aa r X i v : . [ nu c l - t h ] J u l Cluster Radioactivity in Super Heavy Nuclei
M. Warda, ∗ A. Zdeb,
1, 2 and L. M. Robledo Katedra Fizyki Teoretycznej, Uniwersytet Marii Curie–Sk lodowskiej, Lublin, Poland Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, Spain (Dated: July 3, 2018)Cluster radioactivity is an exotic nuclear decay observed in actinides where a light nucleus isemitted while the remaining heavy mass residue is the doubly magic
Pb or a nucleus in itsneighborhood. We have investigated this type of decay in heavier nuclei up to Lv ( Z = 116) withina microscopic theory. It has been found that super asymmetric fission with Pb as heavy fragmentmay be dominant decay channel in some super heavy nuclei. This reaction is closely related withcluster radioactivity.
PACS numbers: 25.85.Ca, 23.70.+j, 27.90.+b
Introduction.
In a seminal experiment carried out byRose and Jones back in 1984 [1] a new type of nucleardecay was discovered. Immersed in an enormous α decaybackground produced by the mother nucleus Ra a fewevents producing C were observed. The phenomenonreceived the name of cluster radioactivity (CR) due tothe intermediate mass of the light fragment emitted. Inthe following years, 20 other cluster emitters have beendiscovered, see reviews in Refs. [2, 3]. In this type ofdecay light nuclei ranging from C to Si are emittedby light actinides from
Fr to
Cm. The remainingheavy mass fragment in all these reactions is either thedoubly magic
Pb or one of its neighbours in the chartof nuclides. For this reason the phenomenon is also calledlead radioactivity. The dominant double magic structureof the heavy fragment clearly shows the strong influenceof shell effects on CR. As compared to other decay chan-nels, CR is an exotic process: typical branching ratiosto the dominant α emission are as low as 10 − − − and consequently the half-lives range from 10 s to 10 s. For nuclei heavier than Cm spontaneous fission be-comes a competing channel increasing the difficulty todetect CR products among the numerous fission frag-ments present in the background. As a consequence, noCR have been observed in nuclei around mass A = 250.CR is usually described in the spirit of the Gamowmodel of α decay. In this model it is assumed that a pre-formed cluster made of several nucleons tunnels througha barrier created by the nuclear and Coulomb potentials[4–7]. The exponential dependence of the tunneling prob-ability with the parameters of the barrier leads to a mod-ified Geiger-Nutall law [8–11] relating half-lives for CR tothe Q value of the reaction. This approach presents twomain disadvantages: first, it requires a model to estimatethe preformation probability of the cluster and second,a local fit of additional model parameters is required toreproduce observed decay half-lives. An alternative tothis model is to treat CR as a very asymmetric fissionprocess to be described with the tools of the traditional ∗ [email protected] fission decay model. This was the approach followed inRef. [12] to predict the existence of CR a few years beforeits experimental discovery. The most important ingredi-ent of any fission model is the analysis of the changes inenergy of the nucleus as it changes deformation in its wayto scission [13, 14]. In the context of CR, it was shown[15] that a super asymmetric fission valley can be foundon the potential energy surface (PES) spanned by thequadrupole and octupole moments. It leads to a scissionpoint with Pb or a nucleus of a very similar mass asone of the fission fragments. In the light actinides thefission barrier associated to this channel reaches a heightof 25 MeV which is the right order of magnitude to re-produce CR half-lives.Several authors have already suggested that SH ele-ments may decay through CR [16–19]. Calculations per-formed within phenomenological approaches and semi-empirical formulae show a trend to predict shorter half-lives for CR in the SH region. Therefore, the competitionwith other disintegration channels becomes relevant andmight have an impact on the very limits of the periodictable [20] or in the r-process nucleo-synthesis [21].The aim of this work is to study the possible existenceof CR in SH nuclei within a fully microscopic theory. Tothis end we have used the selfconsistent Hartree-Fock-Bogoliubov model along with the Gogny D1S interactionto calculate the PES and the collective inertias requiredfor the evaluation of CR half-lives. As relevant collec-tive coordinates the axially symmetric quadrupole andoctupole moments are used. This is a well-establishedquantum mechanics approach that allows to describe avery rich variety of nuclear shapes not limited by thenumber of deformation parameters. This type of calcula-tions has been successfully applied both for the descrip-tion of CR [22] and fission in heavy [23] and SH nuclei[24, 25].One of the common features of fission and CR is thatthe Z and N values of their fragments approximatelyconserve the N/Z ratio of the mother nucleus. There-fore, to investigate the possibility to observe CR in heav-ier nuclei, one has to focus on possible emitters with the
N/Z value of 126 /
82 = 1 .
537 corresponding to the al-ways present heavy fragment of
Pb. Therefore, wehave chosen for our studies a set of even-even isotopes,one for each element, with an
N/Z ratio close to 1 . Ra,
Th,
U, and
Pu,where CR has already been observed. Heavier isotopesinclude
Cm,
Cf,
Fm,
No,
Rf,
Sg,
Hs,
Ds,
Cn,
Fl, and
Lv. The final part of thischain belongs to the region of super heavy (SH) elementsclose to the isotopes experimentally produced in hot fu-sion reaction [26]. It is worth mentioning that
Cnis the heaviest isotope that has been observed decayingthrough fission both at GSI and in Dubna [27, 28]. Theseare difficult experiments due to the low production rateand a total number of only 28 fission events have beenreported for this isotope.
Results.
A typical example of a nucleus where CR hasbeen observed is the light actinide
U. In the PES ofthis nucleus, shown in Fig. 1a, the asymmetric fissionpath, typical for heavy nuclei up to Fm is easily found.It goes from the ground state through the first symmetricfission barrier, which is 6.6 MeV high, to reach the fis-sion isomer. Then it crosses the second asymmetric sad-dle and gets to the scission point. On the same PES an-other valley can be noticed. It begins at the ground stateand heads directly towards octupole deformed shapes.Along most of the whole fission path in this valley bothquadrupole and octupole moments increase simultane-ously up to Q = 45 b and Q = 51 b / . In the saddlepoint the energy reaches 26.6 MeV. The scission point islocated at the saddle point. From there on the PES cor-responds to the Coulomb energy of the two fragments asthey drift away and the energy decreases hyperbolicallywith the increasing distance between the fragments. Thenuclear matter density distributions before and after scis-sion are plotted on the inset of Fig. 1a. From them it iseasy to find that mass and shape of the heavy fragmentcorresponds to the spherical doubly magic lead isotope.The two fragments are well defined far before the ruptureof the neck. The fragments’ matter distribution provesthat this super asymmetric fission channel describes CR.The same topology of the PES has been also found in allcluster emitters observed in the light actinides [15].The CR valley in No depicted in Fig. 1 (b) showsthe same characteristics as in
U but it is shifted to-wards larger quadrupole moments. The scission pointis located at higher quadrupole and octuple deformationmaking the fission barrier broader. The height of the CRfission barrier is reduced to 14.2 MeV. Again, the massof a heavy fragment at scission corresponds to A = 208.In the super-heavy Cn shown in Fig. 1 (c) the sametype of very asymmetric fission valley can be found aswell. Again, it is shifted towards higher quadrupole mo-ments. In this nucleus the traditional symmetric fissionbarrier has a two humped structure and the CR valleystarts at the minimum located between them instead ofthe ground state. Moreover, the CR fission barrier hassubstantially changed its shape: it is much lower than inprevious cases reaching a height of only 1.5 MeV at thesaddle point. Starting from Q = 60 b this fission path Q ( b / ) U Q ( b / ) U Q ( b / ) U Q ( b / ) U Q ( b / ) U Q ( b / ) No Q ( b / ) No Q ( b / ) No Q ( b / ) No Q ( b / ) No
0 20 40 60 80 Q [b] 0 20 40 60 Q ( b / ) Cn
0 20 40 60 80 Q [b] 0 20 40 60 Q ( b / ) Cn
96 6 330 0 0 20 40 60 80 Q [b] 0 20 40 60 Q ( b / ) Cn
0 20 40 60 80 Q [b] 0 20 40 60 Q ( b / ) Cn
0 20 40 60 80 Q [b] 0 20 40 60 Q ( b / ) Cn FIG. 1. PES of (a)
U, (b)
No, and (c)
Cn. Con-stant energy lines are plotted every 3 MeV. Fission paths aremarked with yellow dashed lines. Insets show pre- and post-scission configurations. drops down below the ground state energy. The scissionpoint is located at Q = 128 b and Q = 92 b / withan energy 12.6 MeV below ground state.To study the properties of CR in the SH region we havecomputed PESs spanned in the Q − Q collective spacefor all aforementioned isotopes and we have found thatthe CR valley exists in all considered nuclei. In Fig. 2 (a)a bunch of CR fission paths is shown. Its characteristicpattern presents in the PES smoothly evolves in goingfrom light actinides to SH nuclei. The only sharp mod-ification takes place in the starting point of the paths Ra Lv Sg Q ( b / ) -20 0 20 40 0 20 40 60 80 100 120 140 160 Ra Lv Hs E ( M e V ) Q (b) FIG. 2. (a) CR fission paths in quadrupole and octupolecoordinates for isotopes:
Ra,
Th, U, Pu,
Cm,
Cf,
Fm,
No,
Rf,
Sg,
Hs,
Ds,
Cn,
Fl,and
Lv. (b) Fission barriers of aforementioned nuclei.Squares correspond to scission configuration calculated withEq. (1) with Q values obtained from experimental masses[29], wherever possible, and liquid drop systematics [30] for Sg and heavier nuclei. They start at the secondminimum, not at the ground state, as it was describedabove in the case of
Cn. In these isotopes the CRfission barrier is comprised of two differentiated parts:first a reflection symmetric hump followed by a secondoctupole deformed barrier. In all considered isotopes theCR scission configuration contains a spherical heavy massfragment: the double magic
Pb. This means that inSH elements one may expect decay of the same nature asin light actinides.A similar structure of fission paths in this regionwas also found with the covariant density theory [31]and Skyrme energy density functional [32] in a non-relativistic setup. The asymmetric fragment mass dis-tribution predicted here is not in contradiction withsymmetric fission barriers predicted in macroscopic-microscopic models [33–35]. The first saddle point is re-flection symmetric and much higher than the octupoledeformed second one.The height and shape of fission barriers in super asym-metric fission channel are the crucial features for the understanding of CR decay lifetimes in heavier nuclei.The CR barriers in the considered nuclei are presentedin Fig. 2 (b). In the first five isotopes they have similarheights of around 25 MeV. The scission point is graduallyshifted towards higher quadrupole moments, due to theincreasing size of the cluster in this configuration. Start-ing from
Cf, the energy of the scission point graduallydecreases. In the mass region between 250 and 270 thefission barriers are still very high (8 −
20 MeV) and verybroad. Tunneling probability across them is too low tomake it possible to observe the CR channel. The situa-tion changes substantially already in
Hs. The scissionpoint is below the ground state energy and the second,asymmetric, fission barrier is lower than 5 MeV, which isless than the energy of the first symmetric saddle point.In
Ds and heavier nuclei almost the whole asymmetricpart of the fission path is below the ground state energy.In this way, the fission barrier consists basically of thefirst, symmetric, hump which is relatively narrow andeasy to tunnel. The asymmetric part of fission barrierhas little influence on the half-lives but it is crucial forthe asymmetry of the fragment mass distribution.The reduction of the CR fission barrier height in theregion of SH elements came as a surprise. However, itcan be easy explained on the basis of a simple analysisof the Coulomb repulsion energy of two charged spheresat the scission point. The scission configuration consistsof a spherical
Pb heavy fragment and a lighter clustercreated from the remaining nucleons of the mother nu-cleus. The separation of the fragments is given by thesum of their radii increased by a tip distance d . This ex-tra spacing between daughter nuclei comes from the neckconnecting pre-fragments before scission. After the neckrupture, the distance between the position of half-densityof the fragments in the post-scission configuration varyfrom 2.5 fm to 4.7 fm. The energy of the scission config-uration can be estimated as the Coulomb energy of twopoint charges minus the Q value of the decay: E = k Z − e r + r A − + d − Q . (1)Here Z and A are the charge and mass number of themother nucleus and the fragment’s radius is estimatedusing the traditional r A F = 1 . A / F fm expression. Theresults obtained for the considered nuclei with an averageconstant tip distance value d = 3 . Q − Q collective space are calculated using the per-turbative cranking approximation for pre-scission config-uration. After scission, the reduced mass is taken anda constant zero point energy is used. Huge values forthe half-lives, longer than 10 s are obtained in the ha l f - li f e l og ( T / / s ) proton number Z HFBexp clusterexp alphaexp sf FIG. 3. CR half-lives of considered isotopes compared withexperimental data of CR, α emission and spontaneous fission[36]. -15-10-5 0 5 10 0 20 40 60 80 100 120 140 E ( M e V ) Q (b) symmetrictriaxialsuper asymmetric Cn FIG. 4. Fission barrier of
Cn plotted as a function ofquadrupole moment. Z = 96 −
106 region. However, starting at
Hs we ob-serve their substantial reduction. In nuclei with
A >
Cn is comparable with exper-imental data for fission we will present a more detailedstudy of this representative isotope.In the previous paper [24] the main decay channel of
Cn was claimed to be asymmetric fission. Here wehave shown that this mode is rather a super asymmet-ric fission mode directly connected with the standard CRobserved in the actinides. The other decay channels areless favored. Predictions of α radioactivity give 4 or-ders of magnitude longer half-lives. Symmetric fission in Cn is also highly suppressed as can be seen in Fig. 1(c) and Fig. 4. The second symmetric barrier has thesame width as the asymmetric one but it is as high as5.6 MeV whereas the asymmetric one is only 1.5 MeV.Non-axial deformation may reduce the height of the fis- sion barrier of SH nuclei but it does not affect the mainconclusion. First fission barrier is soft against γ defor-mation and its height stays practically unchanged (it isdiminished only by 0.4 MeV) after including the triaxialdegree of freedom. The second symmetric barrier is re-duced as a consequence of triaxiality by around 2.2 MeVonly with γ = 6 ◦ . However, this effect is too weak tomake the symmetric fission channel the most favorable.The same conclusion applies to all the isotopes in the re-gion Z = 110 −
114 and N = 170 −
176 centered around
Cn that were previously identified as nuclei with asym-metric fission [24, 25]. All those nuclei should decay inthe same mode as
Cn. The CR fission valley also ex-ists in heavier elements, but decay through this channelis suppressed by α emission. From the other hand, inlighter systems the symmetric fission mode is dominant.We may conclude that super asymmetric fission, closelyrelated with CR in the actinides, can be found in someSH nuclei as the dominant decay mode. Conclusions.
We have shown that asymmetric fissionin SH nuclei has the same nature as CR in light actinides.The dominant decay channel of isotopes around
Cn issuper asymmetric fission with doubly magic
Pb as theheavy mass fragment. Lighter fragment corresponding tothe SH nuclei discussed here would be Ni, Zn, Ge,and Se in the fission of
Ds,
Cn,
Fl, and
Lv,respectively. It is important to note the existence of themagic numbers Z = 28 and N = 50 in those light frag-ments that reinforces the strong influence of the magicstructure of the heavy fragment.The super asymmetric fission mode in SH nuclei dis-cussed here differs from the asymmetric one observed inthe Pu-Fm region, cf. Fig. 1 (a). In the actinides, theheavy mass fragment is formed by the shell structure ofthe lighter doubly magic Sn that produces a peak inthe mass yield at A H = 140 [23]. The distinction be-tween asymmetric and super asymmetric fission not onlyconcerns the numerical values of most probable fragmentmasses. Qualitatively different shapes of the nucleus canbe determined before scission in both modes. In CR, theneck is short and narrow whereas in asymmetric fission inthe actinides it is much longer and thicker [23]. As necknucleons are shared between fragments at scission [37]the fragment mass yield in CR are expected to be muchnarrower than in asymmetric fission. The same conclu-sion may be deduced from the fact that the CR fissionvalley is very narrow in comparison with the asymmetricone. The variety of available fission shapes at the scissionline is substantially reduced [38].The same mechanism invoked here for spontaneousfission applies also to fusion-fission and quasifission ob-served in this region of SH nuclei [39–43]. In all the re-actions the possible fragment mass asymmetry is forcedby the shell structure of Pb.It is worth to note that the predicted fission fragmentsproduced by CR in SH nuclei lay out of the region ofthe fission products in the actinides i.e. A = 60 − A L ∼
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