters, respectively. Predicted half-lives were not corrected for negative parity and shape coexistence effects.