-
The total energy of a deformed nucleus is the sum of the GLDM energy and the shell and pairing energies. Within this GLDM the macroscopic energy of a deformed nucleus is defined as[19]
$$ \begin{array}{l} E(r)=E_{{\rm{V}}}+E_{{\rm{S}}}+E_{{\rm{C}}}+E_{{\rm{prox}}}+E_{{\rm{shell}}}+E_{{\rm{pairing}}}, \end{array} $$ (1) where the different terms are respectively the volume, surface, Coulomb, nuclear proximity and rotational energies. For one-body shapes, the volume
$ E_{{\rm{V}}} $ , surface$ E_{{\rm{S}}} $ and Coulomb$ E_{{\rm{C}}} $ energies are given by$$ \begin{array}{l} E_{{\rm{V}}}=-a_{{\rm{v}}}(1-k_{{\rm{v}}}I^{2})A \; {\rm{MeV}}, \end{array} $$ (2) $$ \begin{array}{l} E_{{\rm{S}}}=a_{{\rm{s}}}(1-k_{{\rm{s}}}I^{2})A^{2/3}(S/4\pi R_{0}^{2}) \; {\rm{MeV}}, \end{array} $$ (3) $$ \begin{array}{l} E_{{\rm{C}}}=0.6e^{2}(Z^{2}/R_{0})B_{{\rm{C}}} \; {\rm{MeV}}. \end{array} $$ (4) The set of parameters
$a_{\rm v}=15.494$ ,$k_{\rm v}=1.8$ ,$ a_{{\rm{s}}}=17.9439 $ (or 18.18), and$ k_{{\rm{s}}}=1.8 $ (or 3.1) has been used. It is well known that the parameters ($ a_{{\rm{s}}} $ and$ k_{{\rm{s}}} $ ) in the liquid drop and macroscopic microscopic model were determined from a least-squares fit to the experimental data. There are some differences in the fitting results of different research groups. To study the influence of different parameter on the potential barrier, different parameter values were adopted in our calculations.The
$ B_{{\rm{C}}} $ is the Coulomb shape dependent function,$ S $ is the surface and$ I $ is the relative neutron excess.$$ \begin{array}{l} B_{{\rm{C}}}=0.5\int(V(\theta)/V_{0})(R(\theta)/R_{0})^{3}\sin\theta {\rm{d}}\theta, \end{array} $$ (5) where
$ V(\theta) $ is the electrostatic potential at the surface and$ V_{0} $ the surface potential of the sphere. The effective sharp radius$ R_{0} $ has been chosen as$$ \begin{array}{l} R_{0}=1.28A^{1/3}-0.76+0.8A^{-1/3} \; {\rm{fm}}. \end{array} $$ (6) This formula proposed is derived from the droplet model and the proximity energy and simulates rather a central radius for which
$ R_{0}/A^{1/3} $ increases slightly with the mass. When the fragment are separated[51],$$ \begin{array}{l} E_{{\rm{V}}}=-a_{{\rm{v}}}[(1-k_{{\rm{v}}}I_{1}^{2})A_{1}+(1-k_{{\rm{v}}}I_{2}^{2})A_{2}]\; {\rm{MeV}}, \end{array} $$ (7) $$ \begin{array}{l} E_{{\rm{S}}}=a_{{\rm{s}}}[(1-k_{{\rm{s}}}I_{1}^{2})A_{1}^{2/3}+ (1-k_{{\rm{s}}}I_{2}^{2})A_{2}^{2/3}]\; {\rm{MeV}}, \end{array} $$ (8) $$ \begin{array}{l} E_{{\rm{C}}}=0.6e^{2}Z_{1}^{2}/R_{1}+0.6e^{2}(Z_{2}^{2}/R_{2})+ e^{2}Z_{1}Z_{2}/r \; {\rm{MeV}}. \end{array} $$ (9) To ensure volume conservation,
$ R_{1} $ and$ R_{2} $ read$$ \begin{array}{l} R_{1}=R_{0}(1+\beta^{3})^{-1/3}, \end{array} $$ (10) $$ \begin{array}{l} R_{2}=R_{0}\beta(1+\beta^{3})^{-1/3}, \end{array} $$ (11) where
$$ \beta=\frac{1.28A_{1}^{1/3}-0.76+0.8A_{1}^{-1/3}}{1.28A_{2}^{1/3}-0.76+0.8A_{2}^{-1/3}}. $$ (12) The discontinuity of a few MeV appearing at the contact point due to the difference between
$ A_{1}/Z_{1} $ and$ A_{2}/Z_{2} $ has been linearized from the contact point to the sphere since it originates form discarding the charge rearrangement in the nuclear matter which occurs progressively.The surface energy comes from the effects of the surface tension forces in a half space. When a neck or a gap appears between separated fragments an additional term called proximity energy must be added to take into account the effects of the nuclear forces between the close surface. It moves the barrier top to an external position and strongly decreases the pure Coulomb barrier:
$$ E_{{\rm{prox}}}(r)=2\gamma\int\nolimits_{h_{{\rm{min}}}}^{h_{{\rm{max}}}}\varPhi[D(r, h)/b]2 {\text{π}} h{\rm{d}}h , $$ (13) where
$$ \begin{array}{l} \gamma=0.9517\sqrt{(1-k_{{\rm{s}}}I_{1}^{2})(1-k_{{\rm{s}}}I_{2}^{2})}\; \; {\rm{MeV}} {\rm{fm}}^{-2} .\end{array} $$ (14) $ r $ is the distance between the mass centres,$ h $ is the transverse distance varying from the neck radius or zero to the height of the neck border,$ D $ is the distance between the opposite surfaces in consideration and$ b $ is the surface width fixed at 0.99 fm.$ \varPhi $ is the proximity function. The surface parameter$ \gamma $ is the geometric mean between the surface parameters of the two fragments.The shape dependent shell corrections have been determined within the Droplet Model expressions[55]:
$$ \begin{array}{l} E_{{\rm{shell}}}=E_{{\rm{shell}}}^{{\rm{sphere}}}\big(1-2.6\alpha^{2}\big){\rm{e}}^{-\alpha^{2}} \;, \end{array} $$ (15) where, the range a has been chosen to be
$ 0.32r_{0} $ .$\alpha^{2}=(\delta R)^{2}/a^{2}$ , the factor$\alpha^{2}$ is the Myers-Swiatecki measure for the deformation of the nucleus. The attenuating factor$({\rm e}^{-\alpha^{2}} )$ makes the whole shell correction energy decrease from maximum to zero with increasing distortion of the nucleus. The distortion is the root-mean-square value of the deviation of the radius vector$ R(\theta.\phi) $ , specifying the nuclear surface,$$ (\delta R)^{2}=\frac{\int\int(R-R_0)^2{\rm{d}}\varOmega}{\int\int {\rm{d}}\varOmega}. $$ (16) The calculation method of
$ E_{{\rm{shell}}}^{{\rm{sphere}}} $ is taken from Ref.[56]. The shape dependent pairing energy has been calculated with the following expressions of the finite-range droplet model[57].In the unified fission model, the decay constant of the parent nucleus is simply defined as,
$$ \lambda=\upsilon_{0}P \;, $$ (17) The assault frequency
$ \upsilon_{0} $ has been taken as,$$ \begin{array}{l} \upsilon_{0}=1.0\times10^{21} {\rm{s}}^{-1}. \end{array} $$ (18) In principle, the assault frequency is very complicated, it is intimately linked to nuclear structure information. In the present work, for the sake of simplicity, the assault frequency parameters of fixed values are adopted in our calculations. For all the above results, our calculations for three decay processes were performed with one set of parameters. In fact, the assault frequency of three cluster processes is very different.
The barrier penetrability
$ P $ is calculated within the action integral$$ P=\exp\Big[-\frac{2}{\hbar}\int\nolimits_{R_{{\rm{in}}}}^{R_{{\rm{out}}}}\sqrt{2B(r)(E(r)-Q)}{\rm{d}}r\Big], $$ (19) with
$E(R_{{\rm{in}}})=E(R_{{\rm{out}}})=Q_{{\rm{exp}}}$ .The inertia
$ B(r) $ has been chosen as[59]$$ B(r)=\mu\Bigg\{1+k \times f(r)\frac{17}{15}\exp\bigg\{-\frac{128}{51}\Big[\big(r-R_{{\rm{in}}}\big)/R_{0}\Big]\bigg\}\Bigg\}, $$ (20) where
$$ f(r)=\left\{ \begin{array}{ll} \Bigg(\dfrac{R_{{\rm{cont}}}-r}{R_{{\rm{cont}}}-R_{{\rm{in}}}}\Bigg)^{3} & r \leqslant R_{{\rm{cont}}}, \\ 0 & r\geqslant R_{{\rm{cont}}}, \end{array} \right. $$ where
$ R_{{\rm{cont}}} $ is the sum of$ R_{1} $ and$ R_{2} $ .Accurate knowledge of the
$ B(r) $ ,$ E(r) $ and$ Q $ values is crucial for the calculation, since the WKB penetrabilities are very sensitive to them. The partial half-life is related to the decay constant$ \lambda $ by$$ T_{1/2}=\frac{\ln 2}{\lambda}. $$ (21) -
The effect of the coefficient of surface asymmetry term (
$ k_{{\rm{s}}} $ ) on$ \alpha $ decay half-lives of heavy nuclei are investigated within GLDM. The numerical results are given in Table 1, in which the second column denotes experimental$ Q $ values. The results calculated by the GLDM are given in the third, fourth, and fifth columns when the coefficients of surface asymmetry are different. The experimental$ \alpha $ decay half-lives are given in the sixth column. As can be seen from the Table 1, the calculated half-lives agree precisely with the experimental data and the ratio between them is approximately within a factor of 4, except for the case of$ \alpha $ decay from odd$ A $ nuclei$ ^{235}{\rm{U}} $ ,$ ^{231}{\rm{Pa}} $ and$ ^{237}{\rm{Np}} $ . In addition, from the Table 1 we can see that the half-life of$ \alpha $ decay is not a sensitive dependency on the coefficient of surface asymmetry term ($ k_{{\rm{s}}} $ ).Table 1. Comparison between experimental and theoretical
$ \alpha $ decay half-lives of heavy nuclei.AZ ${Q} _{\alpha}$(Exp.) $ T_{1/2} $(Cal.)/s $ T_{1/2} $(Cal.)/s $ T_{1/2} $(Cal.)/s $ T_{1/2} $(Exp.)/s /MeV (1−0I2) (1−1.8I2) (1−3.1I2) $ ^{226}{\rm{Th}} $ 6.450 $ 1.0\times10^{3} $ $ 1.2\times10^{3} $ $ 1.5\times10^{3} $ $ 1.2\times10^{3} $ $ ^{228}{\rm{Th}} $ 5.519 $ 4.2\times10^{7} $ $ 5.9\times10^{7} $ $ 8.4\times10^{7} $ $ 6.0\times10^{7} $ $ ^{230}{\rm{Th}} $ 4.769 $ 2.3\times10^{12} $ $ 3.6\times10^{12} $ $ 6.0\times10^{12} $ $ 2.4\times10^{12} $ $ ^{232}{\rm{Th}} $ 4.081 $ 7.3\times10^{17} $ $ 1.3\times10^{18} $ $ 2.2\times10^{18} $ $ 4.4\times10^{17} $ $ ^{231}{\rm{Pa}} $ 5.149 $ 2.2\times10^{10} $ $ 3.3\times10^{10} $ $ 5.0\times10^{10} $ $ 9.9\times10^{11} $ $ ^{230}{\rm{U}} $ 5.992 $ 1.1\times10^{6} $ $ 1.4\times10^{6} $ $ 1.9\times10^{6} $ $ 1.7\times10^{6} $ $ ^{232}{\rm{U}} $ 5.413 $ 1.8\times10^{9} $ $ 2.4\times10^{9} $ $ 3.5\times10^{9} $ $ 2.2\times10^{9} $ $ ^{233}{\rm{U}} $ 4.908 $ 3.1\times10^{12} $ $ 4.6\times10^{12} $ $ 7.5\times10^{12} $ $ 5.0\times10^{12} $ $ ^{234}{\rm{U}} $ 4.857 $ 6.9\times10^{12} $ $ 1.1\times10^{13} $ $ 1.7\times10^{13} $ $ 7.7\times10^{13} $ $ ^{235}{\rm{U}} $ 4.678 $ 1.3\times10^{14} $ $ 2.0\times10^{14} $ $ 3.4\times10^{14} $ $ 2.2\times10^{16} $ $ ^{236}{\rm{U}} $ 4.572 $ 7.7\times10^{14} $ $ 1.3\times10^{15} $ $ 2.2\times10^{15} $ $ 7.4\times10^{14} $ $ ^{237}{\rm{Np}} $ 4.958 $ 4.0\times10^{12} $ $ 6.5\times10^{12} $ $ 1.1\times10^{13} $ $ 6.8\times10^{13} $ $ ^{236}{\rm{Pu}} $ 5.866 $ 3.7\times10^{7} $ $ 5.1\times10^{7} $ $ 7.1\times10^{7} $ $ 9.0\times10^{7} $ $ ^{238}{\rm{Pu}} $ 5.593 $ 1.2\times10^{9} $ $ 1.8\times10^{9} $ $ 2.7\times10^{9} $ $ 2.8\times10^{9} $ From the decay dynamics studies it is known that spontaneous decay half-lives are very sensitive to the details of the potential barrier. In order to illustrate the influence of
$ k_{{\rm{s}}} $ on the barriers, the potential barrier governing the$ ^{28}{\rm{Mg}} $ emission from$ ^{234}{\rm{U}} $ is displayed in Fig.1(b). The potential barrier is constructed by a GLDM, taking into account the nuclear proximity, the mass asymmetry, the accurate nuclear radius, the phenomenological shell and pairing correction. The dashed and solid curve show the potential barrier with the different$ k_{{\rm{s}}} $ value. From Fig.1 that the potential barrier changes when$ k_{{\rm{s}}} $ changed from 1.8 to 3.1. This will directly affect the half-lives of cluster radioactivity. One can see clearly from Fig.1 that the influence of the coefficient of surface asymmetry is strongly dependent on the charge asymmetry$\eta^{}_{Z} = (Z_{1}-Z_{2})/(Z_{1}+Z_{2})$ for the same parent nucleus during the rearrangement process. With the decrease of the chagre asymmetry$\eta^{}_{Z}$ , the influence of the change of the macroscopic energy coefficient on the potential is more obvious.Figure 1. The change of the potential barrier caused by the asymmetry of surface
$ k_{{\rm{s}}} $ from 1.8 to 3.1,$ k_{{\rm{s}}} $ values are 1.8 (solid line) and 3.1(dashed line) descriptions for$ \alpha $ decay (a), cluster radioactivity (b), and spontaneous cold fission process (c) of the$ ^{234}{\rm{U}} $ parent nucleus.$ r $ is the distances between the mass centers.We have systematically calculated the cluster radioactivity half-lives from
$ ^{226}{\rm{Th}} $ to$ ^{242}{\rm{Cm}} $ by using GLDM taking into account the influence of$ k_{{\rm{s}}} $ and inertia coefficient on half-lives. The detailed results are listed in Table 2, in which the first column denotes the parent nuclei and the second column denotes$ Q $ values. The results calculated by the GLDM considering the coefficient of surface asymmetry$ k_{{\rm{s}}} $ of different are listed in the third and fourth columns. One can see that the cluster emission half-lives is increased along with the increasing of$ k_{{\rm{s}}} $ from 1.8 to 3.1, while the other parameters is fixed. In this sense, the calculated cluster radioactivity half-lives depend on the coefficient of surface asymmetry$ k_{{\rm{s}}} $ . In order to consider the influence of inertia coefficient on half-lives, in the sixth column of Table 2 are shown the corresponding results of half-lives of the cluster radioactivity in which were calculated by different values of inertia coefficient. Because surface energy depends on deformation, the change of surface energy coefficient also has influence on potential barrier. If we choose the surface energy coefficient as the latest fitting result$a_{\rm s}=18.18$ [60], we can find that the change of the surface energy coefficient makes the half-lives of the radioactivity of the cluster have certain influence by comparing the results of fifth and sixth columns. It can be found from the third to the sixth columns in Table 2 that the deviations between the experimental data and the calculated values are less than$ 10^{2} $ for the most nuclei. Through the analysis, it is found that, although the surface energy coefficient ($ a_{\rm s} $ ), the surface asymmetry coefficient ($ k_{{\rm{s}}} $ ) and the inertia coefficient ($ B(r) $ ) have influence on the half lives of the cluster radioactivity, but the change of these values will not make the uncertainty of half-lives more than two order of magnitude.Table 2. The Q values and the half-lives of cluster radioactivity. The second column denotes Q values extracted from AME2012[58]. The third and fourth columns indicate, respectively, the theoretical half-lives taking into account different macroscopic energy coefficient. The fifth and sixth columns indicate the influence of surface energy and inertia coefficient on half-lives of cluster radioactivity. The experimental data[28] are shown in the last column.
Emitter and cluster $ Q $(Exp.) $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(Exp.)/s /MeV (1−1.8I2) $k = 4.0$ (1−3.1I2) $k = 4.0$ (1−1.8I2) $k = 8.0$ (1−1.8I2) $k = 8.0$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 18.180~0$ $a_{ {\rm{s} } } = 17.943~9$ $ ^{226}{\rm{Th}} $$ \rightarrow ^{14}{\rm{C}} $+$ ^{212}{\rm{Po}} $ 30.546 $ 2.0\times10^{17} $ $ 1.4\times10^{18} $ $ 2.3\times10^{17} $ $ 4.1\times10^{17} $ $>2.0\times10^{15} $ $ ^{226}{\rm{Th}} $$ \rightarrow ^{18}{\rm{O}} $+$ ^{208}{\rm{Pb}} $ 45.728 $ 2.3\times10^{17} $ $ 2.0\times10^{18} $ $ 3.5\times10^{17} $ $ 7.3\times10^{17} $ $>2.0\times10^{15} $ $ ^{228}{\rm{Th}} $$ \rightarrow ^{20}{\rm{O}} $+$ ^{208}{\rm{Pb}} $ 44.724 $ 7.4\times10^{19} $ $ 1.9\times10^{21} $ $ 8.6\times10^{19} $ $ 1.9\times10^{20} $ $ 7.5\times10^{20} $ $ ^{230}{\rm{Th}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{206}{\rm{Hg}} $ 57.761 $ 3.8\times10^{23} $ $ 8.7\times10^{24} $ $ 9.1\times10^{23} $ $ 2.1\times10^{24} $ $ 4.4\times10^{24} $ $ ^{232}{\rm{Th}} $$ \rightarrow ^{26}{\rm{Ne}} $+$ ^{206}{\rm{Hg}} $ 55.914 $ 6.1\times10^{27} $ $ 2.9\times10^{29} $ $ 1.5\times10^{28} $ $ 3.7\times10^{28} $ $>1.6\times10^{29} $ $ ^{231}{\rm{Pa}} $$ \rightarrow ^{23}{\rm{F}} $+$ ^{208}{\rm{Pb}} $ 51.860 $ 2.4\times10^{22} $ $ 8.3\times10^{23} $ $ 3.0\times10^{22} $ $ 8.3\times10^{22} $ $ 1.0\times10^{26} $ $ ^{230}{\rm{U}} $$ \rightarrow ^{22}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 61.387 $ 6.3\times10^{19} $ $ 6.5\times10^{20} $ $ 1.7\times10^{20} $ $ 3.5\times10^{20} $ $>1.6\times10^{18} $ $ ^{230}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{206}{\rm{Pb}} $ 61.351 $ 2.1\times10^{20} $ $ 4.0\times10^{21} $ $ 3.3\times10^{20} $ $ 9.7\times10^{20} $ $>1.6\times10^{18} $ $ ^{232}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{204}{\rm{Hg}} $ 74.319 $ 1.2\times10^{24} $ $ 2.7\times10^{25} $ $ 4.3\times10^{24} $ $ 1.1\times10^{25} $ $>4.5\times10^{22} $ $ ^{232}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 62.310 $ 1.8\times10^{18} $ $ 6.2\times10^{19} $ $ 2.1\times10^{18} $ $ 6.4\times10^{18} $ $ 2.5\times10^{20} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{209}{\rm{Pb}} $ 60.485 $ 4.3\times10^{21} $ $ 1.0\times10^{23} $ $ 7.5\times10^{21} $ $ 2.1\times10^{22} $ $ 6.8\times10^{24} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{25}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 60.728 $ 2.3\times10^{21} $ $ 9.0\times10^{22} $ $ 3.3\times10^{21} $ $ 9.3\times10^{21} $ $ 2.0\times10^{23} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{205}{\rm{Hg}} $ 74.226 $ 1.1\times10^{24} $ $ 2.9\times10^{25} $ $ 3.7\times10^{24} $ $ 9.5\times10^{24} $ $>3.9\times10^{27} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{210}{\rm{Pb}} $ 58.826 $ 5.7\times10^{24} $ $ 1.1\times10^{26} $ $ 1.7\times10^{25} $ $ 3.7\times10^{25} $ $ 1.6\times10^{25} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{26}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 59.415 $ 1.3\times10^{24} $ $ 5.7\times10^{25} $ $ 1.9\times10^{24} $ $ 6.3\times10^{24} $ $ 7.9\times10^{25} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{206}{\rm{Hg}} $ 74.110 $ 1.1\times10^{24} $ $ 3.0\times10^{25} $ $ 4.1\times10^{24} $ $ 9.8\times10^{24} $ $ 3.5\times10^{25} $ $ ^{235}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{211}{\rm{Pb}} $ 57.363 $ 3.1\times10^{27} $ $ 5.5\times10^{28} $ $ 1.2\times10^{28} $ $ 2.4\times10^{28} $ $ 2.8\times10^{27} $ $ ^{235}{\rm{U}} $$ \rightarrow ^{25}{\rm{Ne}} $+$ ^{210}{\rm{Pb}} $ 57.708 $ 1.4\times10^{27} $ $ 3.5\times10^{28} $ $ 4.2\times10^{27} $ $ 9.7\times10^{27} $ $ 2.8\times10^{27} $ $ ^{235}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{207}{\rm{Hg}} $ 72.426 $ 7.0\times10^{26} $ $ 1.7\times10^{28} $ $ 3.3\times10^{27} $ $ 7.6\times10^{27} $ $>2.8\times10^{28} $ $ ^{236}{\rm{U}} $$ \rightarrow ^{30}{\rm{Mg}} $+$ ^{206}{\rm{Hg}} $ 72.275 $ 5.1\times10^{27} $ $ 2.6\times10^{29} $ $ 1.9\times10^{28} $ $ 5.3\times10^{28} $ $ 3.8\times10^{27} $ $ ^{237}{\rm{Np}} $$ \rightarrow ^{30}{\rm{Mg}} $+$ ^{207}{\rm{Tl}} $ 74.790 $ 9.5\times10^{24} $ $ 4.6\times10^{26} $ $ 2.7\times10^{25} $ $ 7.7\times10^{25} $ $>3.7\times10^{27} $ $ ^{236}{\rm{Pu}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{208}{\rm{Pb}} $ 79.669 $ 2.5\times10^{18} $ $ 9.1\times10^{19} $ $ 3.7\times10^{18} $ $ 1.3\times10^{19} $ $ 4.7\times10^{21} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{210}{\rm{Pb}} $ 75.911 $ 3.6\times10^{24} $ $ 8.6\times10^{25} $ $ 1.5\times10^{25} $ $ 3.5\times10^{25} $ $ 5.0\times10^{25} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{30}{\rm{Mg}} $+$ ^{208}{\rm{Pb}} $ 76.796 $ 2.6\times10^{23} $ $ 1.2\times10^{25} $ $ 6.1\times10^{23} $ $ 1.9\times10^{24} $ $ 4.7\times10^{25} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{32}{\rm{Si}} $+$ ^{206}{\rm{Hg}} $ 91.187 $ 1.2\times10^{24} $ $ 4.7\times10^{25} $ $ 6.0\times10^{24} $ $ 1.7\times10^{25} $ $ 1.9\times10^{24} $ It is well known that the symmetry energy coefficient of finite nuclei is usually extracted by directly fitting the measured nuclear masses with different versions of the liquid drop mass formula. Some different forms for describing the mass dependence of symmetry energy coefficients of finite nuclei, which divide the symmetry energy of a nucleus into the volume and surface contributions, were proposed in Refs. [60-61]. The mass dependence of symmetry energy of the nucleus has been adopted in the GLDM model,
$$ \begin{split} E_{{\rm{sym}}} &=a_{\rm v}k_{\rm v}I^{2}A+a_{\rm s}k_{{\rm{s}}} I^{2}A^{2/3} \\& =(a_{\rm v}k_{\rm v}+a_{\rm s}k_{{\rm{s}}} A^{-1/3})I^{2}A \;. \end{split} $$ (22) The symmetry energy coefficient
$ a_{{\rm{sym}}} $ is expressed,$$ a_{{\rm{sym}}}=c_{{\rm{sym}}}\bigg(1-\frac{\kappa}{A^{1/3}}\bigg) \;, $$ (23) where
$ c_{{\rm{sym}}}=a_{{\rm{v}}}k_{{\rm{v}}}=27.889 $ ,$\kappa=(a_{{\rm{s}}}k_{{\rm{s}}} )/(a_{\rm v}k_{{\rm{v}}})=0.643~k_{{\rm{s}}}$ . Because of the volume conservation, the contribution of the symmetry energy coefficient from the volume energy ($ c_{{\rm{sym}}} $ ) is always zero when the shape evolution from one body to two separated fragments is adopted unified way. Therefore, the contribution of the surface part of the symmetry energy coefficient is crucial for the reasonable description of the potential barrier and half-lives. In the framework of the GLDM model, if the coefficient of surface asymmetry$ k_{{\rm{s}}} $ is 1.8, the symmetry energy coefficient is about 22.89. This value is very close to the latest results of the symmetric energy coefficient of finite heavy nuclei[60-61]. Although the difference of the symmetry energy coefficients of finite nuclei given by different theoretical work is very small, the proportion of surface terms (the symmetry energy coefficient of a nucleus into the volume and surface contributions) in the symmetry energy coefficient is not completely determined. For example, in the work of Danielewicz et al.[60], the contribution of the surface term to the symmetry energy coefficient is approximately 17% for the present study region. However, in the work of Wang et al.[61], the contribution of the surface in the symmetry energy coefficient is approximately 25%. In the present work, the contribution of the surface terms to the symmetry energy coefficient is changed from 18% to 26% when the surface asymmetry coefficient$ k_{{\rm{s}}} $ is changed from 1.8 to 3.1. It can be seen from the present results that the difference of surface terms has an certain influence on the cluster radioactivity process.In order to unify the view of
$ \alpha $ decay, cluster radioactivity and cold fission, it is quite natural to interpret cold fission as a cold rearrangement process with fragments in their ground states, in an analogous way with the cluster radioactivity. Thus the cold fission process reduces to the penetration of a potential barrier. The numerical results are given in Table 3, one can see that in the cold fission process correspond to one of the fragments close to the double magic nucleus$ ^{132}{\rm{Sn}} $ , in which the second column denotes$ Q $ values. The results calculated by the GLDM considering different surface asymmetry coefficient$ k_{{\rm{s}}} $ are listed in the third and fourth columns. The experimental cold fission half-lives are given in the last column[62]. From the fission dynamics studies it is known that fission half-lives are very sensitive to the details of the potential barrier and the inertia coefficient[59]. From the Table 3 of the results show that the uncertainty of surface asymmetry and inertia coefficients have a significant effect on the theoretical calculation of the cold fission half-lives. The GLDM model is to reach a reasonable calculation for the half-lives for cold fission processes. By demonstrating the detailed results, we would like to point out the sensitivity of the calculated cold fission half-lives to the macroscopic energy coefficient and the inertia coefficient.Table 3. Same as Table 2, but for cold fission process.
Emitter and cluster $ Q $(Exp.) $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(Exp.)/s /MeV (1−1.8I2) $k = 4.0$ (1−3.1I2) $k = 4.0$ (1−1.8I2) $k = 8.0$ (1−1.8I2) $k = 8.0$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 18.180~0$ $a_{ {\rm{s} } } = 17.943~9$ $ ^{226}{\rm{Th}} $$ \rightarrow ^{92}{\rm{Sr}} $+$ ^{134}{\rm{Te}} $ 188.600 $ 6.0\times10^{30} $ $ 7.2\times10^{33} $ $ 2.1\times10^{32} $ $ 1.6\times10^{33} $ $ ^{228}{\rm{Th}} $$ \rightarrow ^{94}{\rm{Sr}} $+$ ^{134}{\rm{Te}} $ 188.153 $ 6.1\times10^{30} $ $ 1.5\times10^{34} $ $ 1.9\times10^{32} $ $ 1.5\times10^{33} $ $ ^{230}{\rm{Th}} $$ \rightarrow ^{96}{\rm{Sr}} $+$ ^{134}{\rm{Te}} $ 186.330 $ 7.9\times10^{32} $ $ 8.1\times10^{37} $ $ 3.2\times10^{34} $ $ 2.6\times10^{35} $ $ ^{232}{\rm{Th}} $$ \rightarrow ^{100}{\rm{Zr}} $+$ ^{132}{\rm{Sn}} $ 188.374 $ 2.3\times10^{33} $ $ 1.8\times10^{37} $ $ 9.5\times10^{34} $ $ 8.0\times10^{35} $ $ ^{231}{\rm{Pa}} $$ \rightarrow ^{97}{\rm{Y}} $+$ ^{134}{\rm{Te}} $ 192.089 $ 7.7\times10^{32} $ $ 2.1\times10^{36} $ $ 3.7\times10^{34} $ $ 2.9\times10^{35} $ $ ^{230}{\rm{U}} $$ \rightarrow ^{94}{\rm{Sr}} $+$ ^{136}{\rm{Xe}} $ 196.889 $ 2.3\times10^{30} $ $ 2.7\times10^{33} $ $ 9.3\times10^{31} $ $ 7.6\times10^{32} $ $ ^{232}{\rm{U}} $$ \rightarrow ^{98}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 198.439 $ 1.6\times10^{32} $ $ 3.0\times10^{35} $ $ 8.4\times10^{33} $ $ 6.7\times10^{34} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{99}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 197.080 $ 6.9\times10^{33} $ $ 1.5\times10^{37} $ $ 4.2\times10^{35} $ $ 3.3\times10^{36} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{100}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 197.064 $ 3.8\times10^{33} $ $ 1.3\times10^{37} $ $ 2.3\times10^{35} $ $ 1.9\times10^{36} $ $\approx 1.0\times10^{30}$ $ ^{235}{\rm{U}} $$ \rightarrow ^{101}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 196.627 $ 6.5\times10^{33} $ $ 3.1\times10^{37} $ $ 3.9\times10^{35} $ $ 3.3\times10^{36} $ $ ^{236}{\rm{U}} $$ \rightarrow ^{104}{\rm{Mo}} $+$ ^{132}{\rm{Sn}} $ 199.346 $ 5.1\times10^{32} $ $ 3.3\times10^{36} $ $ 2.5\times10^{34} $ $ 2.0\times10^{35} $ $ ^{237}{\rm{Np}} $$ \rightarrow ^{103}{\rm{Nb}} $+$ ^{134}{\rm{Te}} $ 202.434 $ 1.7\times10^{33} $ $ 6.7\times10^{36} $ $ 9.8\times10^{34} $ $ 8.0\times10^{35} $ $ ^{236}{\rm{Pu}} $$ \rightarrow ^{102}{\rm{Mo}} $+$ ^{134}{\rm{Te}} $ 207.540 $ 7.1\times10^{32} $ $ 9.9\times10^{35} $ $ 5.3\times10^{34} $ $ 4.1\times10^{35} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{104}{\rm{Mo}} $+$ ^{134}{\rm{Te}} $ 209.056 $ 1.2\times10^{32} $ $ 3.8\times10^{35} $ $ 7.5\times10^{33} $ $ 6.3\times10^{34} $
Unified Description of the Competition Between α Decay, Cluster Radioactivity and Cold Fission
-
摘要: 基于推广的液滴模型计算了原子核不同自发衰变过程的半衰期。通过考虑亲近势、质量不对称度、准确的核半径、唯象包含了壳修正和对修正的推广液滴模型构建衰变势垒随中心距离的变化。目前的理论计算很好的再现了alpha衰变和结团放射性的半衰期,并能合理的描述冷裂变的半衰期。计算表明目前版本的推广液滴模型可以在统一框架下合理的描述不同原子核的自发衰变过程的寿命。本文也讨论了电荷不对称度和质量惯量对不同衰变半衰期的影响。Abstract: Half-lives of spontaneous nuclear decay processes are calculated by a generalized liquid drop model (GLDM). The potential barrier is constructed by a GLDM, taking into account the nuclear proximity, the mass asymmetry, the accurate nuclear radius, the phenomenological shell and pairing correction. The GLDM model is to continue reproducing the experimental data for
$\alpha$ decay and cluster radioactivity, as well as to reach a reasonable calculation for the half-lives for cold fission processes. These comparisons show that the GLDM is useful tools to investigate these different decay processes in an unified theoretical framework. The influence of macroscopic energy coefficient on the potential barrier and half-lives are strongly dependent on the charge asymmetry ($\eta^{}_{Z} = $ $ (Z_{1}-Z_{2})/(Z_{1}+Z_{2})$ ) for the same parent nucleus during the rearrangement process. The influence of inertia coefficient on half lives also depend on the mass asymmetry$\eta^{}_{Z}$ .-
Key words:
- alpha decay /
- cluster radioactivity /
- cold fission /
- generalized liquid drop model
-
Figure 1. The change of the potential barrier caused by the asymmetry of surface
$ k_{{\rm{s}}} $ from 1.8 to 3.1,$ k_{{\rm{s}}} $ values are 1.8 (solid line) and 3.1(dashed line) descriptions for$ \alpha $ decay (a), cluster radioactivity (b), and spontaneous cold fission process (c) of the$ ^{234}{\rm{U}} $ parent nucleus.$ r $ is the distances between the mass centers.Table 1. Comparison between experimental and theoretical
$ \alpha $ decay half-lives of heavy nuclei.AZ ${Q} _{\alpha}$(Exp.) $ T_{1/2} $(Cal.)/s $ T_{1/2} $(Cal.)/s $ T_{1/2} $(Cal.)/s $ T_{1/2} $(Exp.)/s /MeV (1−0I2) (1−1.8I2) (1−3.1I2) $ ^{226}{\rm{Th}} $ 6.450 $ 1.0\times10^{3} $ $ 1.2\times10^{3} $ $ 1.5\times10^{3} $ $ 1.2\times10^{3} $ $ ^{228}{\rm{Th}} $ 5.519 $ 4.2\times10^{7} $ $ 5.9\times10^{7} $ $ 8.4\times10^{7} $ $ 6.0\times10^{7} $ $ ^{230}{\rm{Th}} $ 4.769 $ 2.3\times10^{12} $ $ 3.6\times10^{12} $ $ 6.0\times10^{12} $ $ 2.4\times10^{12} $ $ ^{232}{\rm{Th}} $ 4.081 $ 7.3\times10^{17} $ $ 1.3\times10^{18} $ $ 2.2\times10^{18} $ $ 4.4\times10^{17} $ $ ^{231}{\rm{Pa}} $ 5.149 $ 2.2\times10^{10} $ $ 3.3\times10^{10} $ $ 5.0\times10^{10} $ $ 9.9\times10^{11} $ $ ^{230}{\rm{U}} $ 5.992 $ 1.1\times10^{6} $ $ 1.4\times10^{6} $ $ 1.9\times10^{6} $ $ 1.7\times10^{6} $ $ ^{232}{\rm{U}} $ 5.413 $ 1.8\times10^{9} $ $ 2.4\times10^{9} $ $ 3.5\times10^{9} $ $ 2.2\times10^{9} $ $ ^{233}{\rm{U}} $ 4.908 $ 3.1\times10^{12} $ $ 4.6\times10^{12} $ $ 7.5\times10^{12} $ $ 5.0\times10^{12} $ $ ^{234}{\rm{U}} $ 4.857 $ 6.9\times10^{12} $ $ 1.1\times10^{13} $ $ 1.7\times10^{13} $ $ 7.7\times10^{13} $ $ ^{235}{\rm{U}} $ 4.678 $ 1.3\times10^{14} $ $ 2.0\times10^{14} $ $ 3.4\times10^{14} $ $ 2.2\times10^{16} $ $ ^{236}{\rm{U}} $ 4.572 $ 7.7\times10^{14} $ $ 1.3\times10^{15} $ $ 2.2\times10^{15} $ $ 7.4\times10^{14} $ $ ^{237}{\rm{Np}} $ 4.958 $ 4.0\times10^{12} $ $ 6.5\times10^{12} $ $ 1.1\times10^{13} $ $ 6.8\times10^{13} $ $ ^{236}{\rm{Pu}} $ 5.866 $ 3.7\times10^{7} $ $ 5.1\times10^{7} $ $ 7.1\times10^{7} $ $ 9.0\times10^{7} $ $ ^{238}{\rm{Pu}} $ 5.593 $ 1.2\times10^{9} $ $ 1.8\times10^{9} $ $ 2.7\times10^{9} $ $ 2.8\times10^{9} $ Table 2. The Q values and the half-lives of cluster radioactivity. The second column denotes Q values extracted from AME2012[58]. The third and fourth columns indicate, respectively, the theoretical half-lives taking into account different macroscopic energy coefficient. The fifth and sixth columns indicate the influence of surface energy and inertia coefficient on half-lives of cluster radioactivity. The experimental data[28] are shown in the last column.
Emitter and cluster $ Q $(Exp.) $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(Exp.)/s /MeV (1−1.8I2) $k = 4.0$ (1−3.1I2) $k = 4.0$ (1−1.8I2) $k = 8.0$ (1−1.8I2) $k = 8.0$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 18.180~0$ $a_{ {\rm{s} } } = 17.943~9$ $ ^{226}{\rm{Th}} $$ \rightarrow ^{14}{\rm{C}} $+$ ^{212}{\rm{Po}} $ 30.546 $ 2.0\times10^{17} $ $ 1.4\times10^{18} $ $ 2.3\times10^{17} $ $ 4.1\times10^{17} $ $>2.0\times10^{15} $ $ ^{226}{\rm{Th}} $$ \rightarrow ^{18}{\rm{O}} $+$ ^{208}{\rm{Pb}} $ 45.728 $ 2.3\times10^{17} $ $ 2.0\times10^{18} $ $ 3.5\times10^{17} $ $ 7.3\times10^{17} $ $>2.0\times10^{15} $ $ ^{228}{\rm{Th}} $$ \rightarrow ^{20}{\rm{O}} $+$ ^{208}{\rm{Pb}} $ 44.724 $ 7.4\times10^{19} $ $ 1.9\times10^{21} $ $ 8.6\times10^{19} $ $ 1.9\times10^{20} $ $ 7.5\times10^{20} $ $ ^{230}{\rm{Th}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{206}{\rm{Hg}} $ 57.761 $ 3.8\times10^{23} $ $ 8.7\times10^{24} $ $ 9.1\times10^{23} $ $ 2.1\times10^{24} $ $ 4.4\times10^{24} $ $ ^{232}{\rm{Th}} $$ \rightarrow ^{26}{\rm{Ne}} $+$ ^{206}{\rm{Hg}} $ 55.914 $ 6.1\times10^{27} $ $ 2.9\times10^{29} $ $ 1.5\times10^{28} $ $ 3.7\times10^{28} $ $>1.6\times10^{29} $ $ ^{231}{\rm{Pa}} $$ \rightarrow ^{23}{\rm{F}} $+$ ^{208}{\rm{Pb}} $ 51.860 $ 2.4\times10^{22} $ $ 8.3\times10^{23} $ $ 3.0\times10^{22} $ $ 8.3\times10^{22} $ $ 1.0\times10^{26} $ $ ^{230}{\rm{U}} $$ \rightarrow ^{22}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 61.387 $ 6.3\times10^{19} $ $ 6.5\times10^{20} $ $ 1.7\times10^{20} $ $ 3.5\times10^{20} $ $>1.6\times10^{18} $ $ ^{230}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{206}{\rm{Pb}} $ 61.351 $ 2.1\times10^{20} $ $ 4.0\times10^{21} $ $ 3.3\times10^{20} $ $ 9.7\times10^{20} $ $>1.6\times10^{18} $ $ ^{232}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{204}{\rm{Hg}} $ 74.319 $ 1.2\times10^{24} $ $ 2.7\times10^{25} $ $ 4.3\times10^{24} $ $ 1.1\times10^{25} $ $>4.5\times10^{22} $ $ ^{232}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 62.310 $ 1.8\times10^{18} $ $ 6.2\times10^{19} $ $ 2.1\times10^{18} $ $ 6.4\times10^{18} $ $ 2.5\times10^{20} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{209}{\rm{Pb}} $ 60.485 $ 4.3\times10^{21} $ $ 1.0\times10^{23} $ $ 7.5\times10^{21} $ $ 2.1\times10^{22} $ $ 6.8\times10^{24} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{25}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 60.728 $ 2.3\times10^{21} $ $ 9.0\times10^{22} $ $ 3.3\times10^{21} $ $ 9.3\times10^{21} $ $ 2.0\times10^{23} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{205}{\rm{Hg}} $ 74.226 $ 1.1\times10^{24} $ $ 2.9\times10^{25} $ $ 3.7\times10^{24} $ $ 9.5\times10^{24} $ $>3.9\times10^{27} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{210}{\rm{Pb}} $ 58.826 $ 5.7\times10^{24} $ $ 1.1\times10^{26} $ $ 1.7\times10^{25} $ $ 3.7\times10^{25} $ $ 1.6\times10^{25} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{26}{\rm{Ne}} $+$ ^{208}{\rm{Pb}} $ 59.415 $ 1.3\times10^{24} $ $ 5.7\times10^{25} $ $ 1.9\times10^{24} $ $ 6.3\times10^{24} $ $ 7.9\times10^{25} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{206}{\rm{Hg}} $ 74.110 $ 1.1\times10^{24} $ $ 3.0\times10^{25} $ $ 4.1\times10^{24} $ $ 9.8\times10^{24} $ $ 3.5\times10^{25} $ $ ^{235}{\rm{U}} $$ \rightarrow ^{24}{\rm{Ne}} $+$ ^{211}{\rm{Pb}} $ 57.363 $ 3.1\times10^{27} $ $ 5.5\times10^{28} $ $ 1.2\times10^{28} $ $ 2.4\times10^{28} $ $ 2.8\times10^{27} $ $ ^{235}{\rm{U}} $$ \rightarrow ^{25}{\rm{Ne}} $+$ ^{210}{\rm{Pb}} $ 57.708 $ 1.4\times10^{27} $ $ 3.5\times10^{28} $ $ 4.2\times10^{27} $ $ 9.7\times10^{27} $ $ 2.8\times10^{27} $ $ ^{235}{\rm{U}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{207}{\rm{Hg}} $ 72.426 $ 7.0\times10^{26} $ $ 1.7\times10^{28} $ $ 3.3\times10^{27} $ $ 7.6\times10^{27} $ $>2.8\times10^{28} $ $ ^{236}{\rm{U}} $$ \rightarrow ^{30}{\rm{Mg}} $+$ ^{206}{\rm{Hg}} $ 72.275 $ 5.1\times10^{27} $ $ 2.6\times10^{29} $ $ 1.9\times10^{28} $ $ 5.3\times10^{28} $ $ 3.8\times10^{27} $ $ ^{237}{\rm{Np}} $$ \rightarrow ^{30}{\rm{Mg}} $+$ ^{207}{\rm{Tl}} $ 74.790 $ 9.5\times10^{24} $ $ 4.6\times10^{26} $ $ 2.7\times10^{25} $ $ 7.7\times10^{25} $ $>3.7\times10^{27} $ $ ^{236}{\rm{Pu}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{208}{\rm{Pb}} $ 79.669 $ 2.5\times10^{18} $ $ 9.1\times10^{19} $ $ 3.7\times10^{18} $ $ 1.3\times10^{19} $ $ 4.7\times10^{21} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{28}{\rm{Mg}} $+$ ^{210}{\rm{Pb}} $ 75.911 $ 3.6\times10^{24} $ $ 8.6\times10^{25} $ $ 1.5\times10^{25} $ $ 3.5\times10^{25} $ $ 5.0\times10^{25} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{30}{\rm{Mg}} $+$ ^{208}{\rm{Pb}} $ 76.796 $ 2.6\times10^{23} $ $ 1.2\times10^{25} $ $ 6.1\times10^{23} $ $ 1.9\times10^{24} $ $ 4.7\times10^{25} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{32}{\rm{Si}} $+$ ^{206}{\rm{Hg}} $ 91.187 $ 1.2\times10^{24} $ $ 4.7\times10^{25} $ $ 6.0\times10^{24} $ $ 1.7\times10^{25} $ $ 1.9\times10^{24} $ Table 3. Same as Table 2, but for cold fission process.
Emitter and cluster $ Q $(Exp.) $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(cal.)/s $ T_{1/2} $(Exp.)/s /MeV (1−1.8I2) $k = 4.0$ (1−3.1I2) $k = 4.0$ (1−1.8I2) $k = 8.0$ (1−1.8I2) $k = 8.0$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 17.943~9$ $a_{ {\rm{s} } } = 18.180~0$ $a_{ {\rm{s} } } = 17.943~9$ $ ^{226}{\rm{Th}} $$ \rightarrow ^{92}{\rm{Sr}} $+$ ^{134}{\rm{Te}} $ 188.600 $ 6.0\times10^{30} $ $ 7.2\times10^{33} $ $ 2.1\times10^{32} $ $ 1.6\times10^{33} $ $ ^{228}{\rm{Th}} $$ \rightarrow ^{94}{\rm{Sr}} $+$ ^{134}{\rm{Te}} $ 188.153 $ 6.1\times10^{30} $ $ 1.5\times10^{34} $ $ 1.9\times10^{32} $ $ 1.5\times10^{33} $ $ ^{230}{\rm{Th}} $$ \rightarrow ^{96}{\rm{Sr}} $+$ ^{134}{\rm{Te}} $ 186.330 $ 7.9\times10^{32} $ $ 8.1\times10^{37} $ $ 3.2\times10^{34} $ $ 2.6\times10^{35} $ $ ^{232}{\rm{Th}} $$ \rightarrow ^{100}{\rm{Zr}} $+$ ^{132}{\rm{Sn}} $ 188.374 $ 2.3\times10^{33} $ $ 1.8\times10^{37} $ $ 9.5\times10^{34} $ $ 8.0\times10^{35} $ $ ^{231}{\rm{Pa}} $$ \rightarrow ^{97}{\rm{Y}} $+$ ^{134}{\rm{Te}} $ 192.089 $ 7.7\times10^{32} $ $ 2.1\times10^{36} $ $ 3.7\times10^{34} $ $ 2.9\times10^{35} $ $ ^{230}{\rm{U}} $$ \rightarrow ^{94}{\rm{Sr}} $+$ ^{136}{\rm{Xe}} $ 196.889 $ 2.3\times10^{30} $ $ 2.7\times10^{33} $ $ 9.3\times10^{31} $ $ 7.6\times10^{32} $ $ ^{232}{\rm{U}} $$ \rightarrow ^{98}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 198.439 $ 1.6\times10^{32} $ $ 3.0\times10^{35} $ $ 8.4\times10^{33} $ $ 6.7\times10^{34} $ $ ^{233}{\rm{U}} $$ \rightarrow ^{99}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 197.080 $ 6.9\times10^{33} $ $ 1.5\times10^{37} $ $ 4.2\times10^{35} $ $ 3.3\times10^{36} $ $ ^{234}{\rm{U}} $$ \rightarrow ^{100}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 197.064 $ 3.8\times10^{33} $ $ 1.3\times10^{37} $ $ 2.3\times10^{35} $ $ 1.9\times10^{36} $ $\approx 1.0\times10^{30}$ $ ^{235}{\rm{U}} $$ \rightarrow ^{101}{\rm{Zr}} $+$ ^{134}{\rm{Te}} $ 196.627 $ 6.5\times10^{33} $ $ 3.1\times10^{37} $ $ 3.9\times10^{35} $ $ 3.3\times10^{36} $ $ ^{236}{\rm{U}} $$ \rightarrow ^{104}{\rm{Mo}} $+$ ^{132}{\rm{Sn}} $ 199.346 $ 5.1\times10^{32} $ $ 3.3\times10^{36} $ $ 2.5\times10^{34} $ $ 2.0\times10^{35} $ $ ^{237}{\rm{Np}} $$ \rightarrow ^{103}{\rm{Nb}} $+$ ^{134}{\rm{Te}} $ 202.434 $ 1.7\times10^{33} $ $ 6.7\times10^{36} $ $ 9.8\times10^{34} $ $ 8.0\times10^{35} $ $ ^{236}{\rm{Pu}} $$ \rightarrow ^{102}{\rm{Mo}} $+$ ^{134}{\rm{Te}} $ 207.540 $ 7.1\times10^{32} $ $ 9.9\times10^{35} $ $ 5.3\times10^{34} $ $ 4.1\times10^{35} $ $ ^{238}{\rm{Pu}} $$ \rightarrow ^{104}{\rm{Mo}} $+$ ^{134}{\rm{Te}} $ 209.056 $ 1.2\times10^{32} $ $ 3.8\times10^{35} $ $ 7.5\times10^{33} $ $ 6.3\times10^{34} $ -
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