COLLISIONS OF TRANSACTINIDES: SUPERHEAVY NUCLEI AND GIANT...

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International Journal of Modern Physics EVol. 16, No. 4 (2007) 969–981c© World Scientific Publishing Company

COLLISIONS OF TRANSACTINIDES:

SUPERHEAVY NUCLEI AND GIANT NUCLEAR MOLECULES

VALERY ZAGREBAEV

Flerov Laboratory of Nuclear Reaction, JINR, Dubna, 141980, Moscow region, Russia

[email protected]

WALTER GREINER

Frankfurt Institute for Advanced Studies, J. W. Goethe-Universitat, Frankfurt, Germany

[email protected]

Received 25 May 2006

Low energy collisions of very heavy nuclei (238U+238U, 232Th+250Cf and 238U+248Cm)have been studied within the realistic dynamical model based on multi-dimensionalLangevin equations. Large charge and mass transfer was found due to the “inversequasi-fission” process leading to formation of survived superheavy long-lived neutron-rich nuclei. In many events lifetime of the composite system consisting of two touchingnuclei turns out to be rather long; sufficient for spontaneous positron formation fromsuper-strong electric field, a fundamental QED process.

1. Introduction

The mechanism of strongly damped collisions between very heavy nuclei was stud-

ied extensively about twenty years ago (see, for example, review 1 and references

therein). Among others, there had been great interest in the use of heavy-ion trans-

fer reactions with actinide targets to produce new nuclear species in the trans-

actinide region 2–7. The cross sections were found to decrease very rapidly with

increasing atomic number of surviving target-like fragments. However, Fm and Md

neutron-rich isotopes have been produced at the level of 0.1 µb. It was observed also

that nuclear structure (in particular, the closed neutron shell N=126) may strongly

influence nucleon flow in dissipative collisions with actinide targets 8.

Renewed interest from our side to collision of transactinide nuclei is conditioned

by the necessity to clarify much better than before the dynamics of heavy nuclear

systems at low excitation energies and by a search for new ways for production of

neutron rich superheavy (SH) nuclei and isotopes. SH elements obtained in “cold”

fusion reactions with Pb or Bi target 9 are situated along the proton drip line being

very neutron-deficient with a short half-life. In fusion of actinides with 48Ca more

neutron-rich SH nuclei are produced 10 with much longer half-life. But they are

still far from the center of the predicted first “island of stability” formed by the

969

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970 V. Zagrebaev & W. Greiner

neutron shell around N=184. In the “cold” fusion, the cross sections for formation

of SH nuclei decrease very fast with increasing charge of the projectile and become

less than 1 pb for Z≥112. On the other hand, heaviest transactinide, Cf, which can

be used as a target in the second method, leads to the SH nucleus with Z=118

being fused with 48Ca. Using the next nearest elements instead of 48Ca (e.g., 50Ti,54Cr, etc.) in fusion reactions with actinides is expected less encouraging, though

experiments of such kind are planned to be performed. In this connection other

ways to the production of SH elements in the region of the “island of stability”

should be searched for.

At incident energies around the Coulomb barrier in the entrance channel the

fusion probability is about 10−3 for mass asymmetric reactions induced by 48Ca

and much less for more symmetric combinations used in the “cold synthesis”. Deep

inelastic (DI) scattering and quasi-fission (QF) 11,12,13 are the main reaction chan-

nels here, whereas the fusion probability [formation of compound nucleus (CN)] is

extremely small. Note that QF phenomena are revealed in, and could be important

also for comparatively light fusing systems 13,14. It is the QF process that inhibits

fusion by several orders of magnitude for heavy nuclear systems. To estimate such

a small quantity for CN formation probability, first of all, one needs to be able to

describe well the main reaction channels, namely DI and QF. Moreover, the quasi-

fission processes are very often indistinguishable from the DI scattering and from

regular fission, which is the main decay channel of excited heavy CN.

Recently a new model has been proposed 15 for simultaneous description of all

these strongly coupled processes: DI scattering, QF, fusion, and regular fission. In

this paper we develop further this model and apply it for analysis of low-energy

dynamics of heavy nuclear systems formed in nucleus-nucleus collisions at the en-

ergies around the Coulomb barrier. Among others there is the purpose to find an

influence of the shell structure of the driving potential (in particular, deep valley

caused by the double shell closure Z=82 and N=126) on formation of CN in mass

asymmetric collisions and on nucleon rearrangement between primary fragments in

more symmetric collisions of actinide nuclei. In the first case, discharge of the sys-

tem into the lead valley (normal or symmetrizing quasi-fission) is the main reaction

channel, which decreases significantly the probability of CN formation. In collisions

of heavy transactinide nuclei (U+Cm, etc.), we expect that the existence of this

valley may noticeably increase the yield of surviving neutron-rich superheavy nu-

clei complementary to the projectile-like fragments (PLF) around lead (“inverse”

or anti-symmetrizing quasi-fission reaction mechanism).

Direct time analysis of the collision process allows us to estimate also the lifetime

of the composite system consisting of two touching heavy nuclei with total charge

Z>180. Such long-living configurations (if they exist) may lead to spontaneous

positron emission from super-strong electric fields of giant quasi-atoms by a static

QED process (transition from neutral to charged QED vacuum) 16,17.

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Collisions of Transactinedes 971

2. The Model

The distance between the nuclear centers R (corresponding to the elongation of a

mono-nucleus), dynamic spheroidal-type surface deformations β1 and β2, mutual

orientations of deformed nuclei ϕ1 and ϕ2, and mass asymmetry η = A1−A2

A1+A2

are

probably the relevant degrees of freedom in fusion-fission dynamics. Taken together

there are 7 degrees of freedom shown in Fig. 1. For the separation stage in the exit

channel neck formation should be included to describe properly mass and energy

distributions of the fission fragments. To avoid increasing the number of degrees of

freedom we take into account the necking process in a phenomenological way 15.

Fig. 1. Degrees of freedom used in the model.

The interaction potential of separated nuclei is calculated rather easily within

the folding procedure with effective nucleon-nucleon interaction or parameterized,

e.g., by the proximity potential 18. Of course, some uncertainty remains here, but

the height of the Coulomb barrier obtained in these models coincides with the em-

pirical Bass parametrization 19 within 1 or 2 MeV. After contact the mechanism

of interaction of two colliding nuclei becomes more complicated. For fast colli-

sions (E/A ∼ εFermi or higher) the nucleus-nucleus potential, Vdiab, should reveal

a strong repulsion at short distances preventing the “frozen” nuclei from penetra-

tion into each other and formation a nuclear matter with double density (diabatic

conditions, sudden potential 20). For slow collisions (near-barrier energies), when

nucleons have enough time to reach equilibrium distribution (adiabatic conditions),

the nucleus-nucleus potential energy, Vadiab, is quite different. The calculation of

the multidimensional adiabatic potential energy surface for heavy nuclear system

is a very complicated physical problem, which is not yet solved in full. In this con-

nection the two-center shell model 21 seems to be most appropriate for calculation

of the adiabatic potential energy surface.

Dynamic deformations of colliding spherical nuclei and mutual orientation of

statically deformed nuclei significantly affect their interaction changing the height

of the Coulomb barrier for more than 10 MeV. It is caused mainly by a strong

dependence of the distance between nuclear surfaces on the deformations and ori-

entations of nuclei, see Fig. 2. For very heavy nuclear systems, like 232Th+250Cf,

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there is no potential pocket typical for lighter systems, the potential is repulsive

everywhere. However, the potential energy is not so much steep in the region of

contact point and two nuclei may keep in contact for a long time increasing their

deformations and transferring nucleons to each other (see below).

Fig. 2. Potential energy surface for the nuclear system formed by 232Th+250Cf as a function ofR and η (β = 0.22) for nose-to-nose (a) and side-by-side (b) orientations, and as a function ofR and β (η = 0.037) (c). Typical trajectories are shown by the thick curves with arrows. Beforenuclei reach the region of nuclear friction forces, they are in their ground states (zero temperature)and no fluctuations occur. However when nuclei come in contact they are already well excited and

fluctuations play a significant role.

A choice of dynamic equations for the considered degrees of freedom is also not

so evident. The main problem here is a proper description of nucleon transfer and

change of the mass asymmetry which is a discrete variable by its nature. More-

over, the corresponding inertia parameter µη, being calculated within the Werner-

Wheeler approach, becomes infinite at contact (scission) point and for separated

nuclei. In Ref. 15 the inertialess Langevin type equation for the mass asymmetry

dη

dt=

2

ACN

D(1)A (η) +

2

ACN

√

D(2)A (η)Γ(t), (1)

has been derived from the corresponding master equation for the distribution func-

tion ϕ(A, t). Here Γ(t) is the normalized random variable with Gaussian distribution

and D(1)A , D

(2)A are the transport coefficients. A is the number of nucleons in one

of the fragments and η = (2A − ACN )/ACN . Assuming that sequential nucleon

transfers play a main role in mass rearrangement, i.e. A′ = A ± 1, we have

D(1)A = λ(A → A + 1)− λ(A → A− 1), D

(2)A =

1

2λ(A → A + 1) + λ(A → A− 1).(2)

The macroscopic transition probability λ(A → A ± 1) is defined in the following

way

λ(±) = λ0

√

ρ(A ± 1)/ρ(A)Ptr(R, β, A → A ± 1). (3)

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Here Ptr(R, β, A → A ± 1) is the probability of one nucleon transfer depending on

the distance between the nuclear surfaces. This probability goes exponentially to

zero at R → ∞ and it is equal to unity for overlapping nuclei. In our calculations we

used the semiclassical approximation for Ptr proposed in Ref. 22. Eq. (1) along with

(3) defines a continuous change of mass asymmetry in the whole space (obviously,dηdt

→ 0 for far separated nuclei). Finally we solve numerically a set of 13 coupled

Langevin type equations for 7 degrees of freedom R, ϑ, β1, β2, ϕ1, ϕ2, η shown in

Fig. 1.

The double differential cross-sections of all the processes are calculated as follows

d2ση

dΩdE(E, θ) =

∫ ∞

0

bdb∆Nη(b, E, θ)

Ntot(b)

1

sin(θ)∆θ∆E. (4)

Here ∆Nη(b, E, θ) is the number of events at a given impact parameter b in which

the system enters into the channel η with kinetic energy in the region (E, E + ∆E)

and center-of-mass outgoing angle in the region (θ, θ + ∆θ), Ntot(b) is the total

number of simulated events for a given value of impact parameter. The expression

(4) describes the energy, angular, charge and mass distributions of the primary

fragments formed in the reaction. Subsequent de-excitation of these fragments via

fission or emission of light particles and gamma-rays was taken into account within

the statistical model leading to the final fragment distributions. The sharing of the

excitation energy between the primary fragments was assumed to be proportional

to their masses. Neutron emission during an evolution of the system was also taken

into account. It was found that the pre-separation neutron evaporation does not

influence significantly the final distributions.

3. Deep Inelastic Scattering

At first we applied the model to describe available experimental data on low-energy

damped collision of very heavy nuclei 23, 136Xe+209Bi, where the DI process should

dominate due to expected prevalence of the Coulomb repulsion over nuclear attrac-

tion and the impossibility of CN formation. The colliding nuclei are very compact

with almost closed shells and the potential energy has only one deep valley corre-

sponding to the entrance channel (η ∼ 0.21) giving rather simple mass distribution

of the reaction fragments. In that case the reaction mechanism depends mainly

on the nucleus-nucleus potential at contact distance (which determines the graz-

ing angle), on the friction forces at this region (which determine the energy loss)

and on nucleon transfer rate at contact. In Fig. 3 the angular, energy and charge

distributions of the Xe-like fragments are shown comparing with our calculations

(histograms). In accordance with experimental conditions only the events with the

total kinetic energy in the region of 260 ≤ E ≤ 546 MeV and with the scattering

angles in the region of 40o ≤ θc.m. ≤ 100o were accumulated. The total cross section

corresponding to all these events is about 2200 mb (experimental estimation 23 is

2100 mb). Due to the rather high excitation energy sequential fission of the primary

heavy fragments may occur in this reaction (mainly those heavier than Bi). In the

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experiment the yield of the heavy fragments was found to be about 30% less com-

paring with Xe-like fragments. Our calculation gives 354 mb for the cross section of

sequential-fission, which is quite comparable with experimental data. Mass distribu-

tion of the fission fragments is shown in Fig. 3(c) by the dotted histogram. This it is

a contamination with sequential fission products of heavy primary fragments lead-

ing to the bump around Z=40 in the experimental charge distribution in Fig. 3(c).

Note that such a good agreement with experimental data of all the calculated DI

reaction properties was never obtained before in dynamic calculations.

Fig. 3. Angular (a), energy-loss (b) and charge (c) distributions of the Xe-like fragments inthe 136Xe+209Bi reaction at Ec.m.

= 569 MeV 23. Histograms are the theoretical predictions.The low-Z wing of the experimental charge distribution is due to incompletely removed events ofsequential fission of the heavy fragment. Dotted histogram indicates the calculated total yield ofsequential fission fragments.

4. Formation of Superheavy Nuclei in Low-Energy Collisions

of Transactinide Nuclei

Reasonable agreement of our calculations with experimental data on low-energy DI

and QF reactions induced by heavy ions stimulated us to study the reaction dy-

namics of very heavy transactinide nuclei. The purpose was to find an influence of

the shell structure of the driving potential (in particular, deep valley caused by the

double shell closure Z=82 and N=126) on nucleon rearrangement between primary

fragments. In Fig. 4 the potential energies are shown depending on mass rearrange-

ment at contact configuration of the nuclear systems formed in 48Ca+248Cm and232Th+250Cf collisions. The lead valley should evidently reveal itself in all heavy

nuclear systems. In the case of 48Ca+248Cm discharge of the system into the lead

valley (normal or symmetrizing quasi-fission) is the main reaction channel, which

decreases significantly the probability of CN formation. In collisions of heavy nuclei

(Th+Cf, U+Cm and so on) we expect that the existence of this valley may notice-

ably increase the yield of surviving neutron-rich superheavy nuclei complementary

to the projectile-like fragments around 208Pb (“inverse” or anti-symmetrizing quasi-

fission process).

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Fig. 4. Potential energy at contact “nose-to-nose” configuration for the two nuclear systemsformed in 48Ca+248Cm (a) and 232Th+250Cf (b) collisions. Spheroidal deformation is equal 0.2for both cases.

mass number

P d TLFfission fragments

LF an

cro

ss s

ectio

n

( m

b/u

nit )

232 250Th + Cf, E = 800 MeVc.m.

8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 010-1

100

101

102

103

primary

survived

Fig. 5. Mass distributions of primary (solid histograms), survived and sequential fission fragments(hatched areas) in the 232Th+250Cf collision at 800 MeV center-of-mass energy.

Using the same parameters of nuclear viscosity and nucleon transfer rate we

calculated the yield of primary and survived fragments formed in the 232Th+250Cf

collision at 800 MeV center-of mass energy. Low fission barriers of the colliding

nuclei and of the most of reaction products conjointly with rather high excitation

energies of them in the exit channel must lead to very low yield of survived heavy

fragments. Indeed, sequential fission of the projectile-like and target-like fragments

dominate in these collisions, see Fig. 5. At first sight, there is no chances to get

survived superheavy nuclei in such the reactions. However, as mentioned above, the

yield of the primary fragments should be increased due to the QF effect (lead val-

ley) comparing with gradual monotonic decrease typical for damped mass transfer

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reactions. Secondly, with increasing neutron number the fission barriers increase on

average (also there is the closed sub-shell at N=162). Thus we may expect non-

negligible yield (at the level of 1 pb) of survived superheavy neutron rich nuclei

produced in these reactions.

Fig. 6. Mass (a) and charge (b) distributions of primary (solid histograms) and surviving (dashedhistograms) fragments in the 232Th+250Cf collision at 800 MeV center-of-mass energy. Thinsolid histogram in (b) shows the primary fragment distribution in the hypothetical reaction248Cm+250Cf.

Results of much longer calculations are shown in Fig. 6, where the mass and

charge distributions of surviving fragments obtained in the 232Th+250Cf collision

at 800 MeV are presented. The pronounced shoulder can be seen in the mass dis-

tribution of the primary fragments near the mass number A=208. It is obviously

explained by the existence of a noticeable valley in the potential energy surface

[see Fig. 4(b) and Fig. 2 (a)], which corresponds to the formation of doubly magic

nucleus 208Pb (η = 0.137). The emerging of the nuclear system into this valley

resembles the well-known quasi-fission process and may be called “inverse (or anti-

symmetrizing) quasi-fission” (the final mass asymmetry is larger than the initial

one). For η > 0.137 (one fragment becomes lighter than lead) the potential energy

sharply increases and the mass distribution of the primary fragments decreases

rapidly at A<208 (A>274). The same is true for the charge distribution at Z<82

(Z>106). As a result, in the charge distribution of the surviving heavy fragments,

Fig. 6 (b), there is also a shoulder at Z∼106 and the yield of nuclei with Z>107

was found in this reaction at the level of less than 1 pb. This result differs sharply

from those obtained in Ref. 26, where the reactions of such kind have been analyzed

within the parametrized diffusion model and the yield of heavy primary fragments

was found to diminish monotonically with increasing charge number. The authors

of Ref. 26 concluded, however, that the “fluctuations and shell effects not taken

into account may considerably increase the formation probabilities”. Such is indeed

the case.

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Fig. 7. Yield of superheavy nuclei in collisions of 238U+238U (dashed), 238U+248Cm (dotted)and 232Th+250Cf (solid lines) at 800 MeV center-of-mass energy. Solid curves in upper part showisotopic distribution of primary fragments in the Th+Cf reaction. In the case of U+U collisionexperimental data 3 are also shown for Z=98 (circles), 99 (squares) and 100 (triangles).

The estimated isotopic yields of survived SH nuclei in the 232Th+250Cf,238U+238U and 238U+248Cm collisions at 800 MeV center-of-mass energy are shown

in Fig. 7. Thus, as we can see, there is a real chance for producing of the long-lived

neutron-rich SH nuclei in such reactions. As the first step, chemical identification

and study of the nuclei up to 274107Bh produced in the reaction 232Th+250Cf may

be performed. The yield of SH elements in damped reactions with transactinides

depends strongly on beam energy and on nuclear combination which should be cho-

sen carefully. In particular, in Fig. 6 the estimated yield of the primary fragments

obtained in the hypothetical reaction 248Cm+250Cf (both constituents are radioac-

tive) is shown, demonstrating a possibility for production of SH neutron-rich nuclei

up to the element 112 (complementary to lead fragments in this reaction).

5. Giant Quasi-Atoms and Spontaneous Positron Formation

One more effect, which may be expected in the damped collisions of transactinides,

is a formation of giant quasi-atoms with a nucleus consisting of two closely located

heavy nuclei. If the lifetime of the composite nuclear system with total charge

Z>173 is rather long (∼ 10−20 s) we may expect a spontaneous positron emission

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from super-strong electric field of quasi-atoms by a static QED process (transition

from neutral to charged QED vacuum) 16,17 (see Fig. 8). About twenty years ago

a search for such effect were carried out and narrow line structures in the positron

spectra were first reported at GSI. Unfortunately these results were not confirmed

later, neither in ANL, nor in new experiments performed at GSI 24,25 (see also

hierarchical references therein). These negative finding, however, were contradicted

by Jack Greenberg (private communication and supervised thesis at Wright Nuclear

Structure Laboratory, Yale university). Thus the situation remains unclear, while

the experimental efforts in this field have ended. We hope that a new analysis,

performed within a realistic dynamical model, may shed additional light on this

problem and also answer the principal question: are there some reaction features

(triggers) testifying to a long reaction delay? If they are, a new experiment could

be planned to detect the positrons in the specific reaction channels.

E

mc-

2

+mc 2

Dirac see

po

si tro

nyie

l d

1s2p

200

10

20

30

40

400 600 800 1000 1200

t

tdelay

e+E(e+ ), KeVe-

Z1

U + Cm

Z Z1+ 2Z2

Fig. 8. Schematic figure of spontaneous decay of the vacuum and spectrum of the positronsformed in supercritical electric field (Z1 + Z2 > 173).

The time analysis of the reactions studied shows that in spite of non-existing at-

tractive potential pocket the system consisting of two very heavy nuclei may hold in

contact rather long in some cases. During this time it moves over multidimensional

potential energy surface with almost zero kinetic energy (result of large nuclear

viscosity), a typical trajectory is shown in Fig. 2. The total reaction time distribu-

tion is shown in Fig. 9 for the 238U+248Cm collision. We found that the dynamic

deformations of colliding nuclei are mainly responsible here for the time delay of

the nucleus-nucleus collision. Ignoring the dynamic deformations in the equations

of motion significantly decreases the reaction time, whereas the nucleon transfer

influences the time distribution not so strongly.

As mentioned above, the lifetime of a giant composite system more than 10−20 s

is quite enough to expect for spontaneous e+e− production from the strong electric

field as a fundamental QED process (“decay of the vacuun”) 16,17. The cross section

for long events (τ > 10−20 s) was found to be maximal just at the beam energy

ensuring two nuclei to be in contact.

Formation of the background positrons in these reactions forces one to find some

additional trigger for the longest events. Such long events correspond to the most

damped collisions with formation of mostly excited primary fragments decaying by

May 24, 2007 9:26 WSPC/143-IJMPE 00643


Fig. 9. Reaction time distributions for the 238U+248Cm collision at 800 MeV center-of-massenergy. Thick solid histograms at both panels correspond to all events with energy loss more than

30 MeV. Thin histogram at the left panel shows the effect of switching-off dynamic deformations.At the right panel (b) thin solid, dashed and dotted histograms show reaction time distributionsin the channels with formation of primary fragments with Eloss > 200 MeV, Eloss > 200 MeV andθc.m.

< 70o and A ≤ 210, correspondingly. Hatched areas show time distributions of events withformation of the primary fragments with A ≤ 220 (light gray), A ≤ 210 (gray), A ≤ 204 (dark)having Eloss > 200 MeV and θc.m.

< 70o.

Fig. 10. Energy-time (a) and angular-time (b) distributions of primary fragments in the238U+248Cm collision at 800 MeV (Eloss > 15 MeV). Landscape is shown in logarithmic scale –lines are drawn over one order of magnitude. Quasi-elastic peak is removed.

fission, see Figs. 10(a). However there is also a chance for production of the primary

fragments in the region of doubly magic nucleus 208Pb, which could survive against

fission due to nucleon evaporation. The number of the longest events depends weakly

on impact parameter up to some critical value. On the other hand, in the angular

distribution of all the excited primary fragments (strongly peaked at the center-

of-mass angle slightly larger than 900) there is the rapidly decreasing tail at small

angles, see Fig. 10(b). Time distribution for the most damped events (Eloss >

150 MeV) in which a large mass transfer occurs and primary fragments scatter in

forward angles (θc.m. < 70o) is more narrow and really shifted to longer time delay,

see hatched area in Fig. 9.

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For the considered case of 238U+248Cm collision at 800 MeV center-of-mass

energy, the detection of the surviving nuclei in the lead region at the laboratory

angles of about 25o and at the low-energy border of their spectrum (around 1000

MeV for Pb) could be a real trigger for longest reaction time.

Acknowledgements

The work was supported by the DFG-RFBR project under Grant No. 04-02-04008.

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