Advances in High Energy Physics
Volume 2016 (2016), Article ID 8729893, 8 pages
http://dx.doi.org/10.1155/2016/8729893

Research Article

New Phase Space Calculations for β-Decay Half-Lives

Sabin Stoica,1,2 Mihail Mirea,1,2 Ovidiu Niţescu,2,3 Jameel-Un Nabi,⁴ and Mavra Ishfaq⁴

¹Horia Hulubei Foundation, P.O. MG6, 077125 Magurele, Romania
²Horia Hulubei National Institute of Physics and Nuclear Engineering, P.O. Box MG6, 077125 Magurele, Romania
³Faculty of Physics, University of Bucharest, P.O. Box MG11, 077125 Magurele, Romania
⁴GIK Institute of Engineering Sciences and Technology, Topi 23640, Khyber Pakhtunkhwa, Pakistan

Received 23 June 2016; Accepted 5 October 2016

Academic Editor: Theocharis Kosmas

Copyright © 2016 Sabin Stoica et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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 same -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 -values reported in literature. In several cases, the new -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.

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 spectrum was included by various recipes, for example, by replacing 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 -particle energy and nuclear charge [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 , 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 -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 -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 -particle energy (referred to as throughout this paper) stated in Table of [15]. These -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 -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 Section 2, 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 for β ⁺ Transitions

The probability per unit time that a nucleus with atomic mass and charge decays for an allowed -branch is given by

where is the weak interaction coupling constant, is the momentum of -particle, is the total energy of -particle, and is the maximum -particle energy. , in decay ( is the mass difference between initial and final states of neutral atoms). Equation (1) is written in natural units so that the unit of momentum is , the unit of energy is , and the unit of time is . Shape factors for allowed transitions which appear in (1) are defined as

where are the nuclear matrix elements and the Fermi functions . Thus, for calculating the decay rates, one needs to calculate the nuclear matrix elements and the PSF that can be defined as

For the allowed decays, the Fermi functions are expressed as

where and are the large and the small radial components of the positron radial wave functions evaluated at nuclear radius which can be obtained by solving the Dirac equation:

where is the central potential for the positron and is the relativistic quantum number. We note that (5) is also written in natural units.

An important step in the PSF calculation for 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 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,

where the charge density is

where is the proton (Woods-Saxon) w.f. of the spherical single particle state and is its occupation amplitude. Factor reflects the spin degeneracy.

The screening effect is taken into account by multiplying the expression of with function , which is the solution of the Thomas-Fermi equation: , with and = Bohr radius. It is calculated within the Majorana method [12]. The boundary conditions are and . 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 of -decay process, the potential used to obtain the electron w.f. is

to take into account the fact that decay releases a final negative ion with charge −1. is positive. In our approach, we considered the solution of the Thomas-Fermi equation as a universal function, giving an effective screening. Here, product , for atomic units. The asymptotic potential between an positron and an ionized atom is . In this case, the charge number corresponds to the daughter nucleus, being the charge number of the parent nucleus. Asymptotically, tends to zero.

In this case, the radial solutions of the Dirac equations should be normalized in order to have the following asymptotic behavior:

where is the speed of the light, are the electron mass/energy, is the electron wave number, (with for decays) is the Sommerfeld parameter, is the phase shift, and 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]. and 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 and energy [6, 7]. The screening correction was given by the following replacement [4, 5]:

where and was taken as a -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, , must be greater than the binding energy of K-shell electron in the daughter atom, . The energy carried off by the neutrino is then given by

If the energy requirement is satisfied, electron capture from K-shell is more probable than that from any other shell because of the greater density at the nucleus of K-shell electrons. The total K-shell capture rate can be expressed as

where

where is a constant (with dimensions of ), are specific combinations of nuclear matrix elements, is the large component of the bound-state radial w.f. of the captured K-shell electron (evaluated at the nuclear surface ), is the neutrino energy in units of , and is the “exchange” correction factor for the K-shell. In analogy with (12), L-shell total capture rate will be

where denotes a particular L-subshell. The contribution of pertaining to 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 can be obtained from (13) by the replacement of by . Electron capture from M-, N-, and higher shells may be defined in a similar fashion, but they have negligible contributions in comparison with K- and L-shells.

Hence, for an allowed transition, the PSF expression of electron capture within the approximation stated above can be written as

For quantities, we used the expression

where is value of decay in units, are the binding energies of and electron orbitals of the parent nucleus, and are their radial densities on the nuclear surface. 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 holds.

are the electron bound states, solutions of the Dirac equation (5), and correspond to eigenvalues ( is the radial quantum number). Quantum number is related to total angular momentum . These w.f. are normalized such that

For simplicity, we consider solutions of Dirac equations and that are divided by radial distance . An asymptotic solution is obtained by means of the WKB approximation and by considering that potential is negligibly small:

where

with

In our calculations, we use number nodes and , for orbitals and , respectively, 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 .

The radial density of the bound-state electron w.f. on the nuclear surface is

where is given in fm, being the Bohr radius. For and electron orbitals, we use and , respectively.

For the EC processes, the potential used to obtain the electron w.f. reads

and charge number corresponds to the parent nucleus. 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 and inward from a very large value of radius . 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 and that have the same number of nodes 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 in Q 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 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 for 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.

For 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 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 of -decay (BP) compared with previous calculations. The value of maximum -decay energy is taken from [15] for pure Fermi transitions. The last two columns show our calculated results.

Nucleus
[15] (MeV)
[16]
[15]
[TW]
[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
14412.5

In Table 2, 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]. -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, Ag, where the difference was large (~a factor 10). This is a case where -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 rates

Table 2: Calculated phase space of -decay (BP) for heavy nuclei compared with the ones we calculated using recipe of [3].

Nucleus
[19] (Mev)
[TW]
[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
13.4576

In Table 3, we preset our results for EC for the same set of nuclei. -values for positron decay were taken from [15] for nuclei marked with . For the rest of nuclei, -values were taken from [19]. Together with the PSF values for EC, the electron densities, , their ratios, and binding energies for orbitals and 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, , 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, , and on the final PSF values. For small values of , 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 -values taken from [13, 14]. We propose to use these new computed values of PSF for calculation of -decay rates.

Table 3: Calculated phase space factors for electron capture (assuming exchange corrections to be equal to 1). The value of maximum -decay energy is taken from [15] for pure Fermi transitions. The electron densities, their ratios, and binding energies are also provided for orbitals and , including those given in [2]. Binding energies are given in units of keV.

Nucleus
(MeV)
[2]
[TW]
[2]
[TW]
[2]
[TW]
[2]
[TW]
[TW]
[2]

C⋆
1.9104
0.00031
0.00031
0.04930
0.02867
0.18790
0.62660
0.12600
0.01176
0.00703
0.00640

O⋆
2.83186
0.00075
0.00065
0.05640
0.04420
0.40160
1.03733
0.02440
0.03251
0.03297
0.03786

Ne⋆
3.405
0.00151
0.00118
0.05840
0.05794
0.68540
1.48302
0.03400
0.06659
0.08713
0.11005

Mg⋆
4.131
0.00268
0.00199
0.06660
0.06811
1.07210
2.11143
0.06330
0.15721
0.218
0.29060

Al⋆
4.2331
0.00344
0.00251
0.06990
0.07265
1.30500
2.40715
0.08940
0.14631
0.27558
0.39270

Si⋆
4.839
0.00435
0.00312
0.07290
0.07661
1.55960
2.74689
0.11770
0.18077
0.47240
0.65060

S⋆
5.461
0.00664
0.00467
0.07810
0.08342
2.14550
3.49498
0.18930
0.25934
0.90680
1.27140

Cl⋆
5.4908
0.00807
0.00563
0.08040
0.08628
2.47200
3.91749
0.22920
0.30899
1.10727
1.56600

Ar⋆
6.043
0.00970
0.00675
0.08240
0.08862
2.82240
4.33190
0.27020
0.36199
1.61130
2.28490

K⋆
6.05
0.01156
0.00802
0.08440
0.09079
3.20600
4.77984
0.32630
0.41921
1.92311
2.73480

Ca⋆
6.642
0.01367
0.00947
0.08620
0.09259
3.60740
5.25087
0.37710
0.48351
2.74237
3.90650

Sc⋆
6.4311
0.01600
0.01113
0.08790
0.09430
4.03810
5.73657
0.43780
0.54865
3.02434
4.28930

Ti⋆
6.986
0.01870
0.01300
0.08960
0.09579
4.49280
6.25222
0.50040
0.62068
4.17496
5.92320

V⋆
7.0543
0.02170
0.01512
0.09100
0.09699
4.96640
6.78377
0.56370
0.69826
4.95575
7.02120

Mn⋆
7.6311
0.02870
0.02016
0.09380
0.09920
5.98920
7.92722
0.69460
0.86703
7.74617
10.9103

Fe
2.374
0.0328
0.0232
0.0950
0.0987
7.1120
8.5130
0.8461
0.958
0.859
1.2033

Co⋆
8.2498
0.03730
0.02651
0.09620
0.10077
7.11200
9.14731
0.84610
1.05584
11.91799
16.6144

Ni
2.136
0.0423
0.0303
0.0974
0.1013
8.3328
9.7882
1.0081
1.158
0.907
1.2580

Zn
1.626
0.0538
0.0390
0.0995
0.1025
9.6586
11.157
1.1936
1.380
0.675
0.9261

Ga
5.175
0.0604
0.0410
0.1006
0.1029
10.3671
11.875
1.2977
1.498
7.80
10.613

Br
4.963
0.0935
0.0704
0.1035
0.1048
13.4737
15.000
1.7820
2.021
11.45
15.162

Rb
2.23815
0.1149
0.0883
0.1063
0.1080
15.1997
16.690
2.0651
2.263
9.069
11.744

Y
3.6226
0.1402
0.1091
0.1080
0.1174
17.0384
18.450
2.3725
2.438
9.528
12.114

Nb
6.111
0.170
0.1344
0.1098
0.1059
18.9856
20.421
2.6977
2.994
33.17
41.975

Cd
2.587
0.319
0.2663
0.1159
0.1102
26.7112
28.044
4.0180
4.351
11.66
14.019

In
6.050
0.348
0.2930
0.1168
0.1116
27.9399
29.232
4.2375
4.548
71.05
84.541

Ag
1.345
0.293
0.2423
0.1150
0.1086
25.5140
26.864
3.8058
4.161
2.816
3.4256

Sb
7.920
0.413
0.3526
0.1187
0.1096
30.4912
31.726
4.6983
5.095
146.5
172.43

Sb
3.913
0.413
0.3516
0.1187
0.1096
30.4912
31.726
4.6983
5.095
35.38
41.804

Te
6.070
0.449
0.3844
0.1196
0.1113
31.8138
33.041
4.9392
5.314
93.70
109.93

I
5.729
0.488
0.4121
0.1205
0.1124
33.1694
34.345
5.1881
5.542
91.54
106.40

I
7.780
0.488
0.4215
0.1205
0.1124
33.1694
34.345
5.1881
5.542
169.3
196.75

Xe
4.450
0.529
0.4609
0.1215
0.1123
34.5644
35.705
5.4528
5.822
60.15
69.410

Ba
5.00
0.623
0.5496
0.1234
0.1130
37.4406
38.514
5.9888
6.375
90.65
103.51

Xe
1.617
0.529
0.4599
0.1215
0.1123
34.5644
35.705
5.4528
5.821
7.72
8.9482

Cs
4.824
0.574
0.501
0.1224
0.112
35.9846
37.111
5.7143
6.128
76.88
88.697

Re
2.800
2.69
2.593
0.1448
0.128
71.6764
72.491
12.5267
13.26
22.86
24.152

Bi
2.708
4.88
4.837
0.1561
0.138
90.5259
91.373
16.2370
17.25
228.17
233.83

Table 4: Calculated phase space factors for electron capture, with -values from [13, 14].

Nucleus
[13, 14] (MeV)
[TW]
[2]
[TW]
[3]

C
3.64613
0.07318
2.33265
226.780
226.834

O
5.14131
0.21794
0.12483
1644.76
1643.41

Ne
4.44215
0.27831
0.18733
677.970
677.912

Mg
4.77904
0.61616
0.39020
995.887
995.685

Al
4.00231
0.62642
0.35240
343.398
343.658

Si
5.06645
0.51788
0.71694
1339.344
1339.30

S
6.13834
1.14585
1.61931
3805.276
3803.16

Cl
5.48869
1.10642
1.57889
1994.797
1995.09

Ar
6.05858
1.61963
2.31915
3410.133
3409.96

K
5.91093
1.83565
2.64042
2917.839
2918.62

Ca
6.73867
2.82284
4.07367
5924.355
5929.26

Sc
6.42269
3.01643
4.33609
4470.946
4471.87

Ti
7.01275
4.20702
6.05196
7100.190
7130.06

V
7.04865
4.94781
7.11022
7175.692
7209.06

Mn
7.63042
7.74479
11.0705
10516.941
10744.5

Fe
2.37330
0.8584
1.22082
14942.286
15765.2

Co
8.24017
11.89015
16.8306
8.354
8.43206

Ni
2.13175
0.9029
1.27259
3.444
3.49486

Zn
1.61859
0.6687
0.93259
0.234
0.24131

Ga
5.17225
7.7902
10.7797
1125.644
1131.60

Br
4.96024
11.439
15.4388
835.295
841.531

Rb
2.23696
2.9044
3.84415
4.321
4.41092

Y
3.62067
9.5180
12.3759
120.264
121.864

Nb
6.10809
33.141
42.9337
2502.372
2526.00

Cd
2.58562
11.652
14.4027
11.221
11.5468

In
6.01928
70.333
15.7203
2099.402
2133.61

Ag
1.34679
2.8233
3.53114
0.01027
1.12362

Sb
7.85483
144.059
174.745
8528.505
8918.59

Sb
3.90909
35.311
42.9919
168.122
172.036

Te
6.06682
93.601
113.234
2124.182
2162.53

I
5.72192
91.3148
109.531
1509.977
1547.75

I
7.77260
168.959
202.635
7930.046
8250.78

Xe
4.44315
59.963
71.4659
354.467
361.241

Ba
4.99761
90.562
106.993
685.518
703.098

Xe
1.57992
7.3638
8.82085
0.105
0.11187

Cs
4.79256
75.871
90.4835
542.400
555.411

Re
2.79851
131.273
145.184
16.123
17.2282

Bi
2.70412
227.499
247.263
12.415
13.4798

(a)

(b)

Figure 1: (a) Electron density on the nuclear surface of 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.

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 same -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 -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 -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.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

Sabin Stoica and Mihail Mirea would like to acknowledge the support of the ANCSI-UEFISCDI through Project PCE-IDEI-2011-3-0318, Contract no. 58.11.2011. Jameel-Un Nabi would like to acknowledge the support of the Higher Education Commission Pakistan through the HEC Project no. 20-3099.

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