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).
(a)
(b)
We integrated the realistic charge distribution over the volume of nucleus, in order to find the Coulomb potential
Moreover, we included the screening effect by multiplying the expression of with a function , which is solution of the Thomas-Fermi equationwith , , and being the Bohr radius. The solution was calculated within the Majorana method [22]. In the case of , the effective potential was modified by the screening function asSo, asymptotically we returned the interaction between an ion of charge of the residual nucleus and the emitted electron/proton. The Coulomb potential in atomic units is and is negative/positive for decay. As mentioned in [22], the Thomas-Fermi equation (13) is a universal equation which does not depend on or other physical constants. The boundaries of the screening function are and =0. Then, the effective potential is for =0 and is suppressed according to the variation of the universal screening function when increases in order to reach the asymptotic behavior for . The effective screening function is displayed in Figure 2. Such a procedure of including the screening effect was also used previously in the computation of PSFs for double-beta decay [25–27] and a similar behavior for the effective screening function was obtained. Essentially, effective Coulomb interactions obtained in the present work manifest the same asymptotic behavior as those obtained in [25, 26] where the Thomas-Fermi equation with a fixed boundary at infinity is solved.
We solved (10) in a screened Coulomb potential, with an accurate numerical method presented in [28, 29]. The method allows control of truncation errors of the solutions and the only remaining uncertainties were due to unavoidable round-off errors and due to the distortion of the potential introduced by the interpolating spline. Detailed information about this method can be found in our previous article [16]. Further, the integration of the PSF values was performed accurately with Gauss-Legendre quadrature in 32 points.
Because we compare our results with those of Gove and Martin [17] we mention selected features of their method of calculation of the PSFs. They obtained the radial electron functions (, ) as solutions of Dirac equations for a point-nucleus and with an unscreened Coulomb spherical potential. The obtained functions were approximate, expressed in terms of functions. According to their prescriptions [17], the finite nuclear size and screening effects were treated as corrections to these approximate functions.
In order to illustrate the differences that appear between our calculation and the approximate method of [17], we compare the PSF results for three virtual cases as discussed in [17]. As seen from Table 1 for and Table 2 for , the differences between the two sets of PSF values may not be so obvious in the decimal logarithm scale, but in half-lives calculation where the absolute PSF values are used, the differences may be relevant as we will see in the next section.
2.2.2. Phase Space Factors for Electron Capture (EC)
Electron capture is a process which competes with positron decay. It is an alternate decay mode for the unstable nuclei that do not have enough energy to decay by positron emission. Considering the fact that the electron capture from the M-, N-, and higher shells has negligible contributions in comparison with the K- and L- ones, we can write the PSF expression of electron capture for an allowed transition as
For the quantities we used the expressionwhere is the Q-value of the decay in units, is the binding energies of the and electron orbitals of the parent nucleus, and is their radial densities on the nuclear surface. represent the values of the exchange correction. In our method we consider these exchange corrections to be unity, for the nuclei considered, the estimated error in doing that being under 1%. The relation holds.
are the electron bound states and solutions of the Dirac equation (10) and correspond to the eigenvalues ( is the radial quantum number). The quantum number is related to the total angular momentum . These wave functions are normalized such that
For the processes, the potential used to obtain the electron wave functions readsand the charge number corresponds to the parent nucleus. is negative. More details about the numerical procedure can be found in [16].
3. Results and Discussion
Half-lives were computed using (1) and (2). The NMEs were calculated using (3) (for Fermi transitions) and (4) (for GT transitions) within the pn-QRPA formalism. For the PSF calculation we used two different recipes. One is our newly calculation recipe [16] and the other one is the conventional computation using the prescription of [17]. We again state that the same set of NMEs was used in both types of half-life calculations. For the bound states, the Dirac equation is solved by assuming a Hartree self-consistent Coulomb field. The finite size effect is introduced within a nuclear charge distribution of a Woods-Saxon form. In the model, the screening is implicitly taken into account. In the calculations of this work, the Coulomb field is obtained by considering a nuclear charge distribution obtained within a shell model and a screening effect is introduced by modifying the potential. Therefore, some differences in the electron binding energies and in the wave functions arise between the two recipes. The electron energies and their radial densities on the nuclear surface for and orbitals recalculated in this work with recently improved numerical codes following the recipe of [16] are listed in Table 3 and compared with those of [30].
The differences between the results obtained for a Coulomb field obtained from a constant charge density inside a spherical nucleus and those obtained with a field constructed with a more realistic charge density corrected with a screening function can offer an estimation of the role played by the ingredients of our model. The radial distributions of the electron are displayed in Figure 3 for these two treatments in the case the parent nucleus . The radial distributions in the case of a pure Coulomb field, plotted with thin lines, are more confined in the vicinity of the nucleus; therefore the amplitudes of the wave function are larger. These differences are translated in the values and of the electron binding energies , as it can be seen in Table 4. Adding the screening effect in our calculations makes our results very close to those of .
Table 5 presents a comparison between the measured and calculated half-lives for /EC decay of twenty medium and heavy nuclei of interest. Entries in third column are calculated using the pn-QRPA method for the NMEs, while the PSFs are calculated by the by method [17]. The fourth column shows the calculated half-lives using our new recipe of PSFs [16] and labeled (this work). Most of the nuclei shown in this table are the same as those presented in Table of [16]. All half-lives are given in units of seconds. Q-values for the reaction were taken from [23, 24]. It is seen from Table 5 that the newly calculated half-lives are systematically larger than those computed using the PSFs of [17]. The last column displays the percentage deviation (PD) of the two calculated half-lives. We calculated the PD between the two computed half-lives using the formula
Table 5 shows that the PD increases to a maximum value of 4.05 % for the nucleus. The case of EC on requires special mention. For this nucleus the pn-QRPA predicts couple of high excited transitions to the daughter nucleus and the available Q-value of these two transitions is lower than the binding energy of the K-shell electron. Using calculation, after the EC process of a K-shell electron, the neutrino may have a negative energy which is not physical. Accordingly we only calculate L-shell EC in this case. In the calculated /EC decay half-lives, the smallest difference was noted for . In Table 6 we show the state-by-state transitions for two cases: and . Shown are also the adopted NMEs using the pn-QRPA model, the calculated PSFs (separately for both EC and -decay reactions), partial half-lives (PHL), Q-values, and branching ratios . The branching ratio ‘’ for each transition was calculated using the formulawhere is the total decay half-life and is the calculated partial half-life of the corresponding transition. For the nucleus we note calculation of much smaller PSF for EC decay to daughter energies using our recipe. The PSFs calculated from recipe is on average within 3 % smaller. This in turn led to a 4 % larger calculated half-life value for using our recipe.
We would like to comment further on the entries in the first two columns of Table 6. These entries are model dependent. The excited states in daughter nuclei (shown in the first column) and NMEs (presented in the second column) were calculated using the pn-QRPA model. The computed excited states satisfied the selection criteria for allowed transitions within the chosen model. A different nuclear model can change the entries in the first two columns and, as stated earlier, is not the focus of current study. Q-values are presented in the third and the sixth columns of Table 6 using following relation:andHere and are masses of parent and daughter nuclei, respectively, whereas is the calculated energy levels in the daughter nucleus (model dependent).
Table 7 shows the comparison of measured and calculated half-lives for decay cases. The Q-values were taken from [23, 24]. The comparison between the two calculations is much better for the decay half-lives than those corresponding to . Table 8 shows the state-by-state calculation of PHL for (largest PD=3.82 %) and (PD=2.04 %). The values of (calculated as in (21)), PSF, NME, and branching ratios are also given in Table 8. We note an overall agreement between our calculated PSF values and those using the recipe in the case of both analyzed nuclei.
The difference between the two calculated half-lives as well as their mutual comparison with the experimental data is done intuitively in a graphical way in Figure 4 for few selected decay cases. We display the ratio between the experimental half-lives and the theoretical ones, , where stands for the calculation recipe, or . With solid lines the ratios calculated with recipe are represented, while with dotted lines the conventional computations are displayed. We note that systematically our half-lives are larger than the , improving the agreement with the measured data for most of the cases. From Figure 4 it can be remarked that the ratios are in general closer to the value 1 than the ones. This effect is highlighted in Figure 4 by the link between the dotted line and the solid line. In Figure 5, the ratios corresponding to decay are displayed in the same manner as in Figure 4. It is noted that no appreciable improvement is brought in calculation of decay half-lives except for a few cases in which the experimental data are undervalued by calculations, as also evident from Table 7. Overall, we note a good agreement of the new theoretical half-lives with the experimental ones. It is again remarked that this comparison could have improved further with a more reliable set of NMEs or choice of better model parameters for the calculation of NMEs (not the subject of current paper).
The differences between the and the present results can be explained by the use of a more rigorous approach in our case for the free states in the PSF computation but also are due to the differences between our potential and the one used by . Regarding only the free states, in the method the screening correction was introduced empirically, by modifying the solutions of the Dirac equation with a function evaluated at the nuclear surface. This function depends on the difference between the effective potential and the point like nucleus Coulomb interaction. On the other hand, in our calculation the screening is introduced by considering an effective Coulomb potential. The Dirac equation is solved numerically for this effective Coulomb potential up to large values of , where the wave functions are well approximated by its asymptotic form. Later, the wave functions are normalized by comparing their value with the asymptotic forms. This normalization determines the value of each wave function on the surface of the nucleus. Further, in the calculation, the nuclear finite size of the nucleus is simulated by additional corrections to the Fermi functions, while in our calculation, the effective Coulomb field is built up from the proton density of the nucleus, as also mentioned before. Regarding the bound states, the method uses tabulated values of the energies and of the radial densities that are obtained by solving the Dirac equation within a more sophisticated self-consistent Coulomb potential.
4. Summary and Conclusion
The aim of present work was to investigate the effect of the incorporation of new PSF values, computed with a more precise and rigorous method, on the theoretical half-lives for and EC decay of unstable nuclei. The newly calculated -decay half-lives were systematically larger than those given in the previous calculations. The mean percentage deviation is larger for the decay (%) as compared to the /EC decay rates (%). For the adopted set of NMEs, in general the half-lives computed with newly PSFs are closer to the measured ones than the half-lives calculated with PSFs with approximate method (i.e., using approximate electron wave functions) [17] for free states. Although the largest uncertainty in the computation of -decay half-lives comes from the NMEs, introduction of the newly PSF values may improve the comparison with experiment and should be taken into account for accurate predictions.
In near future we would be presenting the effect of calculation of newly computed PSF on stellar beta decay rates and comment on its astrophysical implications.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
J.-U. Nabi would like to acknowledge the support of the Higher Education Commission Pakistan through Projects nos. 5557/KPK/NRPU/R&D/HEC/2016 and 9-5(Ph-1-MG-7)/PAK-TURK/R&D/HEC/2017 and Pakistan Science Foundation through Project no. PSF-TUBITAK/KP-GIKI (02). M. Ishfaq wishes to acknowledge the support provided by Scientific and Technological Research Council of Turkey (TUBITAK), Department of Science Fellowships and Grant Programs (BIDEB), 2216 Research Fellowship Program for International Researchers (21514107-115.02-124287). S. Stoica, M. Mirea, and O. Niţescu would like to acknowledge the support of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, through Projects PN-III-P4-ID-PCE-2016-0078 and PN-5N/2018.