Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 77–80, 2002

77

Lifetime of Heavy Composite Systems Formed by Fusion between Heavy Nuclei
Toshiki Maruyama,∗,a Aldo Bonasera,b Massimo Papa,b,c and Satoshi ChibaaaJapan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japanb
Istituto Nazionale di Fisica Nucleare Laboratorio Nazionale del Sud, Via S. Sofia 44, Catania 95123,
Italyc
Istituto Nazionale di Fisica Nucleare Sezione di Catania, Corso Italia 57, Catania 95129, Italy
Received: December 19, 2001; In Final Form: December 19, 2001
We investigate the formation and the decay of heavy systems which are above the fission barrier. By using a
microscopic simulation of constrained molecular dynamics (CoMD) on Au + Au collision, we observe composite
states stay for very long time before decaying by fission.

1. Introduction

In this paper we apply CoMD to the investigation of
Au + 197 Au collisions at low energies where fusion-fission or
deep-inelastic process may occur. In the following we give a
brief review of the model.5
197

Collisions between heavy nuclei at relatively low energy region have attracted strong interests of heavy-ion physicists for 3
different reasons, namely the creation of super heavy elements(SHEs), fission dynamics of very heavy systems, and creationof electron-positron pair due to the strong Coulomb field of thecomposite heavy nuclei as a verification of the QED process. In
these processes, the lifetime of the composite system, created by
the fusion of the projectile and the target, which decays eventually by fission is the key issue to understand the underlying reaction mechanisms and to estimate the probability of occurrenceof these processes.
SHEs are produced in two ways: one is “cold fusion” whichis complete fusion below the classical barrier, and the other is“hot fusion” which allows several neutrons to be emitted. Even
though the name is “hot”, such reactions are still at very low
energy near the barrier and the total mass number is very close
to the aimed one. As far as the formation of SHE is concerned,
the “fusion” of very heavy nuclei where the fission barrier nomore exists is found to be ineffective.1, 2
In the study of fission dynamics of heavy systems including
the spontaneous fission and the fusion-fission of heavy composite, the competition of neutron emission between the fission andthe fission delay have been discussed intensively. However almost all the discussion are done for mass regions where the classical fission barrier exists.
Sometime ago many physicists paid attention to the low energy collision of very heavy nuclei in regard to the spontaneous
positron emission from strong electric fields.3 If a molecule stateof, say, U and U is formed and stays sufficiently long time, thebinding energy of an electron can exceed the electron mass andmight create electron-positron pair by a static QED process. Unfortunately no clear evidence of static positron creation was observed below Coulomb energy region. They have pointed out4
the importance of nuclear reaction which causes the time delayof separation of two nuclei. Although there increases the background component of positrons from nuclear excitation, whichin this case is not interested in, the electron-positron from thestatic QED process is also expected to increase. However, thereaction mechanism of very heavy nuclei has not been discussed
by fully dynamical models.
In this paper we discuss the possibility of molecule-like statesof heavy nuclei and the time scale of very heavy composite system formed by the fusion-fission or deep inelastic processes. To
investigate these problems theoretically we use a recently developed constrained molecular dynamics (CoMD) model.5 Thismodel has been proposed to include the Fermionic nature of constituent nucleons by a constraint that the phase space distribution
should always satisfy the condition f ≤ 1.
∗Corresponding author. E-mail: maru@hadron02.tokai.jaeri.go.jp.

2. Constrained Molecular Dynamics Model
The CoMD model mainly consists of two parts: classicalequation of motion of many-body system, and stochastic process which includes constraint of Pauli principle and the twobody collisions. We write the distribution function of the system as a sum of one-body distribution function neglecting theantisymmetrizationf (r, p) = ∑ fi (r, p),ifi (r, p) =(1)
(r − Ri)2 (p − Pi )
1
·
exp
−
−
2σr 2
2σ p 2(2πσr σ p )3


2

. (2)

The equation of motion of Ri and Pi are derived using the timedependent variational principle which gives:
Ṙi =

∂H
,
∂Pi

Ṗi = −

∂H
.
∂Ri

(3)

In our approach the total energy H for A particles with mass m
consists of the kinetic energy and the effective interactions:
3σ2p
P2i
+A
+V .
2m
2m
i

H =∑

(4)

The second term arises from the Gaussian width in p-space.
However in the following considerations we omit such a constant term.
The effective interaction V we adopt is written as
V = V vol +V (3) +V sym +V surf +V Coul .

(5)

By defining the superimposition integral ρi j as:Zρi j ≡ d 3 ri d 3 r j ρi (ri )ρ j (r j )δ(ri − r j ) ,Zρi ≡ d 3 p fi (r, p) ,

(6)
(7)

the terms in eq 5 can be written as:
V vol =

t0
ρi j ,
2ρ0 i,∑j=it3
ρµij ,(µ + 1)(ρ0 )µ i,∑
j=i
asym
V sym =
[2δτi ,τ j − 1]ρi j ,2ρ0 i,∑
j=i
V (3) =

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

(8)
(9)
(10)

78

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002
V surf =
V

Coul

Cs
∇2Ri (ρi j ) ,2ρ0 i,∑
j=i

1
=
2

∑

i, j=i(i, j∈protons)e2|Ri − R j |Maruyama
(11)


erf

|Ri − R j |

2σ2r


.

(12)

In the above relations the coordinate τi represents the nucleon
isospin degree of freedom.
In this paper we use the parameter set for wave packet widthand effective interactions: σr = 1.15 fm, σ p /h̄ = 0.4748 fm−1 ,t0 = −301.1 MeV, t3 = 242 MeV, µ = 7/6, asym = 26.4 MeV,
Cs = −0.165 MeV fm2 , and ρ0 = 0.165 fm−3 , which reproduces
reasonably well the fusion cross section of Ca + Ca reactions,while in Reference 5 we have used another set of parameters.
The Pauli principle is taken into account in two ways: oneis the Pauli blocking of the final state of two-body collision andthe other is the constraint which brings into the system the Fermi
motion in a stochastic way. The starting point of the constraint
is the requirement:(for all i) ,Z
f i ≡ ∑ δτi ,τ j δsi ,s j
f j (r, p) d 3 r d 3 p ,fi ≤ 1

j

(13)
(14)

h3

where si is the spin coordinate of the nucleon i. The integral is
performed in a hypercube of volume h3 inthe phase-space cen2πh̄h̄tered around the point (Ri , Pi ) with size σr σ p σr and σ2πσpr σpin the r and p spaces respectively.
At each time step and for each particle i the phase space occupation f i is checked. If f i has a value greater than 1 an ensemble Ki of nearest particles (including the particle i) is determinedwithin the distances 3σr and 3σ p in the phase space. Then wechange randomly the momenta of the particles belonging to theensemble Ki in such a way that for the newly generated sample
the total momentum and the total kinetic energy is conserved(“many-body elastic scattering”). The new sample is accepted
only if it reduces the phase space occupation f i (Ref. 5).
To handle the Pauli-blocking in the collision term is straightforward from the constraint. In fact for each NN collision weevaluate the occupation probability after the elastic scattering.
If such functions are both less than 1 the collision is accepted,
rejected otherwise. We note that for the results discussed here
and especially at the lowest energies the collision term is of little
importance.
3. Collision of Au + Au System
To simulate the collision of two 197 Au nuclei, we prepare theground state by applying the frictional cooling method together
with the constraint of CoMD. The ground states we obtain havebinding energy of 8.4 MeV/nucleon and the root mean squaret =3000 fm/c
A max

t =200 fm/c

400

t =2000 fm/c
A max

Au+Au
t = 0 fm/c
10 MeV/u b=6 fm

radius of 5.34 fm. They are rather stable for 1000 fm/c. For
instance our 197 Au ground states evaporates 3.1 nucleons during1000 fm/c. The collision events are performed for impact parameter b of 0 and 6 fm for incident energy in laboratory system
of Elab = 5 ∼ 35 MeV/nucleon.
Figure 1 shows a typical event of CoMD calculation withincident energy Elab = 10 MeV/nucleon with impact parameterb = 6 fm. The two nuclei form a quite deformed compound system, they keep such a deformation almost 2500 fm/c and finallyfission takes place. The system does not show much rotation
since the angular momentum per nucleon is not so large and theelongated shape makes the moment of inertia larger than that in
the initial stage. Therefore the reaction mechanism we are observing here may be in-between the deep inelastic and molecularresonance.
There are many observables which distinguish the reaction
mechanism. The largest fragment mass is one of such welldefined observables which can easily be measured experimentally. Figure 2 shows the time dependences of the largest cluster
mass for the impact parameters b = 0 and 6 fm calculated by
CoMD and QMD. In CoMD calculations we see at the beginning the largest cluster mass Amax = 197 which corresponds toprojectile and target mass number. Within about 50 fm/c, Amax
becomes 394 except for the incident energy Elab = 5 MeV which
is below the barrier where two nuclei never contact. At incident energies above the barrier, the formed large system will
decay into smaller fragments by different modes according to
the energy and angular momentum. At higher incident energies(Elab ≥ 30 MeV/nucleon) the largest cluster mass changes suddenly at the early stage and continuously decreases in time. This
indicates multifragmentation for head-on collisions and deep inelastic reaction for peripheral collisions followed by the emission of nucleons and small fragments. At lower incident energies (Elab ≤ 20 MeV/nucleon) there is a sudden change of thelargest cluster mass at very late time, which indicates a fission of
the system. One should note that in our calculation of Au + Au
system there is almost no event where the system decays only
by emitting particles or light fragments, i.e., pure incomplete
fusion. The instability due to the Coulomb repulsion plays the
major role in the decay process.
Here we should note that the plotted largest fragment mass are
obtained by only one event for each incident energy and impact
parameter. Therefore the fission time includes large amount ofstatistical error. In Figure 2(b) the same quantity as Figure 2(a)is displayed for QMD calculation. These QMD calculations are
based on the same code as CoMD switching off the procedure
for the constraint. The difference between the CoMD and the
QMD is clear and dramatic. At low energy collisions there areno fission process and the system decays only by emitting nucleons and light fragments. At higher energies there is some sudden

300
200
100

b=0 fm0400(b) QMD

b=6 fm
5 MeV/u
10 MeV/u
15 MeV/u
20 MeV/u
25 MeV/u
30 MeV/u

300
200
100
0

Figure 1. Snapshot of 197 Au + 197 Au at Elab = 10 MeV/nucleon b = 6 fm.
The time indicated in each panel is not from the contact of two nuclei
but indicates only that of the simulation.

(a) CoMD

b=0 fm10

b=6 fm2

t [fm/c]

10

3

10

2

t [fm/c]

10

3

Figure 2. The time-dependence of the largest fragment mass Amax .
From the top (a) CoMD and (b) QMD. The left panels show cases ofhead-on collision and the right b = 6 fm.

Lifetime of Heavy Composite Systems Formed by Fusion

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002change of the largest fragment mass even in QMD calculation.
This is not a fission but passing through for head-on collisions
or deep inelastic process for peripheral collisions.

Assuming a very simple form of the time-dependent fission
width Γ(t) = Γf θ(t − Td ), the averaged fission time Tfiss can beobtained by the survival probability of the compound systemagainst two-body process Psurv asPsurv = exp [−(t − Td )Γf /h̄] ,Tfiss ≡ Td + h̄/Γf ,

0

10

20
E lab [MeV/u]

30

Figure 4. The “fission” lifetime of composite system for CoMD and
QMD calculation.

is no classical barrier for fission. Therefore the fission time gets
shorter with increase of the incident energy.
In QMD calculation what is in marked contrast to the CoMD
calculation is that the “fission time” has no maximum energy
and shows monotonic decrease. This is due to the lack of thePauli principle which suppresses two nuclei from overlapping atvery low energies (above the Coulomb barrier).
For peripheral collision (b = 6 fm), the lifetime of very heavy
composite is shorter than the head-on collisions. But theincident-energy dependence is very similar to the b = 0 fm cases.
Though the mechanism is much more dynamical, eq 15,16 fit
well again.
Nevertheless, the super heavy composite system formed by
the head-on collision of Au + Au may survive rather long time of
103 ∼ 104 fm/c.
An interesting aspect of the long-lived very heavy system is,
as mentioned before, the spontaneous positron-electron production from the strong electric field as a static QED process. The
total charge of Au + Au system may be still smaller than the necessary charge (Z ∼ 170) for this process. However, the nuclear
reaction of, e.g. U + U system, should be qualitatively the sameas what we observe in Au + Au system. Although the background positrons should be larger, one can get longest lifetime

0.0
1.0

QMD
10 MeV/u
15 MeV/u
20 MeV/u
25 MeV/u

0.5

2

10

3

t [fm/c]

10

4Figure 3. The survival probability of large fragments (A > 350) which
decay by fission-like mode. The abscissa indicates the time after thecontact of two nuclei. From the top the results are obtained for head-on
collisions by CoMD and QMD. The histograms indicate results fromsimulations and smooth curves are the fits by eq 15,16.

mass asymmetry

0.5

0.6

‘‘fission’’ fragment mass

survival probability
survival probability

b=0 fm
500

CoMD

1.0

10

CoMD
QMD

1000

(15)
(16)

where Td is the delay time and Γf is the “fission width” after thedelay time. The probability Psurv (t) is obtained directly by thesimulation. This fitting can apply well only for fission-like process in our calculation. Figure 3 shows the survival probabilityof a large fragment with A > 350 which decays by fission-likemode or emitting large fragments (A > 30). The histograms are
directly obtained by the simulation and the curves are fitting by
eq 15,16. For all the calculations the fitting works well, particularly the effect of delay time. The assumption of constant fission
width after the delay time, on the other hand, is not completely
supported because of poor statistics and still existing dynamical
effects. One should note that the fitting by eq 15,16 is just to extract the “fission” time of the super heavy composite. Especiallythe time scale of QMD results is obviously different from thatof fission process.
The extracted fission time Tfiss are plotted in Figure 4. With
effective interaction used in Reference 5 the calculated fission
time is about an order longer than this calculation. However inboth cases of CoMD calculations the longest lifetime of heavy
composite is found at Elab = 10 MeV/nucleon.
For lower incident energies (just above the Coulomb barrier)the system might not form a fully thermalized single composite
but might be quasi separated in the phase space, which makes
the system split easily. For higher energies, the fully thermalized
system needs some fluctuations to reseparate even though there

0.0

T fiss [fm/c]

4. Lifetime of Produced Composite Systems

1500

79

0.0
300

b=0 fmb=6 fm

0.4
0.2
10

20
E lab [MeV/u]

30

10

20
E lab [MeV/u]

30

200
100
0

Figure 5. The mass asymmetry of the fission fragments and the fragment mass. The dashed line indicates the initial mass of projectile andtarget nuclei. Error bars indicate statistical standard deviation.

80

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002of strong electric field (stronger than the case of Rutherford or
molecular trajectory) around Elab = 10 MeV/nucleon at some impact parameter and the production of positrons from the static
QED process should be largest around that energy.

Maruyama
as detection of e+ e− formation at around 10 MeV/nucleon andmeasurement of the energy averaged angular distribution and/or
excitation function for binary processes are encouraged.
References

5. Asymmetric Fission
As mentioned above, production of SHE is one of the mostimportant subject in the heavy-ion collision problem. Besides
cold- and hot-fusion, mass transfer in collision of very heavy
nuclei was tried before. One could produce, e.g. up to Fm(Z = 100) in U + U system, or Md (Z = 101) in U + Cm system,by such a mechanism.1, 2 The incident energy, however, was very
close to the Coulomb barrier and the reaction was rather gentle
with the transfer of ∼20 nucleons. In our CoMD calculation for
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and there happens the transfer of much more nucleons though
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process in CoMD calculation and the fission fragment mass A1
and A2 (< A1 ). The mass-asymmetry increases with the incident energy. For b = 0 the asymmetry amounts to about 0.1 at
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SHE production. The new 4π detectors can accumulate lots ofstatistics plus they can make coincidence studies to see if thefragments come from fission.
6. Summary
In summary, we have discussed the formation and decay ofsuper heavy composite in the Au + Au collisions. The CoMD
calculation which takes into account the Fermionic nature of thenucleon many-body system can describe well the low-energy
dynamics including fusion, fission, deep inelastic, emission of
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