Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O.Box 47416-416, Babolsar, Iran Received Date:2021-07-07 Available Online:2021-12-15 Abstract:A systematic study on the α-decay half-lives of nuclei in the range $93\leqslant Z \leqslant 118$ is performed by employing various versions of proximity potentials. To obtain more reliable results, deformation terms are included up to hexadecapole ($\beta_{4}$) in the spherical-deformed nuclear and Coulomb interaction potentials. First, the favored α-decay processes in this region are categorized as even-even, odd A, and odd-odd nuclei. Second, they are grouped into two transitions: ground state to ground state and ground state to isomeric states. Owing to the comparison of their root-mean-square deviations (RMSD's), $Bass 77$ and $Ngo 80$ have the lowest values and better reproduce experimental data. Moreover, by considering preformation probability within the cluster formation model, the results validate the significant reduction in root-mean-square deviations obtained for different versions of proximity. Hence, the deviation between the calculated and experimental data is detracted.
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II.THEORETICAL FRAMEWORKThe α-decay half-life of a parent nucleus can be determined using $ T_{1/2} = \ln2/P_{\alpha}\nu_{0} P $. Here, $ P_{\alpha} $ is the preformation probability, which will be discussed by the next section, and the assault frequency $ \nu_0 $ is related to the oscillation frequency ω [50]
where $ R_{n}^2 = \dfrac{3}{5} R_{0}^2 $ [51] and, due to the topic under study in this paper, the global quantum number G is equal to 22 [52]. The α-decay penetration probability in a different orientation, using the WKB semi-classical approximation, is defined as
where $ {\mu} = m(A_{\alpha}+A_{d})/A_{\alpha} A_{d} $ is the reduced mass in which $ A_{\alpha} = 4 $ and $ A_{d} $ is the daughter nucleus. The $ r_{a} $ and $ r_{b} $ are the turning points, which are obtained from $ V_{\rm T}(r_{a} ) = Q_{\alpha} = V_{\rm T} (r_{b}) $. The total interaction potential $ V_{\rm T} = V_{N}+V_{\rm C}+V_{l} $ between the α-particle and daughter nucleus is defined as the sum of the nuclear, Coulomb, and centrifugal potential, which has a deformation and orientation dependence. The nuclear term is introduced in detail in Ref. [37], which includes different modifications of $ Prox.77 $ that are indicated in Table 1 and other versions of proximity potentials. For details of the deformation effect [42, 43], the mean curvature radius $ \overline{R} $ with azimuthal angle ? between the principal planes of curvature of two deformed nuclei is given by
Proximity version
$\gamma_0$
$\kappa_s$
Proximity version
$\gamma_0$
$\kappa_s$
$Prox.66 $
1.01734
1.79
$Prox.81-III $
1.2502
2.4
$Prox.76 $
1.460734
4
$Prox.88 $
1.2496
2.3
$Prox.79 $
1.2402
3
$Prox.95 $
1.25284
2.345
$Prox.81-I $
1.1756
2.2
$Prox.03-I $
1.08948
1.983
$Prox.81-II $
1.27326
2.5
$ModProx.88 $
1.65
2.3
Table1. Prox.77 and its different modifications corresponding to Table 1 of Ref. [37].
where ? is equal to zero, as nuclei are considered to be in the same plane in this study. $ R_{i1}(\alpha_{i}) $ and $ R_{i2}(\alpha_{i}) $$ (i = 1,2) $ are the radii of curvature in the principal planes of each of the two nuclei
where $ r_{0i} = 1.28 A_{i}^{1/3}-0.76+0.8 A_{i}^{-1/3} $. Here, $ \alpha_i $ is the angle between the radius vector and the symmetry axis of the $ i^{\rm th} $ nuclei. In this study we consider one of two nuclei to be spherical, Therefore, it has no deformation parameters. The rotational effect of two nucleus systems can be calculated using the l-dependent centrifugal potential, which is equal to $ \hbar^{2} l(l+1)/2\mu r^{2} $. l is the orbital angular momentum carried by the α-particle. The values that are used to calculate this potential are determined by utilizing the permitted transitions between the parent nucleus and the daughter nucleus. The Coulomb potential between spherical-deformed and oriented, taken from Ref. [53], is given as follows:
This potential has high accuracy and provides time-saving during calculation. Because the deformation parameters from M?ller $ et \;al $. [54] agree well with the existing magnitude of experimental deformation [55], $ \beta_{\lambda} $ values are taken from [54] for all calculations of the α-decay half-lives of deformed nuclei in this study.
III.RESULTS AND DISCUSSIONWe have systematically studied a wide atomic range of 108 transuranium nuclei, from $ Z = 93 $ onward, using 23 versions of spherical-deformed potentials for both the short-range attractive term and the Coulomb repulsive term of the potential barrier, in which the α-particle and the residual nuclei are considered spherical and deformed, respectively. It is interesting to see how the deformation affects the proximity potentials and the corresponding penetration probability in the α-decay process. Forinstance, the penetration probability for 253Fm → 249Cf+α (with $ \beta_{2} = 0.250 $ and $ \beta_{4} = 0.039 $) is plotted in each direction in Fig. 1 for seven selected versions. As shown, the nuclear potential for decay through $ \theta = 0^{\circ} $ is deeper than $ \theta = 90^{\circ} $. Due to the proximity theory (The force between two gently curved surfaces as a function of the separation degree of freedoms is proportional to the interaction potential per unit area), we realize that a nucleus contains a thicker surface at $ \theta = 0^{\circ} $and the stronger interaction in this area is due to the overlap of the nucleons. Although properties such as the Pauli exclusion principle, spin and parity, and isospin asymmetry play a role in the formation of a particle before emission, we can implicitly expect the probability of α-particle formation in this area to be more likely than others. Also, from this figure, we can see that the $ Q_{\alpha} $ line does not cross the total potential curve in some direction, such as prox. 66; as long as it appears in all directions, we are not able to calculate the penetration probability integral. Figure1. (color online) The penetration probability of the eight selected versions of proximity potentials with respect to the angles of the emitted α-particle.
The height and position of the barrier potential of the α-particle in a deformed nucleus depend on the angle at which the α-particle is emitted. The average penetrability is obtained using
$ P(\theta) = {\frac{1}{2} \int_{0}^{\pi} P \sin(\theta){\rm d}\theta }. $
(9)
For the probability of the α-particle formation before penetration, the cluster-formation model is a proposed energy-dependent theory used to calculate the α preformation factor. The basic assumption in the CFM formalism is that the nucleons around the surface contribute to the preformation of the α-particle. In this model, the α-preformation factor is defined as $ P_{\alpha} = {E_{f\alpha} / E} $, where $ E_{f\alpha} $ is the formation energy of the α-cluster and E is the total energy of a considered system [31-35]. In order to obtain an accurate mass excess, which is vitally important to obtain the precise binding energies, data are provided by [56]. We computed α-decay half-lives and evaluated a quantitative analysis using the root-mean-square deviation
where N is the number of contributed nuclei, and experimental values of half-lives are given by [57], taking into account their branching ratios. Since the parent nucleus can decay from the ground state (gs) to different possible states that satisfy the spin-parity rule
According to this equation, different values of l can transfer from the initial state to the final states; however, it is the minimum possible value of angular momentum that is of interest. Due to this value, the favored transition will occur provided $ l = 0 $. If not, transition to the daughter nucleus is hindered. All considered nuclei in this study are favored, and 22 nuclei involve decays to more than one level with the same spin and parity. The total 141 favored α-decays contain 54 even-even (e-e), 81 odd A, and 6 odd-odd (o-o) nuclei. The experimental data and the details of their isomeric states and energy levels are taken from [57]. The RMSDs of these groups are shown in columns two to four of Table 2, and each of their corresponding values after taking the preformation factor under consideration is presented in columns five to seven. Comparing the RMSDs of different proximity versions for e-e nuclei shows that $ Bass 77 $ and $ Ngo 80 $ have the least values. When Pα is taken under consideration, $ Ngo 80 $ obtains better results than $ Bass 77 $. The deviation between the calculated and experimental values for e-e elements are shown in Fig. 2(a). We can see that these versions reproduce the experimental values very well.
Proximity Potentials
$P_{\alpha}^{1}$
$P_{\alpha}^{\rm CFM}$
e-e
odd A
o-o
e-e
odd A
o-o
$Prox. 66$
1.3660
2.6930
2.7633
0.7961
2.0759
2.0655
$Prox.76 $
2.0879
3.3049
3.4166
1.3441
2.5680
2.6440
$Prox.79 $
1.8156
3.0700
3.1693
1.0950
2.3686
2.4177
$Prox81-I $
1.7723
3.0382
3.1384
1.0559
2.3430
2.3899
$Prox81-II $
1.9395
3.1830
3.2929
1.2057
2.4640
2.5300
$Prox81-III $
1.9064
3.1546
3.2624
1.1756
2.4401
2.5022
$Prox.88 $
1.9165
3.1641
3.2737
1.1846
2.4482
2.5124
$Prox.95 $
1.9177
3.1650
3.2741
1.1857
2.4488
2.5129
$Prox03_I $
1.5815
2.8749
2.9624
0.8957
2.2122
2.2341
$ModProx88 $
2.4822
3.6703
3.8121
1.7189
2.8951
3.0165
$Prox.00 $
1.8242
3.0916
3.1951
1.1032
2.3868
2.4410
$Prox.00DP $
3.5506
4.6520
4.8360
2.7690
3.8184
4.0094
$Prox.2010 $
3.6436
4.7335
4.9148
2.8613
3.8966
4.0867
$Bass1977 $
0.8281
2.2505
2.2620
0.6547
1.8153
1.6867
$Bass1980 $
2.0303
3.2768
3.4024
1.2881
2.5461
2.6317
$CW1976 $
2.8746
4.0475
4.2094
2.1009
3.2450
3.3981
$BW1991 $
2.0596
3.3127
3.4370
1.3153
2.5778
2.6639
$Ngo1980 $
1.0474
2.4430
2.4907
0.5963
1.9144
1.8443
$Denisov $
4.3656
5.4038
5.6299
3.5799
4.5480
4.7921
$Denisov DP$
5.4896
6.4895
6.7427
4.7003
5.6155
5.8963
$AW95 $
1.8899
3.0318
3.1703
1.1799
2.3319
2.4164
$Dutt2011 $
1.9200
3.1811
3.2933
1.1870
2.4637
2.5308
$Guo2013 $
2.4293
3.6335
3.7766
1.6674
2.8625
2.9820
Table2.RMSDs of ${\alpha}-$decay half-lives of even-even, odd A, and odd-odd nuclei, when considering $P_{\alpha}.$
Figure2. (color online) Deviation between the calculated and experimental α-decay half-lives for (a) even-even (b) odd A (c) odd-odd nuclei.
For the other two groups in Table 2, $ Bass 77 $ and $ Ngo 80 $ are the most suitable versions; the deviation of the calculated values are represented in Fig. 2(b) and c for odd A and o-o nuclei, respectively. $ Prox.66 $ is not able to reproduce the half-life of just 290Fl. It is useful to notice that the experimental data can also impact the results. For instance, the 264Hs, which is recognizable in Fig. 2 a as the only A = 264, has an experimental α-decay half-life of 1 s, as reported in NUBASE2020 [57], while it is 1.08 ms in NUBASE2016 [58]; its α-decay half-life is $ 0.365 $ ms and $ 0.178 $ ms and after using Pα it becomes 2.07 ms and 1.01 ms for $ Bass 77 $ and $ Ngo 80 $, respectively. To investigate the validity of the formalisms used in estimating the ground state (gs) to ground state or isomeric states (is), these 108 nuclei are grouped with respect to the energy level of the daughter nucleus that whether they are in their ground state or its isomeric states. Among the total 141 favored decays, 59 transitions are gs-gs and 82 transitions are gs-is, which their RMSD values are given in Table 3. Similar to Table 2, their corresponding values after preformation values are taken under consideration are shown. For gs-gs nuclei, $ Bass 77 $ before and after using Pα are almost equal, whereas $ Ngo 80 $ with Pα is the outlying version. Figure 3 a shows the power of formalisms in producing experimental data of gs-gs nuclei. Furthermore, also $ Bass 77 $ and $ Ngo 80 $ can best satisfy the gs-is elements among all. Figure 3 b shows the deviation between these suitable versions and the experimental values.
Proximity potentials
$P_{\alpha}^{1}$
$P_{\alpha}^{\rm CFM}$
g.s.-g.s.
g.s.-i.s.
g.s.-g.s.
g.s.-i.s.
$Prox. 66$
1.3865
2.7512
0.9531
2.0824
$Prox.76$
2.0324
3.3974
1.3535
2.6262
$Prox.79$
1.7798
3.1522
1.1483
2.4105
$Prox81-I $
1.7383
3.1197
1.1165
2.3828
$Prox81-II $
1.8912
3.2718
1.2351
2.5151
$Prox81-III $
1.8608
3.2421
1.2107
2.4890
$Prox.88 $
1.8696
3.2522
1.2177
2.4979
$Prox.95 $
1.8708
3.2531
1.2185
2.4987
$Prox03_I $
1.5683
2.9469
1.0010
2.2375
$ModProx88 $
2.3980
3.7802
1.6789
2.9771
$Prox.00$
1.7879
3.1749
1.1545
2.4302
$Prox.00DP $
3.4408
4.7824
2.6785
3.9327
$Prox.2010 $
3.5346
4.8635
2.7704
4.0112
$Bass1977 $
0.9399
2.2789
0.9338
1.7658
$Bass1980 $
1.9770
3.3699
1.3058
2.6031
$CW1976 $
2.7592
4.1765
2.0167
3.3506
$BW1991 $
1.9986
3.4094
1.3222
2.6390
$Ngo1980$
1.1017
2.4897
0.8530
1.8920
$Denisov$
4.2357
5.5503
3.4625
4.6825
$Denisov DP $
5.3404
6.6522
4.5584
5.7699
$AW95 $
1.9032
3.0921
1.2804
2.3529
$Dutt2011 $
1.8701
3.2712
1.2166
2.5154
$Guo2013 $
2.3325
3.7493
1.6175
2.9484
Table3.RMSDs of ${\alpha}-$decay half-lives of ground state to ground state and isomeric states of nuclei, when considering $P_{\alpha}$.
Figure3. (color?online) Deviation between calculated and experimental α-decay half-lives for $ a) $ ground state to ground state and $ b) $ ground state to isomeric states.
For the purpose of comparing our calculated half-lives with ones obtained from the analytical formula, a recent form of the universal decay law (UDL) [59] is used. The UDL for α and cluster decay modes is introduced as
where $ A = A_{c} A_{d}/(A_{c}+A_{d}) $ and the constant $ a = 0.4314 $, $ b = -0.4087 $, and $ c = -25.7725 $ are determined by fitting to the experimental values of both α and cluster decays. The RMSD values from the UDL are equal to 0.6769, 1.8092, 1.7120, 0.9650, and 1.7550 for e-e, odd A, o-o, gs-gs, and gs-is nuclei, respectively. It is noticeable that the higher the probability of α-particle formation inside the parent nucleus, the more unstable the nucleus. Consequently, because the formation of α-particles is more difficult near magic numbers, they are more stable; this issue is implied in computation. Furthermore, as the preformation factor increases the half-life values, eventually, almost all versions considerably reduce the RMSDs. This is interpreted to reproduce the half-lives.