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Accepted Manuscript
Systematic investigation of cluster radioactivity for uranium isotopes
W.M. Seif, Laila. H. Amer
PII:
DOI:
Reference:
S0375-9474(17)30434-7
https://doi.org/10.1016/j.nuclphysa.2017.10.004
NUPHA 21122
To appear in:
Nuclear Physics A
Received date:
Revised date:
Accepted date:
30 August 2017
17 October 2017
18 October 2017
Please cite this article in press as: W.M. Seif, Laila.H. Amer, Systematic investigation of cluster radioactivity for uranium
isotopes, Nucl. Phys. A (2017), https://doi.org/10.1016/j.nuclphysa.2017.10.004
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Systematic investigation of cluster radioactivity for
uranium isotopes
W. M. Seif
1
1,*
and Laila. H. Amer 2
Cairo University, Faculty of Science, Department of Physics, Giza 12613, Egypt
2
Taiz University, Faculty of Science, Department of Physics, Taiz, Yemen
*
E-mail: wseif@sci.cu.edu.eg
ABSTRACT
The most probable cluster decays that can be observed for 217-238U
isotopes are investigated. We identified the more-probable decays that
commonly manifest themselves via cold valleys in the driving potentials with
respect to the mass number and the atomic number, individually. The
calculations are performed using the Skyrme-SLy4 nucleon-nucleon
interaction, within the frame work of the performed cluster model. Among the
indicated favored decays that involve emitted light clusters heavier than Įparticle, twenty six decay modes display calculated half-life less than 1022
years, with branching ratio larger than 10-14%. The estimated branching ratio
for the Į-decay of 237U, that did not observed yet, is B=2.1ۭ10−10%
(TĮ=8.7ۭ109 years). The indicated most probable decays that did not
observed yet include the 22Ne decay of 232U, 25Ne and 32Si decays of 233U,
24
Ne and 29Mg decays of 235U, and the 34Si and 30Mg decay modes of 238U,
with 10−14<B(%)<10−7.
I. INTRODUCTION
The recent increased sophistication in the element production and observation
facilities demands a wider range of theoretical investigations on the different
decay modes of heavy, superheavy and exotic nuclei. Cluster decays are rare
decay modes lie between Ƚ decay and spontaneous fission. They were first
theoretically predicted by Săndulescu, Poenaru and Greiner in 1980 [1] and
subsequently observed by Rose and Jones four years later [2]. About 35 decays
involve the light emitted clusters 14C, 15N, 18,20O, 22-26Ne, 23F, 28,30Mg, and 32,34Si
have been detected from parent nuclei between 221Fr and 242Cm [2-9]. The
observed daughter nucleus in these decays was the doubly magic nucleus ଶ଴଼„, or
one of its neighbors. The detected cluster decays exhibit very long half-lives, up to
1028 s, and very small branching ratios, down to 10í16 % [10]. Cluster decays
involving different C, O, Ne and Mg isotopes, as emitted light clusters, have been
1
theoretically indicated from 112,114Ba, 116,118Ce, 120,122Nd and 124,126Sm trans-tin
nuclei [11]. Based on the cold valley description of fission, the cluster
radioactivity has been studied for neutron-deficient isotopes from Ba to Gd [12]
and other trans-tin nuclei [13,14], for actinides [15,16], and for superheavies [17].
New decay modes with emitted clusters of mass number A > 28 were theoretically
indicated in these studies. The behavior of the calculated half-lives of the decays
involving 14C, 20O, 20Ne and 24Ne emitted clusters for 7436 cluster decay modes
from nuclei with 85 ” Z ” 122 has been used to predict the magic neutron (proton)
numbers up to N=200 (Z =114,116) [18].
There are twenty seven known uranium (Z=92) isotopes, 217-243U. Their
half-lives range from 700 ns (221U) to 4.5ۭ109 years (238U). Except the heavy
237,239,241-243
U isotopes, all the U isotopes show observed Į decays. Decay modes
involving emitted clusters heavier than Ƚ-particle have been experimentally
observed from 230U (22,24Ne (branching ratio § 5ۭ10-12 %)), 232U (24Ne (9ۭ10-10
%) and 28Mg (൏5ۭ10-12 %)), 233U (24Ne (9ۭ10-10 %) and 28Mg (൏10-13 %)), 234U
(28Mg (10-11 %) and 24,26Ne (9ۭ10-12 %)), 235U (28Mg (8ۭ10-10 %) and 24,25Ne
(8ۭ10-10 %)), and 236U (Ne + Mg (൏4ۭ10-10 %)) isotopes [9,19,20]. The partial
half-lives for the observed cluster decays of 230-236U rapidly vary with the shell
and other structure influences [10]. The 22Ne and 24Ne cluster decay modes of
230
U were experimentally detected with the same half-life, T22,24Ne൐1.6ۭ1018 s.
The same situation also exists for the 24Ne and 26Ne cluster decays of 234U (T24,26Ne
= 8.6ۭ1025 s), and for the 24Ne and 25Ne decays of 235U (T24,25Ne=2.8ۭ1027 s).
Some models such as the cluster model of Ƚ decay and its extension to the cluster
decay process [21] failed to give a consistent half-lives for a few of these decays.
For instance, the mentioned model did not give a satisfactory explanation of the
cluster decay half-life of 235U. This is due to the sensitivity of the different models
to the experimental released energy and the spin-parity configuration of the
participating nuclei. Also, the model dependence of some calculated quantities
such as the cluster preformation probability is one of the main reasons. Further,
the experimentally predicted half-lives of some decays are obtained by indirect
measurements [21], which yield large uncertainty.
In the present work, we investigate all the possible cluster decays that did
not observed yet from the U isotopes. The favored cluster-decay modes are
indicated through the cold valleys in the calculated driving potentials of binary
fragmentations. The most probable decays to be observed are predicted by
calculating the half-lives and the branching ratios of the obtained favored decays,
in the framework of the performed cluster model. In the next section, the
theoretical framework of the approaches used to calculate the driving potential
and the half-life is outlined. The results are presented and discussed in Sec. III.
Finally, a brief summary and conclusions are given in Sec. IV.
2
II. THEORETICAL FRAMEWORK
The performed cluster model [22,23] has been successfully applied in the Į
and cluster decay studies [24-26] of heavy nuclei. It is based on Gamow two-step
mechanism including preformation and tunneling. The emitted light cluster is
assumed to be initially formed as a distinct entity inside the parent nucleus, beside
the core daughter nucleus. This takes place with a definite preformation
probability Sc. Once formed, the light cluster tries to tunnel through the produced
cluster-core barrier by knocking against it with an assault frequency Ȟ. Finally, it
penetrates the Coulomb barrier with a penetration probability P. The decay width
of this tunneling process can be then defined as
߁ ൌ ԰ߥܲǤሺͳሻ
The driving potential for a binary fragmentation process involving the light
emitted cluster and the daughter nucleus can be obtained in terms of their
interaction poetical and the corresponding released energy ܳ௖ ሺ‡ሻ as
ܸௗ ሺ‫ݎ‬ሻ ൌ ߣܸே ሺ‫ݎ‬ሻ ൅ ܸ஼ ሺ‫ݎ‬ሻ െ ܳ௖ Ǥሺʹሻ
The renormalization factor Ȝ of the nuclear potential is usually required to fit
elastic and inelastic scattering, and fusion data [27]. Here, it is inserted to
guarantee a quasi-stationary state in the Į- and cluster-decay processes [28]. The
method used to determine Ȝ is discussed below.
In the framework of the Hamiltonian energy density approach [29], the nuclear
interaction ܸே ሺ‫ݎ‬ሻ between the emitted cluster (c) and the daughter nucleus (D),
under the frozen density approximation, is given by [27,30,31]
ܸே ሺ‫ݎ‬ሻ ൌ න൛‫ܪ‬ൣߩ௣௖ ሺ‫ݔ‬റሻ ൅ ߩ௣஽ ሺ‫ݎ‬ǡ ‫ݔ‬റሻǡ ߩ௡௖ ሺ‫ݔ‬റሻ ൅ ߩ௡஽ ሺ‫ݎ‬ǡ ‫ݔ‬റሻ൧ െ ‫ܪ‬௖ ൣߩ௣௖ ሺ‫ݔ‬റሻǡ ߩ௡௖ ሺ‫ݔ‬റሻ൧
െ ‫ܪ‬஽ ൣߩ௣஽ ሺ‫ݔ‬റሻǡ ߩ௡஽ ሺ‫ݔ‬റሻ൧ൟ ݀‫ݔ‬റǤሺ͵ሻ
Here, r (fm) defines the separation distance between the centers of mass of the
cluster and daughter nuclei. H, Hc and HD represent the energy density functionals
of the composite system, emitted cluster, and daughter nucleus,
respectively.ߩ௜௝ ሺ݅ ൌ ‫݌‬ǡ ݊Ǣ ݆ ൌ ܿǡ ‫ܦ‬ሻ are the protons (p) and neutrons (n) frozen
density distributions. The Skyrme energy functional consists of kinetic and
nuclear (Sky) terms,
‫ܪ‬൫ߩ௜ ǡ ߬௜ ǡ ‫ܬ‬റ௜ ൯ ൌ
԰ଶ
෍ ߬௜ ൫ߩ௜ ǡ ߘሬറ ߩ௜ ǡ ߘ ଶ ߩ௜ ൯ ൅ ‫ܪ‬ௌ௞௬ ൫ߩ௜ ǡ ߬௜ ǡ ‫ܬ‬റ௜ ൯ǤሺͶሻ
ʹ݉
௜ୀ௡ǡ௣
߬௜ ሺ݅ ൌ ‫݌‬ǡ ݊ሻ and ‫ܬ‬റ௜ in Eq. (4) represent proton (neutron) kinetic energy and the
spin-orbit densities, respectively. They can be obtained using the extended
Thomas-Fermi approximation [30,32] in terms of the local densities and their
௘௙௙
gradients, in addition to form factors ݂௜ ሺ‫ݎ‬റሻ ൌ ݉௜ Ȁ݉௜ ሺ‫ݎ‬റሻ of the proton and
neuron effective masses [27]. The nuclear energy-density part is given as [33],
3
‫ܪ‬ௌ௞௬ ൫ߩ௜ ǡ ߬௜ ǡ ‫ܬ‬റ௜ ൯ ൌ
‫ݐ‬଴
‫ݔ‬଴
ͳ
቎ቀͳ ൅ ቁ ߩଶ െ ൬‫ݔ‬଴ ൅ ൰ ෍ ߩ௜ଶ ቏
ʹ
ʹ
ʹ
௜ୀ௣ǡ௡
‫ݐ‬ଷ ߩఊ
‫ݔ‬ଷ
ͳ
൅
቎ቀͳ ൅ ቁ ߩଶ െ ൬‫ݔ‬ଷ ൅ ൰ ෍ ߩ௜ଶ ቏
ʹ
ͳʹ
ʹ
௜ୀ௣ǡ௡
‫ݔ‬ଵ
‫ݔ‬ଶ
ͳ
൅ ቐቂ‫ݐ‬ଵ ቀͳ ൅ ቁ ൅ ‫ݐ‬ଶ ቀͳ ൅ ቁቃ ߩ߬
ʹ
ʹ
Ͷ
ͳ
ͳ
൅ ൤‫ݐ‬ଶ ൬‫ݔ‬ଶ ൅ ൰ െ ‫ݐ‬ଵ ൬‫ݔ‬ଵ ൅ ൰൨ ෍ ߩ௜ ߬௜ ቑ
ʹ
ʹ
௜ୀ௣ǡ௡
൅
ͳ
‫ݔ‬ଵ
‫ݔ‬ଶ
ቐቂ͵‫ݐ‬ଵ ቀͳ ൅ ቁ െ ‫ݐ‬ଶ ቀͳ ൅ ቁቃ ሺߘሬറ ߩሻଶ
ʹ
ʹ
ͳ͸
ͳ
ͳ
െ ൤͵‫ݐ‬ଵ ൬‫ݔ‬ଵ ൅ ൰ ൅ ‫ݐ‬ଶ ൬‫ݔ‬ଶ ൅ ൰൨ ෍ ሺߘሬറ ߩ௜ ሻଶ ቑ
ʹ
ʹ
௜ୀ௣ǡ௡
൅
ܹ଴
ቌ‫ܬ‬റ ή ߘሬറ ߩ ൅ ෍ ‫ܬ‬റ௜ ή ߘሬറ ߩ௜ ቍ
ʹ
௜ୀ௣ǡ௡
൅
ͳ
቎ሺ‫ ݐ‬െ ‫ݐ‬ଶ ሻ ෍ ‫ܬ‬റ௜ଶ െ ሺ‫ݐ‬ଵ ‫ݔ‬ଵ ൅ ‫ݐ‬ଶ ‫ݔ‬ଶ ሻ‫ܬ‬റଶ ቏Ǥሺͷሻ
ͳ͸ ଵ
௜ୀ௣ǡ௡
The Skyrme-SLy4 parameters ( ߛ , ‫ݐ‬௠ ሺ݉ ൌ Ͳǡͳǡʹǡ͵ሻ, ‫ݔ‬௠ , W0) of the nucleonnucleon (NN) force can be found in [33]. More details about the method of
calculations are given in [26,27]. The Coulomb part of the interaction potential
(ܸ஼ ሺ‫ݎ‬ሻ) can be also obtained using Eq. (3), in terms of the direct (‫ܪ‬஼ௗ௜௥ ) and
exchange (‫ܪ‬஼௘௫௖௛ ) parts of the Coulomb energy-density functional
‫ܪ‬஼௢௨௟ ൫ߩ௣ ൯ ൌ ‫ܪ‬஼ௗ௜௥ ൫ߩ௣ ൯ ൅ ‫ܪ‬஼௘௫௖௛ ൫ߩ௣ ൯
ଵ
ସ
ߩ௣ ሺ‫ݎ‬റ ᇱ ሻ
͵݁ ଶ ͵ ଷ
݁ଶ
ଷ
ᇱ
݀‫ݎ‬
റ
െ
ൌ ߩ௣ ሺ‫ݎ‬റሻ න
൬ ൰ ቀߩ௣ ሺ‫ݎ‬റሻቁ Ǥሺ͸ሻ
ᇱ
ȁ‫ݎ‬റ െ ‫ݎ‬റ ȁ
ʹ
Ͷ ߨ
The exchange part of the Coulomb energy is given in this expression in the Slater
approximation [34]. Because the proton-proton Coulomb interaction is of finite
range, we can use the multipole expansion method to calculate the direct part of
the Coulomb potential [35,36]. The proton and neutron density distributions of the
participating nuclei are obtained by self-consistent Hartree-Fock calculations [37],
in terms of the used Skyrme-SLy4 NN interaction.
4
Returning to the decay width, Eq. (1), the assault frequency Ȟ and the
penetrability P can be obtained, respectively, using the one-dimensional WentzelKramers-Brillouin approximation as
ோమ
ߥ ൌ ܶ ିଵ ൌ ቎ න
ோభ
and
ିଵ
ʹߤ
݀‫ݎ‬቏
԰݇ሺ‫ݎ‬ሻ
ǡሺ͹ሻ
ோయ
ܲ ൌ ‡š’ ቆെʹ න ݇ሺ‫ݎ‬ሻ݀‫ݎ‬ቇǤሺͺሻ
ோమ
The wave number k reads
݇ሺ‫ݎ‬ሻ ൌ ඨ
ʹߤ
ȁܸ ሺ‫ݎ‬ሻ െ ܳ௖ ȁǤሺͻሻ
԰ଶ ்
The reduced mass of the cluster (mc) and daughter (mD) system is ߤ ൌ
௠೎ ௠ವ
௠೎ ା௠ವ
.
ܴ௜ୀଵǡଶǡଷ ሺˆሻ represent the three turning points along the path of the interacting
nuclei at which ்ܸ ሺ‫ݎ‬ሻȁ௥ୀோ೔ ൌ ܳ௖ . The total potential ்ܸ ሺ‫ݎ‬ሻ consists of the nuclear
ܸே ሺ‫ݎ‬ሻand Coulomb ܸ஼ ሺ‫ݎ‬ሻ parts (Eq. (2), in addition to the centrifugal ܸκ ሺ‫ݎ‬ሻ part.
The centrifugal potential can be expressed in terms of the angular momentum Ɛ
carried by the emitted light cluster as, ܸκ ሺ‫ݎ‬ሻ ൌ κሺκ ൅ ͳሻ԰ଶ Ȁʹߤ‫ ݎ‬ଶ Ǥ Ɛ conserves the
spin J and parity ʌ in the decay process. If one of the involved nuclei is even(Z)even(N) nucleus of ground state Jʌ = Ͳା . Ɛ is obtained by the conditions, |JP ࡳ
గ
represent the spin-parity assignments
JD(c)|” Ɛ ” |JP+JD(c)| and ʌP = ʌD(c) ( ࡳ 1)Ɛ . ‫ܬ‬௉ǡ஽ሺ௖ሻ
of the parent P and the other non-even-even nucleus (D or c) participating in the
decay process. According to the principle of least action, the emitted light cluster
is expected to carry the minimum angular momentum Ɛmin that verifies the given
conditions. If all the nuclei participating in the decay are even-even, then Ɛ=0.
The renormalization factor Ȝ appearing in Eq. (2) can be obtained by applying the
Bohr-Sommerfeld quantization conditions [38,39],
ோమ
ߨ
න ݇ሺ‫ݎ‬ǡ ߠሻ݀‫ ݎ‬ൌ ሺʹ݊ ൅ ͳሻ Ǥ
ʹ
ோభ
The quantum number n gives the number of internal nodes for the quasibound
radial wave function of the cluster-daughter system [40]. Because the total
potential based on the Skyrme NN interaction is characterized by a relatively
deep internal pocket, the motion of formed cluster within the obtained pocket can
be well described by a small n, n=0 or 1 [41]. The preformation factor Sc of a
given cluster of mass number Ac in a parent nucleus can be estimated in terms of
the Į-preformation factor SĮ inside it as [42,43],
ܵ௖ ൌ ሺܵఈ ሻ
஺೎ ିଵ
ଷ ǤሺͳͲሻ
5
The Į-preformation factor can be obtained from the relation between the observed
half-life and the calculated decay width [24,44]. Taking account of the shell and
paring (ap) effects, as well as the influence of the difference between the spin
and/or parity of the involved nuclei (aƐ), an empirical formula is proposed in
[24,26] to estimate the Į-preformation factor as,
మ
మ
ࣛ݁ ି଴Ǥ଴଴ଷሺ௓ି௓బ ି௓೘ ሻ ݁ ି଴Ǥ଴଴଺ሺேିேబ ିே೘ ሻ െ ܽ௣
ǡ
ܵఈ ൌ
ܽκ
ͲǤͲͲͶͲሺܼ െ ܼ଴ ሻଵȀଷ ˆ‘”‘††ሺܼሻ െ ‡˜‡ሺܰሻ—…Ž‡‹ǡሺͳͳሻ
ܽ௣ ൌ ቐͲǤͲͲͷ͸ሺܰ െ ܰ଴ ሻଵȀଷ ˆ‘”‡˜‡ሺܼሻ െ ‘††ሺܰሻ—…Ž‡‹ǡ
ͲǤͲͲͺͺሺܼ െ ܼ଴ ൅ ܰ െ ܰ଴ ሻଵȀଷ ˆ‘”‘††ሺܼሻ െ ‘††ሺܰሻ—…Ž‡‹Ǥ
Z0 (N0) in Eq. (11) defines the fully occupied shells and subshells of protons
(neutrons) in the parent nucleus. Zm (Nm) represents the number of protons
(neutrons), outside the shell closures, at which SĮ reaches a local maximum value.
The values of the dimensionless parameter ࣛ (Z0,N0) that vary with the shell
closures are given in [26,45]. For the present investigated cases, we have
ࣛ(Z0=82,N0=102)=0.078, ࣛ(82,126)=0.105, and Zm=Nm=12 [26]. aƐ (Ɛmin0) is
given in terms of the minimum allowed value of the transferred angular
momentum and the mass number of the parent nucleus A as ܽκ ൌ ‫ܣ‬ଶ Ȁ
ሺ͵ͳͻͲඥκ௠௜௡ ሻ െ ܽκ଴ , where ܽκ଴ ൌ ͵Ǥͻ for A>152 [26] and ܽκ଴ ൌ ͳǤͲ for A ” 152
[45]. For the favored decay modes at which Ɛ=0, aƐ is simply unity. Finally, the
partial half-life Tc of a parent nucleus against a particular cluster decay mode is
related to the decay width ߁௖ and the preformation factor Sc by the relation,
ܶ௖ ൌ
III.
԰ ݈݊ ʹ
Ǥሺͳʹሻ
ܵ௖ ߁௖
RESULTS AND DISCUSSION
Towards investigating the most probable cluster decays of 217-238U, we first
calculated the driving potential for all possible binary fragmentation of each
isotope. The driving potential at the Coulomb barrier Vd (Rb), Eq. (2) with Ȝ=1, is
calculated using the energy density formalism (Eqs. (3 and 4)) based on the
Skyrme-SLy4 NN interaction (Eq. (5)) for its nuclear part. The direct and
exchange parts of the Coulomb potential are calculated by folding the proton
densities with the Coulomb energy-density functional, Eq. (6). The cluster decay
energy Qc is calculated using the mass excess [46] of the involved nuclei.
For each mass number of the light cluster Ac, the preferred binary
fragmentation channel is singled out by determining the minimum driving
potential for the cluster isobars. The possibility to have an allowed spontaneous
binary fragmentation or a non-spontaneous one is checked through the released
energy Qc. Only the exoergic channels of Qc൐0 are considered. As examples, we
6
show in Fig. 1 the minimized driving potentials, at the Coulomb barrier, for the
binary fragmentation process of 219U (Fig. 1(a)), 226U (Fig. 1(b)), 234U (Fig. 1(c)),
235
U (Fig. 1(d)) and 237U (Fig. 1(e)) isotopes, as functions of Ac. We considered
the light clusters of Ͷ ൑ ‫ܣ‬௖ ൑ ͸Ͳ. The light clusters corresponding to the first few
local minima in the cold valleys of Vd, as a function of Ac, are labeled in Fig. 1.
Such channels exhibit relatively large Qc values and low driving potential.
Consequently, relatively large penetration probabilities and short half-lives are
expected for the cluster decays through these channels. This indicates moreprobable spontaneous decay channels. As experimentally confirmed, the Ƚparticle appears as the most favored light cluster of all investigated 217-238U
isotopes. It is always indicated by the global minimum of the calculated driving
potential, Fig 1. Other than the Ƚ-particle, we have 270 more-probable binary
fragmentations with light clusters of Ͷ ൏ ‫ܣ‬௖ ൑ ͸Ͳ, as indicated from minimizing
Vd with respect to Ac.
To differentiate between the emitted light clusters according to their atomic
numbers Zc, we examined the minimization of the driving potential with respect to
Zc. For each Zc, the preferred fragmentation channel is initially singled out by
determining the minimum value of Vd for the different Zc isotopes. Figures 2(a)2(f) show the charge-minimized driving potential for the binary fragmentation of
the 219,226,234,235,237U isotopes, respectively, as functions of Zc. Labeled in Fig. 2 are
examples for favorable light clusters manifesting themselves via chargeminimization, but they didn't appear as favorable clusters in the minimization with
respect to Ac (Fig. 1). The charge-minimization of the driving potential indicated
234 more-probable binary fragmentations of 217-238U, with light clusters heavier
than Ƚ-particle. Among these binary fragmentation modes, 102 modes did not
appear as favored light clusters in the minimization with respect to Ac. Generally,
the obtained mass- and charge-favored light clusters confirm the significant role
of the closed shell effect in determining the more-probable cluster decay modes.
Most of the indicated more-probable decay channels include a light cluster and/or
a daughter nucleus having closed-shell structure. This is clearly seen from the
copious appearance of O (Z=8) and Ca (Z=20) isotopes as candidate emitted
clusters from the most investigated isotopes. Also, the isotones of N=8 (14C, 15N,
16
O), N=20 (34Si, 36S, 38Ar), and N=28 (48Ca, 50Ti, 46Ar) abundantly appear as
favorable light fragments. Moreover, one or two Ne isotopes appear as preferred
light clusters for all investigated isotopes. The Ne isotopes leave Pb isotopes
(Z=82) as produced daughter nuclei. Also, the heavy nuclei participating in many
of the indicated favored channels are isotones of N=126, namely 214Ra, 212Rn,
210
Po, 208Pb, 206Hg, and 204Pt. Except 10B that appears once, none of the obtained
light clusters and consequently the involved heavy fragments participating in the
obtained favored channels has odd atomic number. Few of them have odd number
of neutrons. This confirms the role of the pairing effect in the cluster decay
process. The experimentally observed 233U(24Ne)209Pb, 234U(24Ne)210Pb,
235
U(28Mg)207Hg, 236U(24Ne)212Pb and 236U(28Mg)208Hg cluster decays are
7
indicated by the minimized driving potential with respect to Ac, but not by the
charge minimization. On the contrary, the observed 234U(26Ne)208Pb,
236
U(26Ne)210Pb and 236U(30Mg)206Hg cluster decays are indicated by the chargeminimized driving potential, but not by the minimization with respect to Ac.
As a next step, we estimated the half-lives of the cluster decay modes that
would take places through the indicated favored channels. The large number of
more-probable cluster decays that indicated by the characteristics of the driving
potentials versus Ac and Zc are reduced to 92 favored decays with half-lives
shorter less than 1030 s, other than the Ƚ-decays. The most isotopes appearing as
light clusters in these ninety two decays are 22-26Ne and 26,28,30Mg. The isotopes of
Ne and Mg appear in twenty three and twenty two favored decays, respectively.
The heavy daughter nuclei involved in these decays are 198-213Pb and 195-202,204209
Hg isotopes. The light clusters 8Be, 12,14C, 16,18O, and 30,32,34Si appear in
thirteen, eleven, eleven, and nine favored decays, respectively. These decays
include 209-221Ra, 205-212,214,216,218Rn, 201,202,203,205-211Po, and 192,194,196,198,200-204Pt
daughter nuclei, respectively. While 36S appears twice as emitted cluster, 48Ca
appears once. The daughter nuclei in these decays are 186,187Os and 182Hf,
respectively.
Listed in Table I are the light clusters (column 2) and the heavy daughter
nuclei (column 3) participating in the identified Ƚ and favored cluster decay
channels of the 217-238U isotopes. In Table I, we mainly focused on the indicated
decay modes that yield larger branching ratio than 10-20 %, with respect to the total
half-life time of the radioactive parent isotope. The ground-state spin and parity ‫ܬ‬஽గ
of the involved daughter nuclei [19,20] are listed in column 4. The spin-parity
assignments of the parent nuclei and emitted light clusters that have ‫ܬ‬௖గ ് Ͳା are
shown next to them. The spin-parity assignments of the participating nuclei have
been used to determine the minimum angular momentum (κ௠௜௡ ) that should be
transferred by the emitted light cluster in allowed decays. The Q-values [46] of
the presented decay modes are given in column 5 of Table I. The cluster
preformation probability and the calculated half-lives of the decay modes are
listed in columns 6 and 7, respectively. For comparison, the available
experimental half-lives [19,20] of the observed Ƚ- and cluster-decay modes for the
investigated isotopes are given in the last column of Table I. The Ƚ preformation
probability (Ƚ) is extracted from the experimental half-life and the calculated
decay width, Eq. (12). The preformation probabilities for the clusters heavier than
Į-particle are obtained in terms of SĮ using Eq. (10). For 231U and 233U, the
estimated Ƚ from the experimental half-lives are 0.0023 and 0.0004, respectively.
These small values are not consistent with the systematic behavior of the
preformation probability in neighboring nuclei, in which Ƚ is of the order of 10-2.
The uncertain small values of SĮ seriously affect the estimated cluster
preformation probability because of the power law (Eq. (10)) that relates them.
This yields extremely small values of Sc for 231,233U. Also, the 237U isotope has no
observed Ƚ-decay and then we cannot estimate its SȽ using Eq. (12). For these
8
231,235,237
U isotopes, the value of SĮ is obtained using the empirical relation given
by Eq. (11).
Towards more precise prediction of the decay modes that can be
experimentally inspected in future, we calculated the branching ratios B (%) of the
indicated decay modes. The smallest predicted branching ratio of an
experimentally observed cluster decay of an U isotope is that of the 28Mg-decay of
233
U, B൏1.3ۭ10-13 %. Among the cluster decay modes indicated in Table I,
twenty seven decays exhibit branching ratio Bcal (%) larger than 10-14. One of
these decays is the 12C decay mode of the very short-lived 223U (21 ߤs) isotope.
Further, the branching ratio of the indicated Į-decay of the 237U isotope is
2.1ۭ10−10 %. This Į-decay mode is not observed yet. Excluding the 12C decay
mode of 223U, the indicated decays with Bcal൐10-14% are presented in Table II.
The three 8Be-decays indicated for 222,223,224U yield B of the order of 10-13 %, 10-13
% and 10-14 %, respectively, but 8Be isotope has extremely shorter half-life (T1/2=
81.9ۭ10−18 s) than the calculated half-lives of these decays (Table I). As seen in
Table II, the calculated branching ratios are in agreement with the experimentally
observed ones. The largest difference is obtained for the decay modes of 235U, due
to the above-mentioned uncertainty of the preformation probability inside it. Even
for this case, the values of Bcal are less than the observed ones. This generally
supports the indicated cluster decays in Table II that are not observed yet.
The calculated half-life of the 12C-decay indicated for 224U (Table I) is
extremely longer than its total half-life (0.94 s). The same situation also exists
for the 16O-decay indicated for 226U, which has total half-life of 0.269 s. The
relatively long total half-lives start to appear at 230U. Simultaneously, the observed
cluster decays start to appear as well. The calculated partial half-life for the 22Nedecay of 230U and its branching ratio are in good agreement with the observable
ones. The first indicated cluster decay that is not observed yet is the 22Ne-decay of
232
U (T22Ne=1.78ۭ1025 s, B=1.2ۭ10−14 %). The 24Ne and 28Mg decays of this
isotope are already observed. In addition to the observed 24Ne and 28Mg decays of
233
U, its 25Ne (T25Ne=8.84ۭ1023 s, B=5.7ۭ10−10 %) and 32Si (T32Si=3.57ۭ1021 s,
B=1.4ۭ10−7 %) decays are indicated here. The 32Si was observed as an emitted
cluster from 238Pu, with B=1.4ۭ10−14 %. A decay mode involving 32Si is also
indicated here for 234U (T32Si=7.47ۭ1028 s, B=10−14 %). In addition to its observed
24,26
Ne and 28Mg decays, the 234U isotope exhibits a spontaneous fission mode of
B=1.6ۭ10−9 %. Regarding the 235U isotope, in addition to its observed 25Ne and
28
Mg decays, corresponding 24Ne (T24Ne=6.44ۭ1029 s, B=6.2ۭ10−14 %) and 29Mg
(3.66ۭ1030 s, 1.1ۭ10−14 %) decays are indicated here as proposed decays to be
observed. Its 26Ne (6.77ۭ1030 s, 5.9ۭ10−15 %) decay mode shows also
comparable half-life and branching ratio with the indicated decays.
The available experimental data do not differentiate between the various
cluster decay modes of 236U. A branching ratio less than 4.0ۭ10−10 and a partial
9
half-life larger than 1.85ۭ1026 s were generally assigned for its Ne and Mg
decays. The present calculations determine four specific clusters to be observed
from 236U and their probability relative to each other. While the largest probability
is obtained for the 30Mg cluster, the less probability is obtained for 26Ne. The
difference between their branching ratios is two orders of magnitude. The
branching ratios for the 24Ne and 28Mg decays are intermediate between them.
There is no Ƚ-decay has been experimentally detected from the 237U isotope.
It has total half-life of 6.75 days, which is comparable with the half-life of 231U. It
decays principally through ߚ− decay. The present calculations show a detectable
Ƚ-decay mode of 237U. It can decay to 233Th with a half-life of 8.7 ۭ109 years. The
corresponding branching ratio is 2.1ۭ10−10 %. The branching ratios of the other
cluster decays of 237U are extremely small. Finally, we have four indicated cluster
decays for the 238U isotope. It may decay to 214Pb, 210Hg, 208Hg, or 204Pt by
emitting 24Ne (B=1.2ۭ10−13 %), 28Mg (5.4ۭ10−13 %), 30Mg (8.9ۭ10−12 %), or 34Si
(7.2ۭ10−9 %), respectively. The shortest half-life among these decay modes is
that of the 34Si (6.19ۭ1019 years) decay. The 34Si nucleus was detected as emitted
light particle in decay modes of 240Pu ( ൏1.3ۭ10−11 %), 241Am (൏ ͹.4ۭ10−14 %)
and 242Cm (1.1ۭ10−14 %). The half-life of 34Si itself is 2.77 s. It decays through ߚ−
decay mode.
IV. SUMMARY AND CONCLUSIONS
We investigated the most probable cluster-decay channels of the 217-238U
isotopes, in the frame work of the performed cluster model. The driving potential,
at the Coulomb barrier, of all possible binary fragmentations has been calculated
for each investigated isotope. Only the exoergic fragmentation processes (Q>0)
has been considered. The driving potentials have been microscopically calculated
based on the Skyrme-SLy4 NN force and folded Coulomb interaction. For each
mass number Ac (atomic number Zc) of emitted clusters, the preferred
fragmentation channel has been singled out by determining the minimum driving
potential for the different isobars (isotopes). The minimized driving potentials
have been displayed as functions of Ac (Zc). We then identified the more-probable
binary fragmentations corresponding to the obtained cold valleys with respect to
Ac and Zc, individually. The minimization of the driving potential with respect to
Ac (Zc) indicated 270 (234) more-probable binary fragmentations of the 217-238U
isotopes, with light clusters heavier than Į-particle. The favored channels that
commonly manifested themselves via local minima with respect to both Ac and Zc
are 132 channels. The clusters and daughter nuclei participating in the indicated
favored channels confirm the significant role of the closed-shell and nucleonparing effects in the decay process.
We calculated the half-lives and branching ratios for the indicated favored
cluster-decay modes. Among the indicated decays, 74 decays show calculated
10
half-lives less than 1022 years. Twenty six of them exhibit branching ratio Bcal (%)
larger than 10-14. This value represents the lower limit for the experimentally
observed cluster decays of U isotopes. The entire experimentally observed cluster
decays of U appear in these indicated channels. In addition, the detectable
branching ratio of the Į-decay of 237U is estimated to be 2.1ۭ10−10. This Į-decay
mode did not observed yet. Also, four specific cluster decays to be observed from
236
U, and their probability with respect to each other, are specified. The available
experimental data do not differentiate between these decay modes. The indicated
most probable decays that did not observed yet include: (1) The 22Ne decay mode
of 232U (T22Ne=5.6ۭ1017 years, B=1.2ۭ10−14 %) that leaves 210Pb daughter
nucleus; (2) The 25Ne (T25Ne=2.8ۭ1016 years, 5.7ۭ10−10 %) and 32Si
(T32Si=1.1ۭ1014 years, 1.4ۭ10−7 %) decay modes of 233U, producing 208Pb and
201
Pt daughter nuclei, respectively; (3) The 24Ne (T24Ne=2.0ۭ1022 years, 6.2ۭ10−14
%) and 29Mg (T29Mg= 1.2ۭ1023 years, 1.1ۭ10−14 %) decays of 235U, leaving 211Pb
and 206Hg daughter nuclei, respectively; (4) The Į-decay of 237U (TĮ=8.7ۭ109
years, 2.1ۭ10−10 %) to 233Th; and (5) 34Si (T34Si = 6.2ۭ1019 years, 7.2ۭ10−9 %)
and 30Mg (T30Mg=5.0ۭ1022 years, 8.9ۭ10−12 %) decay modes of 238U, leaving
204
Pt and 208Hg daughter nuclei, respectively.
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FIGURES AND TABLES CAPTIONS:
Fig. 1: The driving potential at the Coulomb barrier for the binary
fragmentation of (a) 219U, (b) 226U, (c) 234U, (d) 235U, and (e) 237U, as a
function of the mass number of the light fragment (Ac). For each Ac,
the preferred light cluster was identified by minimizing Vd with respect
to the different isobars. The clusters corresponding to the first few
relatively deep minima are labeled.
Fig. 2: The same as Fig. 1, but the charge-minimized driving potential is drawn
as a function of the atomic number of the light fragment (Zc). For each
Zc, the preferred light cluster was identified by minimizing Vd with
respect to the different isotopes. Examples for clusters that appear here
as local minima, but they didn't appear as favorable clusters in the
minimization with respect to Ac (Fig. 1) are labeled.
Table I: The calculated partial half-lives (column 7) of the favored cluster decay
modes of the 217-238U isotopes (column 1), which were indicated by
minimizing the driving potentials with respect to Ac and Zc. Only the
decay modes yielding branching ratios larger than 10-20 % are presented.
Shown in columns 2, 3, 4, and 5, respectively, are the emitted light
clusters, the produced daughter nuclei, the spin and parity assignments ࡶ࣊ࡰ
[19,20] of the daughter nuclei, and the Q-values of the decay modes. The
curve ( ) and square [ ] brackets indicate uncertain spin and/or parity, and
non-experimental values, respectively [19]. The spin-parity assignments
of the parent nuclei and those of the none even(Z)-even(N) emitted
clusters, which have ‫ܬ‬௖గ ് Ͳା , are given next to them. The sixth column
gives the Ƚ and cluster preformation factors SĮ(c). The method of
estimating SĮ(c) is explained in the text. The last column gives the
available observed partial half-lives of the listed decay modes [19,20]. (*)
This value is obtained using Eq. (11).
Table II: The calculated branching ratios Bcal (%), larger than 10-14, of the favored
cluster decays indicated in Table I. The Bexp (%) of the experimentally
observed decays [19,20] are given in the last column.
Table I
13
Isotope
217
U
[1/2−]
218
U
0+
219
U
[9/2+]
220
U
0+
221
U
[9/2+]
Cluster Daughter ࡶ࣊
ࡰ
4
[5/2 ]
8.425
7.10ۭ10
Be
209
Ra
5/2−
16.178
9.69ۭ10-06
5.24ۭ1018
He
214
Th
0+
8.786
1.20ۭ10-02
6.00ۭ10-03
B‡
210
Ra
0+
16.510
1.53ۭ10-07
2.66ۭ1019
He
215
Th
(1/2−)
9.940
1.50ۭ10-02
5.50ۭ10-05
211
Ra
5/2(−)
17.516
2.58ۭ10-07
3.93ۭ1016
8
4
8
Be
4
U
0+
10.404
1.58ۭ10-02
6.00ۭ10-08
Be
212
Ra
0+
18.187
6.26ۭ10-05
1.97ۭ1012
He
217
Th
[9/2+]
[9.840]
2.27ۭ10-02
7.00ۭ10-07
213
Ra
1/2−
19.180
1.46ۭ10-04
8.48ۭ1009
Rn
5/2−
33.409
9.38ۭ10-07
1.63ۭ1013
Be
C
209
He
218
Th
0+
9.422
2.97ۭ10-02
4.70ۭ10-06
214
Ra
0+
19.185
2.73ۭ10-04
1.41ۭ1009
Rn
0+
33.825
2.51ۭ10-06
8.08ۭ1011
8
Be
12
223
C
210
He
219
Th
[9/2+]
8.940
1.64ۭ10-01
2.10ۭ10-05
215
Ra
[9/2+]
18.364
1.47ۭ10-02
2.58ۭ1009
Rn
1/2−
34.595
1.33ۭ10-03
1.11ۭ1008
Po
5/2−
47.722
1.19ۭ10-04
2.67ۭ1012
U
4
3/2−
8
Be
12
C
211
O
207
He
220
Th
0+
8.621
1.77ۭ10-02
9.00ۭ10-04
216
Ra
0+
17.841 8.17ۭ10-05
7.36ۭ1012
Rn
0+
34.373 3.77ۭ10-07
7.35ۭ1011
Po
0+
47.920 1.74ۭ10-09
9.35ۭ1016
16
224
U
0+
4
8
Be
12
C
212
O
208
He
221
Th
(7/2+)
8.015
2.52ۭ10-02
6.10ۭ10-02
217
Ra
(9/2+)
16.548 1.86ۭ10-04
1.45ۭ1016
Po
1/2−
48.481 1.02ۭ10-08
2.41ۭ1015
16
225
U
[5/2+]
4
8
Be
16
226
U
0+
4
O
209
He
222
Th
0+
7.700
3.74ۭ10-02
2.69ۭ10-01
218
Ra
0+
15.736 4.68ۭ10-04
1.00ۭ1018
Po
0+
48.019 7.32ۭ10-08
1.09ۭ1015
8
Be
16
227
U
(3/2+)
228
U
4
O
210
He
223
Th
(5/2)+
7.211
219
Ra
(7/2)+
14.685 3.69ۭ10-05
4.89ۭ1022
3.80ۭ10-02
5.46ۭ1002
8
4
1.60ۭ10
-02
0+
8
4
-02
Th
8
4
−
216
He
12
222
Tcal. (s)
SĮ(c)
Th
He
8
4
Q (MeV)
213
Be
He
224Th
0+
6.804
14
1.26ۭ10-02
66.00
Texp (s)
1.60ۭ10-02
6.00ۭ10-03
5.50ۭ10-05
6.00ۭ10-08
7.00ۭ10-07
4.70ۭ10-06
2.10ۭ10-05
9.00ۭ10-04
6.10ۭ10-02
2.69ۭ10-01
66.00
5.46ۭ1002
8
0+
Be
14
4
4
0+
Ne
206
Pb
0+
61.032
1.14ۭ10-10
6.62ۭ1020
He
225Th
(3/2+)
6.475
2.70ۭ10-02
1.74ۭ1004
O
211
Po
9/2+
44.426
1.29ۭ10-09
5.14ۭ1023
Ne
207
Pb
1/2−
61.687
1.05ۭ10-11
1.12ۭ1021
He
226Th
0+
5.993
5.29ۭ10-02
1.75ۭ1006
0+
28.342
2.94ۭ10-06
2.97ۭ1024
Pb
0+
61.388
1.16ۭ10-09
2.15ۭ1019
216
Ne
208
26
Mg
204
Hg
0+
72.519
2.30ۭ10-11
4.34ۭ1024
28
Mg
202
Hg
0+
73.979
3.24ۭ10-12
1.52ۭ1024
Pt
0+
85.598
6.44ۭ10-14
4.12ۭ1027
Hf
0+
121.890 1.00ۭ10-20
3.93ۭ1026
Th
1/2+
5.576
1.15ۭ10-02 * 1.91ۭ1009
198
Ca
182
He
227
208
Pb
0+
60.710
6.00ۭ10-15
5.36ۭ1025
Ne
207
Pb
1/2−
62.211
1.36ۭ10-15
3.53ۭ1024
He
228
Th
0+
5.414
4.97ۭ10-02
2.17ۭ1009
Rn
0+
26.373
2.24ۭ10-06
4.33ۭ1029
Ne(5/2+)
4
14
9.73ۭ1009
2.17ۭ1009
Ne
210
Pb
0+
57.364
7.49ۭ10-10
1.78ۭ1025
24
Ne
208
Pb
0+
62.311
1.01ۭ10-10
5.52ۭ1019
2.44ۭ1020
Mg
204
Hg
0+
74.320
1.85ۭ10-12
8.78ۭ1023
4.35ۭ1022
Pt
0+
85.290
3.38ۭ10-14
1.23ۭ1028
Th
5/2+
4.909
3.28ۭ10-02
5.02ۭ1012
5.02ۭ1012
6.98ۭ1024
Si
200
He
229
5/2+
24
Ne
209
Pb
9/2+
60.486
4.19ۭ10-12
3.46ۭ1023
Ne(1/2+)
208
Pb
0+
60.728
1.34ۭ10-12
8.84ۭ1023
Hg
1/2−
74.226
4.39ۭ10-14
4.33ۭ1025
Pt (5/2−)
84.738
4.61ۭ10-16
3.57ۭ1021
0+
4.858
4.46ۭ10-02
7.74ۭ1012
7.74ۭ1012
28
Mg
32
0+
> 1.6ۭ1018
22
4
U
1.75ۭ1006
218
U
234
1.74ۭ1004
C
32
25
Rn
Si
28
233
5.00ۭ1020
1.52ۭ1020
24
0+
7.01ۭ10-07
C
4
U
30.521
8.96ۭ10-09
48
232
0+
45.958
32
23
Rn
0+
14
U
8.66ۭ1023
Po
22
231
4.86ۭ10-04
210
22
U
14.009
O
18
230
0+
214
22
U
Ra
C
18
229
220
4
205
Si
201
He
230
Th
3.87ۭ1027
24
Ne
210
Pb
0+
58.826
4.42ۭ10-11
8.14ۭ1024
8.60ۭ1025
26
Ne
208
Pb
0+
59.416
15
5.55ۭ10-12
6.94ۭ1025
8.60ۭ1025
28
Mg
32
235
74.111
6.98ۭ10-13
2.82ۭ1024
Pt
0+
84.916
1.10ۭ10-14
7.47ۭ1028
Th
5/2+
4.678
2.00ۭ10-02 *
4.01ۭ1014
202
He
231
4
7/2−
24
3.50ۭ1025
2.22ۭ1016
Ne
211
Pb
9/2+
57.364
9.43ۭ10-14
6.44ۭ1029
Ne(1/2+)
210
Pb
0+
57.709
2.56ۭ10-14
2.33ۭ1030
Ne
209
Pb
9/2+
58.056
6.95ۭ10-15
6.77ۭ1030
Mg
207
Hg
[9/2+]
72.426
5.12ۭ10-16
5.83ۭ1029
Mg(3/2+)
206
Hg
0+
72.469
1.39ۭ10-16
3.66ۭ1030
Pt
[1/2−]
84.628
2.78ۭ10-18
7.83ۭ1032
Th
0+
4.573
5.20ۭ10-02 *
7.39ۭ1014
7.39ۭ1014
25
26
28
32
236
0+
Hg
Si
U
29
206
U
0+
4
Si
203
He
232
Ne
212
Pb
0+
55.945
1.43ۭ10-10
7.61ۭ1028
൐1.85ۭ1026
26
Ne
210
Pb
0+
56.696
2.00ۭ10-11
3.49ۭ1029
൐1.85ۭ1026
28
Mg
208
Hg
0+
70.735
2.78ۭ10-12
1.60ۭ1028
൐1.85ۭ1026
30
Mg
206
Hg
0+
72.276
3.87ۭ10-13
3.19ۭ1027
൐1.85ۭ1026
S
196
Os
0+
93.565
2.03ۭ10-17
4.64ۭ1033
He
233
Th
(1/2)+
4.234
7.72ۭ10-02
2.75ۭ1017
[1/2-]
84.979
5.81ۭ10-13
2.03ۭ1027
234Th
0+
4.270
6.96ۭ10-02
1.41ۭ1017
Ne
214
Pb
0+
53.442
1.34ۭ10-09
1.15ۭ1032
28
Mg
210
Hg
0+
67.698
3.83ۭ10-11
2.59ۭ1031
30
Mg
208
Hg
0+
69.463
6.49ۭ10-12
1.59ۭ1030
Pt
0+
85.186
1.86ۭ10-13
1.95ۭ1027
Os
0+
93.987
8.99ۭ10-16
2.21ۭ1034
0+
106.689 4.36ۭ10-18
1.11ۭ1035
U
1/2+
4
238
4
U
0+
൐2.78ۭ1027
24
40
237
൐2.78ۭ1027
34
Si
He
24
34
Si
40
46
203
204
S
198
Ar
192
Pt
W
Table II:
16
1.41ۭ1017
Isotope
Cluster
224
U
12
226
U
16
230
U
232
U
U
212
O
210
Po
2.5ۭ10−14
22
Ne
208
Pb
8.1ۭ10−12
22
Ne
210
Pb
1.2ۭ10−14
24
Ne
208
Pb
3.9ۭ10−9
8.9ۭ10−10
Mg
204
Hg
2.5ۭ10−13
൏5.0ۭ10−12
͹.2ۭ10−11
Ne
209
Pb
1.9ۭ10−9
25
Ne
208
Pb
5.7ۭ10−10
Mg
205
Hg
1.2ۭ10−11
32
U
Si
201
24
Ne
26
28
236
U
U
4.8ۭ10−12
൏1.3ۭ10−13
Pt
1.4ۭ10−7
210
Pb
9.5ۭ10−11
9.0ۭ10−12
Ne
208
Pb
1.1ۭ10−11
9.0ۭ10−12
Mg
206
Hg
2.7ۭ10−10
1.4ۭ10−11
32
235
1.2ۭ10−13
Rn
24
28
234
Bexp (%)
C
28
233
Bcal (%)
Daughter
Si
202
24
Ne
25
Pt
1.0ۭ10−14
211
Pb
6.2ۭ10−14
Ne
210
Pb
1.7ۭ10−14
§ 8.0ۭ10−10
28
Mg
207
Hg
6.9ۭ10−14
8.0ۭ10−10
29
Mg
206
Hg
1.1ۭ10−14
24
Ne
212
Pb
9.7ۭ10−13
൏ 4.0ۭ10−10
26
Ne
210
Pb
2.1ۭ10−13
൏ 4.0ۭ10−10
28
Mg
208
Hg
4.6ۭ10−12
൏ 4.0ۭ10−10
30
Mg
206
Hg
2.3ۭ10−11
൏ 4.0ۭ10−10
He
233
237
U
4
238
U
24
Ne
214
28
Mg
30
Mg
34
Si
2.1ۭ10−10
Th
Pb
1.2ۭ10−13
210
Hg
5.4ۭ10−13
208
Hg
8.9ۭ10−12
204
7.2ۭ10−9
Pt
17
Vd(MeV)
60
50
40
30
20
10
0
(a) 219U
4He 8Be
Vd(MeV)
0
5
60
50
40
30
20
10
0
4He
Vd(MeV)
60
50
40
30
20
10
0
15
20
8Be 12C
16O
25
30
25
30
Ac
35
40
45
50
55
60
65
5
10
15
20
Ac
35
40
45
50
55
60
65
40
45
50
55
60
65
40
45
50
55
60
65
45
50
55
60
65
(c) 234U
14C 18O
10Be
28Mg
24Ne
4He
0
Vd(MeV)
10
(b) 226U
0
60
50
40
30
20
10
0
5
10
15
20
25
30
Ac
35
(d) 235U
28Mg
24Ne
10Be 14
C
18O
4He
0
Vd(MeV)
16O
12C
60
50
40
30
20
10
0
5
10
15
20
25
30
Ac
35
(e) 237U
10Be
14C 19O
24Ne
28Mg
34Si
4He
0
5
10
15
20
25
30
Fig.1
Ac
35
40
Vd(MeV)
60
50
40
30
20
10
0
(a)
219U
22Ne
0
50
10
15
Zc
20
25
30
(b) 226U
40
Vd(MeV)
5
30
20
10
14C
0
0
5
10
15
20
25
30
15
20
25
30
Zc
60
(c) 234U
Vd(MeV)
50
40
30
22O
20
26Ne
10
0
Vd(MeV)
0
60
50
40
30
20
10
0
5
ZC
(d) 235U
22O
0
60
5
29Mg
26Ne
10
ZC
15
20
25
30
20
25
30
(e) 237U
50
Vd(MeV)
10
40
30
26Ne
20
30Mg
10
0
0
5
10
ZC
15
Fig. 2
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