We present results for
The precise knowledge of the
Accurate estimates of half-lives of neutron-rich nuclei have gained much interest in the recent past. This is primarily because of their key role in
This paper is organized in the following format. Section
In (
In (
In this paper the NMEs calculation was performed using the pn-QRPA model. We used the Nilsson model [
The formalism for the PSF calculation for
The above formula determines the PSFs for both the Fermi and Gamow-Teller allowed transitions, by substituting
Ideally, the central potential
(a) Realistic proton density for
We integrated the realistic charge distribution over the volume of nucleus, in order to find the Coulomb potential
Moreover, we included the screening effect by multiplying the expression of
The effective screening function
We solved (
Because we compare our results with those of Gove and Martin [
In order to illustrate the differences that appear between our calculation and the approximate method of
Comparison of PSF results calculated with our recipe (TW) and Gove and Martin recipe (
Z | A |
|
|
|
---|---|---|---|---|
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 | 3.206 |
Comparison of PSF results calculated with our recipe (TW) and (
Z | A |
|
|
|
---|---|---|---|---|
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 |
Electron capture is a process which competes with positron decay. It is an alternate decay mode for the
For the
For the
Half-lives were computed using (
The first four columns give the electron binding energies
Nucleus |
|
|
|
|
|
|
---|---|---|---|---|---|---|
|
6.63991 | 0.7451126 | 0.0327063 | 0.0898416 | 0.0328 | 0.0950 |
|
7.95277 | 0.9181003 | 0.0411267 | 0.0943036 | 0.0423 | 0.0974 |
|
9.20973 | 1.1092315 | 0.0530401 | 0.0953901 | 0.0538 | 0.0995 |
|
13.0121 | 1.6760343 | 0.0917936 | 0.1019465 | 0.0935 | 0.1035 |
|
14.6718 | 1.9336296 | 0.1136202 | 0.1042520 | 0.1149 | 0.1063 |
|
16.4688 | 2.0221700 | 0.1389702 | 0.1272602 | 0.1402 | 0.1080 |
|
18.3994 | 2.4889101 | 0.1686632 | 0.1121010 | 0.170 | 0.1098 |
|
26.1177 | 3.9008765 | 0.3182812 | 0.1109908 | 0.319 | 0.1159 |
|
24.95904 | 3.558919 | 0.2900978 | 0.1193451 | 0.293 | 0.1150 |
|
29.99173 | 4.140248 | 0.4109690 | 0.1416423 | 0.413 | 0.1187 |
|
29.99173 | 4.140248 | 0.4101592 | 0.1416413 | 0.413 | 0.1187 |
|
31.18294 | 4.70109 | 0.4488353 | 0.12019908 | 0.449 | 0.1196 |
|
32.50419 | 4.937340 | 0.4894928 | 0.1210013 | 0.488 | 0.1205 |
|
32.50419 | 4.937340 | 0.4893257 | 0.1210012 | 0.488 | 0.1205 |
|
33.95443 | 5.055330 | 0.5289785 | 0.1300301 | 0.529 | 0.1215 |
|
36.81175 | 4.975840 | 0.6244052 | 0.1662902 | 0.623 | 0.1234 |
|
33.95440 | 5.055328 | 0.5280176 | 0.1300293 | 0.529 | 0.1215 |
|
35.30411 | 5.151703 | 0.5764054 | 0.1394027 | 0.574 | 0.1224 |
|
71.29588 | 12.23093 | 2.7488862 | 0.1490819 | 2.69 | 0.1448 |
|
90.39904 | 16.18943 | 5.0324494 | 0.1587003 | 4.88 | 0.1561 |
The differences between the results obtained for a Coulomb field obtained from a constant charge density inside a spherical nucleus and those obtained with a field constructed with a more realistic charge density corrected with a screening function can offer an estimation of the role played by the ingredients of our model. The radial distributions of the electron
The results
Method |
|
|
|
|
---|---|---|---|---|
Ref. [ |
3.60740 | 0.37710 | 0.01367 | 0.08620 |
|
3.762904 | 0.357179 | 0.01308474 | 0.0825267 |
No screening | 5.454032 | 1.198887 | 0.01434885 | 0.1877057 |
The radial distributions
Table
Comparison of measured, calculated half-lives and percentage deviation (PD) for
Nucleus |
|
|
|
PD (%) |
---|---|---|---|---|
|
2.98E+04 | 1.29E+04 | 1.30E+04 | 0.77 |
|
5.25E+05 | 4.26E+05 | 4.44E+05 | 4.05 |
|
3.31E+04 | 9.80E+03 | 1.01E+04 | 2.97 |
|
5.83E+04 | 1.62E+04 | 1.66E+04 | 2.41 |
|
1.65E+04 | 5.00E+03 | 5.12E+03 | 2.34 |
|
9.21E+06 | 1.25E+07 | 1.27E+07 | 1.57 |
|
5.26E+04 | 4.25E+04 | 4.32E+04 | 1.62 |
|
3.30E+02 | 2.35E+02 | 2.42E+02 | 2.89 |
|
3.57E+06 | 2.45E+04 | 2.52E+04 | 2.78 |
|
4.00E+00 | 3.92E+00 | 4.04E+00 | 2.97 |
|
4.00E+02 | 2.42E+02 | 2.47E+02 | 2.02 |
|
1.02E+02 | 9.55E+01 | 9.77E+01 | 2.25 |
|
3.48E+02 | 9.98E+01 | 1.02E+02 | 2.16 |
|
2.91E+00 | 9.49E-01 | 9.73E-01 | 2.47 |
|
5.90E+01 | 2.01E+01 | 2.05E+01 | 1.95 |
|
2.40E+01 | 1.73E+01 | 1.76E+01 | 1.70 |
|
2.76E+03 | 1.58E+03 | 1.61E+03 | 1.86 |
|
9.84E+01 | 5.35E+02 | 5.42E+02 | 1.29 |
|
2.30E+05 | 3.67E+05 | 3.80E+05 | 3.42 |
|
1.32E+06 | 1.47E+06 | 1.52E+06 | 3.46 |
Table
State-by-state comparison of calculated PSF (for
|
NME |
|
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|---|---|
|
|||||||||||
|
|||||||||||
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 |
|
|||||||||||
|
|||||||||||
|
|||||||||||
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 | 60.30 |
We would like to comment further on the entries in the first two columns of Table
Table
Same as Table
Nucleus |
|
|
|
PD (%) |
---|---|---|---|---|
|
6.53E-01 | 4.45E-01 | 4.57E-01 | 2.63 |
|
2.02E-01 | 2.28E-01 | 2.37E-01 | 3.82 |
|
7.10E+00 | 8.04E+00 | 8.25E+00 | 2.55 |
|
2.90E+00 | 8.45E+00 | 8.73E+00 | 3.21 |
|
6.78E+02 | 1.90E+02 | 1.94E+02 | 2.06 |
|
6.00E+00 | 5.93E+00 | 6.08E+00 | 2.47 |
|
8.73E+00 | 6.35E+00 | 6.53E+00 | 2.76 |
|
2.73E+02 | 5.61E+02 | 5.74E+02 | 2.26 |
|
1.20E+01 | 6.27E+01 | 6.46E+01 | 2.94 |
|
1.75E+00 | 6.27E+00 | 6.47E+00 | 3.09 |
|
7.57E+04 | 3.89E+04 | 4.00E+04 | 2.75 |
|
1.45E+02 | 1.00E+02 | 1.03E+02 | 2.91 |
|
1.18E+01 | 4.53E+01 | 4.67E+01 | 3.00 |
|
8.44E+02 | 4.69E+02 | 4.84E+02 | 3.10 |
|
1.36E+01 | 1.36E+00 | 1.40E+00 | 2.86 |
|
6.36E+02 | 7.97E+02 | 8.19E+02 | 2.69 |
|
1.15E+01 | 2.44E+01 | 2.51E+01 | 2.79 |
|
8.11E+02 | 4.74E+01 | 4.85E+01 | 2.27 |
|
5.60E+01 | 7.93E+01 | 8.13E+01 | 2.46 |
|
6.84E+02 | 8.64E+03 | 8.82E+03 | 2.04 |
State-by-state comparison of calculated PSF (for
|
NME |
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|
|
||||||||
|
||||||||
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 |
|
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|
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|
||||||||
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 | 0.002 |
The difference between the two calculated half-lives as well as their mutual comparison with the experimental data is done intuitively in a graphical way in Figure
Ratios
Same as Figure
The differences between the
The aim of present work was to investigate the effect of the incorporation of new PSF values, computed with a more precise and rigorous method, on the theoretical half-lives for
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.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
J.-U. Nabi would like to acknowledge the support of the Higher Education Commission Pakistan through Projects nos. 5557/KPK/NRPU/R&D/HEC/2016 and 9-5(Ph-1-MG-7)/PAK-TURK/R&D/HEC/2017 and Pakistan Science Foundation through Project no. PSF-TUBITAK/KP-GIKI (02). M. Ishfaq wishes to acknowledge the support provided by Scientific and Technological Research Council of Turkey (TUBITAK), Department of Science Fellowships and Grant Programs (BIDEB), 2216 Research Fellowship Program for International Researchers (21514107-115.02-124287). S. Stoica, M. Mirea, and O. Niţescu would like to acknowledge the support of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, through Projects PN-III-P4-ID-PCE-2016-0078 and PN-5N/2018.