Cold valleys in cluster radioactivity

Ş Mişicu and W Greiner

Institut für Theoretische Physik, J.W.v.-Goethe Universität, Robert-Mayer-Str 8-10, 60325 Frankfurt am Main, Germany

Received 22 August 2003
Published 18 September 2003

1. Introduction

The phenomenon of cluster radioactivity (CR) was predicted towards the end of 1970s by Sandulescu et al [1] and Rose and Jones [2] confirmed this a few years later. Since then the ¹⁴C decay of many other isotopes of Ra nuclei, and many other heavier cluster decays, has been observed, e.g. ²⁰O from ²²⁸Th, ^24,26Ne from ^230,232Th and ^232,234U, ²³F from ²³¹Pa, ^28,30Mg from ²³⁸Pu and ^32,34Si from ²³⁸Pu and ²⁴¹Am (see for a review [3]).

The possibility to have a cluster process is related to its exotermicity:

Here M(A_i, Z_i) are the atomic masses. For the elements found in the second half of the periodic system, which have average binding energy per nucleon smaller than the lighter elements, this condition is fulfilled for a large range of nuclear decays. However, in majority of cases the possibility of decay is hindered by the small barrier penetrability. This quantity reaches large values only in two cases: α-decay and spontaneous fission of heavy nuclei. An exception is given by high penetrabilities (values which can be sometimes even larger than those corresponding to the α-decay) of the daughter nuclei ¹⁴C, ²⁴Ne, etc, when the complementary nucleus is close to the double-magic ²⁰⁸Pb. This fact is connected to the much larger value of the ratio Q_C/B_C relative to the value Q_α/B_α, where B_C,α are the heights of the corresponding barriers.

The concept of cold valley was introduced in relation to the structure of minima in the so-called driving potential. The driving potential is defined as the difference between the interaction potential and the decay energy of the reaction

The driving potential also depends on the charge Z₁, mass A₁, the distance between the centres-of-masses of the two nuclei R, mutual orientations ω_1,2 and the quadrupole (β₂), octupole (β₃) and hexadecupole deformations (β₄) through the heavy ion potential V defined as

The ansatz (1) corresponds to the s-channel, i.e. relative angular momentum l =  0.

The concept of cold valleys was developed in connection with occurrence of fusion valleys as doorways to the synthesis of superheavy elements [4, 5]. The occurrence of the cold valleys for mass asymmetric combinations is due to the shell effects. It was advocated in [6], using the frame of the fragmentation theory, that due to the existence of different mass-asymmetry valleys for the same compound system, a new, highly asymmetric fission mode appears in which one of the fragments is close to the double-magic nucleus ²⁰⁸Pb. This new type of decay is nothing else but the cluster radioactivity.

In two of our very recent publications we studied the consequences of the existence of these valleys for the formation, quasi-fission and decay of some superheavy nuclei [7] and cold fission of ²⁵²Cf [8], while taking into account several multipole deformations of the target (heavy fragment) and projectile (light fragment), as well various reciprocal orientations. There are two cases of particular orientations which are at the opposite extremes of the barriers range. The orientation giving low barriers takes place when the symmetry axes of the two nuclei are aligned (β₁ = β₂ =  0 in equation (2)). In this case one speaks about the nose-to-nose or pole-pole (p-p) orientation. The orientation producing high barriers corresponds to the nuclei colliding with their bellies, the symmetry axes being parallel (β₁ = β₂ = π/2 in equation (2)). We call this orientation belly-to-belly or equator-equator (e-e).

The goal of this paper is to perform a similar analysis for different mass regions of nuclei which are expected to exhibit CR using the deformations for the values computed in the frame of the macroscopic-microscopic model by Möller et al [9].

2. Cluster radioactivity

2.1.  Cluster radioactivity in ¹¹⁴Ba

Except the heavy nuclei region of CR, mentioned in the introduction, another cluster emitter nucleus is suspected, i.e. the neutron-deficient isotope ¹¹⁴Ba (N≈Z) which emits the cluster ¹²C [10]. Similar to the heavy nuclei region where the daughter nucleus is the double-magic ²⁰⁸Pb or a neighbouring nucleus, the daughter of ¹¹⁴Ba is the nucleus ¹⁰²Sn which is close to the double-magic ¹⁰⁰Sn. The inspection of the corresponding driving potential in figure 1 confirms this prediction. Along with the ¹²C minimum there are two other comparable minima, one to its left, ⁸Be, and the second to its right, ¹⁶O. The prevalence of the ¹²C-channel compared to the other two is given by the magicity of the partner ¹⁰²Sn.

It is interesting to note that all minima are corresponding to N = Z clusters, and that from the most asymmetric minimum, α + ¹¹⁰Xe up to the splitting ²⁸Si + ⁸⁶Mo these minima are separated by rather higher barriers and can be joined by a continuous chain of α particles, the splitting ¹²C + ¹⁰²Sn being a relative minimum on this curve (see the dashed line in figure 1). This would mean that if we imagine the reverse process and we collide ²⁸Si with ⁸⁶Mo it is energetically more favourable to transfer α particles in order to reach the compound nucleus (CN) than to initiate single nucleon, di-neutron or di-proton transfer. Eventually the dinuclear system will land on the `α-like valley'<sup>12</sup>C + <sup>102</sup>Sn and a part of the flux will be lost by quasi-fission and the rest of the flux will move further in the direction of the compound nucleus by crossing a sequence of `α-like barriers'.

2.2.  Cluster radioactivity in radium and actinides

The cold valleys in the case of CR are strongly dependent on the deformation of the clusters and daughter nuclei. As shown in [11], quadrupole and hexadecupole deformations, especially of the daughter nucleus could play an important role in reaching the experimental values of the decay rate.

In the case of ²²²Ra the most pronounced valley corresponds to the case ¹⁴C + ²⁰⁸Pb, both nuclei being spherical (see figure 2). However, beyond the cluster mass number A₁ =  16 the deformation causes the deviation of the e-e driving potential from the p-p one. The existence of the valley corresponding to the emission of the cluster ¹⁴C and the double-magic daughter nucleus ²⁰⁸Pb is confirmed in experiment [12]. Similarly for the isotopes Z =  221, 223, 224 and 226 of radium.

At the other extreme of observed cluster emitters, i.e. ²⁴²Cm, we are inferring from figure 3 that the cluster valleys are shifted to larger cluster masses. In the case of p-p configuration the deepest valley corresponds to the splitting ⁴⁰S + ²⁰²Hg and the second important valley is the recently reported splitting ³⁴Si + ²⁰⁸Pb [13]. This last valley is less favourable in the frame of the cold valley picture of decay because of the sphericity of both cluster, ³⁴S, and daughter nucleus ²⁰⁸Pb which sensitively rises the barrier compared to the case of the prolate (β₂ =  0.254) cluster ⁴⁰S, emitted at the equator of the oblate (β₂≈ - 0.09) daughter nucleus ²⁰²Hg. As one can see from figure 3, if the cluster ⁴⁰Si would be emitted at the pole of ²⁰²Hg, with its symmetry axis perpendicular to the axis joining the centres of the two nuclei (e-e configuration), then the corresponding valley would completely lose its importance and the only observed cluster decay should be ³⁴Si + ²⁰⁸Pb. Of course, the spectroscopic factors may invert the importance of these two valleys by favouring the clusterization of the double-magic ²⁰⁸Pb. It is therefore necessary that the investigation of the cluster emission in ²⁴²Cm looks for both splittings and not only of ³⁴Si + ²⁰⁸Pb.

In the case of Th, where cluster radioactivity has been observed, we draw in figure 4 the driving potential in p-p configuration for four of its even-even isotopes. For all four investigated isotopes we indicated the corresponding α-decay which is the predominant cluster emission mode. For ²²⁶Th the first two deepest cluster valleys are given by the combinations ¹⁴C + ²¹²Po and ¹⁸O + ²⁰⁸Pb, and are actually experimentally confirmed cluster emission channels. We note here that ²²⁶Th is the only isotope of Th where one observes the emission of ¹⁴C. For ²²⁸Th the deepest minimum is encountered for the splitting ²⁰O + ²⁰⁸Pb which is the only observed cluster emission channel up to now [14]. The splittings, ¹⁴C + ²¹⁴Po and ²²Ne + ²⁰⁶Hg, although not observed, are also corresponding to deep valleys, and we put a question mark (?) to stress that their possible observation in the future should not be excluded. Adding two more neutrons (mother nucleus ²³⁰Th), we find that the deepest minimum moves two cluster charge numbers further and corresponds to the observed splitting ²⁴Ne + ²⁰⁶Hg [15]. However the neighbouring splittings ²⁰O + ²¹⁰Pb and ²²O + ²⁰⁸Pb have also close values of the minima but they are not recorded in experiment. Also the ¹⁴C emission channel should be favourable for ²³⁰Th according to the cold valley approach but no experimental evidence was provided to date. The last isotope of thorium, ²³²Th, is the most rich in comparable deep valleys. If we leave aside the valley of ¹⁴C + ²¹⁸Po, then the deepest one is the experimentally confirmed splitting ²⁴Ne + ²⁰⁸Hg. Also observed was the neighbouring splitting ²⁶Ne + ²⁰⁶Hg. The other minima corresponding to the emission of ^20,22O and ^28,30Mg are not yet experimentally registred.

One may conclude that the reason of the decrease in the importance of ¹⁴C emission when moving from ²²⁶Th to ²³²Th is due to the corresponding daughter nucleus, i.e. polonium. By adding more and more neutrons to Po, we get away from the neutron magic shell and most likely the spectroscopic factors are decreasing. The probability of reaching different cold valleys from the ground state (cluster preformation) should come into play.

If one checks the deformations of the emitted clusters one finds that that the deepest valleys for ^228,230,232Th are corresponding to high quadrupole and/or hexadecupole distortions which indicate a possible subcluster structure, as in the cases of ²⁰O + ²⁰⁸Pb, ²⁴Ne + ²⁰⁶Hg and ²⁴Ne + ²⁰⁸Hg. This can be guessed from figure 5 where the density contours of the clusterization is given for the decays ²⁴Ne + ²⁰⁸Hg and ²⁸Mg + ²⁰⁴Pt in p-p and e-e orientations.

It is also worthwhile to note for the case of thorium that the cluster radioactivity cold valleys are grouped in `super-valleys' which are broader when moving to larger neutron number of the mother nucleus. For the emitter <sup>228</sup>Th we have a `super-valley' which ranges from cluster mass A₁ =  18 to 24 whereas for ²³²Th it ranges from A₁ =  18 to 30.

For uranium also we take four even-even isotopes which are known cluster emitters (see figure 6). The mother nucleus ²³⁰U has a very pronounced valley centred on the splitting ²²Ne + ²⁰⁸Pb, which is actually the only case of observed CR channel for this nucleus. The reason for obtaining such a deep minimum is due not only to the double magicity of the daughter nucleus, but also to the large cluster's quadrupole (β₂ =  0.326) and especially hexadecupole (β₄ =  0.225) deformations. For the next isotope, ²³²U, the minimum shifts on the cluster ²⁴Ne, which is oblate (β₂ =  - 0.215), and in order to overcome a smaller barrier, it should be emitted with its symmetry axis perpendicular to the fission axis. Adding two more neutrons, we obtain two comparable minima for ²⁴Ne + ²¹⁰Pb and ²⁸Mg + ²⁰⁶Hg. In between we have the combination ²⁶Ne + ²⁰⁸Pb which although is not giving a minimum in the driving potential it shares the same valley as ²⁸Mg + ²⁰⁶Hg. It is important to stress that the clusters of all these three splittings, which were also reported in [16], have large quadrupole deformations and especially hexadecupole deformations. The last investigated isotope of uranium, ²³⁶U has less pronounced minima, but like in the previous cases, two cluster decay channels were observed in experiment, corresponding to two deep minima, ²⁸Mg + ²⁰⁸Hg and ³⁰Mg + ²⁰⁶Hg. In the first case we deal with a prolate cluster and in the second with an oblate cluster, both endowed also with a sensitive negative hexadecupole deformation.

In fact for the two isotopes ^234,236U we obtained that the deepest minima correspond to ¹⁴C because it enters, in combination with Rn isotopes which are increasingly deformed. However, the emission of ¹⁴C was not observed in any isotope of uranium which means that we deal with a hindrance in the preformation of such combinations, the main responsible, in our opinion, being the daughter nucleus which prefers to acquire a spherical or almost spherical shape, in the neighbourhood of the double-magic ²⁰⁸Pb. Everything is going on as if the heavier (daughter) nucleus, is transferring nucleons, or even clusters, until it reaches the valley containing spherical or almost spherical isotopes of Pb and Hg. At the same time the cluster is mainly acquiring quadrupole and hexadecupole deformation, having in `almost' cases a clear clusterized structure indicates that a sequential transfer of alphas is taking place. For the case of the cluster ²²Ne we can consider for example the transfer of two alphas to the spherical ¹⁴C which are leading to a large hexadecupole deformations with the two alphas arranged at the opposite ends of ¹⁴C.

For the investigated isotopes of uranium we should also remark that when increasing the number of neutrons the gradual weakening of the cluster valley and the increase in importance of the minimum corresponding to the splitting ⁴⁰S + ^A - 40Os. However, we expect in this case like for ¹⁴C a hindrance in the preformation.

In the case of plutonium (figure 7) we remark for its lowest mass number isotope, ²³⁶Pu, where CR is observed, that very deep minima are obtained for the splitting ²⁸Mg + ²⁰⁸Pb. This is also the only case of emitted heavy cluster confirmed by experiment. For ²³⁸Pu the experimentally observed splittings ²⁸Mg + ²¹⁰Pb, ³⁰Mg + ²⁰⁸Pb and ³²Si + ²⁰⁶Hg are entering in competition with the splitting ⁴⁰S + ¹⁹⁸Pt. However, it is expected that this last splitting should have a smaller spectroscopic factor compared to the first two splittings, due to the lack of magicity in the proton and neutron numbers of the cluster and daughter nucleus. The next even isotope, ²⁴⁰Pu, exhibits a deep minimum on the observed decay mode ³⁴Si + ²⁰⁶Hg. However, the minimum on ⁴⁰S + ²⁰⁰Pt is deeper.

To date, no cluster emission was observed for ²⁴⁴Pu. However, a pronounced minimum is obtained in the present calculations for the cluster ⁴⁰S. For ²⁴⁴Pu we note the occurrence of two close minima, ³⁸Si + ²⁰⁶Hg and ⁴⁰S + ²⁰⁴Pt in which both daughter nuclei posses a magic neutron number (N =  126).

Results for californium isotopes are displayed in figure 8. The first isotope of californium that we considered is ²⁴⁸Cf. The deepest minimum corresponds to the splitting ⁴⁰S + ²⁰⁸Pb. The cluster is predicted by the macroscopic-microscopic model [9] to be prolate deformed (β₂ =  0.254) which sensitively lowers the barrier. Together with that comes the double magicity of ²⁰⁸Pb. For ²⁵⁰Cf, there are two deep minima in competition in the same region, i.e. ⁴⁰S + ²¹⁰Pb and ⁴²S + ²⁰⁸Pb. Adding two more neutrons causes the broadening and shifting of the valley further to the right. Minima for the spherical clusters ⁴⁶Ar, ⁴⁸Ca and ⁵⁰Ca are occurring this time due to the magic neutron shell, N =  126, of the corresponding daughter nuclei, ²⁰⁶Hg and ²⁰⁴Pt A third minimum comes from the splitting ⁴²S + ²¹⁰Pb. The last investigated isotope of californium, ²⁵⁴Cf, has the same feature as the previous one, i.e. predominance of the neutron magic number.

We note also in the case of californium that the weakening and spreading of the CR valley may be due to the increase in the number of neutrons.

For fermium we investigated the even isotopes with mass numbers A =  254-260 and extended the range of cluster mass numbers up to A₁ =  70. The Fe-Cr valley, already mentioned in a previous work on superheavy elements [7], is very stable for all four isotopes and centred on ⁶²Cr (see figure 9). Obviously in this case the cluster mechanism remarked for the known cluster emitters is apparently not essential, both partners of the splitting having no magic proton or neutron numbers. This is a typical example of a valley where the dictating criterion is the quadrupole and hexadecupole deformation [7, 17, 18]. Besides that we expect small preformation factors compared to lighter clusters, due to the increasing mass and lack of magicity. In what concerns valleys of lighter clusters we easily identify for ^254,256Fm, a calcium valley, where the magicity is a property of both cluster and daughter nuclei. For ^258,260Fm this valley is washed out and the ⁴⁰S gains in importance.

2.3.  Cluster radioactivity in the superheavy island

The exploration of the CR in the superheavy island did not receive much attention to date^Note1. A reason for this is the instability of nuclei in this region. The most important decay channel in the formation is the quasi-fission, the α-decay and spontaneous fission, the only claimed signatures of the CN formation occurring far less frequently.

In such circumstances the importance of CR channels, if observed, consists in checking the validity of the dinuclear picture of the fusion process [19]. In this picture the synthesis of the CN nucleus takes place by sequential transfer of nucleons and clusters until the projectile is completely swallowed by the target. Thus, if for example a medium mass projectile, e.g. ⁴⁸Ca, bombards an actinide target, e.g. ²³⁸U, ²⁴⁴Pu or ²⁴⁸Cm then, the quantum flux along the mass asymmetry coordinate will be split in a part moving towards symmetry and which eventually will decay through quasi-fission and another part will move in the direction of CN formation. A part of this flux can be, however, lost again through quasi-fission during the passage of the Bussinaro-Gallone barrier which separates the point of injection with ⁴⁸Ca from the CN point. In this case good candidates for quasi-fission channels are the valleys corresponding to CR. In figure 10 we display the driving potential for the superheavy nuclei ²⁸⁶112, ²⁹²114 and ²⁹⁶116. When the dinuclear system is driven from the injection point (in the valley corresponding to ⁵⁰Ca) towards decreasing A₁ then, part of the flux is likely to be lost through quasi-fission in the first encountered cold valley (⁴⁰S) and to a smaller extent in the CR valleys, found on the top of the `Bussinaro-Gallone mountain', which correspond to the typical `magic' clusters:²⁸Mg, ^22,24Ne and ¹⁴C. From the recent experiment from Dubna we know however that none of these cases of clusterization is recorded [21]. This can be an indication whether that the flux crossing the Bussinaro-Gallone barrier is so small that the events related to decay are under the experimental threshold or that the dinuclear picture of fusion is not working and that the fusion takes place through a more involved path in the R-η space, at large target-projectile overlap, which avoids the CR valleys found at small target-projectile overlap.

3. Conclusions

The various predictions of cluster emission likelihood made in this paper have to be completed by a study of preformation factors. In order to determine the decay rates, it is not enough to have the relative barrier heights. According to the empirical law of Blendowske and Walliser [22], the logarithm of the spectroscopic factor has a linear decrease with the mass of the emitted cluster. To date the experimental and phenomenological theoretical models [3] are pointing to a deviation from this line beyond ³⁴Si. In any case, the spectroscopic factor decreases sensitively with mass number and this is one of the reasons why for mother nuclei with Z larger than 96, when the most favourable channels predicted by the cold valley are shifted beyond cluster mass number 40, emitted clusters are not observed, although the enter in combination with ²⁰⁸Pb.

Recently [23] we presented a scheme based on the algebraic shell model to obtain a rapid evaluation of the spectroscopic factor. It is shown that when one considers the clusterization of various mother nuclei (from Rn to No), although the spectroscopic factors deviate from the empirical behaviour and present oscillations due to shells, their values are many orders of magnitudes smaller than those for ^28,30Mg when one approaches ⁴⁸Ca and the corresponding decrease of driving potential barriers observed in figures 4-9 cannot compensate this drastic reduction.

The calculation of the driving potential in actinides demonstrated that the deepest minimum of this quantity along the cluster mass coordinate is a good indication for the most favourable cases of CR. The same happens for ²⁵²Cf, where the highest experimental mass-distribution cold fission yields are falling in the valley produced by high quadrupole and hexadecupole deformations of the emitted fragments. This is in our opinion a manifestation of the same decay mechanism in two different fragments mass regions.

As for the recently synthesized superheavy nuclei we expect to see, in case the dinuclear scenario of fusion is valid, quasi-fission events when the lighter partner has a mass intermediate between α and ⁴⁸Ca. The pure CR phenomeneon is however, in light of the relatively high-lying valleys, less probable in superheavy nuclei compared to actinides. A possibility would be to look for the emission of Be, C, O, Ne, etc clusters from superheavy nuclei with Z =  118, 120, 122, 124, etc, where the daughter nucleus is predicted to be double magic, i.e. Z =  114 and N =  184 or N =  186. However, there is no definite experimental indication to date on the synthesis of these nuclei.

Acknowledgments

SM would like to acknowledge the financial support from the European Community through a Marie Curie fellowship. We are also indebted to professors A Sandulescu and D Poenaru for useful discussions.

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Notes

Note1

 There is, to our knowledge, a very recent study which is based also on the concept of cold valley [20] and where the heavy cluster emission from the superheavy nucleus ²⁷⁷110(Ds) and its α-chain products, ²⁷³Hs and ²⁶⁹Sg, is investigated and concluded that in addition to α-decay and fission, ¹⁴C, ³⁴Si and ⁵⁰Ca clusters present the optimal cases of CR. The obtained half-lives are, however, a huge order of magnitude larger than the expected CN life-times.