Heavy clusters in cold nuclear rearrangements
in fusion and fission
(
)
P. ARMBRUSTER
Gesellschaft f¨ur Schwerionenforschung mbH - Planckstr. 1, D64291 Darmstadt, Germany (ricevuto il 19 Settembre 1997; approvato il 15 Ottobre 1997)
Summary. — The experimental evidence for the appearance of cluster aspects in the dynamics of large rearrangement processes, as fusion and fission, is presented. Clus-ters in the sense as used in the following are strongly bound, doubly magic neutron rich nuclei as48 Ca28, 78 Ni50, 132 Sn82, and 208
Pb126, the spherical nuclei
Z= 114 – 126 and N =184, and nuclei with closed shellsN =28;50;82, and 126, andZ =28;50, and
82. As with increasing nucleon numbers, the absolute shell corrections to the binding energies increase, the strongest effects are to be observed for the higher shells. The
132
Sn cluster manifests itself in low energy fission (FAISSNERH. and WILDERMUTH K., Nucl. Phys., 58 (1964) 177). The208
Pb cluster gave us the new radioactivity (ROSE M.J. and JONES G. A., Nature, 307 (1984) 245) and the first superheavy elements (SHE) (ARMBRUSTERP., Ann. Rev. Nucl. Part. Sci., 35 (1985) 135-94; M ¨UNZENBERG G., Rep. Progr. Phys., 51 (1988) 57). I will discuss experiments concerning the stability of clusters to intrinsic excitation energy in fusion and fission (ARMBRUSTERP., Lect. Notes Phys., 158 (1982) 1), and the manifestation of clusters in the fusion entrance channel (ARMBRUSTERP., J. Phys. Soc. Jpn., 58 (1989) 232). The importance of com-pactness of the clustering system seems to be equally decisive in fission and fusion. Finally, I will cover the importance of clusters for the production of SHEs.
PACS 25.70.Jj – Fusion and fusion-fission reactions. PACS 25.85 – Fission reactions.
PACS 01.30.Cc – Conference proceedings.
1. – Where are we in the production of the superheavy elements ?
Since 1981 six superheavy elements with deformed nuclei were discovered at GSI, Darmstadt. The first three elements bohrium (Bh), hassium (Hs) and meitnerium (Mt) were produced in the eighties [1-3], whereas we synthesized the last three recently af-ter all parts of our experimental equipment were improved and the effective luminosity
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and
Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
Fig. 1. – The decay chain of277 112 produced at 344 MeV in70 Zn (208 Pb, n)277 112 [6]. increased to a value of 0.1 pb,1 d ,1
[4-6]. We detected three isotopes of element 110, altogether 17 atoms [4, 6], one isotope of element 111 [5], altogether 4 atoms, and finally, one isotope of element 112, altogether 2 atoms [6]. Figure 1 shows one of the decay chains of277112. The short alpha-half-lives for the decays of277112 and273110 compared to the decay of269Hs are a direct demonstration of the deformed neutron shell at N = 162.
Calculations give for these nuclei a barrel-like deformation ( 4 <0) [7, 8]. South-west of
208Pb a lighter analogue of such barrel-like nuclei is found around188W112.
Regarding the production cross-sections for the two methods used in heavy-element research, actinide-based 4n and 5n fusion reactions and Pb- and Bi-based 1n reactions for both, a steady decrease from nobelium to higher atomic numbers by a factor of 4 per element is found (see fig. 6b). The two methods show a similar downward trend, with the 1n-reactions being advantageous for elements beyond seaborgium (Sg). Figure 2 presents our cross-sections as excitation functions for even-even compound nuclei. For elements 108 to 112 the 1n cross-sections are shown. For the five heaviest elements only the 1n channel is established, whereas for the lighter elements also 2n- and 3n-reactions were observed. Besides the strong decrease of their maximum values the excitation functions for all elements are comparable for the cases investigated. They show for the 1n channel a maximum at (122) MeV and a halfwidth of about 5 MeV. The maxima are slightly below
the fusion barriers calculated by using the Bass potential [9]. The excitation energy of 12 MeV corresponds to the energy of the maximum of the 1n-channel.
2. – Shell corrections and fission barriers
The increasing shell corrections to the binding energies between U and the SHEs com-pensate the decrease of the macroscopic fission barriers [10]. Minimum values of fission barriers are reached for the even-even isotopes for nobelium and rutherfordium, which fission spontaneously. For heavier elements alpha-decay dominates over spontaneous fis-sion. We have determined fission barriers and barrier curvatures for a sequence of heavy nuclides withN,Z =48running up to264Hs [11, 12]. The analysis of the data
corrob-orated the new elements to be purely shell-stabilized and to have large fission barriers. Calculations showed that260Sg has a single high and narrow barrier [7]. The barrier exit point close to the superdeformed 2:1 configuration has an elongation of only about 3 fm. Beyond Sg the liquid-drop barrier has fallen below the zero-point energy (BLD=
0.5 MeV) and shell corrections alone dominate. This is the SHE-region and there we find the discovered isotopes of elements 107 to 112 and the new deformed neutron shell at
N = 162. Going further up towards the spherical neutron shell atN = 184, the shell
correction energies might become even stronger [8]. The landscape of shell correction energies between the small values near225U (U 0) and the next doubly closed shell
nucleus is a smoothly descending slope with small local ponds near252Fm and270Hs. We find gentle transitions from one region to the other with the general trend to deeper shell corrections. We may expect about 20 SHEs between Sg andZ =126with about 400
iso-topes. Until now, six SHEs with 14 isotopes were detected, the rest is elusive. Not the ground-state properties of SHEs, but the vanishing stability of shell corrections to heat-ing and the production mechanism prevent us from penetratheat-ing deeper into the region of purely shell-stabilized nuclei.
3. – Clusters and temperature
Evidences from fusion reactions. In comparison to a liquid drop-like nuclear entity the
quantal ordering of the constituents of a nucleus may increase its binding energy. This mi-croscopic correction to the binding energy depends on the shape of the nucleus and it is restricted to a narrow, well-defined region of deformation parameters. We observe spher-ical and deformed microscopspher-ically stabilized nuclei. The stabilization is equivalent to a dip in the multi-dimensional deformation space of the potential energy surface (PES). Moving out of this dip is equivalent to passing a barrier. If the liquid drop PES does not protect
any more against fission, as in the case of SHEs, the only protection against immediate fragmentation of the nucleus are the locally restricted minima in the PES, which are due to the microscopic correction to the binding energy, the necessary condition for the exis-tence of SHE. Destroying the microscopic quantal ordering removes the stability against fission. With increasing excitation energy of the nucleonic system particle-hole excitations reduce the microscopic corrections, and the theory predicts that shell corrections are ex-ponentially damped with a1=e-damping energy of 18 MeV [13]. This value is confirmed
experimentally for shell stabilized deformed EVRs produced in actinide-based fusion re-actions [14] and the analysis of their cross-sections corroborates the predicted damping energy.
A very different temperature dependence is observed for heavy systems where spher-ical closed-shell nuclei are produced. Besides the spherspher-ical SHEs atN = 184there is
a second region with large shell correction energies and small liquid drop barriers, the spherical nuclei atN =126, which were investigated in fusion reactions of different
en-trance channel asymmetry leading to Th-compound nuclei [15]. Via 1n- to 4n-reactions Th-isotopes were synthesized using targets of Hf-, Yb-, and Zr- isotopes and their cor-responding projectiles Ar, Ca, and Sn [16, 17]. Figure 3a shows the survival probability for 4n-reactions leading to spherical Th-isotopes around theN =126shell. The analysis
of the data showed that the large shell corrections to the fission barriers do not increase the production cross-sections. Down to the lowest investigated excitation energies of 20 MeV the fused compound systems behaved like systems without shell corrections to the barriers.
Our early findings were corroborated in two further recent experiments at GSI [18, 19]. At relativistic energies the production of fragmentation products along N = 126
becomes possible using collisions of238U projectiles and varying lighter target atoms. At the end of a long cascade of evaporated nucleons, the strong increase of the fission barrier by the ground-state shell correction energy should give to theN = 126EVRs a higher
survival probability which was not found, see fig. 3b [18]. In a further experiment beams of fragmentation products passed a Pb target and in secondary collisions with Pb atoms the fragmentation products are exposed to the virtual electromagnetic field between the collision partners. The electromagnetic response function of the fissile fragment shows a maximum near the energy of the GDR at 11 MeV and the mean excitation of the fissioning nuclei should be as low as 11 MeV (T =0:7MeV). The expected decrease of the fission
cross-section for the shell stabilized spherical fragments withN=126was not found, see
fig. 3c [19].
All our independent and different experiments showed that the increase of the fission barrier of the shell-stabilized nuclei withN =126does not increase the survival
proba-bility against fission down toT =0:7MeV. Even at excitation energies of an 1n- channel
(12 MeV) spherical EVRs may hardly profit from the shell stabilization of their ground state. The collective enhancement of their level densities at the saddle point by rotational levels leading to higher fission probabilities compensates the reduced fission probability via a higher fission barrier [20, 21]. This compensation is found for spherical nuclei only. In nuclei with deformed ground states a collective enhancement of the level densities is present at the deformations of the ground state as well as of the saddle point. The dif-ference of level densities for ground state and saddle point deformations is small and the survival probability against fission is hardly affected.
The purely shell-stabilized SHE with spherical nuclei predicted at Z = 116 – 126, N =184most probably behave as their shell stabilized spherical counterparts atN =126.
b) c) e.m. fission cross-section in lead
Fig. 3. – a) Maxima of the 4n excitation functions as a function of the neutron number for Th evap-oration residues (full points). A factor1=(
–2 15
2
)was applied in order to remove trivial
entrance-channel effects for different target-projectile combinations and to make the scale of the ordinate equal to the survival probability(,n=,tot)times the transmission coefficient of the fusion
bar-rier for a low angular momentum [15]. Calculated survival probability(,n=,tot) for zero
an-gular momentum: Standard evaporation calculation (solid line), calculation without shell effects, and a damping energy of 6 MeV (dashed line). b) Production cross-sections of Th, Ac, and Ra iso-topes from relativistic238
U-fragmentation reactions [18]. Calculations taking into account shell cor-rections (full line) and shell corcor-rections + collective enhancement (dashed-dotted line) are shown. c) Cross-sections for electromagnetic fission (E
= 11 MeV) in 450 MeV A collisions on Pb of frag-mentation products from relativistic238
U-fragmentation reactions [19]. Calculations as in b): shell corrections (dashed line), shell corrections + collective enhancement (solid line).
Fig. 4. – a) Mass yields of the light fission fragment group for234
U at a light fission product kinetic energy of 114.1 MeV, equivalent to 5 MeV excitation energy [27]. b) Mass yield curve of spontaneous fission of258
Fm [28] and thermal neutron fission of257
Fm [29].
collision systems using rare earth elements [22], both would lead to hot compound sys-tems (E
> 30MeV). They loose stability against fission by the collective enhancement
of level densities at the saddle point, and, thus, may show reduced survival probabilities by several orders of magnitude. As at excitation energies close to the fission barrier the enhancement should disappear, SHE with spherical nuclei should be synthesized there. Radiative capture and 1n reactions should be most favorable.
Evidences from fission. The second large-range rearrangement process dominated
by shell corrections in the reaction partners is asymmetric fission. It is observed up to a range of excitation energies of about 40 - 50 MeV equivalent to a temperature of 1.5 MeV and disappears with an exponential damping constant close to the theoretical value of 18 MeV, as discussed for actinide-based fusion. But for spherical shell-stabilized clusters a small resistance to dissolution is also found in low-energy fission.
Calculations of the PESs including shell corrections for fissioning nuclei [23, 24] re-vealed several fission paths. Asymmetric fission proceeds via two channels, a spherical configuration connected to the spherical clustersZ =50andN =82in the heavy
frag-ment (standard I) or a strongly deformed configuration in the heavy fragfrag-ment atN 88
(standard II). The standard I-channel is observed best in spontaneous fission of heavy plu-tonium isotopes [25]. It is disappearing rapidly if the excitation energy is increased. We showed in relativistic heavy ion reactions investigating the GDR-fission of238U-projectiles at 12 MeV excitation energy [26], the standard I-channel to be reduced by a factor of 2 compared to241Pu (nth, f) at 6 MeV excitation energy. The standard I-channel seems to disappear with a small damping energy of about 6 MeV. Its flux is absorbed by the
stan-dard II-channel which disappears with a damping energy of 18 MeV.
Looking at the small part of fission products having high kinetic energies in thermal neutron fission of235U or233U we find in a narrow range of kinetic energies a fission mass distribution with a predominant yield at134Te [27]. The yields at the high kinetic energies are primary yields as no neutrons are emitted. ’Cold fission’ was carefully analyzed in several systems by measuring isotopic and mass yields in the complementary light fission product group. Figure 4a shows the mass distribution of light fission products for234U. The predominant yield for the pair100Zr/134Te observed at 114 MeV disappears exponen-tially with the damping energy of 6 MeV. A small heating destroys the appearance of the
N =82cluster and the system evolves into another fission path. The pair90Kr54/144Ba88
carrying the deformed shellN = 88[23] is still present at the highest fission yields as
part of the standard II-channel and disappears as the latter with a damping energy close to the theoretical value of 18 MeV [13].
The spontaneous fission of258Fm, which shows a narrow symmetric mass distribution [28], is interpreted as a breaking into two129Sn-nuclei, that is into twoZ = 50-clusters
(fig. 4b). In comparison, the mass distribution in thermal neutron fission of257Fm [29], which is a fission of the same nucleus, but with an excitation energy of 6 MeV, is still symmetric, however spread out over a broad mass range. This small excitation energy in the rearranging system of nucleons is sufficient to force the system into another path, and to dissolve theZ=50cluster configuration.
4. – Cluster aspects in the fusion entrance channel
Theoretical predictions. It is not possible to treat the many theoretical efforts
unbi-ased. My bias is their applicability to experimental data on the heaviest collision systems. Swiatecki [30] introduced the extra-push model, a macroscopic model assuming the dissi-pation losses to be due to the macroscopic motion of the walls and windows defining the po-tential of the collision system in respect to the Fermi motion of the nucleons (wall-window friction). The model in its latest version [31] as already in its first version [30] predicts an approximately parabolic dependence of the extra push energy compensating the dissipa-tive losses on a scaling parameter (x,xthr), withxthe fissility of the fused system and
xthr= 0.72. The same threshold value was predicted by Sierk and Nix [32] already in a
paper of 1974. In a later paper of the same authors [33] a slightly higher threshold value ofxthr= 0.74. and a slope value, as found in ref. [31], were published. The ratio of
disrup-tive Coulomb forces and attracdisrup-tive surface tension forces governs the amalgamation of two nuclei into one. For a monosystem this ratio is given by the fissility parameter. For a two-touching-sphere configuration, Bass [34] defined a corresponding parameter making use of the proximity force. As the amalgamating nuclei already at the Coulomb barrier are in closer contact than the two-touching-sphere configuration, some average out of the fissilities of the mono and binary systems presents the appropriate fissilityxfor an
asym-metric collision system. Blocki et al. [31] determined a weighting parameter 1/3 for the arithmetic average of the two limiting fissilities,x = 2=3xmono+1=3xbinary. Attempts
to include fluctuations in the framework of macroscopic approaches to fusion have been made by Fr¨obrich and coworkers [35] and by Aguiar [36].
Macroscopic models neglect nuclear structure and give a guideline. One of the early microscopic approaches to calculate the distributions of fission fragments goes back to N¨orenberg [37]. Since then many further efforts were made. Among others, Berger et
Fig. 5. – ElementZ
1combined with element Z
2giving the amalgamated element ( Z
1 +Z
2). These
quantities span a net of lines as indicated in an asymmetry (Z 1 ,Z 2 =Z 1 +Z 2) vs. ( Z 1 +Z 2) plot. The
elements indicated between Ne and Cf are given in distances of 8 atomic numbers. The thick line is the on-set of fusion limitation in the entrance channel,x= 0.72 [31]. The points indicate systems
where EVRs were detected. Filled diamonds: actinide-based reactions, filled triangles: Pb-based reactions, filled circles: all other systems. Collision systems with 2 and more closed shells in the partners are indicated as lines or hatched regions.
Brosa et al. [24] developed scission models using the Strutinsky method. N¨orenberg in-troduced the assumption of ’dissipative diabatic dynamics’ to include nuclear structure to fusion reactions [39], and applied this model to fusion of heavy nuclei [40]. Until now, there is no calculation which allows to predict cross-sections at the level of the experiments. A revival of theory is mandatory in order to accompany further experimental efforts.
Experimental evidence of entrance channel limitation from fusion reactions. The
onset of entrance channel limitation atxthr= 0.72 together with the asymmetry scaling
parameter defines a line separating unhindered fusion from hindered combinations (fig. 5). The asymmetry of the collision system=(Z1,Z2)=(Z1+Z2)is plotted vs. the element
number of the fused system (Z1+Z2), and the borderline of the two regions is given. The
elements starting with Ne up to Cf in distances of 8 units in atomic number are chosen as a grid for orientation. For given values ofZ1andZ2the small variations of the fissility
parameter for the different stable isotopes are neglected. Successful fusion reactions are indicated with full symbols. Three groups are seen. Fusion of symmetric systems with
Fig. 6. – a) EVRs cross-sections at the Bass barrier for nearly symmetric collision systems. Tri-angles [15], filled diamonds [17], filled circles 49, squares 50. b) The production cross-sections via 1n Pb- and Bi-based reactions for elementsZ= 102 – 112 [1-6] in comparison to
238
U-based reac-tions [45, 46].
(Z1+Z2) = 80 – 92 [15-17, 41-43], of Pb- and Bi-based systems with (Z1+Z2) = 98 – 112
[1-6], and of actinide-based systems with (Z1+Z2) = 104 – 108 [44-46] was established
down to cross-section limits of a few picobarns. Figure 6a shows the fusion cross-sections measured for nearly symmetric collision systems. The cross-sections are given at the unshifted Bass reference barrier. BetweenZ1Z2= 1600 and 2100, orxthr= 0.72 and
0.81, the sections drop by seven orders of magnitude. Figure 6b presents the cross-sections for SHE synthesis.
We measured for the entrance channel limited symmetric systems the barrier shifts [15-17, 41-43]. There is no doubt that between fissilities of 0.68 and 0.82, barrier shifts are observed. Some systems show the phenomenon at smaller fissilities, others at larger val-ues than predicted. The comparison of systems having closed shell nuclei, as90Zr,86Kr, and124Sn, to transitional soft nuclei as 96Zr, 100Mo,104Ru, and110Pd as collision part-ners, shows a nuclear structure dependence. For soft collision partners barrier shifts are observed between 0.68 and 0.76, whereas for the closed-shell collision partners (N =50
andZ = 50) they are observed for higher fissilities between 0.72 and 0.78. The nuclear
structure of the collision partners determines decisively the on-set of entrance channel limitation. The systems with closed shell nuclei as a partner show smaller dissipative losses than the charged liquid drops underlying the theoretical models. For collision sys-tems with open valence nucleons the dissipative losses in the very early stages of approach to the barrier are of special importance.
As barrier fluctuations are not contained in the models [31, 33], only mean trajectories are predicted. In these models there is fusion above the barrier and any kind of fusion below this barrier is neglected. As EVRs are amply found already below the extra push barrier B, the models fail to make numerical predictions on the fusion probability at the energies which are of highest relevance for the production of heavy elements. To under-stand the fusion processes which lead to the heavy elements, just those processes have to be understood which are assumed to be negligible. The extra push losses are the sum of a great number of statistically independent single energy loss processes which micro-scopically are equivalent to nucleons changing orbits or to collective excitations changing
Fig. 7. –p(BB) as a function of the effective fissility for nearly symmetric collision systems [15-17,
41-43], for Pb/Bi-based systems [1-6] and for actinide-based systems [44-46].
the character. These losses are avoided in the fusion processes actually observed near the barrierBB. The fusion probabilitiesp(BB) are deduced from experiments, and are shown
in fig. 7. For all the nearly symmetric systems we investigated, the fusion probability at the Bass barrier as a function of the weighted fissility was determined.p(BB) is the
prob-ability to penetrate the unshifted barrierBB, which is for unhindered fusion by definition
50%. How to extractp(BB) from the measured cross-sections (fig. 6a) is described in
ref. [17]. We interpretp(BB) as the probability of penetrating the barrier without losing
energy via dissipative processes, that is penetrating by avoiding barrier shifts or ’extra push’ [47]. For a given value ofp(BB) the closed shell systems withZ =50andN =50
show a small shift to largerx-values in comparison to the soft collision systems.
Thep(BB) values for Pb- and Bi-based reactions are normalized top(BB) = 0.5 for
208Pb(48Ca,xn). We assume the observed decrease in the cross-section (fig. 6b) is caused by a decrease of the fusion probability alone. The values ofp(BB) thus obtained (fig. 7) are
lower values, because,n=,
fis assumed to stay constant. The value of
p(BB) for the
pro-duction of277112 is410 ,6
. We observe that the Pb- and Bi-based systems separate from the symmetric collision systems. This finding may be interpreted as a nuclear-structure dependence of the threshold where the extra push phenomenon sets in, as also observed for symmetric closed shell systems. For the Pb- and Bi-based systemsp(BB)-values
de-viate from the 50% barrier penetration limit forx>0.79. Compared to the macroscopic
predictionx= 0.72 this is a delayed on-set byx= 0.07. With this scaling the closed
shell collision partners show unhindered fusion up to a ratio of Coulomb to nuclear forces of nearly 0.80 reached in the reaction208Pb(48Ca, n)255No. The slope of p(BB) is
un-changed compared to symmetric systems. A factor of 3 is lost forx= 0.01. This factor
reflects the fluctuations in the barrier penetrability, which as the analysis showed, have to increase with growing barrier shifts [47, 43]. All systems show similar fluctuations, but the cluster nuclei withstand better the destruction by the increasing dissipative heating.
Extensive studies at Dubna [44-46] on fusion of heavy elements for fissilitiesx>0.72
using actinide targets allow to enter the region where these most asymmetric collision systems finally also show reduced barrier penetration in the entrance channel. The sys-tem22Ne/238U with its fissility ofx= 0.72 is assumed to have ap(BB)-value of 0.5. For
the 5n-reaction, a fission loss in the evaporation cascade of 510 ,6
assumed to stay constant in the region of isotopes investigated, as fission barriers are high. The cross section of the systems investigated forx >0.72 decreases rapidly, and
the fusion probability, as shown in fig. 7, shows a similar slope as the Pb-based systems. The onset atx= 0.72 of entrance channel hindrance as observed for symmetric systems,
explains the small cross-sections observed in the synthesis reactions for Sg and Hs.p(BB)
for the reaction238U (34S, 5n)267Hs atx= 0.80 has a value of only 210 ,5
.
The macroscopic hindrance of fusion in the entrance channel is found to be a common phenomenon to all fusion reactions, and reduces, once it is active, the cross-section per ele-ment by a factor of three to four. The critical fissility where the hindrance sets in depends on the nuclear structure. Collision systems with closed shell nuclei (Z = 50;82;N = 50;82;126) show a delayed on-set of entrance-channel limitation.
5. – Clusters and compactness
Which shapes are connected to reactions showing spherical shell-nuclei? An indication of which scission configurations are involved in fission is given by the total kinetic energies of the fragments. A conversion of the latter into distances between the nascent fragments is possible by interpreting kinetic energies as mutual Coulomb energy at scission. The total kinetic energy for the258Fm mass split in two129Sn-nuclei is 238 MeV [28] corre-sponding to a distance of 15.1 fm between the two fragments. The interaction distance between two touching spherical nuclei is about 1 fm smaller. The scission configuration in the258Fm case needs a small deformation which might be a dynamical octupole defor-mation of the spherical Sn-nuclei. Also for cold fission of234U it was shown [27] that the distance corresponding to scission for the pair134Te/100Zr is larger by about 1 fm than the nuclear interaction distance between two-touching spheres. The134Te/100Zr config-uration is the most compact configconfig-uration of all mass splits possible, but even this split is more elongated than the interaction distance of a two-center sphere configuration. At least one nucleus at scission is deformed. The total kinetic energies of fission fragments for cold fission, e.g., of258Fm, nearly reach the reactionQ-values. Here, the smallest
scission distances theoretically possible would be reached. The most compact scission configurations are observed for nuclei with Z = 50and N = 82. Clusters like 129Sn
and134Te in fission are born from compact scission configurations, but we note that all scission configurations are more elongated than the interaction distance in the fusion en-trance channel. The fusion enen-trance configuration is more compact than the fission exit configuration. Fusion and fission cannot be reversed, even in the case of ’cold fission’.
Calculations on the fission paths for258Fm reproducing the fission properties are avail-able [48, 49]. The path leading to129Sn-clusters in258Fm spontaneous fission, and the shapes as they evolve in a parameter space including high even and odd multipoles for the deformation were presented. The fragments were allowed to deform, with the curvature of the end tips kept fixed at the value of the spherical end products [48]. Measuring the distance between the centers of mass of the two parts in units of the radius of the equiv-alent sphere, the ground state is at 0.85R0, the fission barrier peaks at 1.0 R0, and its
exit point is at 1.15R0. To reproduce the spontaneous fission half-life of 0.4 ms a narrow
barrier is demanded. The barrier exit point is far away from scission, located at 2.0R0.
The system elongates by more than 6 fm before it fragments. Where do the fragment shells ofN =82andZ=50start to act? Figure 8a shows for the case of264Fm, a double
132Sn cluster, a two-center level diagram for protons. The levels correspond to shapes of the intersecting spheres. All the way between the barrier exit point and scission the shell gaps are seen. At 1.15R0 the132Sn clusters would overlap by about 3 fm. The system
Fig. 8. – Single-particle levels of the two-center systems: a)132 Sn/132 Sn, b) 86 Kr/136 Xe and c)64 Ni/208
Pb. The gaps at Z = 250,N = 50=82, andZ =28=82 are very prominent. They
are damped-out approaching the fission barrier exit points [48, 50].
needs to find a shape where a part of the nucleons is outside the overlap region in order to avoid compression. A slightly necked-in configuration evolves.
Comparing the barrier exit points for258Fm and260Sg we find the same distance of about 1.15R0, which corresponds to the shape of fission isomers with their 2:1 axis
ra-tio. The level diagram of Fig. 8a and the analysis of the 258Fm case suggest that the cluster configurations are present at configurations as compact as the second minimum. Also for ’cold fission’ the134Te configuration starts to be preformed already at this early stage, as is shown in a corresponding diagram for the86Kr/136Xe system in fig. 8b. As discussed in the case of258Fm the system may meet several bifurcation points to other fission modes before scission [24, 48]. On the way to scission preformed spherical clusters may be dissolved again, as happened to a part of the yield in258Fm spontaneous fission. The chance to take such a path involving larger deformations and leading to deformed clusters increases with growing intrinsic excitation energy.
6. – The208Pb cluster key to SHE production
The mean fissility, as defined earlier, of the reaction208Pb(54Cr,2n)260Sg ofx= 0.84
equals the fissility of the258Fm system. 258Fm and260Sg are both deformed nuclei with a ground-state deformation corresponding to 0.85R0. The fission barrier exit points for
both systems are calculated at about 1.15R0[7, 48]. Both systems have about equal
dis-tances of 14 fm at the interaction point (r0= 1.4 fm) of the incoming fusion partners and
Fig. 9. – Stages corresponding to the fusion path towards element 112 in the70
Zn/208
Pb collision system.
Pb- and Bi-based fusion reactions show their maximal cross-sections at an excitation energy of 12 MeV. The excitation energies and the temperature of the compound system are well below the limit ofT =1:5MeV where shell correction energies for deformed
nu-clei become ineffective. During the approach and amalgamation phase the system carries a total energy of about 12 MeV above the ground state of the compound system, but even if the system moves in a similar PES in fission of258Fm and fusion of Sg, there remains this difference in intrinsic energy. The Sn clusters emerge from the258Fm ground state, whereas the Pb cluster is embedded in a moderately heated collision system. Figure 8c shows the two-center proton level diagram of64Ni on208Pb used to synthesize element
Z =110according to M¨oller [50]. Indeed, the Ni- (Z=28) and Pb-clusters open a
chan-nel of reduced level densities reaching a distance of 1.15R0. The fusion and fission paths
are reversible for the two systems between 1.15R0(barrier exit) and the early stages of
approach or the late stages of separation, respectively. The static two-center PES-models in fission and fusion [24, 51] are applicable here, as both the dissipation and the inertia parameters are of minor importance on this part of the rearrangement process. Where to store and how to release the 12 MeV of total energy without disturbing the Pb cluster remains a problem for the last stages of fusion. It has to be envisaged that the nucleons from the light partner carry most of the excitation energy and the Pb cluster stays stable. In the final stage ((1.2 – 0.85)R0) all cluster structures gradually may dissolve. Here,
the dissipative dynamics of the macroscopic models [31] takes over, and finally also in the cluster supported cold fusion processes, the increasing Coulomb forces prevent more and more the formation of EVRs, as is indicated by the steep decrease of the fusion probabil-ity,p(BB), in fig. 7. The small total energy in the collision system certainly does not allow
Fig. 10. – The perspectives to go beyond element 112 via cold fusion reactions. Some isotopes at cross-sections below 10,36
cm2
of elements 113/114 may be synthesized continuing our experiments with Zn and Ge beams.48
Ca/248
Cm and the radioactive beam reactions84
Ge and94
Kr on208
Pb are indicated.
to destroy the underlying doubly-closed shell structure of208Pb, but it may be enough to trigger the transfer of nucleons to higher orbits of208Pb in order to finally achieve complete fusion.
Figure 9 presents the sequence of stages for a collision system leading to element 112 at different distances of approach. For the actinide-based systems with high fissilities, the dissipative losses set in, as for their analogue, the soft symmetric collision systems presented in fig. 7, already at distances of about 1.5R0. In accordance with the fissility
weighting of the macroscopic models, here the system has passed 1/3 of the distance to the compound nucleus. For the cluster-based fusion process with its low intrinsic excitation energy of only 12 MeV the dissipative losses are avoided in the early stages (see fig. 8c). About 2/3 of the distance between the first contact between the collision partners and the compound nucleus formation has been passed at the barrier exit point (1.15R0), where
the cluster configuration starts to dissolve. The clustering in the entrance channel gives an explanation for the delayed on-set of dissipative losses, which is equivalent to a higher weight given to the fissility of the binary system. An increasing weight of this fissility would let coincide all systems of fig. 7 in onep(BB)-dependence. The clustering explains
why fusion using actinides as targets is so difficult beyond fissilities of 0.80, and why there is an access to SHE using Pb- and Bi-based systems even at mean fissilities of 0.90.
The compound system carrying 12 MeV of excitation energy has to cool down by emis-sion of a neutron to an energy below the fisemis-sion barrier, to be protected against immediate reseparation. As the macroscopic energies defined by the liquid drop PES are repelling in all stages of the collision, the system stays between the ground state and the saddle point deformations ((0.85 – 1.00)R0) during a short-time interval of about 210
,20
s. The neu-tron has to be emitted then. Otherwise the two collision partners reseparate and reach distances where no shell corrections protect against immediate fission. Mean neutron emission times atE
= 12 MeV are a few times 10,17
s. The emission probability in the short interval of 210
,20
s is less than 10,3
channel which is equal to 210 ,2
will be reduced to a value of 210 ,5
. Besides the small barrier penetration factor (fig. 7), it is this reduction of,
n =,
fby pre-compound emission,
which determines the small production cross-sections observed.
Figure 10 gives the perspectives to go beyond element 112 via cold fusion reactions. Some isotopes at cross-sections below 1 pb of elements 113 - 114 may be synthesized con-tinuing our experiments with Zn- and Ge-beams. In208Pb(76Ge,n)283114169a deformed nucleus ofZ = 114could be reached at a cross-section of 0.1 pb. Our future
experi-ments are planned also to test this synthesis reaction. The next qualitative step to pass atN = 172the region of deformed nuclei and to enter the region of spherical nuclei of
SHEs requires beams of selenium and higher elements. N = 184would be reached at Z=124using beams of100Mo. The use of radioactive beams of94Kr would allow to reach N =184already atZ = 118. The use of actinide targets and symmetric systems does
no allow to keep the fused system cool enough to fully exploit the shell-dips in the PES. Even for Pb- and Bi-based reactions, it is an open question whether the reduced collective enhancement of the level density for spherical nuclei would not bring the cross-sections to a level which we cannot reach in a foreseeable future. Also the large number of va-lence nucleons of the radioactive94Kr nuclei used in future radioactive beam experiments may introduce dissipative losses, which may largely compensate the gain in cross-section expected for neutron-rich projectiles.
Whether we reach the N = 184spherical nuclei is not a question of their nuclear
structure. We showed that there are purely shell stabilized nuclei, barrel-like deformed nuclei atN = 162. Why should there be no next spherical shell beyond 208Pb close to Z=114,126withN =184? The shell structure creating shell closures is a microscopical
ordering phenomenon in the macroscopic PES, which will even repeat for still higher nucleon numbers. But the heavier the system, the smaller the dips in the PES, which becomes steeper and steeper and more repelling. The macroscopic ratio of Coulomb and nuclear forces given by the fissility grows going to higher elements and finally will prevent further production of higher elements, not for principle reasons, but for practical reasons of production yields. We have shown that there are purely shell-stabilized elements with deformed nuclei. The bold idea of SHE beyond the liquid drop limits was verified. How far towards higher shells the idea may still be expanded stays open. Whether long half lives are reached is a question of inferior importance. In any case, large factors of stabilization of 19 orders of magnitude in half-lives up to seconds were already demonstrated. To show this also for spherical nuclei, possibly with even larger stabilization, remains a challenge not so much of nuclear structure physics, but a challenge for reaction physics to explain and measure small cross-sections in the range below 10,36
cm2.
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