# New Phase Space Calculations forβ-Decay Half-Lives

Authors: Sabin Stoica; Mihail Mirea; Ovidiu Niţescu; Jameel-Un Nabi; Mavra Ishfaq
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

We revisit the computation of the phase space factors (PSF) involved in the positron decay and EC processes for a large number of nuclei of experimental interest. To obtain the electron/positron wave functions, we develop a code for solving accurately the Dirac equation with a nuclear potential derived from a realistic proton density distribution in the nucleus. The finite nuclear size (FNS) and screening effects are included through recipes which differ from those used in previous calculations. Comparing our results with previous calculations, performed with the sameQ-values, we find a close agreement for positron decays, while, for the EC process, there are relevant differences. For the EC process, we also find that the screening effect has a notable influence on the computed PSF values especially for light nuclei. Further, we recomputed the same PSF values but using the most recent Q-values reported in literature. In several cases, the new Q-values differ significantly from the older ones, leading to large differences in the PSF values as compared with previous results. Our new PSF values can contribute to more reliable calculations of the beta-decay rates, used in the study of nuclei far from the stability line and stellar evolution.


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## Body

## 1. Introduction

The phase space factors for beta decay and electron capture were calculated since a long time [1–3] and were considered to be evaluated with sufficient accuracy. However, in those works, the distortion of the electron wave functions (w.f.) by the Coulomb field of the nucleus was taken into account through Fermi functions which were expressed in terms of approximate radial solutions of the Dirac equation at the nuclear surface. Also, other corrections were introduced in the calculations in approximate ways. Thus, the screening effect on β spectrum was included by various recipes, for example, by replacing VZ potential with a momentum dependent screening (for low energy positrons) [3] and by modifying the electron radial w.f. [4, 5]. Also, the finite size of the nucleus (FNS) was taken into account by adding to the Fermi functions obtained in the “point-nucleus” approximation corrections that depend on β-particle energy and nuclear charge Z [6, 7]. Also, for the nuclear radius, older formula has been used [3, 8]. For the EC process, the electron bound-state radial w.f. were also obtained as approximate solution of the Dirac equation evaluated at the nuclear surface. They were improved by including exchange and overlap corrections, which were obtained within a relativistic HF approach.In this work, we revisit the computation of the PSF involved in the positron decay and electron capture (EC) processes for light and heavy nuclei of experimental interest. The Dirac equation is solved numerically with a Coulomb potential derived from a realistic proton distribution in the nucleus which includes the FNS correction. The numerical procedure follows the power series method described in [9] and is similar to that described in [10, 11]. The screening effect was introduced by using a screened Coulomb potential, obtained by multiplying the Coulomb potential by function ϕ(x), solution of the Thomas-Fermi equation obtained by the Majorana method [12]. The accuracy imposed in our numerical algorithms used to solve the Dirac equation always exceeds the convergence criteria given in those references. Also, a more efficient procedure to identify the electron bound states without ambiguity was developed.In order to make a comparison between the actual PSF values found in literature and ours, the same PSF are also computed with the approach described in [2, 3] and using the same Q-values. For positron decays, our results are in close agreement with the other previous results, while, for the EC process, we found significant differences. For these processes, we also find that the screening effect has a notable influence on the computed PSF values for light nuclei. Further, we recomputed the same PSF values using updated Q-values, reported recently in literature [13, 14], which, for several light nuclei, differ significantly from the older ones. As an example we cite the maximum β-particle energy (referred to as W0 throughout this paper) stated in Table 2 of [15]. These W0-values differ considerably from those given in [13, 14]. One reason for this big difference could be that Wilkinson and Macefield, in order to compare their calculation with those performed earlier by Towner and Hardy [16], restricted their phase space to only pure Fermi transitions. In other words, the Gamow-Teller window was not accessed in phase space calculation of [15]. Thus, in this paper, we propose new PSF values computed with a more accurate method and using updated Q-values, for a large number of nuclei of experimental interest. Our calculations can be useful for more reliable computation of the beta-decay rates of nuclei far from the stability line, as well as for better understanding of the stellar evolution.Our work is further motivated by similar calculations done for the double beta-decay (DBD) process. The PSF for DBD were also considered for a long time to be computed with enough accuracy and were used as such for predicting DBD lifetimes. However, recently, they were recalculated with improved methods, especially for positron and EC decay modes [17, 18], and several differences were found as compared to previous calculations where approximate electron/positron w.f. were used.The paper is organized as follows. In Section2, we present briefly the two approaches used to compute our PSF values. Our results are reported in Section 3. Here, we compare them with experimental data and previous results and discuss the differences. Finally, we summarize the main points and present our conclusions in Section 4.


## 2. Formalism

Following essentially the formalism from [3], we give here the necessary equations which we use to calculate the PSF.


### 2.1. Phase Space Factors forβ+ Transitions

The probability per unit time that a nucleus with atomic massA and charge Z decays for an allowed β-branch is given by(1)λ0=g22π3∫1W0pWW0-W2S0Z,WdW,where g is the weak interaction coupling constant, p is the momentum of β-particle, W=p2+1 is the total energy of β-particle, and W0 is the maximum β-particle energy. W0=Q-1, in β+ decay (Q is the mass difference between initial and final states of neutral atoms). Equation (1) is written in natural units ħ=m=c=1 so that the unit of momentum is mc, the unit of energy is mc2, and the unit of time is ħ/mc2. Shape factors S0(Z,W) for allowed transitions which appear in (1) are defined as(2)S0Z,W=λ1Z,WM0,12,where M0,1 are the nuclear matrix elements and the Fermi functions λ1(Z,W). Thus, for calculating the β+ decay rates, one needs to calculate the nuclear matrix elements and the PSF that can be defined as(3)FBP=∫1W0pWW0-W2λ1WdW.For the allowedβ decays, the Fermi functions are expressed as(4)λ1Z,W=g-12+f122p2,where g-1Z,W and f1Z,W are the large and the small radial components of the positron radial wave functions evaluated at nuclear radius R which can be obtained by solving the Dirac equation:(5)ddrgκW,rfκW,r=-κrW+V+1-W+V+1-κrgκW,rfκW,r,where V is the central potential for the positron and κ=l-j2j+1 is the relativistic quantum number. We note that (5) is also written in natural units.An important step in the PSF calculation forβ+ decay is the method of obtaining the positron continuum radial functions. For this, we develop a new method (code) of solving the Dirac equation, which is adapted from the method used previously for the computation of PSF for DBD process [18, 20].We solved (5) in a nuclear potential Vr derived from a realistic proton density distribution in the nucleus. This is done by solving the Schrodinger equation with a Woods-Saxon potential. In this case,(6)VZ,r=αħc∫ρer′→r→-r′→dr′→,where the charge density is(7)ρer→=∑i2ji+1vi2Ψir→2,where Ψi is the proton (Woods-Saxon) w.f. of the spherical single particle state i and vi is its occupation amplitude. Factor 2ji+1 reflects the spin degeneracy.The screening effect is taken into account by multiplying the expression ofV(r) with function ϕ(r), which is the solution of the Thomas-Fermi equation: d2ϕ/dx2=ϕ3/2/x, with x=r/b,b≈0.8853a0Z-1/3 and a0 = Bohr radius. It is calculated within the Majorana method [12]. The boundary conditions are ϕ0=1 and ϕ∞=0. As mentioned above, the screening effect is taken into account by a method developed in [12]. The possible ways in which the screening function modifies the Coulomb potential depend on the specific mechanism and its boundary conditions.For the case ofβ+-decay process, the potential used to obtain the electron w.f. is(8)rVβ+Z,r=rVZ,r+1×ϕr-1to take into account the fact that β decay releases a final negative ion with charge −1. V(Z,r) is positive. In our approach, we considered the solution of the Thomas-Fermi equation as a universal function, giving an effective screening. Here, product αħc=1, for atomic units. The asymptotic potential between an positron and an ionized atom is rVβ+=-1. In this case, the charge number Z=Z0-1 corresponds to the daughter nucleus, Z0 being the charge number of the parent nucleus. Asymptotically, ϕ(r) tends to zero.In this case, the radial solutions of the Dirac equations should be normalized in order to have the following asymptotic behavior:(9)gkϵ,rfkϵ,r~ħe-iδkprϵ+mec22ϵsin⁡kr-lπ2-ηln⁡2kr+δkϵ-mec22ϵcos⁡kr-lπ2-ηln⁡2kr+δk,where c is the speed of the light, me/ϵ are the electron mass/energy, k=p/ħ is the electron wave number, η=Ze2/ħv (with Z=±Z for β∓ decays) is the Sommerfeld parameter, δκ is the phase shift, and V is the Coulomb interaction energy between the electron and the daughter nucleus.On the other side, we also calculated the PSF for positron decays with the method described in [3]. gk and fk functions were calculated by solving the Dirac equation for a point-nucleus unscreened Coulomb potential, for which the equation has analytical solutions. The finite nuclear size and screening effects were introduced as corrections, after the recipe described in [3]. The finite-size correction was introduced by means of an empirical deviation that depended on atomic mass Z and energy W [6, 7]. The screening correction was given by the following replacement [4, 5]:(10)g-12Z,W⟶pW′p′Wg-12Z,W′,f-12Z,W⟶pW′p′Wf12Z,W′,where W′=W+V0,p′=W′2-1 and V0 was taken as a p-dependent screening potential. For further details of this formalism, we refer to [3]. No electromagnetic corrections were undertaken in this calculation of PSF.


### 2.2. Phase Space Factors for Electron Capture (EC)

Electron capture is always an alternate decay mode for radioactive isotopes that do not have sufficient energy to decay by positron emission. This is a process which competes with positron decay. In order for electron capture leading to a vacancy in, say,K-shell, the atomic mass difference between initial and final states, Q, must be greater than the binding energy ofK-shell electron in the daughter atom, ϵK. The energy carried off by the neutrino is then given by(11)qK=Q-ϵK.If the energy requirementQ>ϵK is satisfied, electron capture fromK-shell is more probable than that from any other shell because of the greater density at the nucleus ofK-shell electrons. The totalK-shell capture rate can be expressed as(12)λEC,K0=λK0BK,where(13)λK0=g2M0,124π2qK2gK2,where g2 is a constant (with dimensions of t-1), M0,1 are specific combinations of nuclear matrix elements, gK is the large component of the bound-state radial w.f. of the capturedK-shell electron (evaluated at the nuclear surface RA), qK is the neutrino energy in units of mc2, and BK is the “exchange” correction factor for theK-shell. In analogy with (12),L-shell total capture rate will be(14)λEC,Li0=λLi0BLi,where Li denotes a particularL-subshell. The contribution of L1 pertaining to 2s1/2 orbital is the most important, so we keep in our calculations only the contribution of this subshell to the calculated our PSF. The expressions for λL10 can be obtained from (13) by the replacement of qK,gK by qL1,gL1. Electron capture fromM-,N-, and higher shells may be defined in a similar fashion, but they have negligible contributions in comparison withK- andL-shells.Hence, for an allowed transition, the PSF expression of electron capture within the approximation stated above can be written as(15)FECK,L1=π2qK2gK2BK+qL12gL12BL1.ForqK/L1 quantities, we used the expression(16)qK/L1=WEC-ϵK/L1,where WEC is Q value of β+ decay in mec2 units, ϵi are the binding energies of 1s1/2 and 2s1/2 electron orbitals of the parent nucleus, and gi are their radial densities on the nuclear surface. Bi≈1 represent the values of the exchange correction. These are due to an imperfect overlap of the initial and final atomic states caused by the one unit charge difference [21]. In our method, we consider these exchange corrections to be unity, for the nuclei considered, the estimated error in doing that being under 1%. Relation W0=WEC-1 holds.g K / L 1 are the electron bound states, solutions of the Dirac equation (5), and correspond to eigenvalues ϵn (n is the radial quantum number). Quantum number κ is related to total angular momentum jκ=|κ|-1/2. These w.f. are normalized such that(17)∫0∞gn,κ2r+fn,κ2rdr=1.For simplicity, we consider solutions of Dirac equations gn,κ and fn,κ that are divided by radial distance r. An asymptotic solution is obtained by means of the WKB approximation and by considering that potential V is negligibly small:(18)fn,κgn,κ=cħϵ+mec2gn,κ′gn,κ+κr,where(19)gn,κ′gn,κ=-12μ′μ-1-μ,with(20)μ=ϵ+mec2ħ2c2V-ϵ+mec2+κ2r21/2.In our calculations, we use number nodesn=0 and n=1, for orbitals 1s1/2 and 2s1/2, respectively, κ being −1. Numerically, the eigenvalues of the discrete spectrum are obtained by matching two numerical solutions of the Dirac equation: the inverse solution that starts from the asymptotic conditions and the direct one that starts at r=0.The radial density of the bound-state electron w.f. on the nuclear surface is(21)Dn,κ2=1mec23ħca03a0RA2gn,κ2RA+fn,κ2RA,where RA=1.2A01/3 is given in fm, a0 being the Bohr radius. For 1s1/2 and 2s1/2 electron orbitals, we use gK2=D0,-12 and gL12=D1,-12, respectively.For the EC processes, the potential used to obtain the electron w.f. reads(22)rVECZ,r=rVZ,rϕr,and charge number Z=Z0 corresponds to the parent nucleus. V(Z,r) is negative.The numerical solutions of the Dirac equation were obtained within the power series method of [9], by using similar numerical algorithm as that of [10, 11]. The method is able to provide numerical solutions of the Dirac equation for central fields. We provide a grid with values of the potential for different radial distances. The radial w.f. is expanded in an infinite power series that depends on the radial increment and the potential values. The w.f. is calculated step by step in the mesh points. The increment and the number of terms in the series expansion determine the accuracy of the solutions. In our calculations, the increment interval is 10−4 fm and at least 100 terms are taken into account in the series expansion. These values exceed the convergence criteria of [10]. To renormalize the numerical solutions, we made use of the fact that, at very large distances, the behavior of the w.f. must approach that of the Coulomb function. Therefore, the amplitudes and the phase shifts can be extracted by comparing the numerical solution and the analytical ones. For discrete states, the asymptotic behavior of the w.f. gives a guess for the inverse solutions. The eigenvalue is obtained when the direct solutions and the inverse ones match each other. We constructed an adequate procedure to find the bound states of the electron up to an accuracy of 0.3 keV, or lower, by searching solutions up to 130 keV binding energies. In this range of energies, all the possible bound-state energies are found. We calculated the solutions starting outward from r=0 and inward from a very large value of radius r. The bound states should be obtained when both solutions are equal in an intermediate point, for the two components of the wave function. We found these energies by interpolation. We selected radial wave functions fn,κ and gn,κ that have the same number of nodes n=0 or 1.For the PSF computation, all integrals in (5) were performed accurately with Gauss-Legendre quadrature in 32 points. We calculated up to 49 values of the radial functions inQ value energy interval, that were interpolated with spline functions.We also calculated the PSF for EC process using (15) but employing essentially the formalism adopted by [2]. Here, we used the electron radial density (and density ratios) as given in Table 2 of [2]. Exchange corrections were taken as unity. Binding energies were also taken from the same reference.


## 2.1. Phase Space Factors forβ+ Transitions

The probability per unit time that a nucleus with atomic massA and charge Z decays for an allowed β-branch is given by(1)λ0=g22π3∫1W0pWW0-W2S0Z,WdW,where g is the weak interaction coupling constant, p is the momentum of β-particle, W=p2+1 is the total energy of β-particle, and W0 is the maximum β-particle energy. W0=Q-1, in β+ decay (Q is the mass difference between initial and final states of neutral atoms). Equation (1) is written in natural units ħ=m=c=1 so that the unit of momentum is mc, the unit of energy is mc2, and the unit of time is ħ/mc2. Shape factors S0(Z,W) for allowed transitions which appear in (1) are defined as(2)S0Z,W=λ1Z,WM0,12,where M0,1 are the nuclear matrix elements and the Fermi functions λ1(Z,W). Thus, for calculating the β+ decay rates, one needs to calculate the nuclear matrix elements and the PSF that can be defined as(3)FBP=∫1W0pWW0-W2λ1WdW.For the allowedβ decays, the Fermi functions are expressed as(4)λ1Z,W=g-12+f122p2,where g-1Z,W and f1Z,W are the large and the small radial components of the positron radial wave functions evaluated at nuclear radius R which can be obtained by solving the Dirac equation:(5)ddrgκW,rfκW,r=-κrW+V+1-W+V+1-κrgκW,rfκW,r,where V is the central potential for the positron and κ=l-j2j+1 is the relativistic quantum number. We note that (5) is also written in natural units.An important step in the PSF calculation forβ+ decay is the method of obtaining the positron continuum radial functions. For this, we develop a new method (code) of solving the Dirac equation, which is adapted from the method used previously for the computation of PSF for DBD process [18, 20].We solved (5) in a nuclear potential Vr derived from a realistic proton density distribution in the nucleus. This is done by solving the Schrodinger equation with a Woods-Saxon potential. In this case,(6)VZ,r=αħc∫ρer′→r→-r′→dr′→,where the charge density is(7)ρer→=∑i2ji+1vi2Ψir→2,where Ψi is the proton (Woods-Saxon) w.f. of the spherical single particle state i and vi is its occupation amplitude. Factor 2ji+1 reflects the spin degeneracy.The screening effect is taken into account by multiplying the expression ofV(r) with function ϕ(r), which is the solution of the Thomas-Fermi equation: d2ϕ/dx2=ϕ3/2/x, with x=r/b,b≈0.8853a0Z-1/3 and a0 = Bohr radius. It is calculated within the Majorana method [12]. The boundary conditions are ϕ0=1 and ϕ∞=0. As mentioned above, the screening effect is taken into account by a method developed in [12]. The possible ways in which the screening function modifies the Coulomb potential depend on the specific mechanism and its boundary conditions.For the case ofβ+-decay process, the potential used to obtain the electron w.f. is(8)rVβ+Z,r=rVZ,r+1×ϕr-1to take into account the fact that β decay releases a final negative ion with charge −1. V(Z,r) is positive. In our approach, we considered the solution of the Thomas-Fermi equation as a universal function, giving an effective screening. Here, product αħc=1, for atomic units. The asymptotic potential between an positron and an ionized atom is rVβ+=-1. In this case, the charge number Z=Z0-1 corresponds to the daughter nucleus, Z0 being the charge number of the parent nucleus. Asymptotically, ϕ(r) tends to zero.In this case, the radial solutions of the Dirac equations should be normalized in order to have the following asymptotic behavior:(9)gkϵ,rfkϵ,r~ħe-iδkprϵ+mec22ϵsin⁡kr-lπ2-ηln⁡2kr+δkϵ-mec22ϵcos⁡kr-lπ2-ηln⁡2kr+δk,where c is the speed of the light, me/ϵ are the electron mass/energy, k=p/ħ is the electron wave number, η=Ze2/ħv (with Z=±Z for β∓ decays) is the Sommerfeld parameter, δκ is the phase shift, and V is the Coulomb interaction energy between the electron and the daughter nucleus.On the other side, we also calculated the PSF for positron decays with the method described in [3]. gk and fk functions were calculated by solving the Dirac equation for a point-nucleus unscreened Coulomb potential, for which the equation has analytical solutions. The finite nuclear size and screening effects were introduced as corrections, after the recipe described in [3]. The finite-size correction was introduced by means of an empirical deviation that depended on atomic mass Z and energy W [6, 7]. The screening correction was given by the following replacement [4, 5]:(10)g-12Z,W⟶pW′p′Wg-12Z,W′,f-12Z,W⟶pW′p′Wf12Z,W′,where W′=W+V0,p′=W′2-1 and V0 was taken as a p-dependent screening potential. For further details of this formalism, we refer to [3]. No electromagnetic corrections were undertaken in this calculation of PSF.


## 2.2. Phase Space Factors for Electron Capture (EC)

Electron capture is always an alternate decay mode for radioactive isotopes that do not have sufficient energy to decay by positron emission. This is a process which competes with positron decay. In order for electron capture leading to a vacancy in, say,K-shell, the atomic mass difference between initial and final states, Q, must be greater than the binding energy ofK-shell electron in the daughter atom, ϵK. The energy carried off by the neutrino is then given by(11)qK=Q-ϵK.If the energy requirementQ>ϵK is satisfied, electron capture fromK-shell is more probable than that from any other shell because of the greater density at the nucleus ofK-shell electrons. The totalK-shell capture rate can be expressed as(12)λEC,K0=λK0BK,where(13)λK0=g2M0,124π2qK2gK2,where g2 is a constant (with dimensions of t-1), M0,1 are specific combinations of nuclear matrix elements, gK is the large component of the bound-state radial w.f. of the capturedK-shell electron (evaluated at the nuclear surface RA), qK is the neutrino energy in units of mc2, and BK is the “exchange” correction factor for theK-shell. In analogy with (12),L-shell total capture rate will be(14)λEC,Li0=λLi0BLi,where Li denotes a particularL-subshell. The contribution of L1 pertaining to 2s1/2 orbital is the most important, so we keep in our calculations only the contribution of this subshell to the calculated our PSF. The expressions for λL10 can be obtained from (13) by the replacement of qK,gK by qL1,gL1. Electron capture fromM-,N-, and higher shells may be defined in a similar fashion, but they have negligible contributions in comparison withK- andL-shells.Hence, for an allowed transition, the PSF expression of electron capture within the approximation stated above can be written as(15)FECK,L1=π2qK2gK2BK+qL12gL12BL1.ForqK/L1 quantities, we used the expression(16)qK/L1=WEC-ϵK/L1,where WEC is Q value of β+ decay in mec2 units, ϵi are the binding energies of 1s1/2 and 2s1/2 electron orbitals of the parent nucleus, and gi are their radial densities on the nuclear surface. Bi≈1 represent the values of the exchange correction. These are due to an imperfect overlap of the initial and final atomic states caused by the one unit charge difference [21]. In our method, we consider these exchange corrections to be unity, for the nuclei considered, the estimated error in doing that being under 1%. Relation W0=WEC-1 holds.g K / L 1 are the electron bound states, solutions of the Dirac equation (5), and correspond to eigenvalues ϵn (n is the radial quantum number). Quantum number κ is related to total angular momentum jκ=|κ|-1/2. These w.f. are normalized such that(17)∫0∞gn,κ2r+fn,κ2rdr=1.For simplicity, we consider solutions of Dirac equations gn,κ and fn,κ that are divided by radial distance r. An asymptotic solution is obtained by means of the WKB approximation and by considering that potential V is negligibly small:(18)fn,κgn,κ=cħϵ+mec2gn,κ′gn,κ+κr,where(19)gn,κ′gn,κ=-12μ′μ-1-μ,with(20)μ=ϵ+mec2ħ2c2V-ϵ+mec2+κ2r21/2.In our calculations, we use number nodesn=0 and n=1, for orbitals 1s1/2 and 2s1/2, respectively, κ being −1. Numerically, the eigenvalues of the discrete spectrum are obtained by matching two numerical solutions of the Dirac equation: the inverse solution that starts from the asymptotic conditions and the direct one that starts at r=0.The radial density of the bound-state electron w.f. on the nuclear surface is(21)Dn,κ2=1mec23ħca03a0RA2gn,κ2RA+fn,κ2RA,where RA=1.2A01/3 is given in fm, a0 being the Bohr radius. For 1s1/2 and 2s1/2 electron orbitals, we use gK2=D0,-12 and gL12=D1,-12, respectively.For the EC processes, the potential used to obtain the electron w.f. reads(22)rVECZ,r=rVZ,rϕr,and charge number Z=Z0 corresponds to the parent nucleus. V(Z,r) is negative.The numerical solutions of the Dirac equation were obtained within the power series method of [9], by using similar numerical algorithm as that of [10, 11]. The method is able to provide numerical solutions of the Dirac equation for central fields. We provide a grid with values of the potential for different radial distances. The radial w.f. is expanded in an infinite power series that depends on the radial increment and the potential values. The w.f. is calculated step by step in the mesh points. The increment and the number of terms in the series expansion determine the accuracy of the solutions. In our calculations, the increment interval is 10−4 fm and at least 100 terms are taken into account in the series expansion. These values exceed the convergence criteria of [10]. To renormalize the numerical solutions, we made use of the fact that, at very large distances, the behavior of the w.f. must approach that of the Coulomb function. Therefore, the amplitudes and the phase shifts can be extracted by comparing the numerical solution and the analytical ones. For discrete states, the asymptotic behavior of the w.f. gives a guess for the inverse solutions. The eigenvalue is obtained when the direct solutions and the inverse ones match each other. We constructed an adequate procedure to find the bound states of the electron up to an accuracy of 0.3 keV, or lower, by searching solutions up to 130 keV binding energies. In this range of energies, all the possible bound-state energies are found. We calculated the solutions starting outward from r=0 and inward from a very large value of radius r. The bound states should be obtained when both solutions are equal in an intermediate point, for the two components of the wave function. We found these energies by interpolation. We selected radial wave functions fn,κ and gn,κ that have the same number of nodes n=0 or 1.For the PSF computation, all integrals in (5) were performed accurately with Gauss-Legendre quadrature in 32 points. We calculated up to 49 values of the radial functions inQ value energy interval, that were interpolated with spline functions.We also calculated the PSF for EC process using (15) but employing essentially the formalism adopted by [2]. Here, we used the electron radial density (and density ratios) as given in Table 2 of [2]. Exchange corrections were taken as unity. Binding energies were also taken from the same reference.


## 3. Results and Discussion

We perform PSF computations forβ+ decay and the EC process with the method described in the previous section that we call TW (this work), for a large number of nuclei of experimental interest.Forβ+ decays, we found previous PSF results computed with approximate methods [15, 16], for sixteen nuclei of astrophysical interest. In Table 1, we display the PSF values for these nuclei calculated with our new method (TW) and, for comparison, the values taken from [15, 16]. Also, we present the PSF values computed by us using the recipe described in [3]. All calculations were done with W0 value indicated in [15]. One can see that the agreement between TW results and the other results is in general under 1%, except the last two (heavy) nuclei where the differences reach ~3%.Table 1
Calculated phase space ofβ+-decay (BP) compared with previous calculations. The value of maximum β-decay energy is taken from [15] for pure Fermi transitions. The last two columns show our calculated results.
Nucleus W 0 [15] (MeV) F B P [16] F B P [15] F B P [TW] F B P [3] 10C 0.8884 2.361 2.361 2.325 2.326 14O 1.8098 43.398 43.378 42.822 42.814 18Ne 2.383 136.83 136.83 135.19 135.08 22Mg 3.109 427.02 426.88 422.19 421.51 26Al 3.211 483.84 483.68 478.3 477.43 26Si 3.817 1036.8 1035.9 1025.51 1023.059 30S 4.439 1990.2 1987.8 1969.24 1963.9 34Cl 4.468 2014.7 2013.4 1993.13 1987.4 34Ar 5.021 3388.3 3383.8 3351.58 3339.85 38K 5.028 3346.9 3344.9 3312.82 3300.54 38Ca 5.620 5515.9 5510.3 5457.95 5449 42Sc 5.409 4533.5 4531.7 4490.19 4462.21 42Ti 5.964 7025.4 7024.1 6934.9 6853.74 46V 6.032 7285.9 7284.2 7186.04 7091.9 50Mn 6.609 10818 10810 10492.76 10262 54Co 7.227 15956 15951 14988.470 14412.5In Table2, we display our computed PSF with the new method for few heavy nuclei, for which we did not find previous results. For comparison, we computed the same PSF values with the recipe adopted from [3]. W0-values were taken from [19] for both sets of calculations. We found a rather good agreement between the two sets of results, with differences within, generally, a few percent. There was one exception, 105Ag, where the difference was large (~a factor 10). This is a case where W0-value is very small (0.325 MeV), and this might make our numerical routine inaccurate at such small values. However, this discrepancy may not be so significant, as long as the calculated PSF value is small enough to have little contribution to the corresponding beta-decay ratesTable 2
Calculated phase space ofβ+-decay (BP) for heavy nuclei compared with the ones we calculated using recipe of [3].
Nucleus W 0 [19] (Mev) F B P [TW] F B P [3] 52Fe 1.3525 8.3403 8.4132 56Ni 1.1109 3.4439 3.5250 62Zn 0.5974 0.2344 0.2438 66Ga 4.153 1125.6442 1132.5483 76Br 3.9409 835.1982 843.3343 81Rb 1.2161 4.3222 6.8878 88Y 2.6006 120.2644 121.8624 90Nb 5.0893 2503.0555 2533.7049 102Cd 1.565 11.2214 11.5267 103In 5.0005 2100.3727 2136.0153 105Ag 0.325 0.0102 0.1127 107Sb 6.837 8528.5047 8931.8197 113Sb 2.8891 168.1487 172.0209 113Te 5.048 2124.1816 2165.2927 115I 4.7029 1517.2376 1549.2409 116I 6.7547 7913.1790 8272.0244 116Xe 3.235 352.3565 361.4082 120Ba 3.98 678.0918 705.0294 120Xe 0.5587 0.1047 0.1108 126Cs 3.7731 542.4653 563.8184 182Re 1.778 16.123 17.206 205Bi 1.6835 12.3984 13.4576In Table3, we preset our results for EC for the same set of nuclei. Q-values for positron decay were taken from [15] for nuclei marked with ⋆. For the rest of nuclei, Q-values were taken from [19]. Together with the PSF values for EC, the electron densities, gK,L1, their ratios, and binding energies ϵ for orbitals 1s1/2 and 2s1/2 are also given in Table 3. We compare the results performed with the new method (TW) with those calculated using the recipe of [2]. For these transitions, the differences between the two sets of results are significantly larger than for the positron decays, ranging from a few percent to about a mammoth 35%. We attribute these differences in the calculated PSF values mainly to electron densities, gK, whose values, calculated with the “old” and “new” methods, differ significantly from each other. We also checked the influence of the screening effect on the PSF values. We found that while, for the positron decays, this effect is very small, for the EC transitions, there are some differences between the “screened” and “unscreened” PSF values. Figure 1 shows this effect on the electron density, gK, and on the final PSF values. For small values of Z, the results without screening give PSF values that are 10–15% larger than those listed in Table 3. For heavier nuclei, these differences are only up to 2-3%. The screening effect in PSF calculation is more important for light nuclei and leads to a decrease in the PSF values up to 15%. Finally, in Table 4, we present PSF values for EC transitions recomputed with updated Q-values taken from [13, 14]. We propose to use these new computed values of PSF for calculation of β-decay rates.Table 3
Calculated phase space factorsFEC for electron capture (assuming exchange corrections to be equal to 1). The value of maximum β-decay energy is taken from [15] for pure Fermi transitions. The electron densities, their ratios, and binding energies ϵ are also provided for orbitals 1s1/2 and 2s1/2, including those given in [2]. Binding energies are given in units of keV.
Nucleus Q β + (MeV) g K 2 [2] g K 2 [TW] g L 1 2 / g K 2 [2] g L 1 2 / g K 2 [TW] ϵ K [2] ϵ K [TW] ϵ L 1 [2] ϵ L 1[TW] F E C K , L 1 [TW] F E C K , L 1 [2] 10C⋆ 1.9104 0.00031 0.00031 0.04930 0.02867 0.18790 0.62660 0.12600 0.01176 0.00703 0.00640 14O⋆ 2.83186 0.00075 0.00065 0.05640 0.04420 0.40160 1.03733 0.02440 0.03251 0.03297 0.03786 18Ne⋆ 3.405 0.00151 0.00118 0.05840 0.05794 0.68540 1.48302 0.03400 0.06659 0.08713 0.11005 22Mg⋆ 4.131 0.00268 0.00199 0.06660 0.06811 1.07210 2.11143 0.06330 0.15721 0.218 0.29060 26Al⋆ 4.2331 0.00344 0.00251 0.06990 0.07265 1.30500 2.40715 0.08940 0.14631 0.27558 0.39270 26Si⋆ 4.839 0.00435 0.00312 0.07290 0.07661 1.55960 2.74689 0.11770 0.18077 0.47240 0.65060 30S⋆ 5.461 0.00664 0.00467 0.07810 0.08342 2.14550 3.49498 0.18930 0.25934 0.90680 1.27140 34Cl⋆ 5.4908 0.00807 0.00563 0.08040 0.08628 2.47200 3.91749 0.22920 0.30899 1.10727 1.56600 34Ar⋆ 6.043 0.00970 0.00675 0.08240 0.08862 2.82240 4.33190 0.27020 0.36199 1.61130 2.28490 38K⋆ 6.05 0.01156 0.00802 0.08440 0.09079 3.20600 4.77984 0.32630 0.41921 1.92311 2.73480 38Ca⋆ 6.642 0.01367 0.00947 0.08620 0.09259 3.60740 5.25087 0.37710 0.48351 2.74237 3.90650 42Sc⋆ 6.4311 0.01600 0.01113 0.08790 0.09430 4.03810 5.73657 0.43780 0.54865 3.02434 4.28930 42Ti⋆ 6.986 0.01870 0.01300 0.08960 0.09579 4.49280 6.25222 0.50040 0.62068 4.17496 5.92320 46V⋆ 7.0543 0.02170 0.01512 0.09100 0.09699 4.96640 6.78377 0.56370 0.69826 4.95575 7.02120 50Mn⋆ 7.6311 0.02870 0.02016 0.09380 0.09920 5.98920 7.92722 0.69460 0.86703 7.74617 10.9103 52Fe 2.374 0.0328 0.0232 0.0950 0.0987 7.1120 8.5130 0.8461 0.958 0.859 1.2033 54Co⋆ 8.2498 0.03730 0.02651 0.09620 0.10077 7.11200 9.14731 0.84610 1.05584 11.91799 16.6144 56Ni 2.136 0.0423 0.0303 0.0974 0.1013 8.3328 9.7882 1.0081 1.158 0.907 1.2580 62Zn 1.626 0.0538 0.0390 0.0995 0.1025 9.6586 11.157 1.1936 1.380 0.675 0.9261 66Ga 5.175 0.0604 0.0410 0.1006 0.1029 10.3671 11.875 1.2977 1.498 7.80 10.613 76Br 4.963 0.0935 0.0704 0.1035 0.1048 13.4737 15.000 1.7820 2.021 11.45 15.162 81Rb 2.23815 0.1149 0.0883 0.1063 0.1080 15.1997 16.690 2.0651 2.263 9.069 11.744 88Y 3.6226 0.1402 0.1091 0.1080 0.1174 17.0384 18.450 2.3725 2.438 9.528 12.114 90Nb 6.111 0.170 0.1344 0.1098 0.1059 18.9856 20.421 2.6977 2.994 33.17 41.975 102Cd 2.587 0.319 0.2663 0.1159 0.1102 26.7112 28.044 4.0180 4.351 11.66 14.019 103In 6.050 0.348 0.2930 0.1168 0.1116 27.9399 29.232 4.2375 4.548 71.05 84.541 105Ag 1.345 0.293 0.2423 0.1150 0.1086 25.5140 26.864 3.8058 4.161 2.816 3.4256 107Sb 7.920 0.413 0.3526 0.1187 0.1096 30.4912 31.726 4.6983 5.095 146.5 172.43 113Sb 3.913 0.413 0.3516 0.1187 0.1096 30.4912 31.726 4.6983 5.095 35.38 41.804 113Te 6.070 0.449 0.3844 0.1196 0.1113 31.8138 33.041 4.9392 5.314 93.70 109.93 115I 5.729 0.488 0.4121 0.1205 0.1124 33.1694 34.345 5.1881 5.542 91.54 106.40 116I 7.780 0.488 0.4215 0.1205 0.1124 33.1694 34.345 5.1881 5.542 169.3 196.75 116Xe 4.450 0.529 0.4609 0.1215 0.1123 34.5644 35.705 5.4528 5.822 60.15 69.410 120Ba 5.00 0.623 0.5496 0.1234 0.1130 37.4406 38.514 5.9888 6.375 90.65 103.51 120Xe 1.617 0.529 0.4599 0.1215 0.1123 34.5644 35.705 5.4528 5.821 7.72 8.9482 126Cs 4.824 0.574 0.501 0.1224 0.112 35.9846 37.111 5.7143 6.128 76.88 88.697 182Re 2.800 2.69 2.593 0.1448 0.128 71.6764 72.491 12.5267 13.26 22.86 24.152 205Bi 2.708 4.88 4.837 0.1561 0.138 90.5259 91.373 16.2370 17.25 228.17 233.83Table 4
Calculated phase space factorsFEC for electron capture, with Q-values from [13, 14].
Nucleus Q EC [13, 14] (MeV) F EC [TW] F EC [2] F B P [TW] F B P [3] 10C 3.64613 0.07318 2.33265 226.780 226.834 14O 5.14131 0.21794 0.12483 1644.76 1643.41 18Ne 4.44215 0.27831 0.18733 677.970 677.912 22Mg 4.77904 0.61616 0.39020 995.887 995.685 26Al 4.00231 0.62642 0.35240 343.398 343.658 26Si 5.06645 0.51788 0.71694 1339.344 1339.30 30S 6.13834 1.14585 1.61931 3805.276 3803.16 34Cl 5.48869 1.10642 1.57889 1994.797 1995.09 34Ar 6.05858 1.61963 2.31915 3410.133 3409.96 38K 5.91093 1.83565 2.64042 2917.839 2918.62 38Ca 6.73867 2.82284 4.07367 5924.355 5929.26 42Sc 6.42269 3.01643 4.33609 4470.946 4471.87 42Ti 7.01275 4.20702 6.05196 7100.190 7130.06 46V 7.04865 4.94781 7.11022 7175.692 7209.06 50Mn 7.63042 7.74479 11.0705 10516.941 10744.5 52Fe 2.37330 0.8584 1.22082 14942.286 15765.2 54Co 8.24017 11.89015 16.8306 8.354 8.43206 56Ni 2.13175 0.9029 1.27259 3.444 3.49486 62Zn 1.61859 0.6687 0.93259 0.234 0.24131 66Ga 5.17225 7.7902 10.7797 1125.644 1131.60 76Br 4.96024 11.439 15.4388 835.295 841.531 81Rb 2.23696 2.9044 3.84415 4.321 4.41092 88Y 3.62067 9.5180 12.3759 120.264 121.864 90Nb 6.10809 33.141 42.9337 2502.372 2526.00 102Cd 2.58562 11.652 14.4027 11.221 11.5468 103In 6.01928 70.333 15.7203 2099.402 2133.61 105Ag 1.34679 2.8233 3.53114 0.01027 1.12362 107Sb 7.85483 144.059 174.745 8528.505 8918.59 113Sb 3.90909 35.311 42.9919 168.122 172.036 113Te 6.06682 93.601 113.234 2124.182 2162.53 115I 5.72192 91.3148 109.531 1509.977 1547.75 116I 7.77260 168.959 202.635 7930.046 8250.78 116Xe 4.44315 59.963 71.4659 354.467 361.241 120Ba 4.99761 90.562 106.993 685.518 703.098 120Xe 1.57992 7.3638 8.82085 0.105 0.11187 126Cs 4.79256 75.871 90.4835 542.400 555.411 182Re 2.79851 131.273 145.184 16.123 17.2282 205Bi 2.70412 227.499 247.263 12.415 13.4798Figure 1
(a) Electron density on the nuclear surface of1s1/2 state as a function of the atomic number of the parent nucleus. The values calculated with screening are displayed with filled circles and those without screening are plotted with empty symbols. The dashed lines are given to guide the eye. (b) Calculated PSF for EC as a function of the mass number of the parent nucleus. The filled circles display the calculation with screening, while the empty ones the calculation without screening.
(a) (b)


## 4. Summary and Conclusion

In summary, we constructed a new code for computing PSF values for positron decays and EC processes. In our approach, we get positron-free and electron bound w.f. by solving a Dirac equation with a Coulomb-type potential, obtained from a realistic distribution of protons in the daughter nuclei. The FNS and screening effects are addressed as well by our new recipe. Using the sameQ-values, we compare our results with previous calculations where electron/positron w.f. were obtained in an approximate way. For positron decays, the agreement with older results is quite good, while, for EC processes, the differences between “new” and “old” PSF values are as big as 35%. We further found that the screening effect is important for EC processes, especially for light nuclei, having an impact up to 10–15% on the calculated PSF values. Finally, using our new method, we recomputed the PSF for all nuclei using updated Q-values. We hope that these computed PSF values will prove useful in more accurate estimations of the beta-decay rates. We are currently working on the impact of newly computed PSF values on β-decay half-lives and hope to report our findings in the near future.Finally, we mention that our numerical formalism for getting exact electron/positron w.f. for free and bound states is also suited for the treatment of other charged leptons, such as muons, for example, in any central field. So, it can be also used in applications such as the calculation of muon conversion capture rates. In this respect, there are works where the exact muon w.f. are obtained by solving the Dirac equation in a Coulomb-like potential, within the context of genetic algorithms and neural network techniques [22–24]. Within this method, one can also take into account deviations from a pure central Coulomb field by using experimental finite-size charge densities for the attracting nucleons, a procedure which differs from that described in this work.



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