# Study of the Effect of Newly Calculated Phase Space Factor onβ-Decay Half-Lives

Authors: Mavra Ishfaq; Jameel-Un Nabi; Ovidiu Niţescu; Mihail Mirea; Sabin Stoica
Journal: Advances in High Energy Physics
 (2019)

---

## Abstract

We present results forβ-decay half-lives based on a new recipe for calculation of phase space factors recently introduced. Our study includes f
p-shell and heavier nuclei of experimental and astrophysical interests. The investigation of the kinematics of some β-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 β-decay half-lives and is not the subject of this paper. This paper explores if improvements in calculating the β-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.


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## Body

## 1. Introduction

The precise knowledge of theβ-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 β-decay half-lives in agreement with experimental results has been a challenging problem for nuclear theorists [4–8]. Theoretically, the half-life formulas for β-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 β-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 β-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 inr-process nucleosynthesis. Similarly, precise value of β-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 β-decay half-lives. We also extend here our previous PSF calculations (of positron decay and EC reactions) to include β-decay reactions. In order to complete the calculation of β-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 β-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 β-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. Section2 describes the essential formalism for the calculation of PSFs and β-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

β-decay half-lives can be calculated as a sum over all transition probabilities to the daughter nucleus states through excitation energies lying within the Q
β value(1)
T
1
/
2
=
∑
0
≤
E
f
≤
Q
β
1
t
f
-
1
,where the partial half-lives (PHL), t
f, can be calculated using(2)
t
f
=
C
g
A
/
g
v
2
F
A
Z
,
A
,
E
B
G
T
E
f
+
F
V
Z
,
A
,
E
B
F
E
f
.In (2) value of C was taken as 6143 s [18] and g
A, g
v are axial-vector and vector coupling constants of the weak interaction, respectively, having g
A/g
v= -1.2694 [19], while E
f is the final state energy. E
=
Q
β
-
E
f where Q
β is the window accessible to either β
+-, β
--, or EC decay. F
A
/
V are the PSFs. B
G
T and B
F are the reduced transition probabilities for Gamow-Teller and Fermi transitions, respectively, and expressed as(3)
B
F
E
f
=
1
2
I
i
+
1
f
M
F
i
2
,
(4)
B
G
T
E
f
=
1
2
I
i
+
1
f
M
G
T
i
2In (3) and (4), I
i denotes the spin of the parent state, M
F and M
G
T 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 χ for particle-particle and κ for particle-hole interactions. Here, we used the same range for χ and κ as was discussed in [7, 8]. Deformation parameter values β
2 for all cases were taken from Ref. [21]. For pairing gaps we used a global approach Δ
n = Δ
p = 12/A [MeV]. A large model space up to 7ħ
ω 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 forβ
+
/
β
- Transitions

The formalism for the PSF calculation forβ
+
/
β
- 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 β-branch is given by(5)
λ
0
=
G
β
2
2
π
3
∫
1
W
0
p
W
W
0
-
W
2
S
0
Z
,
W
d
W
,where G
β is the weak interaction coupling constant, p is the momentum of β-particle, W = p
2
+
1 is the total energy of β-particle, and W
0 is the maximum β-particle energy. W
0 = Q
-
1 (Q
+
1) in β
+ (β
-) decay. Q is the mass difference between initial and final states of neutral atoms. Equation (5) is written in natural units (ħ
=
m
=
c
=
1), so that the unit of momentum is m
c, the unit of energy is m
c
2, and the unit of time is ħ /m
c
2. The shape factors S
0
(
Z
,
W
) for allowed transitions which appear in (5) are defined as(6)
S
0
Z
,
W
=
λ
1
Z
,
W
M
0,1
2
,where M
0,1 are the NMEs related to the Fermi and Gamow-Teller reduced transition probabilities as(7)
M
0,1
2
=
1
2
I
i
+
1
B
F
,
G
T
,and λ
1
(
Z
,
W
) stands for Fermi functions. For the calculation of the β-decay rates, one needs to calculate the NMEs and the PSFs that can be defined as(8)
F
β
+
/
β
-
=
∫
1
W
0
p
W
W
0
-
W
2
λ
1
W
d
W
.The above formula determines the PSFs for both the Fermi and Gamow-Teller allowed transitions, by substitutingF
V or F
A in (2), respectively. For the allowed β-decay the Fermi functions can be expressed as(9)
λ
1
Z
,
W
=
g
-
1
2
+
f
1
2
2
p
2
,which is just the definition used in [17] (see (3)), for our particular case k
=
1. We note that in the above formula a coefficient 1
/
(
2
p
2
) appears. Usually, this coefficient is included in the proper normalization of the wave functions, as we did. The functions g
-
1
(
Z
,
W
) and f
1
(
Z
,
W
) are the large and small radial components of the positron (or electron) radial wave functions evaluated at the nuclear radius R. They are solutions of the coupled set of differential equations [17](10)
d
d
r
+
κ
+
1
r
g
κ
W
,
r
=
W
+
V
r
+
1
f
κ
W
,
r
d
d
r
+
κ
-
1
r
f
κ
W
,
r
=
-
W
+
V
r
-
1
g
κ
W
,
rwhere V
(
r
) is the central potential for the positron (or the electron) and κ
=
(
l
-
j
)
(
2
j
+
1
) is the relativistic quantum number.Ideally, the central potentialV
(
r
) 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 V
(
r
) from a realistic proton density distribution in the nucleus. The charge density can be written as(11)
ρ
e
r
→
=
∑
i
2
j
i
+
1
v
i
2
ψ
i
r
→
2
,where ψ
i is the proton wave function of the spherical single particle state i and v
i is its occupation amplitude. The wave functions ψ
i were found by solving Schrödinger equation with a Wood-Saxon potential. The (
2
j
i
+
1
) term in (11) reflects the spin degeneracy. As an example, we depict the realistic proton density for X
e
120 in cylindrical coordinates in Figure 1(a). The profile of this proton density for the daughter nucleus X
e
120 (thick line) is compared with a constant density (dot-dashed line) in Figure 1(b).Figure 1
(a) Realistic proton density forX
e
120 represented in cylindrical coordinates. (b) Profile of the realistic proton density for X
e
120 (thick line) compared with that given by a constant density approximation (dot-dashed line).
(a)
(b)We integrated the realistic charge distribution over the volume of nucleus, in order to find the Coulomb potential(12)
V
Z
,
r
=
α
ħ
c
∫
ρ
e
r
′
→
r
→
-
r
′
→
d
r
′
→
.Moreover, we included the screening effect by multiplying the expression ofV
(
r
) with a function ϕ
(
r
), which is solution of the Thomas-Fermi equation(13)
d
2
ϕ
d
x
2
=
ϕ
3
/
2
x
,with x
=
r
/
b, b
≈
0.8853
a
0
Z
-
1
/
3, and a
0 being the Bohr radius. The solution ϕ
(
r
) was calculated within the Majorana method [22]. In the case of β
-
/
β
+, the effective potential V
β
∓ was modified by the screening function ϕ
(
r
) as(14)
r
V
β
∓
Z
,
r
=
r
V
Z
,
r
+
1
×
ϕ
r
-
1
.So, asymptotically we returned the interaction between an ion of charge ±
1 of the residual nucleus and the emitted electron/proton. The Coulomb potential in atomic units is Z
/
r and is negative/positive for β
-
/
β
+ decay. As mentioned in [22], the Thomas-Fermi equation (13) is a universal equation which does not depend on Z or other physical constants. The boundaries of the screening function are ϕ
(
0
)
=
1 and ϕ
(
∞
)=0. Then, the effective potential is V
β
∓
(
r
)
=
V
(
r
) for r=0 and is suppressed according to the variation of the universal screening function when r increases in order to reach the asymptotic behavior V
β
∓
(
r
)
=
-
1
/
r for r
→
∞. 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.Figure 2
The effective screening functionϕ
e
f
f
=
V
β
∓
/
V as function of the dimensionless distance x for Z=54. The dashed and the dot-dashed lines correspond to β
- and β
+ decay, respectively. The full line gives the behavior of the screening function with the boundaries ϕ
(
0
)=1 and ϕ
(
∞
)=0, as given in [22].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 (f, g) 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 Γ 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 ofG
M [17], we compare the PSF results for three virtual cases as discussed in [17]. As seen from Table 1 for β
+ and Table 2 for β
-, 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.Table 1
Comparison of PSF results calculated with our recipe (TW) and Gove and Martin recipe (G
M), for β
+ decay of three virtual nuclei.
Z
A
Q
β
+
(
M
e
V
)
l
o
g
F
β
+
G
M  [17]
l
o
g
F
β
+
T
W
10
20
0.05
-4.643
-4.646
0.50
-0.700
-0.702
5.00
3.575
3.574
50
120
0.05
-6.117
-6.195
0.50
-1.152
-1.175
5.00
3.319
3.311
90
230
0.05
-7.088
-7.272
0.50
-1.330
-1.388
5.00
3.236
3.206Table 2
Comparison of PSF results calculated with our recipe (TW) and (G
M), for β
- decay of three virtual nuclei.
Z
A
Q
β
-
(
M
e
V
)
l
o
g
F
β
-
G
M  [17]
l
o
g
F
β
-
T
W
10
20
0.05
-3.755
-3.793
0.50
-0.366
-0.369
5.00
3.776
3.774
50
120
0.05
-2.776
-2.978
0.50
0.389
0.307
5.00
4.304
4.282
90
230
0.05
-1.857
-2.026
0.50
1.269
1.106
5.00
4.929
4.904


#### 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 theβ
+ 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(15)
F
E
C
K
,
L
1
=
π
2
q
K
2
g
K
2
B
K
+
q
L
1
2
g
L
1
2
B
L
1
.For theq
K
/
L
1 quantities we used the expression(16)
q
K
/
L
1
=
W
E
C
-
ϵ
K
/
L
1
,where W
E
C is the Q-value of the β
+ decay in m
e
c
2 units, ϵ
i is the binding energies of the 1
s
1
/
2 and 2
s
1
/
2 electron orbitals of the parent nucleus, and g
i is their radial densities on the nuclear surface. B
i
≈
1 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 W
0
=
W
E
C
-
1 holds.g
K
/
L
1 are the electron bound states and solutions of the Dirac equation (10) and correspond to the eigenvalues ϵ
n (n is the radial quantum number). The quantum number κ is related to the total angular momentum j
κ
=
|
κ
|
-
1
/
2. These wave functions are normalized such that(17)
∫
0
∞
g
n
,
κ
2
r
+
f
n
,
κ
2
r
d
r
=
1
.For theE
C processes, the potential used to obtain the electron wave functions reads(18)
r
V
E
C
Z
,
r
=
r
V
Z
,
r
ϕ
r
,and the charge number Z
=
Z
0 corresponds to the parent nucleus. V
(
Z
,
r
) is negative. More details about the numerical procedure can be found in [16].


## 2.1. Half-Life Calculation

β-decay half-lives can be calculated as a sum over all transition probabilities to the daughter nucleus states through excitation energies lying within the Q
β value(1)
T
1
/
2
=
∑
0
≤
E
f
≤
Q
β
1
t
f
-
1
,where the partial half-lives (PHL), t
f, can be calculated using(2)
t
f
=
C
g
A
/
g
v
2
F
A
Z
,
A
,
E
B
G
T
E
f
+
F
V
Z
,
A
,
E
B
F
E
f
.In (2) value of C was taken as 6143 s [18] and g
A, g
v are axial-vector and vector coupling constants of the weak interaction, respectively, having g
A/g
v= -1.2694 [19], while E
f is the final state energy. E
=
Q
β
-
E
f where Q
β is the window accessible to either β
+-, β
--, or EC decay. F
A
/
V are the PSFs. B
G
T and B
F are the reduced transition probabilities for Gamow-Teller and Fermi transitions, respectively, and expressed as(3)
B
F
E
f
=
1
2
I
i
+
1
f
M
F
i
2
,
(4)
B
G
T
E
f
=
1
2
I
i
+
1
f
M
G
T
i
2In (3) and (4), I
i denotes the spin of the parent state, M
F and M
G
T 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 χ for particle-particle and κ for particle-hole interactions. Here, we used the same range for χ and κ as was discussed in [7, 8]. Deformation parameter values β
2 for all cases were taken from Ref. [21]. For pairing gaps we used a global approach Δ
n = Δ
p = 12/A [MeV]. A large model space up to 7ħ
ω 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 forβ
+
/
β
- Transitions

The formalism for the PSF calculation forβ
+
/
β
- 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 β-branch is given by(5)
λ
0
=
G
β
2
2
π
3
∫
1
W
0
p
W
W
0
-
W
2
S
0
Z
,
W
d
W
,where G
β is the weak interaction coupling constant, p is the momentum of β-particle, W = p
2
+
1 is the total energy of β-particle, and W
0 is the maximum β-particle energy. W
0 = Q
-
1 (Q
+
1) in β
+ (β
-) decay. Q is the mass difference between initial and final states of neutral atoms. Equation (5) is written in natural units (ħ
=
m
=
c
=
1), so that the unit of momentum is m
c, the unit of energy is m
c
2, and the unit of time is ħ /m
c
2. The shape factors S
0
(
Z
,
W
) for allowed transitions which appear in (5) are defined as(6)
S
0
Z
,
W
=
λ
1
Z
,
W
M
0,1
2
,where M
0,1 are the NMEs related to the Fermi and Gamow-Teller reduced transition probabilities as(7)
M
0,1
2
=
1
2
I
i
+
1
B
F
,
G
T
,and λ
1
(
Z
,
W
) stands for Fermi functions. For the calculation of the β-decay rates, one needs to calculate the NMEs and the PSFs that can be defined as(8)
F
β
+
/
β
-
=
∫
1
W
0
p
W
W
0
-
W
2
λ
1
W
d
W
.The above formula determines the PSFs for both the Fermi and Gamow-Teller allowed transitions, by substitutingF
V or F
A in (2), respectively. For the allowed β-decay the Fermi functions can be expressed as(9)
λ
1
Z
,
W
=
g
-
1
2
+
f
1
2
2
p
2
,which is just the definition used in [17] (see (3)), for our particular case k
=
1. We note that in the above formula a coefficient 1
/
(
2
p
2
) appears. Usually, this coefficient is included in the proper normalization of the wave functions, as we did. The functions g
-
1
(
Z
,
W
) and f
1
(
Z
,
W
) are the large and small radial components of the positron (or electron) radial wave functions evaluated at the nuclear radius R. They are solutions of the coupled set of differential equations [17](10)
d
d
r
+
κ
+
1
r
g
κ
W
,
r
=
W
+
V
r
+
1
f
κ
W
,
r
d
d
r
+
κ
-
1
r
f
κ
W
,
r
=
-
W
+
V
r
-
1
g
κ
W
,
rwhere V
(
r
) is the central potential for the positron (or the electron) and κ
=
(
l
-
j
)
(
2
j
+
1
) is the relativistic quantum number.Ideally, the central potentialV
(
r
) 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 V
(
r
) from a realistic proton density distribution in the nucleus. The charge density can be written as(11)
ρ
e
r
→
=
∑
i
2
j
i
+
1
v
i
2
ψ
i
r
→
2
,where ψ
i is the proton wave function of the spherical single particle state i and v
i is its occupation amplitude. The wave functions ψ
i were found by solving Schrödinger equation with a Wood-Saxon potential. The (
2
j
i
+
1
) term in (11) reflects the spin degeneracy. As an example, we depict the realistic proton density for X
e
120 in cylindrical coordinates in Figure 1(a). The profile of this proton density for the daughter nucleus X
e
120 (thick line) is compared with a constant density (dot-dashed line) in Figure 1(b).Figure 1
(a) Realistic proton density forX
e
120 represented in cylindrical coordinates. (b) Profile of the realistic proton density for X
e
120 (thick line) compared with that given by a constant density approximation (dot-dashed line).
(a)
(b)We integrated the realistic charge distribution over the volume of nucleus, in order to find the Coulomb potential(12)
V
Z
,
r
=
α
ħ
c
∫
ρ
e
r
′
→
r
→
-
r
′
→
d
r
′
→
.Moreover, we included the screening effect by multiplying the expression ofV
(
r
) with a function ϕ
(
r
), which is solution of the Thomas-Fermi equation(13)
d
2
ϕ
d
x
2
=
ϕ
3
/
2
x
,with x
=
r
/
b, b
≈
0.8853
a
0
Z
-
1
/
3, and a
0 being the Bohr radius. The solution ϕ
(
r
) was calculated within the Majorana method [22]. In the case of β
-
/
β
+, the effective potential V
β
∓ was modified by the screening function ϕ
(
r
) as(14)
r
V
β
∓
Z
,
r
=
r
V
Z
,
r
+
1
×
ϕ
r
-
1
.So, asymptotically we returned the interaction between an ion of charge ±
1 of the residual nucleus and the emitted electron/proton. The Coulomb potential in atomic units is Z
/
r and is negative/positive for β
-
/
β
+ decay. As mentioned in [22], the Thomas-Fermi equation (13) is a universal equation which does not depend on Z or other physical constants. The boundaries of the screening function are ϕ
(
0
)
=
1 and ϕ
(
∞
)=0. Then, the effective potential is V
β
∓
(
r
)
=
V
(
r
) for r=0 and is suppressed according to the variation of the universal screening function when r increases in order to reach the asymptotic behavior V
β
∓
(
r
)
=
-
1
/
r for r
→
∞. 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.Figure 2
The effective screening functionϕ
e
f
f
=
V
β
∓
/
V as function of the dimensionless distance x for Z=54. The dashed and the dot-dashed lines correspond to β
- and β
+ decay, respectively. The full line gives the behavior of the screening function with the boundaries ϕ
(
0
)=1 and ϕ
(
∞
)=0, as given in [22].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 (f, g) 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 Γ 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 ofG
M [17], we compare the PSF results for three virtual cases as discussed in [17]. As seen from Table 1 for β
+ and Table 2 for β
-, 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.Table 1
Comparison of PSF results calculated with our recipe (TW) and Gove and Martin recipe (G
M), for β
+ decay of three virtual nuclei.
Z
A
Q
β
+
(
M
e
V
)
l
o
g
F
β
+
G
M  [17]
l
o
g
F
β
+
T
W
10
20
0.05
-4.643
-4.646
0.50
-0.700
-0.702
5.00
3.575
3.574
50
120
0.05
-6.117
-6.195
0.50
-1.152
-1.175
5.00
3.319
3.311
90
230
0.05
-7.088
-7.272
0.50
-1.330
-1.388
5.00
3.236
3.206Table 2
Comparison of PSF results calculated with our recipe (TW) and (G
M), for β
- decay of three virtual nuclei.
Z
A
Q
β
-
(
M
e
V
)
l
o
g
F
β
-
G
M  [17]
l
o
g
F
β
-
T
W
10
20
0.05
-3.755
-3.793
0.50
-0.366
-0.369
5.00
3.776
3.774
50
120
0.05
-2.776
-2.978
0.50
0.389
0.307
5.00
4.304
4.282
90
230
0.05
-1.857
-2.026
0.50
1.269
1.106
5.00
4.929
4.904


### 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 theβ
+ 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(15)
F
E
C
K
,
L
1
=
π
2
q
K
2
g
K
2
B
K
+
q
L
1
2
g
L
1
2
B
L
1
.For theq
K
/
L
1 quantities we used the expression(16)
q
K
/
L
1
=
W
E
C
-
ϵ
K
/
L
1
,where W
E
C is the Q-value of the β
+ decay in m
e
c
2 units, ϵ
i is the binding energies of the 1
s
1
/
2 and 2
s
1
/
2 electron orbitals of the parent nucleus, and g
i is their radial densities on the nuclear surface. B
i
≈
1 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 W
0
=
W
E
C
-
1 holds.g
K
/
L
1 are the electron bound states and solutions of the Dirac equation (10) and correspond to the eigenvalues ϵ
n (n is the radial quantum number). The quantum number κ is related to the total angular momentum j
κ
=
|
κ
|
-
1
/
2. These wave functions are normalized such that(17)
∫
0
∞
g
n
,
κ
2
r
+
f
n
,
κ
2
r
d
r
=
1
.For theE
C processes, the potential used to obtain the electron wave functions reads(18)
r
V
E
C
Z
,
r
=
r
V
Z
,
r
ϕ
r
,and the charge number Z
=
Z
0 corresponds to the parent nucleus. V
(
Z
,
r
) is negative. More details about the numerical procedure can be found in [16].


## 2.2.1. Phase Space Factors forβ
+
/
β
- Transitions

The formalism for the PSF calculation forβ
+
/
β
- 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 β-branch is given by(5)
λ
0
=
G
β
2
2
π
3
∫
1
W
0
p
W
W
0
-
W
2
S
0
Z
,
W
d
W
,where G
β is the weak interaction coupling constant, p is the momentum of β-particle, W = p
2
+
1 is the total energy of β-particle, and W
0 is the maximum β-particle energy. W
0 = Q
-
1 (Q
+
1) in β
+ (β
-) decay. Q is the mass difference between initial and final states of neutral atoms. Equation (5) is written in natural units (ħ
=
m
=
c
=
1), so that the unit of momentum is m
c, the unit of energy is m
c
2, and the unit of time is ħ /m
c
2. The shape factors S
0
(
Z
,
W
) for allowed transitions which appear in (5) are defined as(6)
S
0
Z
,
W
=
λ
1
Z
,
W
M
0,1
2
,where M
0,1 are the NMEs related to the Fermi and Gamow-Teller reduced transition probabilities as(7)
M
0,1
2
=
1
2
I
i
+
1
B
F
,
G
T
,and λ
1
(
Z
,
W
) stands for Fermi functions. For the calculation of the β-decay rates, one needs to calculate the NMEs and the PSFs that can be defined as(8)
F
β
+
/
β
-
=
∫
1
W
0
p
W
W
0
-
W
2
λ
1
W
d
W
.The above formula determines the PSFs for both the Fermi and Gamow-Teller allowed transitions, by substitutingF
V or F
A in (2), respectively. For the allowed β-decay the Fermi functions can be expressed as(9)
λ
1
Z
,
W
=
g
-
1
2
+
f
1
2
2
p
2
,which is just the definition used in [17] (see (3)), for our particular case k
=
1. We note that in the above formula a coefficient 1
/
(
2
p
2
) appears. Usually, this coefficient is included in the proper normalization of the wave functions, as we did. The functions g
-
1
(
Z
,
W
) and f
1
(
Z
,
W
) are the large and small radial components of the positron (or electron) radial wave functions evaluated at the nuclear radius R. They are solutions of the coupled set of differential equations [17](10)
d
d
r
+
κ
+
1
r
g
κ
W
,
r
=
W
+
V
r
+
1
f
κ
W
,
r
d
d
r
+
κ
-
1
r
f
κ
W
,
r
=
-
W
+
V
r
-
1
g
κ
W
,
rwhere V
(
r
) is the central potential for the positron (or the electron) and κ
=
(
l
-
j
)
(
2
j
+
1
) is the relativistic quantum number.Ideally, the central potentialV
(
r
) 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 V
(
r
) from a realistic proton density distribution in the nucleus. The charge density can be written as(11)
ρ
e
r
→
=
∑
i
2
j
i
+
1
v
i
2
ψ
i
r
→
2
,where ψ
i is the proton wave function of the spherical single particle state i and v
i is its occupation amplitude. The wave functions ψ
i were found by solving Schrödinger equation with a Wood-Saxon potential. The (
2
j
i
+
1
) term in (11) reflects the spin degeneracy. As an example, we depict the realistic proton density for X
e
120 in cylindrical coordinates in Figure 1(a). The profile of this proton density for the daughter nucleus X
e
120 (thick line) is compared with a constant density (dot-dashed line) in Figure 1(b).Figure 1
(a) Realistic proton density forX
e
120 represented in cylindrical coordinates. (b) Profile of the realistic proton density for X
e
120 (thick line) compared with that given by a constant density approximation (dot-dashed line).
(a)
(b)We integrated the realistic charge distribution over the volume of nucleus, in order to find the Coulomb potential(12)
V
Z
,
r
=
α
ħ
c
∫
ρ
e
r
′
→
r
→
-
r
′
→
d
r
′
→
.Moreover, we included the screening effect by multiplying the expression ofV
(
r
) with a function ϕ
(
r
), which is solution of the Thomas-Fermi equation(13)
d
2
ϕ
d
x
2
=
ϕ
3
/
2
x
,with x
=
r
/
b, b
≈
0.8853
a
0
Z
-
1
/
3, and a
0 being the Bohr radius. The solution ϕ
(
r
) was calculated within the Majorana method [22]. In the case of β
-
/
β
+, the effective potential V
β
∓ was modified by the screening function ϕ
(
r
) as(14)
r
V
β
∓
Z
,
r
=
r
V
Z
,
r
+
1
×
ϕ
r
-
1
.So, asymptotically we returned the interaction between an ion of charge ±
1 of the residual nucleus and the emitted electron/proton. The Coulomb potential in atomic units is Z
/
r and is negative/positive for β
-
/
β
+ decay. As mentioned in [22], the Thomas-Fermi equation (13) is a universal equation which does not depend on Z or other physical constants. The boundaries of the screening function are ϕ
(
0
)
=
1 and ϕ
(
∞
)=0. Then, the effective potential is V
β
∓
(
r
)
=
V
(
r
) for r=0 and is suppressed according to the variation of the universal screening function when r increases in order to reach the asymptotic behavior V
β
∓
(
r
)
=
-
1
/
r for r
→
∞. 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.Figure 2
The effective screening functionϕ
e
f
f
=
V
β
∓
/
V as function of the dimensionless distance x for Z=54. The dashed and the dot-dashed lines correspond to β
- and β
+ decay, respectively. The full line gives the behavior of the screening function with the boundaries ϕ
(
0
)=1 and ϕ
(
∞
)=0, as given in [22].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 (f, g) 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 Γ 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 ofG
M [17], we compare the PSF results for three virtual cases as discussed in [17]. As seen from Table 1 for β
+ and Table 2 for β
-, 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.Table 1
Comparison of PSF results calculated with our recipe (TW) and Gove and Martin recipe (G
M), for β
+ decay of three virtual nuclei.
Z
A
Q
β
+
(
M
e
V
)
l
o
g
F
β
+
G
M  [17]
l
o
g
F
β
+
T
W
10
20
0.05
-4.643
-4.646
0.50
-0.700
-0.702
5.00
3.575
3.574
50
120
0.05
-6.117
-6.195
0.50
-1.152
-1.175
5.00
3.319
3.311
90
230
0.05
-7.088
-7.272
0.50
-1.330
-1.388
5.00
3.236
3.206Table 2
Comparison of PSF results calculated with our recipe (TW) and (G
M), for β
- decay of three virtual nuclei.
Z
A
Q
β
-
(
M
e
V
)
l
o
g
F
β
-
G
M  [17]
l
o
g
F
β
-
T
W
10
20
0.05
-3.755
-3.793
0.50
-0.366
-0.369
5.00
3.776
3.774
50
120
0.05
-2.776
-2.978
0.50
0.389
0.307
5.00
4.304
4.282
90
230
0.05
-1.857
-2.026
0.50
1.269
1.106
5.00
4.929
4.904


## 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 theβ
+ 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(15)
F
E
C
K
,
L
1
=
π
2
q
K
2
g
K
2
B
K
+
q
L
1
2
g
L
1
2
B
L
1
.For theq
K
/
L
1 quantities we used the expression(16)
q
K
/
L
1
=
W
E
C
-
ϵ
K
/
L
1
,where W
E
C is the Q-value of the β
+ decay in m
e
c
2 units, ϵ
i is the binding energies of the 1
s
1
/
2 and 2
s
1
/
2 electron orbitals of the parent nucleus, and g
i is their radial densities on the nuclear surface. B
i
≈
1 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 W
0
=
W
E
C
-
1 holds.g
K
/
L
1 are the electron bound states and solutions of the Dirac equation (10) and correspond to the eigenvalues ϵ
n (n is the radial quantum number). The quantum number κ is related to the total angular momentum j
κ
=
|
κ
|
-
1
/
2. These wave functions are normalized such that(17)
∫
0
∞
g
n
,
κ
2
r
+
f
n
,
κ
2
r
d
r
=
1
.For theE
C processes, the potential used to obtain the electron wave functions reads(18)
r
V
E
C
Z
,
r
=
r
V
Z
,
r
ϕ
r
,and the charge number Z
=
Z
0 corresponds to the parent nucleus. V
(
Z
,
r
) 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 G
M [17]. We again state that the same set of NMEs was used in both types of half-life calculations. For the G
M 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 G
M 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 K and L
1 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].Table 3
The first four columns give the electron binding energiesϵ
i for the shells K and L
1 and their radial wave function densities g
i at the nuclear surface calculated in this work for different nuclei. The last two columns give the values of [30].
Nucleus
ϵ
K (keV)
ϵ
L
1 (keV)
g
K
2
(
[
ħ
/
m
c
]
-
3
)
g
L
1
2
/
g
K
2
g
K
2
(
[
ħ
/
m
c
]
-
3
) [30]
g
L
1
2
/
g
K
2  [30]
52Fe
6.63991
0.7451126
0.0327063
0.0898416
0.0328
0.0950
56Ni
7.95277
0.9181003
0.0411267
0.0943036
0.0423
0.0974
62Zn
9.20973
1.1092315
0.0530401
0.0953901
0.0538
0.0995
76Br
13.0121
1.6760343
0.0917936
0.1019465
0.0935
0.1035
81Rb
14.6718
1.9336296
0.1136202
0.1042520
0.1149
0.1063
88Y
16.4688
2.0221700
0.1389702
0.1272602
0.1402
0.1080
90Nb
18.3994
2.4889101
0.1686632
0.1121010
0.170
0.1098
102Cd
26.1177
3.9008765
0.3182812
0.1109908
0.319
0.1159
105Ag
24.95904
3.558919
0.2900978
0.1193451
0.293
0.1150
107Sb
29.99173
4.140248
0.4109690
0.1416423
0.413
0.1187
113Sb
29.99173
4.140248
0.4101592
0.1416413
0.413
0.1187
113Te
31.18294
4.70109
0.4488353
0.12019908
0.449
0.1196
115I
32.50419
4.937340
0.4894928
0.1210013
0.488
0.1205
116I
32.50419
4.937340
0.4893257
0.1210012
0.488
0.1205
116Xe
33.95443
5.055330
0.5289785
0.1300301
0.529
0.1215
120Ba
36.81175
4.975840
0.6244052
0.1662902
0.623
0.1234
120Xe
33.95440
5.055328
0.5280176
0.1300293
0.529
0.1215
126Cs
35.30411
5.151703
0.5764054
0.1394027
0.574
0.1224
182Re
71.29588
12.23093
2.7488862
0.1490819
2.69
0.1448
205Bi
90.39904
16.18943
5.0324494
0.1587003
4.88
0.1561The 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(
g
k
2
+
f
k
2
)
r
2 are displayed in Figure 3 for these two treatments in the case the parent nucleus C
a
38. 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 g
i
2 and of the electron binding energies ϵ
i, as it can be seen in Table 4. Adding the screening effect in our calculations makes our results very close to those of G
M.Table 4
The resultsG
M obtained in [30] are compared with the results TW of this work and those obtained without a realistic charge density and without screening correction in the case of C
a
38. The K and L
1 shells are illustrated. The binding energies are denoted as ϵ
i and the radial densities on the surface of the nucleus are denoted as g
i
2.
Method
ϵ
K (keV)
ϵ
L
1 (keV)
g
K
2 ([
ħ
/
m
c
]
-
3)
g
L
1
2
/
g
K
2
Ref. [30]
3.60740
0.37710
0.01367
0.08620
T
W
3.762904
0.357179
0.01308474
0.0825267
No screening
5.454032
1.198887
0.01434885
0.1877057Figure 3
The radial distributions(
g
k
2
+
f
k
2
)
r
2 are plotted for the K and L
1 shells as function of the distance r in terms of atomic units a
0 for C
a
38. The thick lines are obtained with the treatment used in this work. The thin curve is used for the potential without screening and with a constant proton density inside the nucleus. The full line is used for the K shell while the dashed one corresponds to the L
1 orbital.Table5 presents a comparison between the measured and calculated half-lives for β
+/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 (
G
M
) method [17]. The fourth column shows the calculated half-lives using our new recipe of PSFs [16] and labeled (
T
W
) (this work). Most of the nuclei shown in this table are the same as those presented in Table 2 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 (
G
M
) [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(19)
P
D
=
T
1
/
2
T
W
-
T
1
/
2
G
M
T
1
/
2
T
W
×
100
%Table 5
Comparison of measured, calculated half-lives and percentage deviation (PD) forβ
+/EC-decay of selected nuclei. For the case of 205Bi we calculate only L-shell EC.
Nucleus
T
1
/
2
(
E
X
P
)  (s) [23, 24]
T
1
/
2
(
G
M
)  (s) [17]
T
1
/
2
(
T
W
) (s)
PD (%)
52Fe
2.98E+04
1.29E+04
1.30E+04
0.77
56Ni
5.25E+05
4.26E+05
4.44E+05
4.05
62Zn
3.31E+04
9.80E+03
1.01E+04
2.97
76Br
5.83E+04
1.62E+04
1.66E+04
2.41
81Rb
1.65E+04
5.00E+03
5.12E+03
2.34
88Y
9.21E+06
1.25E+07
1.27E+07
1.57
90Nb
5.26E+04
4.25E+04
4.32E+04
1.62
102Cd
3.30E+02
2.35E+02
2.42E+02
2.89
105Ag
3.57E+06
2.45E+04
2.52E+04
2.78
107Sb
4.00E+00
3.92E+00
4.04E+00
2.97
113Sb
4.00E+02
2.42E+02
2.47E+02
2.02
113Te
1.02E+02
9.55E+01
9.77E+01
2.25
115I
3.48E+02
9.98E+01
1.02E+02
2.16
116I
2.91E+00
9.49E-01
9.73E-01
2.47
116Xe
5.90E+01
2.01E+01
2.05E+01
1.95
120Ba
2.40E+01
1.73E+01
1.76E+01
1.70
120Xe
2.76E+03
1.58E+03
1.61E+03
1.86
126Cs
9.84E+01
5.35E+02
5.42E+02
1.29
182Re
2.30E+05
3.67E+05
3.80E+05
3.42
205Bi
1.32E+06
1.47E+06
1.52E+06
3.46Table5 shows that the PD increases to a maximum value of 4.05 % for the N
i
56 nucleus. The case of EC on B
i
205 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 β
+/EC decay half-lives, the smallest difference was noted for F
e
52. In Table 6 we show the state-by-state transitions for two cases: F
e
52 and N
i
56. Shown are also the adopted NMEs using the pn-QRPA model, the calculated PSFs (separately for both EC and β
+-decay reactions), partial half-lives (PHL), Q-values, and branching ratios I
(
β
+
/
E
C
). The branching ratio ‘I’ for each transition was calculated using the formula(20)
I
=
T
1
/
2
t
f
×
100
%
,where T
1
/
2 is the total β decay half-life and t
f is the calculated partial half-life of the corresponding transition. For the nucleus N
i
56 we note calculation of much smaller PSF for EC decay to daughter energies using our recipe. The PSFs calculated from (
G
M
) recipe is on average within 3 % smaller. This in turn led to a 4 % larger calculated half-life value for N
i
56 using our recipe.Table 6
State-by-state comparison of calculated PSF (forβ
+/EC-decay) using recipe of [17] and current prescription (TW). Shown also are the daughter energy levels, nuclear matrix elements NME, Q values, partial half-lives (PHL) for β
+/EC-decay, and branching ratio I
(
β
+
/
E
C
) of the selected nuclei.
E
x(MeV)
NME
Q
E
C (MeV)
F
E
C
(
G
M
) [17]
F
E
C
(
T
W
)
Q
β
+ (MeV)
F
β
+
(
G
M
)  [17]
F
β
+
(
T
W
)
P
H
L
(
G
M
) [17]
P
H
L
(
T
W
)
I
(
β
+
/
E
C
)
(
G
M
)  [17]
I
(
β
+
/
E
C
)
(
T
W
)
52
F
e
0.000
0.02768
2.3733
1.22206
1.20150
1.3512
8.41032
8.31627
1.50147E+04
1.51955E+04
85.701
85.713
0.004
0.00170
2.3693
1.21794
1.19744
1.3472
8.30402
8.21074
2.46699E+05
2.49683E+05
5.2160
5.2160
0.196
0.00325
2.1773
1.02771
1.01077
1.1552
4.29193
4.24688
2.31350E+05
2.34078E+05
5.5620
5.5640
0.291
0.00134
2.0823
0.94007
0.92425
1.0602
2.98166
2.94390
7.60563E+05
7.71098E+05
1.6920
1.6890
0.720
0.00253
1.6533
0.59198
0.58175
0.6312
0.33565
0.32942
1.70770E+06
1.73853E+06
0.7540
0.7490
0.939
0.00087
1.4343
0.44543
0.43734
0.4122
5.69E-02
5.53E-02
9.20851E+06
9.39057E+06
0.1400
0.1390
1.011
0.00350
1.3623
0.40124
0.39435
0.3402
2.53E-02
2.47E-02
2.67938E+06
2.72749E+06
0.4800
0.4780
1.362
0.00255
1.0113
0.22076
0.21663
-0.0107
-
-
7.10887E+06
7.24422E+06
0.1810
0.1800
1.467
0.00052
0.9063
0.17691
0.17374
-0.1157
-
-
4.36772E+07
4.44742E+07
0.0290
0.0290
1.685
0.00017
0.6883
1.01706E-01
9.97773E-02
-0.3371
-
-
2.35223E+08
2.39771E+08
0.0050
0.0050
1.754
0.00009
0.6193
8.22094E-02
8.06130E-02
-0.4027
-
-
5.31896E+08
5.42429E+08
0.0020
0.0020
1.821
0.00972
0.5523
6.54251E-02
6.39581E-02
-0.4697
-
-
6.29314E+06
6.43748E+06
0.2040
0.2020
2.119
0.00282
0.2543
1.35835E-02
1.32041E-02
-0.7677
-
-
1.04562E+08
1.07568E+08
0.0120
0.0120
2.143
0.00572
0.2303
1.10469E-02
1.07735E-02
-0.7917
-
-
6.33834E+07
6.49917E+07
0.0200
0.0200
56
N
i
1.196
0.00038
0.9357
0.24313
0.23335
-0.0862
-
-
4.31594E+07
4.49676E+07
0.986
0.988
1.247
0.00019
0.8847
0.21729
0.20842
-0.1372
-
-
9.85234E+07
1.02717E+08
0.432
0.432
1.252
0.00034
0.8797
0.21476
0.20605
-0.1422
-
-
5.55046E+07
5.78528E+07
0.767
0.768
1.288
0.00009
0.8437
0.19721
0.18939
-0.1782
-
-
2.13703E+08
2.22525E+08
0.199
0.200
1.289
0.00009
0.8427
0.19676
0.18894
-0.1792
-
-
2.31551E+08
2.41134E+08
0.184
0.184
1.299
0.00008
0.8327
0.19227
0.18444
-0.1892
-
-
2.69031E+08
2.80450E+08
0.158
0.158
1.309
0.00001
0.8227
0.18754
0.18000
-0.1992
-
-
1.73974E+09
1.81258E+09
0.024
0.024
1.313
0.00002
0.8187
0.18593
0.17824
-0.2032
-
-
1.22606E+09
1.27894E+09
0.035
0.035
1.318
0.00001
0.8137
0.18344
0.17605
-0.2082
-
-
1.76475E+09
1.83885E+09
0.024
0.024
1.363
0.00000
0.7687
0.16374
0.15695
-0.2532
-
-
4.34936E+10
4.53747E+10
0.001
0.001
1.363
0.00000
0.7687
0.16372
0.15695
-0.2532
-
-
2.36127E+10
2.46320E+10
0.002
0.002
1.373
0.00000
0.7587
0.15927
0.15285
-0.2632
-
-
7.54380E+10
7.86020E+10
0.001
0.001
1.471
0.00000
0.6607
0.12034
0.11558
-0.3612
-
-
1.51481E+10
1.57716E+10
0.003
0.003
1.482
0.00001
0.6497
0.11645
0.11172
-0.3722
-
-
3.31503E+09
3.45531E+09
0.013
0.013
1.503
0.00001
0.6287
0.10904
0.10454
-0.3932
-
-
3.72258E+09
3.88306E+09
0.011
0.011
1.711
0.07180
0.4207
4.82788E-02
4.62700E-02
-0.6012
-
-
1.15477E+06
1.20491E+06
36.85
36.85
1.742
0.13730
0.3897
4.13076E-02
3.95910E-02
-0.6322
-
-
7.05812E+05
7.36415E+05
60.30
60.30We would like to comment further on the entries in the first two columns of Table6. 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:(21)
Q
E
C
=
m
p
-
m
d
-
E
f
,and(22)
Q
B
P
=
m
p
-
m
d
-
E
f
-
2
m
e
c
2
.Here m
p and m
d are masses of parent and daughter nuclei, respectively, whereas E
f is the calculated energy levels in the daughter nucleus (model dependent).Table7 shows the comparison of measured and calculated half-lives for β
- decay cases. The Q-values were taken from [23, 24]. The comparison between the two calculations is much better for the β
- decay half-lives than those corresponding to β
+. Table 8 shows the state-by-state calculation of PHL for S
r
100 (largest PD=3.82 %) and N
d
152 (PD=2.04 %). The values of Q
β
- (calculated as in (21)), PSF, NME, and branching ratios I
(
β
-
) are also given in Table 8. We note an overall agreement between our calculated PSF values and those using the (
G
M
) recipe in the case of both analyzed nuclei.Table 7
Same as Table5 but for β
--decaying nuclei.
Nucleus
T
1
/
2
E
X
P  (s) [23, 24]
T
1
/
2
(
G
M
)  (s) [17]
T
1
/
2
(
T
W
)  (s)
PD (%)
98Sr
6.53E-01
4.45E-01
4.57E-01
2.63
100Sr
2.02E-01
2.28E-01
2.37E-01
3.82
100Zr
7.10E+00
8.04E+00
8.25E+00
2.55
102Zr
2.90E+00
8.45E+00
8.73E+00
3.21
102Mo
6.78E+02
1.90E+02
1.94E+02
2.06
104Mo
6.00E+00
5.93E+00
6.08E+00
2.47
106Mo
8.73E+00
6.35E+00
6.53E+00
2.76
108Ru
2.73E+02
5.61E+02
5.74E+02
2.26
110Ru
1.20E+01
6.27E+01
6.46E+01
2.94
112Ru
1.75E+00
6.27E+00
6.47E+00
3.09
112Pd
7.57E+04
3.89E+04
4.00E+04
2.75
114Pd
1.45E+02
1.00E+02
1.03E+02
2.91
116Pd
1.18E+01
4.53E+01
4.67E+01
3.00
138Xe
8.44E+02
4.69E+02
4.84E+02
3.10
140Xe
1.36E+01
1.36E+00
1.40E+00
2.86
142Ba
6.36E+02
7.97E+02
8.19E+02
2.69
144Ba
1.15E+01
2.44E+01
2.51E+01
2.79
146Ce
8.11E+02
4.74E+01
4.85E+01
2.27
148Ce
5.60E+01
7.93E+01
8.13E+01
2.46
152Nd
6.84E+02
8.64E+03
8.82E+03
2.04Table 8
State-by-state comparison of calculated PSF (forβ-decay) using recipe of [17] and current prescription (TW). Shown also are the daughter energy levels, nuclear matrix elements NME, Q values, partial half-lives (PHL), and branching ratio I
β
- for β
--decay of the selected nuclei.
E
x (MeV)
NME
Q
β
- (MeV)
F
β
-
(
G
M
)  [17]
F
β
-
(
T
W
)
P
H
L
(
G
M
) [17]
P
H
L
(
T
W
)
I
(
β
-
)
(
G
M
)  [17]
I
(
β
-
)
(
T
W
)
100
S
r
0.13300
0.01473
7.50300
86906.3
72966.7
3.12798E+00
3.72554E+00
7.2730
6.3720
0.35700
0.00115
7.14579
69557.1
64334.6
4.99967E+01
5.40553E+01
0.4550
0.4390
0.86400
0.00510
6.63935
49775.9
47317.7
1.57827E+01
1.66026E+01
1.4410
1.4300
1.00300
0.00322
6.50023
45212.0
43184.3
2.75281E+01
2.88207E+01
0.8260
0.8240
1.06300
0.00429
6.44024
43349.3
41485.6
2.15140E+01
2.24805E+01
1.0570
1.0560
1.10400
0.00278
6.39893
42102.4
40346.8
3.42401E+01
3.57299E+01
0.6640
0.6640
1.35000
0.26502
6.15278
35247.6
33999.4
4.28548E-01
4.44280E-01
53.084
53.434
1.41900
0.01521
6.08400
33499.6
32359.0
7.85740E+00
8.13437E+00
2.8950
2.9180
1.62000
0.00002
5.88283
28778.0
27883.9
7.89102E+03
8.14407E+03
0.0030
0.0030
1.64200
0.00000
5.86112
28301.8
27429.7
2.05672E+05
2.12210E+05
0.0000
0.0000
1.66900
0.00080
5.83372
27709.5
26864.9
1.81631E+02
1.87341E+02
0.1250
0.1270
2.03600
0.03084
5.46720
20688.6
20109.3
6.27339E+00
6.45412E+00
3.6260
3.6780
2.19100
0.00096
5.31207
18180.4
17678.5
2.30158E+02
2.36693E+02
0.0990
0.1000
2.21100
0.00002
5.29152
17867.1
17374.4
1.26201E+04
1.29779E+04
0.0020
0.0020
2.21700
0.00126
5.28623
17787.1
17296.8
1.79065E+02
1.84140E+02
0.1270
0.1290
2.33500
0.01496
5.16812
16074.2
15633.6
1.66481E+01
1.71173E+01
1.3660
1.3870
2.40500
0.00704
5.09834
15125.2
14712.6
3.75885E+01
3.86428E+01
0.6050
0.6140
2.48600
0.05959
5.01712
14077.3
13695.1
4.77186E+00
4.90505E+00
4.7670
4.8400
2.55600
0.00015
4.94728
13222.8
12865.6
2.04511E+03
2.10190E+03
0.0110
0.0110
2.59700
0.00001
4.90586
12735.9
12392.7
3.58668E+04
3.68600E+04
0.0010
0.0010
2.71000
0.00051
4.79275
11477.9
11170.8
6.88307E+02
7.07229E+02
0.0330
0.0340
2.73300
0.04131
4.76967
11233.8
10933.7
8.62714E+00
8.86390E+00
2.6370
2.6780
2.92100
0.00092
4.58202
9397.47
9150.19
4.64906E+02
4.77470E+02
0.0490
0.0500
3.04900
0.00007
4.45378
8285.68
8069.38
7.37202E+03
7.56963E+03
0.0030
0.0030
3.19200
0.00721
4.31146
7176.38
6990.87
7.73167E+01
7.93683E+01
0.2940
0.2990
3.29000
0.00845
4.21307
6480.88
6314.67
7.30755E+01
7.49989E+01
0.3110
0.3170
3.29600
0.01583
4.20677
6438.24
6273.21
3.92753E+01
4.03085E+01
0.5790
0.5890
3.36400
0.00058
4.13861
5990.84
5838.16
1.15661E+03
1.18685E+03
0.0200
0.0200
3.43900
0.00029
4.06433
5531.71
5391.47
2.52582E+03
2.59152E+03
0.0090
0.0090
3.45000
0.01045
4.05281
5463.09
5324.69
7.01033E+01
7.19255E+01
0.3250
0.3300
3.47100
0.00612
4.03206
5341.13
5205.96
1.22440E+02
1.25619E+02
0.1860
0.1890
3.49600
0.00482
4.00734
5198.67
5067.29
1.59594E+02
1.63732E+02
0.1430
0.1450
3.56900
0.00134
3.93407
4794.07
4673.52
6.24537E+02
6.40647E+02
0.0360
0.0370
3.68200
0.11627
3.82086
4218.21
4112.92
8.16217E+00
8.37111E+00
2.7870
2.8360
3.88800
0.00006
3.61534
3313.42
3232.01
2.10528E+04
2.15831E+04
0.0010
0.0010
4.05100
0.01897
3.45234
2711.28
2645.88
7.78203E+01
7.97437E+01
0.2920
0.2980
4.08800
0.37148
3.41515
2586.91
2524.78
4.16561E+00
4.26810E+00
5.4610
5.5620
4.12800
0.20740
3.37510
2458.04
2399.32
7.85229E+00
8.04448E+00
2.8970
2.9510
4.16900
0.00354
3.33446
2332.50
2277.04
4.84389E+02
4.96188E+02
0.0470
0.0480
4.22200
0.00034
3.28135
2176.15
2124.70
5.47159E+03
5.60410E+03
0.0040
0.0040
4.30800
0.00556
3.19476
1939.11
1893.63
3.71131E+02
3.80044E+02
0.0610
0.0620
4.31100
0.00722
3.19210
1932.17
1886.87
2.87049E+02
2.93941E+02
0.0790
0.0810
4.40800
0.00101
3.09502
1691.86
1652.46
2.33941E+03
2.39520E+03
0.010
0.010
4.43400
0.00003
3.06860
1630.72
1592.78
8.24004E+04
8.43631E+04
0.000
0.000
4.53600
0.00007
2.96655
1410.81
1378.03
4.15176E+04
4.25053E+04
0.001
0.001
4.56000
0.00318
2.94261
1362.77
1331.13
9.23619E+02
9.45568E+02
0.025
0.025
4.62800
0.00284
2.87494
1233.96
1205.44
1.14035E+03
1.16733E+03
0.020
0.020
4.63100
0.03599
2.87213
1228.83
1200.43
9.05090E+01
9.26499E+01
0.251
0.256
4.68800
0.00938
2.81463
1127.45
1101.55
3.78490E+02
3.87390E+02
0.060
0.061
4.70300
0.01517
2.79959
1102.05
1076.78
2.39492E+02
2.45112E+02
0.095
0.097
4.78400
0.00317
2.71946
974.255
952.119
1.29597E+03
1.32610E+03
0.018
0.018
4.86000
0.33212
2.64278
863.180
843.718
1.39636E+01
1.42857E+01
1.629
1.662
4.87900
0.15890
2.62366
837.107
818.264
3.00956E+01
3.07887E+01
0.756
0.771
4.98800
0.04141
2.51538
700.916
685.342
1.37918E+02
1.41052E+02
0.165
0.168
4.99400
0.00028
2.50876
693.199
677.809
2.02785E+04
2.07390E+04
0.001
0.001
5.02400
0.00000
2.47918
659.521
644.946
2.18678E+06
2.23620E+06
0.000
0.000
5.02400
0.00000
2.47907
659.398
644.827
3.04674E+06
3.11559E+06
0.000
0.000
5.07300
0.00072
2.42998
606.361
593.032
9.17207E+03
9.37823E+03
0.002
0.003
5.07800
0.06105
2.41548
601.620
588.399
1.08997E+02
1.11446E+02
0.209
0.213
5.08800
0.00814
2.41543
591.359
578.371
8.31303E+02
8.49970E+02
0.027
0.028
5.14500
0.00755
2.35801
534.741
523.045
9.90972E+02
1.01313E+03
0.023
0.023
5.18900
0.00337
2.31408
494.418
483.653
2.40044E+03
2.45387E+03
0.009
0.010
5.22000
0.02056
2.28304
467.396
457.267
4.16606E+02
4.25834E+02
0.055
0.056
5.24100
0.01309
2.26217
449.896
440.172
6.79596E+02
6.94608E+02
0.033
0.034
5.31000
0.00883
2.19284
395.410
386.919
1.14639E+03
1.17155E+03
0.020
0.020
5.32100
0.00091
2.18167
387.143
378.840
1.14066E+04
1.16566E+04
0.002
0.002
5.33200
0.00363
2.17111
379.456
371.325
2.90418E+03
2.96777E+03
0.008
0.008
5.38200
0.19989
2.12061
344.315
336.988
5.81626E+01
5.94271E+01
0.391
0.399
5.42200
0.29697
2.08095
318.556
311.801
4.23152E+01
4.32320E+01
0.538
0.549
5.43600
0.00100
2.06672
309.690
303.131
1.28840E+04
1.31628E+04
0.002
0.002
5.44100
0.01497
2.06191
306.737
300.243
8.71971E+02
8.90832E+02
0.026
0.027
5.47800
0.02798
2.02462
284.592
278.578
5.02652E+02
5.13504E+02
0.045
0.046
5.51600
0.00603
1.98731
263.713
258.144
2.51817E+03
2.57250E+03
0.009
0.009
5.58100
0.00030
1.92248
230.309
225.470
5.86286E+04
5.98869E+04
0.000
0.000
5.58200
0.00028
1.92101
229.593
224.770
6.27505E+04
6.40971E+04
0.000
0.000
5.68100
0.00107
1.82168
185.048
181.209
2.02077E+04
2.06358E+04
0.001
0.001
5.68800
0.00137
1.81513
182.369
178.592
1.60203E+04
1.63591E+04
0.001
0.001
5.77800
0.00050
1.72493
148.469
145.430
5.40763E+04
5.52065E+04
0.000
0.000
5.83100
0.34429
1.67201
131.012
128.381
8.87476E+01
9.05664E+01
0.256
0.262
6.01400
0.87770
1.48870
82.5496
80.9479
5.52508E+01
5.63440E+01
0.412
0.421
6.03600
0.00558
1.46719
77.9438
76.4297
9.19758E+03
9.37979E+03
0.002
0.003
6.08500
0.00058
1.41812
68.1847
66.8508
1.01248E+05
1.03268E+05
0.000
0.000
6.21900
0.65507
1.28368
46.2444
45.3545
1.32146E+02
1.34739E+02
0.172
0.176
6.31900
0.15270
1.18405
33.8740
33.2233
7.73910E+02
6.90755E+06
0.029
0.030
6.33400
0.00002
1.16928
32.2838
31.6656
6.77527E+06
6.90755E+06
0.000
0.000
6.42900
0.01233
1.07429
23.3854
22.9594
1.38889E+04
1.41466E+04
0.002
0.002
6.51300
0.01217
0.98992
17.1872
16.8828
1.91431E+04
1.94883E+04
0.001
0.001
6.51600
0.00006
0.98675
16.9821
16.6819
3.95940E+06
4.03065E+06
0.000
0.000
6.57300
0.00003
0.93005
13.6192
13.3789
9.25172E+06
9.41784E+06
0.000
0.000
6.58400
0.00331
0.91930
13.0440
12.8128
9.25986E+04
9.42691E+04
0.000
0.000
6.70900
0.00618
0.79391
7.61027
7.47817
8.50784E+04
8.65813E+04
0.000
0.000
6.72200
0.11568
0.78109
7.17285
7.04917
4.82458E+03
4.90923E+03
0.005
0.005
6.77000
0.01393
0.73306
5.70097
5.60592
5.04157E+04
5.12705E+04
0.000
0.000
6.89500
0.03451
0.60845
2.93440
2.88738
3.95262E+04
4.01699E+04
0.001
0.001
7.08500
0.00065
0.41782
0.80160
0.78904
7.62566E+06
7.74711E+06
0.000
0.000
7.10800
0.28661
0.39513
0.66388
0.65345
2.10383E+04
2.13742E+04
0.000
0.001
7.25000
0.05443
0.25270
0.15179
0.14949
4.84551E+05
4.92007E+05
0.000
0.000
7.30300
0.00180
0.19970
7.12037E-02
7.01109E-02
3.12973E+07
3.17851E+07
0.000
0.000
7.34900
0.20453
0.15433
3.14666E-02
3.09831E-03
6.21998E+05
6.31705E+05
0.000
0.000
7.43700
0.00053
6.55474E-02
2.22245E-03
2.19039E-03
3.40802E+09
3.45789E+09
0.000
0.000
7.50100
0.00000
2.44828E-03
1.11208E-07
1.39749E-07
1.78361E+18
1.41935E+18
0.000
0.000
152
N
d
0.00700
0.00023
1.10500
64.4753
63.0324
2.69504E+05
2.75673E+05
3.206
3.201
0.04300
0.00018
1.06233
57.1811
55.9198
3.98034E+05
4.07012E+05
2.170
2.168
0.04800
0.00034
1.05678
56.0922
54.8558
2.12257E+05
2.17040E+05
4.070
4.066
0.08500
0.00021
1.01958
49.1965
48.1182
3.85716E+05
3.94359E+05
2.240
2.238
0.16100
0.00075
0.94396
37.1767
36.3803
1.42898E+05
1.46027E+05
6.046
6.043
0.16100
0.00142
0.94376
37.1486
36.3528
7.57971E+04
7.74564E+04
11.39
11.39
0.17200
0.00001
0.93269
35.5966
34.8310
9.15658E+06
9.35783E+06
0.094
0.094
0.17500
0.00053
0.93033
35.2726
34.5135
2.13186E+05
2.17876E+05
4.052
4.050
0.19900
0.00037
0.90628
32.0940
31.3982
3.39245E+05
3.46763E+05
2.547
2.545
0.22900
0.00042
0.87600
28.4051
27.7849
3.34891E+05
3.42366E+05
2.580
2.578
0.27400
0.00278
0.83105
23.5294
23.0191
6.12332E+04
6.25907E+04
14.10
14.09
0.32400
0.00006
0.78073
18.8524
18.4532
3.53330E+06
3.60975E+06
0.245
0.244
0.32500
0.00058
0.77950
18.7474
18.3506
3.70394E+05
3.78405E+05
2.332
2.332
0.33100
0.00025
0.77444
18.3211
17.9336
8.65513E+05
8.84212E+05
0.998
0.998
0.33300
0.00003
0.77208
18.1243
17.7411
7.84943E+06
8.01899E+06
0.110
0.110
0.33400
0.00036
0.77115
18.0476
17.6660
6.17153E+05
6.30484E+05
1.400
1.400
0.33500
0.00000
0.77037
17.9829
17.6027
8.69619E+08
8.88405E+08
0.001
0.001
0.34200
0.00008
0.76308
17.3901
17.0228
2.78870E+06
2.84887E+06
0.310
0.310
0.35000
0.00121
0.75540
16.7806
16.4280
1.97513E+05
2.01752E+05
4.374
4.374
0.36800
0.00002
0.73716
15.3978
15.0799
1.05188E+07
1.07405E+07
0.082
0.082
0.37400
0.00000
0.73059
14.9209
14.6152
1.47159E+08
1.50237E+08
0.006
0.006
0.39700
0.00000
0.70821
13.3799
13.1094
2.66595E+08
2.72097E+08
0.003
0.003
0.40200
0.00000
0.70320
13.0521
12.7882
3.55337E+09
3.62670E+09
0.000
0.000
0.45200
0.00003
0.65259
10.0639
9.85865
1.33142E+07
1.35914E+07
0.065
0.065
0.47600
0.00005
0.62948
8.88352
8.70489
9.68863E+06
9.88744E+06
0.089
0.089
0.48800
0.00349
0.61732
8.30575
8.13928
1.38175E+05
1.41001E+05
6.252
6.259
0.51100
0.00796
0.59432
7.29006
7.14343
6.90199E+04
7.04367E+04
12.51
12.529
0.54600
0.00061
0.55945
5.92805
5.80563
1.10052E+06
1.12373E+06
0.785
0.785
0.55500
0.00137
0.54988
5.58944
5.47456
5.23679E+05
5.34669E+05
1.650
1.651
0.60100
0.00004
0.50426
4.16753
4.08228
2.47689E+07
2.52862E+07
0.035
0.035
0.60200
0.00299
0.50322
4.13846
4.05387
3.23348E+05
3.30095E+05
2.672
2.673
0.64600
0.00150
0.45851
3.02789
2.97763
8.84107E+05
9.02058E+05
0.977
0.978
0.65000
0.00629
0.45524
2.95622
2.89755
2.15336E+05
2.19696E+05
4.012
4.017
0.69600
0.00589
0.40853
2.06216
2.02139
3.29366E+05
3.36009E+05
2.623
2.626
0.72000
0.00739
0.38524
1.69907
1.66403
3.18926E+05
3.25589E+05
2.709
2.710
0.74100
0.00000
0.36417
1.41219
1.38322
1.21197E+09
1.23735E+09
0.001
0.001
0.74700
0.00029
0.35817
1.33734
1.31027
1.02657E+07
1.04794E+07
0.084
0.084
0.76400
0.00047
0.34126
1.14121
1.11921
7.42169E+06
7.57038E+06
0.116
0.117
0.78500
0.00071
0.31972
0.92353
0.90484
6.12158E+06
6.24805E+06
0.141
0.141
0.80600
0.00082
0.29870
0.74084
0.72553
6.59923E+06
6.73842E+06
0.131
0.131
0.81200
0.00282
0.29257
0.69283
0.67836
2.04836E+06
2.09207E+06
0.422
0.422
0.82100
0.00469
0.28442
0.63250
0.61922
1.34848E+06
1.37739E+06
0.641
0.641
0.85100
0.01661
0.25449
0.44274
0.43351
5.44359E+05
5.55942E+05
1.587
1.587
0.91700
0.00059
0.18846
0.17078
0.16689
3.99157E+07
4.08462E+07
0.022
0.022
0.92400
0.00261
0.18069
0.14965
0.14621
1.02513E+07
1.04925E+07
0.084
0.084
0.96800
0.00007
0.13735
6.3563E-02
6.1968E-02
9.47034E+08
9.71424E+08
0.001
0.001
0.96800
0.00093
0.13673
6.2679E-02
6.1101E-02
6.89166E+07
7.06975E+07
0.013
0.012
1.00300
0.00039
1.0210E-01
2.5429E-02
2.4708E-02
4.00310E+08
4.11999E+08
0.002
0.002The 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 Figure4 for few selected β
+ decay cases. We display the ratio between the experimental half-lives and the theoretical ones, R
=
T
1
/
2
(
e
x
p
)
/
T
1
/
2
(
X
), where (
X
) stands for the calculation recipe, (
G
M
) or (
T
W
). With solid lines the ratios calculated with (
T
W
) recipe are represented, while with dotted lines the conventional (
G
M
) computations are displayed. We note that systematically our half-lives are larger than the G
M, improving the agreement with the measured data for most of the cases. From Figure 4 it can be remarked that the (
T
W
) ratios are in general closer to the value 1 than the (
G
M
) 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 β
- decay are displayed in the same manner as in Figure 4. It is noted that no appreciable improvement is brought in calculation of β
- 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).Figure 4
RatiosR between experimental [23, 24] and calculated half-lives undergoing β
+ decay for selected cases. Full lines: theoretical half-lives calculated within the (
T
W
) recipe. Dotted lines: theoretical half-lives calculated with the (
G
M
) recipe of [17].Figure 5
Same as Figure4 but for selected β
- decay cases.The differences between the(
G
M
) 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 (
G
M
). Regarding only the free states, in the (
G
M
) 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 r, 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 (
G
M
) 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 (
G
M
) 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 forβ
± and EC decay of unstable nuclei. The newly calculated β-decay half-lives were systematically larger than those given in the previous calculations. The mean percentage deviation is larger for the β
- decay (2.73%) as compared to the β
+/EC decay rates (2.35%). 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 β-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.



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*Source: 5783618-2019-03-26.xml*