Systematic Studies on the β-decay Half-lives of r-process Nuclei

Based on the accurate macroscopic-microscopic mass formula and the experimental data of β-decay half-lives of the nuclei with atomic number ranging from 20 to 190, a systematic formula has been proposed to calculate β-decay half-lives of neutron-rich nuclei. The formula is proved to reproduce the experimental β-decay half-lives of neutron-rich nuclei very well, and then is used to study the r-process nucleosynthesis in models of high-entropy mass outflows. The calculated abundances show a good agreement with the solar r-abundances around the third peak and the rare earth mass region.


Introduction
The rapid neutron-capture process (r-process) of stellar nucleosynthesis is believed to explain the production of about half of the nuclides heavier than iron in the universe [1,2].A large amount of nuclear data, such as mass, decay and reaction rates, are necessary in the studies of r-process.Among them, the β-decay half-lives of a large range of extremely neutron-rich nuclei are one of the most important inputs in the r-process model calculations.However, for the lack of experimental data, the identification of the realistic site and the exact path of the r-process has proved to be very difficult.
In this work, we have systematically investigated the variation of β-decay half-lives with the decay energy Q and nucleon number (Z, N) based on the experimental data.A systematic formula has been proposed to calculate the β-decay half-lives of that neutron-rich nuclei after taking into account the shell effects and pairing effects.The formula is tested for the resent experimental data, and then used to predict the β-decay half-lives of some nuclei far from stability including the ones on the possible r-process path.

Systematic formula for β-decay half-lives
According to Fermi's golden rule [3], the decay constant (or transition probability) of β-decay is defined by To obtain the decay constant, the transition matrix element M i f and phase-space factor f (Z, E m ) should be worked out firstly.f (Z, E m ) is a function of the radius of the nucleus R = r 0 A 1/3 , nuclear charge Z, and the energies of the emitted electron E e .A formula for β-decay half-lives ln(T 1/2 ) should include these three terms: The term (α 2 Z 2 − 5) ln Q is essential for the generalized Sargent law.The atomic masses are the key values to calculate Q-values, thus we select the very recent atomic mass table given by Wang [4] as our input data.The deviations between experimental and calculated Q-values are about 5%.
The nuclear matrix element contains the information of nuclear structure, the shell effects and pairing effects.The pairing effects on β-decay half-lives versus Q-values can be well described by Eq. (2).
The shell effects become strong near the neutron and proton magic-numbers, which can be well characterized by It must be pointed out that a complete correction of shell effects should include all major shell as well as sub-shell closures.However, when taking account of the range of available experimental data, only portions of magic-numbers which have the most remarkable effects are included in Eq. ( 3).Consequently, a systematic formula for β-decay half-lives of neuron-rich nuclei far from stability line can be derived from the relationship between β-decay half-lives and Q-values, nucleon numbers (Z,N), neutron excess (N − Z)/A, with corrections of pairing effects and shell effects.The new formula is expressed as with the parameters a i (i = 1, 2, • • • , 9) determined according to all available experimental data of β-decay half-lives of nuclei far from stability line (δN > 5).The most important term (α 2 Z 2 − 5 − a 7 ) has a vital effect on the calculation of β-decay half-lives.The latter three terms in Eq. ( 4) have relatively small contribution to the total value and have minor fluctuation versus (Z, N).As for the shell correction S (Z, N), it has obvious contribution only near the nucleon magic-numbers but contributes nothing far from nucleon magic-numbers.
Through a least-square fitting to all available experimental data of 350 nuclei far from the β-stability line, we obtain all parameters in Eq. ( 4): a i (i = 1, 2, • • • , 9) = 3.016, 3.879, 1.322, 6.030, 1.669, 11.09, 1.07, -0.935, -5.398, respectively.The average ratio between calculated β-decay half-lives and experimental ones is about 1.69.It reproduces 308 (88%) nuclei of 350 available experimental data within a factor of 3, and 261 (74.6%) nuclei within a factor of 2. To demonstrate the extrapolating capacity of the systematic formula, we have also calculated the β-decay half-lives of the isotopic chains such as Zr, Rb, Sr, Y and compare them with the experimental data [5].The results are shown in Fig. 1.One can see that the agreements are very well.The ratios between calculated β-decay half-lives and experimental ones of these 17 nuclei are within a factor of 1.15.It is interesting to see from Fig. 1 that extrapolations of these isotopic chains have smaller deviations than the other reproduced β-decay half-lives.
The half-lives deduced with the current systematic formula has been used to study the rprocess nucleosynthesis in models of high-entropy mass outflows.The calculated abundances show a good agreement with the solar r-abundances around the third peak and the rare earth mass region.[6] (diamonds), this work (dots), microscopic approaches [7,8] (triangles) and measured ones [5] for Zr, Rb, Sr, Y isotopes.

Summary
Based on the Fermi's theory of β-decay and experimental data, the variation of β-decay halflives with the decay energy Q and nucleon number (Z, N) has been systematically investigated.A empirical formula has been proposed for the calculation of β-decay half-lives of neutron-rich nuclei.The present formula works well in extrapolating the experimentally unknown r-process nuclei and in the model calculations of r-process nucleosynthesis.