Advances in High Energy Physics

Volume 2019, Article ID 5783618, 16 pages

https://doi.org/10.1155/2019/5783618

## Study of the Effect of Newly Calculated Phase Space Factor on *β*-Decay Half-Lives

^{1}GIK Institute of Engineering Sciences and Technology, Topi 23640, Khyber Pakhtunkhwa, Pakistan^{2}University of Bucharest, Faculty of Physics, P.O. Box MG11, 077125 Magurele, Romania^{3}National Institute of Physics and Nuclear Engineering, P.O. Box MG6, 077125 Magurele, Romania^{4}International Centre for Advanced Training and Research in Physics, P.O. Box MG12, 077125 Magurele, Romania

Correspondence should be addressed to Sabin Stoica; or.enpin.yroeht@aciots

Received 14 December 2018; Revised 16 February 2019; Accepted 27 February 2019; Published 26 March 2019

Academic Editor: Roelof Bijker

Copyright © 2019 Mavra Ishfaq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present results for -decay half-lives based on a new recipe for calculation of phase space factors recently introduced. Our study includes -shell and heavier nuclei of experimental and astrophysical interests. The investigation of the kinematics of some -decay half-lives is presented, and new phase space factor values are compared with those obtained with previous theoretical approximations. Accurate calculation of nuclear matrix elements is a prerequisite for reliable computation of -decay half-lives and is not the subject of this paper. This paper explores if improvements in calculating the -decay half-lives can be obtained when using a given set of nuclear matrix elements and employing the new values of the phase space factors. Although the largest uncertainty in half-lives computations come from the nuclear matrix elements, introduction of the new values of the phase space factors may improve the comparison with experiment. The new half-lives are systematically larger than previous calculations and may have interesting consequences for calculation of stellar rates.

#### 1. Introduction

The precise knowledge of the -decay rates represents an important ingredient for understanding the nuclear structure as well as the astrophysical processes like presupernova evolution of massive stars, nucleosynthesis (s-, p-, r-, rp-) processes, etc. [1–3]). That is why the calculation of the -decay half-lives in agreement with experimental results has been a challenging problem for nuclear theorists [4–8]. Theoretically, the half-life formulas for -decay can be expressed as a product of nuclear matrix elements (NMEs), involving the nuclear structure of the decaying parent and of the daughter nuclei, and the phase space factors (PSFs) that take into account the distortion of the electron wave function by the nuclear Coulomb field. Hence, for a precise calculation of the -decay half-lives, an accurate computation of both these quantities is needed. The largest uncertainties come from the NME computation. In literature one can find different calculations of the NMEs for -decay, realized for different types of transitions and final states, and with different theoretical models (e.g., based on gross theory [9], QRPA approaches [2, 5–8, 10–14], and shell model [15]). We would not be discussing calculation of NMEs in this paper. Until recently the PSFs were considered to be calculated with enough precision and, consequently, not much attention was paid to a more rigorous calculation of them. However, recently we recomputed the PSFs for positron decay and electron capture (EC) processes for 28 nuclei of astrophysical interest, using a numerical approach [16]. We solved the Dirac equation (getting exact electron wave functions) with a nuclear potential derived from a realistic proton density distribution in the nucleus. We also included the screening effects. The new recipe for calculation can easily be extended to any arbitrarily heavy nuclei.

Accurate estimates of half-lives of neutron-rich nuclei have gained much interest in the recent past. This is primarily because of their key role in -process nucleosynthesis. Similarly, precise value of -decay half-lives of proton-rich nuclei is a prerequisite for solving many astrophysical problems. In this paper, we study the effect of introducing the new PSF values, obtained with our recently introduced recipe [16] on the calculation of -decay half-lives. We also extend here our previous PSF calculations (of positron decay and EC reactions) to include -decay reactions. In order to complete the calculation of -decay half-lives, we calculate the set of NMEs using the proton-neutron quasi-particle random phase approximation model in deformed basis and a schematic separable potential both in particle-particle and particle-hole channels. Other nuclear models and a set of improved input parameters may result in a better calculation of NMEs. However this improvement is not under the scope of current paper. We calculate both Gamow-Teller and Fermi transitions to ground and excited states, for medium and heavy nuclei of interest. We present first an investigation of the kinematics of -decay half-lives and our PSF values are compared with those obtained with previous theoretical approximations. Later the newly computed half-lives are compared with other previous theoretical predictions and experimental data. We investigate if our new PSF values lead to any improvement in the calculated -decay half-life values. Our present study may be extended to investigate the effect of the new PSF values on stellar decay rates, which we take as a future assignment.

This paper is organized in the following format. Section 2 describes the essential formalism for the calculation of PSFs and -decay half-lives. We present our results in Section 3 where we also make a comparison of the current calculations with experimental data and previous calculation [17]. We conclude finally in Section 4.

#### 2. Formalism

##### 2.1. Half-Life Calculation

-decay half-lives can be calculated as a sum over all transition probabilities to the daughter nucleus states through excitation energies lying within the valuewhere the partial half-lives (PHL), , can be calculated using

In (2) value of C was taken as 6143 s [18] and , are axial-vector and vector coupling constants of the weak interaction, respectively, having /= -1.2694 [19], while is the final state energy. where is the window accessible to either -, -, or EC decay. are the PSFs. and are the reduced transition probabilities for Gamow-Teller and Fermi transitions, respectively, and expressed as

In (3) and (4), denotes the spin of the parent state, and are the Fermi and Gamow-Teller transition operators, respectively. Detailed calculation of the NMEs within the proton-neutron quasi-particle random phase approximation (pn-QRPA) formalism may be found in [7, 8].

In this paper the NMEs calculation was performed using the pn-QRPA model. We used the Nilsson model [20] to calculate single particle energies and wave functions which takes into account the nuclear deformation. Pairing correlations were tackled using the BCS approach. We considered proton-neutron residual interaction in two channels, namely, the particle-particle and the particle-hole interactions. Separable forms were chosen for these interactions and were characterized by interaction constants for particle-particle and for particle-hole interactions. Here, we used the same range for and as was discussed in [7, 8]. Deformation parameter values for all cases were taken from Ref. [21]. For pairing gaps we used a global approach = = 12/ [MeV]. A large model space up to 7 was incorporated in our model to perform half-lives calculations for heavy nuclei considered in this paper.

##### 2.2. Phase Space Factors Calculation

###### 2.2.1. Phase Space Factors for Transitions

The formalism for the PSF calculation for allowed transitions was discussed in detail in our previous paper [16]. Here, we reproduce the main features of the formalism for the sake of completion. The probability per unit time that a nucleus with atomic mass A and charge Z decays for an allowed -branch is given bywhere is the weak interaction coupling constant, is the momentum of -particle, = is the total energy of -particle, and is the maximum -particle energy. = () in () decay. is the mass difference between initial and final states of neutral atoms. Equation (5) is written in natural units (), so that the unit of momentum is , the unit of energy is , and the unit of time is /. The shape factors for allowed transitions which appear in (5) are defined aswhere are the NMEs related to the Fermi and Gamow-Teller reduced transition probabilities asand stands for Fermi functions. For the calculation of the -decay rates, one needs to calculate the NMEs and the PSFs that can be defined as

The above formula determines the PSFs for both the Fermi and Gamow-Teller allowed transitions, by substituting or in (2), respectively. For the allowed -decay the Fermi functions can be expressed aswhich is just the definition used in [17] (see (3)), for our particular case . We note that in the above formula a coefficient appears. Usually, this coefficient is included in the proper normalization of the wave functions, as we did. The functions and are the large and small radial components of the positron (or electron) radial wave functions evaluated at the nuclear radius . They are solutions of the coupled set of differential equations [17]where is the central potential for the positron (or the electron) and is the relativistic quantum number.

Ideally, the central potential from (10) should include the effects of the extended nuclear charge distribution and of the screening by orbital electrons. Unlike the recipe of Gove and Martin [17], where these screening effects were treated as corrections to the wave functions, in our recipe they are included directly in the potential. This was done by deriving the potential from a realistic proton density distribution in the nucleus. The charge density can be written aswhere is the proton wave function of the spherical single particle state and is its occupation amplitude. The wave functions were found by solving Schrödinger equation with a Wood-Saxon potential. The term in (11) reflects the spin degeneracy. As an example, we depict the realistic proton density for in cylindrical coordinates in Figure 1(a). The profile of this proton density for the daughter nucleus (thick line) is compared with a constant density (dot-dashed line) in Figure 1(b).