Fusion process studied with preequilibrium giant dipole resonance in time-dependent
Hartree-Fock theory
C. Simenel1,2, Ph. Chomaz2 and G. de France2
1 DSM/DAPNIA/SPhN, CEA/SACLAY, F-91191 Gif-sur-Yvette Cedex, France and
2 Grand Accélérateur National d’Ions Lourds (GANIL), CEA/DSM-CNRS/IN2P3,
Bvd Henri Becquerel, BP 55027,F-14076 CAEN Cedex 5, France
(Dated: November 1, 2018)
The equilibration of macroscopic degrees of freedom during the fusion of heavy nuclei, like the
charge and the shape, are studied in the Time-Dependent Hartree-Fock theory. The preequilibrium
Giant Dipole Resonance (GDR) is used to probe the fusion path. It is shown that such isovector
collective state is excited in N/Z asymmetric fusion and to a less extent in mass asymmetric systems.
The characteristics of this GDR are governed by the structure of the fused system in its preequi-
librium phase, like its deformation, rotation and vibration. In particular, we show that a lowering
of the preequilibrium GDR energy is expected as compared to the statistical one. Revisiting ex-
perimental data, we extract an evidence of this lowering for the first time. We also quantify the
fusion-evaporation enhancement due to γ-ray emission from the preequilibrium GDR. This cooling
mechanism along the fusion path may be suitable to synthesize in the future super heavy elements
using radioactive beams with strong N/Z asymmetries in the entrance channel.
PACS numbers: 24.30.Cz, 21.60.Jz, 25.70.Gh, 25.70.Jj
I. INTRODUCTION
The fusion of two nuclei occurs at small impact pa-
rameters when the overlap between their wave functions
is big enough to allow the strong interaction to overcome
the Coulomb repulsion. Heavy-ion fusion reactions have
numerous applications, like the study of high spin states
in yrast and super-deformed bands [1] or the formation of
Heavy and Super Heavy Elements (SHE) [2]. Induced by
beams of unstable nuclei, this mechanism will also allow
to produce very exotic species and allow for the study of
isospin equilibration in the fused system.
The fusion process can be schematically divided in
three steps: (i) an approach phase during which each nu-
cleus feels only the Coulomb field of its partner and which
ends up when the nuclear interaction starts to dominate,
(ii) a rapid equilibration of the energy and the angular
momentum transfered from the relative motion to the
internal degrees of freedom, leading to the formation of
a Compound Nucleus (CN) and (iii) a statistical decay
of the CN. Lots of theoretical and experimental efforts
[3] are made to understand step (i). These studies focus
on an energy range located around the fusion barrier. At
these energies the fusion is controlled by quantum tunnel-
ing which is strongly influenced by the couplings between
the internal degrees of freedom and the relative motion of
the two colliding partners. Although the cooling mecha-
nisms involved in (iii) are well known and consist mainly
in light particle and γ-ray emission in competition with
fission for heavy systems, the initial conditions of the
statistical decay depend on the equilibration process (ii)
which is still subject to many debates nowadays. Indeed,
step (ii) is characterized by an equilibration of several
degrees of freedom like the shape [4] or the charge [5]
which can be accompanied by the emission of preequi-
librium particles. Such emission decreases the excitation
energy and the angular momentum. The latter quanti-
ties are crucial and must be determined precisely because
they have a major influence on the CN survival probabil-
ity and therefore on the synthesis of very exotic systems
such as the SHE.
In this paper we study the equilibration of the charges
in fused systems, its interplay with other macroscopic de-
grees of freedom like the shape and the rotation, and its
implications on the statistical decay. To probe theoret-
ically and experimentally this way to fusion, we use the
preequilibrium isovector Giant Dipole Resonance (GDR)
[6, 7, 8, 9, 10]. Giant Resonances are interpreted as the
first quantum of collective vibrations involving protons
and neutrons fluids. The Giant Monopole Resonance
can be described as a breathing mode, an alternation
of compression and dilatation of the whole nucleus. The
GDR corresponds to a collective oscillation of the pro-
tons against the neutrons. The Giant Quadrupole Res-
onance consists in a nuclear shape oscillation between
prolate and oblate deformations. Many other resonances
have been discovered [11, 12]. In particular Giant Res-
onances have been observed in hot nuclei formed by fu-
sion [13, 14]. This demonstrates the survival of ordered
vibrations in very excited systems, which are known to
be chaotic, even if some Giant Resonance characteristics
like the width are affected by the temperature [15, 16].
Moreover, the strong couplings between various collec-
tive modes which occur for Giant Resonances built on
the ground state [17, 18] are still present in fusion reac-
tions [10, 19]. It might therefore be possible to use the
Giant Resonances properties to probe the nuclear struc-
ture of the composite system on its way to fusion.
The choice of the preequilibrium GDR, that is, a GDR
excited in step (ii) before the formation of a fully equi-
librated CN, is motivated by the fact that its properties
strongly depend on the structure of the state on which
http://arxiv.org/abs/0704.0496v3
it is built, for instance the deformation [5]. The idea is
to form a CN with two N/Z asymmetric reactants. Such
a reaction may lead to the excitation of a dipole mode
because of the presence of a net dipole moment in the en-
trance channel. This dipole oscillation should occur be-
fore the charges are fully equilibrated, that is, during the
preequilibrium phase in which the system keeps a mem-
ory of the entrance channel [5, 6, 7, 8, 9, 10, 20, 21]. In
addition, for such N/Z asymmetric reactions, an enhance-
ment of the fast GDR γ-ray emission is expected as com-
pared to the ”slower” statistical γ-ray yield [7, 8, 9, 10].
This is of particular interest since the properties of these
GDR γ-rays characterize the dinuclear system which pre-
cedes the hot equilibrated CN. The first experimental in-
dications on the existence of such new phenomenon have
been reported in [22, 23, 24, 25, 26] for fusion reactions
and in [26, 27, 28, 29, 30, 31, 32, 33] in the case of deep
inelastic collisions.
The paper is organized as follows: In Sec. II we
study the properties of the preequilibrium GDR using the
Time-Dependent Hartree-Fock (TDHF) formalism. In
Sec. III we show how an N/Z asymmetric entrance chan-
nel may increase the fusion-evaporation cross-sections.
Finally, we conclude in section IV.
II. TDHF STUDY OF THE PREEQUILIBRIUM
GIANT DIPOLE RESONANCE
At the early time of the fusion reaction, the system
keeps the memory of the entrance channel. We call this
stage of the collision the preequilibrium phase which ends
when all the degrees of freedom are equilibrated in the
compound system and when the statistical decay starts.
One of these degrees of freedom is the isospin, which
measures the asymmetry between protons and neutrons.
When the two nuclei have different N/Z ratios, the pro-
ton and neutron centers of mass of the total system do
not coincide. As shown in [6, 21], there is a non zero force
between the two kind of nucleons which tends to restore
the initial isospin asymmetry. In such a case, an oscil-
lation of protons against neutrons on the way to fusion
might occur, that is, the so-called preequilibrium GDR
[5, 6, 7, 8, 9, 10, 20].
In fusion reactions the shape of the system changes
drastically during the preequilibrium phase. Studies of
the dynamics in the fusion reaction mechanism requires
sophisticated calculations to extract the preequilibrium
GDR characteristics (energy, width...) and in turn, on
the way to fusion. To achieve this goal, we choose to
use, as in the pioneer work of Bonche and Ngô on charge
equilibration [5], the TDHF approach because it is a fully
microscopic theory which takes into account the quantal
nature of the single particle dynamics. Moreover in the
present study we will restrict ourself to the observation
of one-body observables (e.g. the density ρ(r)) which
are supposed to be well described by such a mean field
approach. However it is clear that an important challenge
is to develop methods going beyond mean field which is
beyond the scope of this paper.
In this section we present quantum calculations
on preequilibrium giant collective vibrations using the
TDHF theory. We shall start with a brief description of
the TDHF theory in Sec. II A. Then we examine the role
of various relevant symmetries in the entrance channel,
namely the N/Z and mass symmetries (Sec. II B-IID).
Finally, in Sec. II E we shall compare our results with
the experimental data obtained by Flibotte et al. [22].
A. TDHF approach
In the TDHF approach [34, 35, 36, 37, 38, 39, 40], each
single particle wave function is propagated in the mean
field generated by the ensemble of particles. The mean
field approximation does not take into account the dissi-
pation due to two-body interactions [41, 42, 43, 44]. How-
ever TDHF takes care of one-body mechanisms such as
Landau spreading and evaporation damping [45]. Quan-
tum effects induced by the single particle dynamics like
shell effects or modification of the moment of inertia [46]
are accounted for properly.
The main advantage of TDHF is its fully microscopic
treatment of the N-body dynamics with the same effec-
tive interaction as the one used for the calculation of the
Hartree-Fock (HF) ground sates of the collision partners.
The consistency of the method for the structure of nuclei
and the nuclear reactions increases its prediction power
and its availability to study the interplay between exotic
structures and reaction mechanisms.
Moreover the TDHF equation is strongly non linear
which is of great importance for reactions around the bar-
rier because it includes couplings between relative motion
and internal degrees of freedom of the collision partners.
Also TDHF provides a good description of collective mo-
tion and can even exhibit couplings between collective
modes [17]. In fact the TDHF theory is optimized for
the prediction of expectation values of one-body observ-
ables and gives their exact evolution in the extreme case
where the residual interaction vanishes. However, the
TDHF prediction of multipole moments in nuclear colli-
sion, for instance, may differ from the correct evolution
because of the omission of the residual interaction. An
improvement of the description would be given by the
inclusion of the effect of the residual interaction on the
dynamics, which would increase considerably the compu-
tational time and is beyond the scope of this paper.
The TDHF theory describes the evolution of the
one-body density matrix ρ(t) of matrix elements
〈rsq|ρ̂|r′s′q′〉 =
ϕ∗i (r
′s′q′)ϕi(rsq), where ϕi(rsq) =
〈rsq|i〉 denotes the component with a spin s and isospin
q of the occupied single particle wave-function ϕi. This
evolution is determined by a non linear Liouville-von
Neumann equation,
ρ− [h(ρ), ρ] = 0 (1)
where h(ρ) is the matrix associated to the self consistent
mean-field Hamiltonian. We have used the code built by
P. Bonche and coworkers [47] with an effective Skyrme
interaction [48] and SLy4d parameters [47]. In its actual
version, TDHF does not account for pairing interactions.
B. N/Z asymmetric reactions
As far as the dipole motion in the preequilibrium phase
is concerned, it is obvious that the main relevant asym-
metry responsible for such a motion is a difference in the
charge-to-mass ratio between the collision partners [6].
The associated experimental signature is an enhancement
of the γ-ray emission in the GDR energy region of the
compound system [22, 23, 24, 25, 26] which is attributed
to a dipole oscillation. Several informations about the
fusion path can be extracted from such a dipole oscilla-
tion and its corresponding γ-ray spectrum. For numerical
tractability we start our study of the fusion process with
a light system: 12Be+28S→40Ca. We first deduce the
γ-ray spectrum from the dipole motion. Then we study
the effects of the deformation of the compound system,
and of the impact parameter on this motion.
1. The preequilibrium GDR γ-ray spectrum
We first consider a central collision at an energy of
1 MeV/nucleon in the center of mass. The expectation
value of the dipole moment Q̂D is defined by
QD = 〈Q̂D〉 =
(Xp −Xn) (2)
whereXp =
〈x̂p〉
andXn =
〈x̂n〉
are the positions
of the proton and neutron centers of mass respectively.
The expectation value of the conjugated dipole moment
P̂D is then associated to the relative velocity between
protons and neutrons, and is defined by the relation
PD = 〈P̂D〉 =
(Pp − Pn) (3)
where Pp =
〈p̂p〉 and Pn =
〈p̂n〉 are the to-
tal proton and neutron moments respectively. These
definitions ensure the canonical commutation relation
Q̂D, P̂D
= i~.
The time evolutions of QD and PD are plotted in
Figs. 1-c and 1-b respectively. The trajectories in both
the (QD,t) and (PD,t) planes exhibit oscillations which
we attribute to the preequilibrium GDR. We also note
that PD(t) oscillates in phase quadrature with QD(t) and
that those oscillations are damped due to the one-body
dissipation. Consequently, the plot of PD as a function
of QD shown in Fig. 1-a is a spiral. The GDR period
extracted from these plots is around 107 fm/c, which
corresponds to an energy of ∼ 11.6 MeV.
FIG. 1: Time evolution of the expectation value of the dipole
moment, QD, and its conjugated moment, PD, in the reaction
12Be+28S→40Ca at an energy of 1 MeV/nucleon in the center
of mass and at zero impact parameter.
During the collision and before the equilibrium is
reached, a fast rearrangement of charges occurs within
the composite system [5], generating the γ-ray emission.
We extract the preequilibrium GDR γ-ray spectrum from
the Fourier transform of the acceleration of the charges
[9, 49]
(Eγ) =
|I(Eγ)|2
where α is the fine structure constant and
I(Eγ) =
The spectrum obtained from Eq. 4 is plotted in Fig. 2
(solid line). In order to have a spectrum without spu-
rious peaks coming from the finite integration time,
we multiply the quantity d
by a gaussian function
[50]. In addition, this function plays a role
of a filter in the time domain. This filter prevents the sig-
nal to be affected by the interaction between the nucleus
and the emitted nucleons which have been reflected on
the box [51]. We choose τ = 320 fm/c in our calculations.
This ensures the fact that the spectra are free of spuri-
ous effects coming from the echo. However this procedure
adds a width Γ ∼ ~
∼ 0.6 MeV. This is a drawback if
one is concerned with detailed spectroscopy. However, in
this paper, we are only interested by the gross properties
of the preequilibrium GDR in order to study the fusion
mechanisms. As we can see in Fig. 2, the preequilibrium
GDR energy is E
GDR = 11.64 MeV, which corresponds
FIG. 2: preequilibrium GDR γ-ray spectrum calculated in
the reaction 12Be+28S → 40Ca (solid line) at an energy of 1
MeV/nucleon in the center of mass and γ-ray spectrum of a
GDR built on the ground state of 40Ca (dotted line).
to the previous value deduced from the GDR oscillation
period.
The energy of the preequilibrium GDR is much lower
than the one of the GDR built on the spherical ground
state of the 40Ca. This situation will be now explored
into more details.
2. Deformation effect
To better characterize the preequilibrium GDR, it is
necessary to compare it with the usual GDR built upon
the CN ground state [20]. This GDR is generated by
applying an isovector dipole boost with a velocity kD on
the 40Ca HF ground state |ψ(t)〉 = exp
−ikDQ̂D
|HF 〉
yielding an oscillation of QD(t) and PD(t) in phase
quadrature as we can see in Fig. 3. The period of
the oscillation is around 80 fm/c which is lower than
in the fusion case and corresponds to a higher energy
(EGDR = 15.5 MeV) as it is shown in the associated
GDR γ-ray spectrum in Fig. 2 (dotted line). The lower
energy obtained for the fusing system reveals a strong
prolate deformation [5, 9, 10, 20]. The two mechanisms
(fusion reaction and dipole boost) are expected to gen-
erate a GDR with quite different dynamical properties.
This can be seen in the density plot projected in the re-
action plane shown in Fig. 4, which shows that in the
case of a fusion reaction, the CN relaxes its initial pro-
late elongation along the collision axis with a time which
is larger than the typical dipole oscillation period of the
FIG. 3: GDR built upon the HF ground state in 40Ca and
excited by an isovector dipole boost: evolution of the expec-
tation value of the associated dipole moment, QD, and its
conjugated moment, PD, as a function of time.
FIG. 4: Density plots projected on the reaction plane for dif-
ferent times in the case of the fusion reaction. Lines represent
isodensities.
Deformation effects can be studied all along the fusion
path [4, 20]. The quadrupole deformation parameter ǫ
is defined by a scaling of the axis from a spherical to a
deformed shape along the x-axis
Rx = R0(1 + α)
Ryz = R0(1− ǫ) (5)
where α is defined by the conservation of the volume of
the nucleus RxRyRz = R
0, which leads to
(2− ǫ)ǫ
(1 − ǫ)2
. (6)
If one neglects high order terms in ǫ, we get the usual
value α ≃ 2ǫ.
The deformation parameter is related to the expecta-
tion values of the monopole and quadrupole moments Q̂0
FIG. 5: Time evolution of the deformation, ǫ, in 40Ca
formed in the 12Be+28S fusion reaction at an energy of 1
MeV/nucleon in the center of mass. The time axis origin is
chosen when the maximum of the fusion barrier is reached.
The average preequilibrium deformation ǫp obtained from the
GDR energy (see Eq. 13) is represented by a dashed line.
and Q̂2 which are expressed by
Q0 = 〈Q̂0〉 =
dr ρ(r)r2 (7)
Q2 = 〈Q̂2〉 =
dr ρ(r)r2
. (8)
We can write Q2 as a function of Q0
Q2 = −
Q0 + 3
dr ρ(r)x2. (9)
Eqs. 5 and 7 lead to
dr ρ(r)x2 = (1 + α)2
Q0. (10)
Using Eqs. 6, 9, 10 and ǫ < 1, we get
ǫ(t) = 1−
2Q2(t)√
5Q0(t)
which, at first order in ǫ, becomes
ǫ(t) =
Q2(t)
5Q0(t)
. (12)
In Ref. [10] we used Eq. 12 to characterize the average
deformation. In Fig. 5 we present the time evolution of
the deformation, ǫ(t), obtained from the more general
FIG. 6: a) Energy of the preequilibrium GDR obtained from
the first oscillation of the dipole moment and b) the defor-
mation parameter, ǫ, obtained from Eq. 13 (dashed line) and
from Eq. 11 (solid line), as a function of the center of mass
energy.
expression of ε given in Eq. 11. We consider a 40Ca
formed in the 12Be+28S fusion reaction at an energy of 1
MeV/nucleon in the center of mass. The important point
here is that the deformation does not relax and strongly
affects the frequency of the oscillations. A lower energy is
expected for the longitudinal collective motion E
GDR in
the fused system as compared to the one simulated in a
spherical 40Ca [5, 7, 8, 9, 10, 20]. Following a macroscopic
model for the dipole oscillation, we expect the energy of
the GDR to evolve with the deformation along the x-axis
(collision axis) as
= (1 − ǫp)2 (13)
where ǫp is the average deformation during the preequi-
librium stage. The frequency of the GDR along the defor-
mation axis fulfills this relation with ǫp ≃ 0.13 in excel-
lent agreement with the observed deformation in Fig. 5.
We have also investigated the effect of the center of
mass energy ECM on the preequilibrium GDR energy
and on the deformation parameter (see Fig. 6). The GDR
energy exhibits small variations (less than 1 MeV) with
the center of mass energy (Fig. 6-a). For ECM < 40
MeV, the increase of E
with ECM is attributed to
the formation of a dinuclear system with a slow neck
dynamics at low energy [9]. The presence of the neck is
in fact expected to slow down the charge equilibration
process, and then to increase the GDR period.
For ECM > 40 MeV, Fig. 6-a a decrease of E
when ECM increases. As illustrated in Fig. 6-b, this is
associated to a larger quadrupole deformation when the
collision is more violent. Consequently, the higher the
center of mass energy, the more prolately deformed the
CN. In Fig. 6-b, the deformation is estimated from Eq. 13
(dashed line) and from Eq. 11 (solid line) at the first max-
imum after one oscillation of ǫ(t) (e.g. at t ∼ 225 fm/c
in the case of ECM = 1 MeV/u as we can see in Fig. 5).
We also observe in this energy domain a good agreement
between the deformations calculated with both methods.
This lowering of the GDR energy due to deformation is
not specific to nuclear physics. Indeed, an energy split-
ting of the isovector dipole mode has been observed in
fissioning atomic clusters due to a strong prolate defor-
mation of the fission phase [52]. In such systems, the use
of LASERs with the ”pulse and probe” technique is ex-
pected to give access to the deformation and also to the
fission time [53].
3. non central collisions
To better mimic the situation of a fusion reaction, we
extended our calculations to non-zero impact parameters.
In fact, a non central collision may excite collective ro-
tational states in the deformed preequilibrated CN. This
rotation may be coupled to the preequilibrium GDR [20].
In particular, the interplay of dipole vibration and defor-
mation can be affected by the rotation. In addition to
the center of mass coordinates with x along the beam
axis and y perpendicular to the reaction plane, we de-
fine a new coordinate system x′, y′, z′, where x′ is the
deformation axis, and y = y′ is the rotation axis (see
Fig. 7). In the head-on collision example studied previ-
ously, those two frames are the same. For symmetry rea-
sons, the dipole oscillation cannot occur along the z = z′
and y = y′ axis.
For non-central collisions, the oscillation is only for-
bidden along the y = y′ axis [5, 20]. In this case the
amplitude of the oscillation along x′ slightly decreases
with the impact parameter. This decrease becomes sig-
nificant at rather large impact parameters as we can see
in Fig. 8 where we have plotted the amplitude of the first
oscillation of the dipole moment along x′ (solid line) as
a function of the impact parameter. This decrease is ac-
companied by an oscillation of the dipole moment along
the z′ axis with a smaller amplitude which increases with
the impact parameter b. Both amplitudes are of the same
order when b ∼ 5 fm.
FIG. 7: Description of the two frames used in non central
collisions.
FIG. 8: Amplitude of the first oscillation of the dipole moment
along x′ (solid line) and along z′ (dashed line) as a function
of the impact parameter, b, in the 12Be+28S fusion reaction
at an energy of 1 MeV/nucleon in the center of mass.
The oscillation along the z′ axis results from a weak
symmetry breaking due to the rotation of the system
[10]. In order to demonstrate this, let us start with the
time-dependent Schrödinger equation in the laboratory
frame R: i~|ψ̇〉 = Ĥ |ψ〉. In the rotating frame R′, the
expression of the wave function is |ψ′〉 = R̂(α)|ψ〉 where
R̂(α) = e−iα(t)Ĵy is a rotation matrix, Ĵy is the generator
of the rotations around y and α(t) is the angle between
the two frames (see Fig. 7). We express the Schrödinger
FIG. 9: Time evolution of the total dipole moment for the
8Be+32S→40Ca reaction at an energy of 1 MeV/nucleon in
the center of mass. At time t = 0 fm/c, the distance between
the centers of mass of the nuclei is 92.8 fm. The arrow indi-
cates the time when the fusion barrier is reached. The dashed
line gives the result of the adiabatic model (cf. Eq. 15).
equation as −~α̇ĴyR̂−1|ψ′〉+i~R̂−1|ψ̇′〉 = ĤR̂−1|ψ′〉 and
we get [10]
i~|ψ̇′〉 =
R̂ĤR̂−1 + ~α̇Ĵy
|ψ′〉. (14)
Eq. 14 is the Schrödinger equation expressed in the
rotating frame R′ of the CN and Ĥ ′ = R̂ĤR̂−1+~α̇Ĵy is
the Hamiltonian expressed in this frame. The last term
induces a motion along the z′ axis from a dipole vibration
along x′. It is quantified by the dipole moment along z′
which is plotted as a dashed line in Fig. 8. This is a
clear manifestation of couplings between rotational and
vibrational motions in nuclei.
In this subsection we have shown that an N/Z asymme-
try in the entrance channel generates a dipole oscillation
during the preequilibrium phase of a fusion reaction. In
the next one we will see that, due to polarization effects,
such a motion also occurs in N/Z symmetric systems al-
though with a smaller amplitude.
C. N/Z symmetric reactions
We now examine the situation of a central collision in-
volving two N = Z nuclei using the example of 8Be+32S
at ECM = 1 MeV/nucleon (
8Be is bound with a strong
prolate deformation in Hartree-Fock calculations with
the SLy4d force). As we can see in Fig. 9, the amplitude
FIG. 10: Schematic representation of the isovector polariza-
tion due to Coulomb repulsion between protons that occurs
before fusion. The protons are represented by a solid line and
the neutrons by a dotted line. Xi is the position of the center
of mass of the nucleus i.
of the dipole oscillations is significantly reduced as com-
pared to the N/Z asymmetric case (cf. Fig. 1-c). In this
latter system (12Be+28S), the dipole oscillations are gen-
erated by the N/Z asymmetry, whereas in the 8Be+32S
reaction, they are only due to the mass asymmetry of
the two collision partners. Indeed, a mass asymmetry
induces a difference in the isovector polarization in the
collision partners. This polarization is due to Coulomb
repulsion between protons of the colliding nuclei before
the fusion starts [5].
To show it, let us use an adiabatic approach in which
we consider that the polarization of a nucleus at a dis-
tance X = X2 −X1 between the centers of mass is gen-
erated by the Coulomb field of its collision partner. Xi
is the position of the center of mass of the nucleus i.
The distance between the proton and neutron centers of
mass in nucleus i is supposed to be small as compared
to X (see Fig. 10). The equality between the external
Coulomb field and the restoring force between protons
and neutrons leads to a dipole moment in the nucleus i
QDi(t) ≃ (−1)i
NiZiZje
AiEGDR
imX(t)
where i 6= j = 1 (for 32S) or 2 (for 8Be). The GDR
energy is calculated in each collision partner from the
dipole response frequency following a small amplitude
dipole boost. We get EGDR = 23.0 MeV for
32S and
EGDRx = 17.2 MeV for
8Be along its deformation axis
which is chosen to be aligned with the collision axis. The
dipole moment in the total system becomes
QD(t) =
N1Z2 −N2Z1
X(t) +QD1(t) +QD2(t). (15)
The first term of the right hand side of Eq. 15 is usually
dominant for a N/Z asymmetric reaction [8]. However, it
vanishes for a N/Z symmetric one. In this case, one is left
with the sum of the dipole moments of the partners. This
simple adiabatic model (dashed line in Fig. 9) gives the
good trend of the total dipole moment up to the vicinity
of the contact point.
After the fusion starts, the dipole moment increases
and oscillates in the preequilibrium system. The adia-
batic model is too simple to describe this phenomenon.
In fact, due to the polarization, the nuclear interaction
acts first on neutrons and then is expected to modify
strongly the dipole moment at the initial stage of the
fusion [5].
The consequence of this polarization in a mass asym-
metric system is a dipole oscillation which can be in-
terpreted, as previously, in term of an excitation of a
preequilibrium GDR. However, the GDR excitation is
very small as compared to the N/Z asymmetric case. Of
course, for a mass and N/Z symmetric reaction no pree-
quilibrium GDR are allowed for symmetry reason [10].
As we will see in the next section, the special case of
an N/Z asymmetric and mass symmetric system exhibits
some interesting behaviors as far as the collective motions
are concerned.
D. mass asymmetry and isoscalar vibrations
In this subsection we study the couplings between the
isovector dipole motion and isoscalar vibrations in the
preequilibrium phase and their dependence on the mass
asymmetry in the entrance channel. The dipole motion
can be coupled to isoscalar vibrations through the non
linearity of the TDHF equation [10, 17, 19]. The presence
of such isoscalar vibrations in the preequilibrium system
depends on the structure of the colliding partners and
on their mass asymmetry. For instance a mass symmet-
ric system has a stronger quadrupole deformation at the
touching point than a mass asymmetric one. In such a
system a quadrupole vibration might appear.
Let us start this study with the time evolution of the
instantaneous dipole period [10] which is very sensitive to
couplings with isoscalar vibrations. We define this period
as being twice the time to describe half a revolution in the
spiral diagram representing the evolution of the system
in the (PD, QD) space. The resulting evolution is plotted
in Fig. 11 for two N/Z asymmetric central collisions:
• the mass asymmetric 12Be+28S reaction at ECM =
1 MeV/nucleon.
• the mass symmetric 20O + 20Mg reaction at
ECM = 1.6 MeV/nucleon.
The center of mass energy has been chosen to obtain
the same ECM/VB ratio for both reactions (VB is the
Coulomb barrier).
The mean values of the GDR period obtained for the
two reactions are different. For the mass symmetric re-
action, this value is ≃ 170 fm/c, whereas in the mass
asymmetric case it is ≃ 105 fm/c, in good agreement
with the one obtained from Fig. 1 (107 fm/c). This dif-
ference is attributed to a larger deformation of the CN
FIG. 11: Time evolution of the GDR period for 20O+20Mg
at 1.6 MeV/nucleon (solid line) and for 12Be+28S at
1 MeV/nucleon (dashed line). Both energies are in the center
of mass.
in the mass symmetric case which, in average, is ǫ ∼ 0.2
(from Eq. 12), as compared to the mass asymmetric sys-
tem (ǫ ∼ 0.13). Note that it is not appropriate to use
Eq. 13, to calculate the deformation from the observed
GDR energy frequency for 20O+20Mg since it is valid
only for small deformations.
The dipole moment time evolution for those two re-
actions (Figs. 1 and 12), shows that unlike 20O+20Mg,
the oscillations in the 12Be+28S system are dominated
by a single energy. This is consistent with the evolution
of the GDR period in Fig. 11 which is rather constant
in the mass asymmetric case whereas it exhibits strong
oscillations in the mass symmetric one. This anharmonic-
ity can also be seen in the GDR γ-ray spectrum of the
20O+20Mg reaction plotted in Fig. 13. Indeed, one can
clearly identify two peaks in this spectrum at 7.7 MeV
and 10.8 MeV.
To better understand what is the origin of the dif-
ferences between the two systems, we have calculated
the evolutions of the monopole Q0 and quadrupole Q2
moments defined by Eqs. 7 and 8 respectively. Those
evolutions are plotted in Fig. 14-a for 20O+20Mg. We
first note that Q2 is always positive, that is, the com-
pound system keeps a prolate deformation. In addition,
Q0 and Q2 exhibit strong oscillations with the same pe-
riod ∼ 165 fm/c. Therefore, we conclude that they have
the same origin which is interpreted as a vibration of
the density around a prolate shape [10]. This mode is
only excited in the mass symmetric channel: the evolu-
tions of Q0(t) and Q2(t) for the mass asymmetric reac-
tion (12Be+28S) at 1 MeV/nucleon in the center of mass
FIG. 12: Evolution of the expectation value of the dipole mo-
ment, QD, and its conjugated moment, PD, in the reactions
20O+20Mg→40Ca at an energy of 1.6 MeV/nucleon in the
center of mass.
FIG. 13: GDR γ-ray spectrum calculated in the
20O+20Mg→40Ca reaction at an energy of 1.6 MeV/nucleon
in the center of mass.
(thick lines in Fig. 14-b) do not show any significant oscil-
lation of these moments. Evolutions of Q0(t) and Q2(t)
at 1.6 MeV/nucleon in the center of mass are also plot-
ted (thin lines in Fig. 14-b). They do not exhibit any
significant oscillation neither. Therefore, the vibrations
observed in Fig. 14-a are not attributed to a difference
in the collision energy but to the mass asymmetry in the
entrance channel.
The monopole and quadrupole oscillations modify the
properties of the dipole mode in a time dependent way
[8, 20]. Let us consider a harmonic oscillator for the
dipole motion with a time dependent rigidity constant.
This is a way to simulate the non linearities of TDHF.
Indeed, the observed oscillation of the density modifies
the restoring force between protons and neutrons. This is
due to the fact that the density enters in the mean field
potential of the TDHF equation (Eq. 1). This restor-
ing force is lower along the deformation axis of a pro-
lately deformed nucleus than in the perpendicular axis.
Thus, variations of the density profile in the TDHF equa-
tion can be modeled by a corresponding variation of the
rigidity constant k(t). In such a model, the evolution
of the dipole moment is given by the differential equa-
tion Q̈D(t) + (k(t)/µ)QD(t) = 0 where µ =
the reduced mass of the system. We note ω0 the av-
erage pulsation related to the rigidity constant given by
k(t)/µ = ω20(1 + η cosωt), where ω is the pulsation of
the density oscillation deduced from Fig. 14-a and η is a
dimensionless constant which quantifies the coupling be-
tween the GDR and the other collective mode associated
to the density vibration. We thus have
Q̈D(t) + ω
0 [1 + η cosωt]QD(t) = 0. (16)
This equation is the so called Mathieu’s equation [10].
It is interesting to show how we can get this equation
from a more microscopic equation like the TDHF one
(Eq. 1) in a one dimensional framework. Following the
way of ref. [54], the Wigner transform of Eq. 1 for a local
self consistent potential V is
V f (17)
where f(x, p, t) =
ds exp(−ip.s/~) ρ(x+ s
, x− s
, t) is
the Wigner transform of the density matrix ρ(x1, x2, t) =
〈x1|ρ̂(t)|x2〉. The upper indices on the derivative opera-
tors in Eq. 17 stand for the function on which the oper-
ator acts. We have of course f = fp + fn where fp and
fn are the Wigner transforms of the proton and neutron
density matrices respectively.
We now apply the Wigner Function Moment (WFM)
method to get a closed system of dynamical equations for
the dipole and its conjugated moments. We calculate the
integrals on the phase space of Eq. 17 with the weights
xτ on the one hand, and pτ on the other hand (τ=1 for
protons and −1 for neutrons). The distance D between
proton and neutron centers of mass can be written as
dx dp x (fp − fn) and we get
(fp − fn)
dx dp x sin
V (fp − fn)
where the time dependence has been omitted for simplic-
ity. The right hand side term is the integral of multiple
p-derivatives of f so it vanishes because fp, fn and all
their p−derivatives vanish for |p| → ∞. With P being
the relative momentum between protons and neutrons
dp dx p (fp − fn) we get
. (18)
We now calculate the integral of Eq. 17 with the
weight pτ . Noting the matter density n(x, t) =
dp f(x, p, t) and the kinetic energy density A(x, t) =
dp p2 f(x, p, t) we have
(Ap −An) = −
(np − nn) .
Using A = 0 for |x| → ∞ we have
Ṗ = −
(np − nn) . (19)
Eqs. 18 and 19 are the system of dynamical equations
of motion we were looking for. It is important to stress
that this system of equations is obtained without approx-
imation for a local potential. To go further, we need an
explicit form of the potential. If we consider for instance
a harmonic oscillator V = kx2/2, we obtain the dipole
moment evolution equation: mD̈ = −k D with the solu-
tion D = D0 cosω0 t, where ω0 =
If a breathing mode occurs at a pulsation ω, then the
density n(x, t) oscillates with the pulsation ω: n(x, t) =
n0(x) [1 + λ(x) cosωt]. Since the potential is self consis-
tent, it also presents oscillations which are a function of
cosωt: V (x, t) ≡ V (x, cosωt). We assume for this poten-
tial the separable form V (x, t) = V0(x) (1 + F [cosωt]),
where V0(x) is the potential when no breathing mode
is excited. Using a harmonic picture for V0, that is,
V0(x) =
2, we get from Eqs. 18 and 19 the equa-
tion for the dipole moment QD =
Q̈D(t) + ω
0 (1 + F [cosωt])QD(t) = 0. (20)
We finally see that the Mathieu’s equation (Eq. 16) ap-
pears to be an approximation of Eq. 20 where only the
linear part of the function F(ξ) ≃ ηξ is conserved.
We have solved the Mathieu’s equation numerically
with a set of parameters suitable for our problem. The
pulsation of the density oscillation is extracted from
Fig. 14-a and we get ω ≃ 7.5 MeV/~. For the pulsation
of the GDR we choose the main peak at ωGDR ≃ 7.7
MeV/~ (see Fig. 12). It is related to the pulsation ω0
by the relation ω0 = rωGDR. The constants r and η are
tuned to reproduce approximatively the TDHF results
period. The parameter r is expected to be close to 1 but
not exactly 1 because of the presence of the oscillating
term which may slightly change the mean value of the
dipole pulsation. The solution of the Mathieu’s equation
oscillates with a time-dependent period which reproduces
the TDHF case quite well with r ≃ 1.1 and η ≃ 0.5 (see
Fig. 15).
FIG. 14: Evolution with time of the monopole (Q0, solid line)
and quadrupole (Q2, dashed line) moments in the reactions
20O+20Mg→40Ca at 1.6 MeV/nucleon (a) and for 12Be+28S
at 1 MeV/nucleon (thick lines) and 1.6 MeV/nucleon (thin
lines) (b). Both energies are in the center of mass.
FIG. 15: Time evolution of the GDR period calculated for the
reaction 20O+20Mg→ 40Ca at an energy of 1.6 MeV/nucleon
in the center of mass (solid line) and its modelization by the
Mathieu’s equation (dashed line).
FIG. 16: Evolution of the expectation value of the dipole
moment, QD, and its conjugated moment, PD, in the case of
the N/Z asymmetric reaction 40Ca+100Mo at a center of mass
energy of 0.83 MeV/nucleon.
In a recent paper [19], following the formalism devel-
oped in a study of non linear vibrations [17], we related η
to a matrix element of the residual interaction coupling
collective states.
As a consequence, the excitation of collective modes
such as the quadrupole and monopole vibrations is cou-
pled to the preequilibrium GDR. Such vibrations occur
only in the mass symmetric reaction we studied. The ef-
fects of this coupling are a reduction of the GDR energy
(estimated around 10 per cent in this case) and an addi-
tional spreading of the resonance line shape due to the
modulation of the dipole frequency.
E. comparison with experiments
As a test case, we have performed TDHF calcula-
tions of the reactions studied by Flibotte et al. [22].
In this paper, two systems have been investigated: an
N/Z asymmetric one (40Ca+100Mo) and an N/Z quasi-
symmetric one (36S+104Pd) at a center of mass energy
of 0.83 MeV/nucleon. These systems have been cho-
sen because they lead to the same composite system
(140Sm). The corresponding dipole evolutions obtained
from TDHF are plotted in Fig. 16 for the N/Z asymmet-
ric reaction and in Fig. 17 for the N/Z quasi-symmetric
one. A dipole oscillation is observed in both reactions
but with a stronger amplitude in the N/Z asymmetric
The preequilibrium GDR γ-ray spectra for those reac-
tions are calculated using Eq. 4 and plotted in Fig. 18-a.
FIG. 17: Evolution of the expectation value of the dipole
moment, QD, and its conjugated moment, PD, in the case of
the N/Z quasi-symmetric reaction 36S+104Pd at a center of
mass energy of 0.83 MeV/nucleon.
The area under the peak associated to the N/Z asymmet-
ric reaction (solid line) is considerably larger than the one
under the N/Z quasi-symmetric one (dashed line).
To estimate the importance of the preequilibrium γ-
ray emission with respect to the statistical decay and its
role on the fusion process, we have calculated the spec-
trum associated to the first chance statistical γ-ray decay.
It is obtained from the γ-ray emission probability in all
directions per energy unit assuming an equilibrated CN
[9, 55, 56]. Its expression is
E4γ e
E2γ − E2GDR
where m is the nucleon mass, ΓGDR and EGDR are the
width and the energy of the statistical GDR respectively,
and T is the temperature of the equilibrated CN. At first
order, the energy of the GDR does not depend on the
temperature [14]. We use the values EGDR = 15 MeV
and ΓGDR = 7 MeV. Following the same method as the
one employed in Ref. [9], we approximate the CN width
ΓCN with the total neutron width
ΓCN ≃ Γn =
2mr20A
T 2e−
T (22)
where Bn = 8.5 MeV is the neutron binding energy and
r0 = 1.2 fm. The temperature T is calculated from the
equation
where a ≃ 1/10 MeV−1 is the level density parameter
and E∗ = 71 MeV is the excitation energy. The resulting
spectrum is plotted in Fig. 18-a (dotted line).
We note that the N/Z asymmetric preequilibrium spec-
trum is comparable in intensity to the first step statistical
one. This fact has already been pointed out by Baran et
al. [9] who got a similar spectrum for the N/Z asymmet-
ric reaction with a semiclassical approach.
Another important conclusion which can be drawn
from Fig. 18-a is the lowering of the GDR γ-ray energy
for the non statistical part as compared to the statistical
one which is attributed to the deformation of the nucleus
(see sec. II B 2). This phenomenon is also reported by
Baran et al. [9]. In fact we get from Fig. 18-a a position
of the peak of about 7.5 MeV for the preequilibrium GDR
while Baran et al. obtained ∼9 MeV. On the experimen-
tal side, the γ-ray spectra are dominated by a statistical
background decreasing exponentially. In addition to this
background, the GDR creates a bump located around
the GDR energy (Fig. 1 of Ref. [22]). To get rid of the
statistical background, the authors of [22] linearized the
γ-ray spectra by dividing them by a theoretical statistical
background. The resulting spectra are plotted in Fig. 2
of Ref. [22]. This procedure is used by the authors to
determine the preequilibrium to statistical ratio for the
GDR component. However it cannot be used to deter-
mine the positions in energy of the peaks because the
division by an exponential background induces a shift in
energy which is different for both contributions (statisti-
cal and preequilibrium) if they are not centered around
the same energy, as expected from Fig. 18-a.
We modified the procedure as follows. First, we assume
that no preequilibrium γ-ray is emitted in the N/Z quasi-
symmetric reaction 36S+104Pd. We then subtract the
total γ-ray spectrum associated to the quasi-symmetric
reaction from the N/Z asymmetric one. These two spec-
tra are plotted in Fig. 1 of Ref. [22]. The result of this
subtraction is the preequilibrium component of the GDR
in the reaction 40Ca+100Mo, and is plotted in Fig. 18-b.
The error bars are both statistical and systematic due to
the graphical extraction of the data. Below 5 MeV the
systematic error is to high to get relevant data. Focusing
on the energy position of the preequilibrium component,
we note a good agreement between TDHF predictions
and experimental data.
To conclude, we extracted from existing data, for the
first time, an experimental observation of the lowering
of the preequilibrium GDR predicted by our TDHF cal-
culations. This analysis shows that the preequilibrium
GDR is, indeed, a powerful experimental tool to study
the fusion path. Another application of N/Z asymmetric
fusion reactions is proposed in the next section.
FIG. 18: a) preequilibrium GDR γ-ray spectrum calcu-
lated in the reactions 40Ca+100Mo (solid line) and 36S+104Pd
(dashed line). The dotted line represents the first chance sta-
tistical γ-ray decay spectrum. b) Experimental data resulting
from the subtraction of the γ-ray spectra obtained by Flibotte
et al. [22] in the reactions 40Ca+100Mo and 36S+104Pd.
III. FUSION/EVAPORATION CROSS
SECTIONS OF HEAVY NUCLEI
As mentioned in [9], the emission of a preequilibrium
GDR γ-ray decreases the excitation energy hence the ini-
tial temperature of the nucleus reaching the statistical
phase. The emission of preequilibrium particles, which
can be controlled in our example by the N/Z asymmetry,
is thus a new interesting cooling mechanism for the for-
mation of Heavy and Super Heavy Elements. For such
nuclei, the statistical fission considerably dominates the
neutron emission and the survival probability of the CN
becomes very small.
SHE must be populated at low excitation energy.
Firstly, because the smaller the excitation energy, the
smaller the fission probability. Secondly, because the
shell corrections decrease with excitation energy [57].
These corrections are responsible for the stability of the
transfermiums nuclei (Z > 100) in their ground state.
The quantum stabilization decreases quite rapidly with
excitation energy until the fission barrier vanishes. Those
two reasons are strong motivations to study the cooling
mechanisms involved in the preequilibrium phase of the
CN formation.
In the following, we expose one cooling mechanism re-
sponsible for the predicted enhancement of the survival
probability in the case of a N/Z asymmetric reaction. As
an illustration, we treat only the γ-emission part of the
preequilibrium GDR decay. Although it may play an im-
portant role, we do not treat the preequilibrium neutron
emission for two reasons:
• Only the direct neutron decay of giant resonances
can be assessed in TDHF. Then, we would be able
to describe only a small part of this neutron emis-
sion, the other parts being the sequential and sta-
tistical decays. Missing the sequential decay would
be a strong limitation of the description.
• We would need not only the number of emitted neu-
trons, but also their energy. Consequently, huge
spatial grid would have to be used in order to per-
form a spatial Fourier transform of the single par-
ticle wave functions, which is out of range of three
dimensional TDHF codes because of computational
limitations.
Let us define PE∗
init.
(E∗) the survival probability at an
excitation energy E∗ of a CN which started its statistical
decay at the energy E∗init. We also note P
surv and P
the final survival probabilities of the CN formed by N/Z
symmetric and asymmetric reactions respectively.
Fig. 19-a illustrates schematically the evolution of the
survival probability (x-axis) when the excitation energy
decreases (y-axis) in a case of an N/Z symmetry in the
entrance channel. In this case, no γ-ray emission is ex-
pected in the preequilibrium phase and the initial ex-
citation energy is always maximum E∗init = E
0 , where
E∗0 = Q + Ecm is the excitation energy when no pree-
quilibrium particles are emitted, Ecm is the center of
mass energy and Q = (M1 +M2 −MCN )c2. During the
statistical decay, the excitation energy decreases mainly
through neutron emission, but at the same time the
survival probability of the compound nucleus decreases
too. For instance, when the excitation energy reaches
E∗1 = E
0−EGDR, the survival probability P1 = PE∗0 (E
at this energy might be small. At the end of the decay,
when the excitation energy is zero, the survival probabil-
ity becomes PSsurv = PE∗0 (0) = P1PE
Fig. 19-b shows the same for an N/Z asymmetric re-
action. In this last case, the nucleus can emit a pree-
quilibrium GDR γ-ray with a probability Pγ . The nu-
clei which emit such a γ-ray begin the statistical decay
at a lower energy Einit = E
1 , whereas those which did
not emit a γ-ray still starts their decay at Einit = E
The probability for the latter case is 1 − Pγ . The sur-
vival probability at the end of the decay then reads
PAsurv = [(1− Pγ)P1 + Pγ ]PE∗1 (0). The ratio of the sur-
FIG. 19: Schematic representation of the CN population dur-
ing the statistical decay in the case of an N/Z symmetric
collision (a) and an N/Z asymmetric reaction (b).
vival probabilities between the N/Z symmetric and asym-
metric cases is
PAsurv
PSsurv
= 1 +
(1− P1) . (24)
We now use a simple model to get an estimate of this
quantity. It is clear that, to get a quantitative predictions
of survival probabilities, the studied mechanism has to
be included in more elaborated statistical models, which
is beyond the scope of this paper. The probability Pγ
can be calculated by integrating Eq. 4 over the energy
range. This can be done for example with a TDHF cal-
culation or using the classical electrodynamic formulae
from Ref. [49]. Following these formulae, we approximate
the probability to emit a preequilibrium GDR γ-ray per
interval of energy by
2e2QD(0)
3π(~c)3
E21 +
(E − E1)2 + ΓGDR
(E + E1)2 +
where E1 =
− ΓGDR2
is the “shifted” energy of
the damped harmonic motion and ΓGDR is the damping
width of the preequilibrium GDR. The initial value of the
dipole moment, QD(0), can be estimated from Eq. 15 at
the touching point and neglecting the polarization of the
collision partners [8]. We get
QD(0) ≃
R1 +R2
(Z1N2 − Z2N1)
where Ri is the radius of nucleus i.
To determine PE∗0 (E
1 ), we need to solve a system of
six equations: Eqs. 22, 23 and
= −Γn(t)
(Bn + T (t)) (25)
= −Γf(t)
P (t) (26)
Γf (t) =
~ω0ωs
Bf (t)
T (t) (27)
Bf (t) ≡ Bf [E∗(t)] = Bf (0)e−
Ed (28)
Eq. 25 gives the evolution of the excitation energy, assum-
ing as in [9] that the CN width can be identified to the
neutron width. This implies that we neglect the statisti-
cal gamma emission. This choice is justified by the fact
that the statistical neutron emission is much more prob-
able than the gamma emission in the excitation energy
domain of interest where the fission dominates, which is
above the neutron emission threshold Bn. Eq. 26 gives
the evolution of the survival probability against fission
P . Eq. 27 gives the evolution of the fission width. The
parameters ω0 and ωS are the oscillator frequencies of the
two parabolas approximating the potential V (x) in the
first minimum and at the saddle point respectively. The
variable x is related to the distance between the mass
centers of the nascent fission fragments (see [58]) and
β = 5× 1021s−1 is the reduced friction. Eq. 28 gives the
evolution of the fission barrier Bf . For SHE, this barrier
has only a quantum nature and vanishes at high excita-
tion energy. Ed ≃ 20 MeV is the shell damping energy
[58]. We consider that a CN with an excitation energy
between E∗1 and E
0 decays only by fission or neutron
emission.
We take here the example of the reac-
tion 124Xe+141Xe→265Hs∗ at the fusion barrier
(Ecm = Bfus), that is, an excitation energy E
0 = 54
MeV. With an energy and a width of the GDR of 13
MeV and 4 MeV respectively, the preequilibrium γ-ray
emission probability is Pγ ≃ 0.05. For the statistical
decay we take Bf [E
∗ = 0] ≃ 8.5 MeV, Bn = 6.5 MeV
and ω0 ≃ ωS ≃ 1 MeV/~. We also get a survival
probability PE∗0 (E
1 ) ≃ 0.01 which is small as compared
to Pγ . Following Eq. 24, the enhancement of the total
survival probability due solely to the N/Z asymmetry in
the entrance channel becomes PAsurv/P
surv ∼ 6.
To conclude, we see that such an effect may be useful
for the formation of Heavy and Super Heavy Elements.
Indeed, based on our conclusions, very asymmetric N/Z
collisions induced by radioactive ion beams that are com-
ing online in several laboratories, should allow the syn-
thesis SHE with a larger cross sections than are obtain-
able with beams of stable isotopes.
IV. CONCLUSION
In this paper we have performed TDHF calculations to
study in some details the properties of the preequilibrium
GDR that can be excited before the formation of a fully
equilibrated CN. We have shown that this probe can be
used to better understand the early stage of the fusion
path, and more precisely the charge equilibration. We
have clarified the role of the N/Z and/or mass asymme-
tries on the GDR excitation. The energy of the preequi-
librium GDR is expected to decrease with excitation en-
ergy, an effect attributed to a strong prolate shape asso-
ciated to the fused system. We presented the first experi-
mental indication of this shift in energy. The calculations
for an N/Z asymmetric collisions at non zero impact pa-
rameters have been performed and revealed couplings be-
tween the dipole oscillations and the CN rotation. Other
couplings between vibrational modes for mass symmetric
reactions have also been studied.
Finally we suggest that the use of N/Z asymmetric fu-
sion reactions is a good choice to synthesize Heavy and
Super Heavy Elements. In that case, the preequilibrium
GDR γ-ray emission cooling mechanism might be well
suited to reach the statistical phase with a low excita-
tion energy yielding a larger survival probability against
fission. The availability of radioactive beams with large
N/Z asymmetry and sufficient intensities for these kind of
studies will be extremely useful to check experimentally
our predictions in the near future.
Acknowledgments
This paper is dedicated to the memory of P. Bonche,
the author of the TDHF code we used. We are grateful
to M. Di Toro for a useful reading of the manuscript. We
also thank V. Baran, M. Colonna, D. Lacroix, D. Boilley
and J. P. Wieleczko for several fruitful discussions, and
P. Schuck for providing a pertinent reference.
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  The equilibration of macroscopic degrees of freedom during the fusion of
heavy nuclei, like the charge and the shape, are studied in the Time-Dependent
Hartree-Fock theory. The pre-equilibrium Giant Dipole Resonance (GDR) is used
to probe the fusion path. It is shown that such isovector collective state is
excited in N/Z asymmetric fusion and to a less extent in mass asymmetric
systems. The characteristics of this GDR are governed by the structure of the
fused system in its preequilibrium phase, like its deformation, rotation and
vibration. In particular, we show that a lowering of the pre-equilibrium GDR
energy is expected as compared to the statistical one. Revisiting experimental
data, we extract an evidence of this lowering for the first time. We also
quantify the fusion-evaporation enhancement due to gamma-ray emission from the
pre-equilibrium GDR. This cooling mechanism along the fusion path may be
suitable to synthesize in the future super heavy elements using radioactive
beams with strong N/Z asymmetries in the entrance channel.

Fusion process studied with preequilibrium giant dipole resonance in time-dependent
Hartree-Fock theory
C. Simenel1,2, Ph. Chomaz2 and G. de France2
1 DSM/DAPNIA/SPhN, CEA/SACLAY, F-91191 Gif-sur-Yvette Cedex, France and
2 Grand Accélérateur National d’Ions Lourds (GANIL), CEA/DSM-CNRS/IN2P3,
Bvd Henri Becquerel, BP 55027,F-14076 CAEN Cedex 5, France
(Dated: November 1, 2018)
The equilibration of macroscopic degrees of freedom during the fusion of heavy nuclei, like the
charge and the shape, are studied in the Time-Dependent Hartree-Fock theory. The preequilibrium
Giant Dipole Resonance (GDR) is used to probe the fusion path. It is shown that such isovector
collective state is excited in N/Z asymmetric fusion and to a less extent in mass asymmetric systems.
The characteristics of this GDR are governed by the structure of the fused system in its preequi-
librium phase, like its deformation, rotation and vibration. In particular, we show that a lowering
of the preequilibrium GDR energy is expected as compared to the statistical one. Revisiting ex-
perimental data, we extract an evidence of this lowering for the first time. We also quantify the
fusion-evaporation enhancement due to γ-ray emission from the preequilibrium GDR. This cooling
mechanism along the fusion path may be suitable to synthesize in the future super heavy elements
using radioactive beams with strong N/Z asymmetries in the entrance channel.
PACS numbers: 24.30.Cz, 21.60.Jz, 25.70.Gh, 25.70.Jj
I. INTRODUCTION
The fusion of two nuclei occurs at small impact pa-
rameters when the overlap between their wave functions
is big enough to allow the strong interaction to overcome
the Coulomb repulsion. Heavy-ion fusion reactions have
numerous applications, like the study of high spin states
in yrast and super-deformed bands [1] or the formation of
Heavy and Super Heavy Elements (SHE) [2]. Induced by
beams of unstable nuclei, this mechanism will also allow
to produce very exotic species and allow for the study of
isospin equilibration in the fused system.
The fusion process can be schematically divided in
three steps: (i) an approach phase during which each nu-
cleus feels only the Coulomb field of its partner and which
ends up when the nuclear interaction starts to dominate,
(ii) a rapid equilibration of the energy and the angular
momentum transfered from the relative motion to the
internal degrees of freedom, leading to the formation of
a Compound Nucleus (CN) and (iii) a statistical decay
of the CN. Lots of theoretical and experimental efforts
[3] are made to understand step (i). These studies focus
on an energy range located around the fusion barrier. At
these energies the fusion is controlled by quantum tunnel-
ing which is strongly influenced by the couplings between
the internal degrees of freedom and the relative motion of
the two colliding partners. Although the cooling mecha-
nisms involved in (iii) are well known and consist mainly
in light particle and γ-ray emission in competition with
fission for heavy systems, the initial conditions of the
statistical decay depend on the equilibration process (ii)
which is still subject to many debates nowadays. Indeed,
step (ii) is characterized by an equilibration of several
degrees of freedom like the shape [4] or the charge [5]
which can be accompanied by the emission of preequi-
librium particles. Such emission decreases the excitation
energy and the angular momentum. The latter quanti-
ties are crucial and must be determined precisely because
they have a major influence on the CN survival probabil-
ity and therefore on the synthesis of very exotic systems
such as the SHE.
In this paper we study the equilibration of the charges
in fused systems, its interplay with other macroscopic de-
grees of freedom like the shape and the rotation, and its
implications on the statistical decay. To probe theoret-
ically and experimentally this way to fusion, we use the
preequilibrium isovector Giant Dipole Resonance (GDR)
[6, 7, 8, 9, 10]. Giant Resonances are interpreted as the
first quantum of collective vibrations involving protons
and neutrons fluids. The Giant Monopole Resonance
can be described as a breathing mode, an alternation
of compression and dilatation of the whole nucleus. The
GDR corresponds to a collective oscillation of the pro-
tons against the neutrons. The Giant Quadrupole Res-
onance consists in a nuclear shape oscillation between
prolate and oblate deformations. Many other resonances
have been discovered [11, 12]. In particular Giant Res-
onances have been observed in hot nuclei formed by fu-
sion [13, 14]. This demonstrates the survival of ordered
vibrations in very excited systems, which are known to
be chaotic, even if some Giant Resonance characteristics
like the width are affected by the temperature [15, 16].
Moreover, the strong couplings between various collec-
tive modes which occur for Giant Resonances built on
the ground state [17, 18] are still present in fusion reac-
tions [10, 19]. It might therefore be possible to use the
Giant Resonances properties to probe the nuclear struc-
ture of the composite system on its way to fusion.
The choice of the preequilibrium GDR, that is, a GDR
excited in step (ii) before the formation of a fully equi-
librated CN, is motivated by the fact that its properties
strongly depend on the structure of the state on which
http://arxiv.org/abs/0704.0496v3
it is built, for instance the deformation [5]. The idea is
to form a CN with two N/Z asymmetric reactants. Such
a reaction may lead to the excitation of a dipole mode
because of the presence of a net dipole moment in the en-
trance channel. This dipole oscillation should occur be-
fore the charges are fully equilibrated, that is, during the
preequilibrium phase in which the system keeps a mem-
ory of the entrance channel [5, 6, 7, 8, 9, 10, 20, 21]. In
addition, for such N/Z asymmetric reactions, an enhance-
ment of the fast GDR γ-ray emission is expected as com-
pared to the ”slower” statistical γ-ray yield [7, 8, 9, 10].
This is of particular interest since the properties of these
GDR γ-rays characterize the dinuclear system which pre-
cedes the hot equilibrated CN. The first experimental in-
dications on the existence of such new phenomenon have
been reported in [22, 23, 24, 25, 26] for fusion reactions
and in [26, 27, 28, 29, 30, 31, 32, 33] in the case of deep
inelastic collisions.
The paper is organized as follows: In Sec. II we
study the properties of the preequilibrium GDR using the
Time-Dependent Hartree-Fock (TDHF) formalism. In
Sec. III we show how an N/Z asymmetric entrance chan-
nel may increase the fusion-evaporation cross-sections.
Finally, we conclude in section IV.
II. TDHF STUDY OF THE PREEQUILIBRIUM
GIANT DIPOLE RESONANCE
At the early time of the fusion reaction, the system
keeps the memory of the entrance channel. We call this
stage of the collision the preequilibrium phase which ends
when all the degrees of freedom are equilibrated in the
compound system and when the statistical decay starts.
One of these degrees of freedom is the isospin, which
measures the asymmetry between protons and neutrons.
When the two nuclei have different N/Z ratios, the pro-
ton and neutron centers of mass of the total system do
not coincide. As shown in [6, 21], there is a non zero force
between the two kind of nucleons which tends to restore
the initial isospin asymmetry. In such a case, an oscil-
lation of protons against neutrons on the way to fusion
might occur, that is, the so-called preequilibrium GDR
[5, 6, 7, 8, 9, 10, 20].
In fusion reactions the shape of the system changes
drastically during the preequilibrium phase. Studies of
the dynamics in the fusion reaction mechanism requires
sophisticated calculations to extract the preequilibrium
GDR characteristics (energy, width...) and in turn, on
the way to fusion. To achieve this goal, we choose to
use, as in the pioneer work of Bonche and Ngô on charge
equilibration [5], the TDHF approach because it is a fully
microscopic theory which takes into account the quantal
nature of the single particle dynamics. Moreover in the
present study we will restrict ourself to the observation
of one-body observables (e.g. the density ρ(r)) which
are supposed to be well described by such a mean field
approach. However it is clear that an important challenge
is to develop methods going beyond mean field which is
beyond the scope of this paper.
In this section we present quantum calculations
on preequilibrium giant collective vibrations using the
TDHF theory. We shall start with a brief description of
the TDHF theory in Sec. II A. Then we examine the role
of various relevant symmetries in the entrance channel,
namely the N/Z and mass symmetries (Sec. II B-IID).
Finally, in Sec. II E we shall compare our results with
the experimental data obtained by Flibotte et al. [22].
A. TDHF approach
In the TDHF approach [34, 35, 36, 37, 38, 39, 40], each
single particle wave function is propagated in the mean
field generated by the ensemble of particles. The mean
field approximation does not take into account the dissi-
pation due to two-body interactions [41, 42, 43, 44]. How-
ever TDHF takes care of one-body mechanisms such as
Landau spreading and evaporation damping [45]. Quan-
tum effects induced by the single particle dynamics like
shell effects or modification of the moment of inertia [46]
are accounted for properly.
The main advantage of TDHF is its fully microscopic
treatment of the N-body dynamics with the same effec-
tive interaction as the one used for the calculation of the
Hartree-Fock (HF) ground sates of the collision partners.
The consistency of the method for the structure of nuclei
and the nuclear reactions increases its prediction power
and its availability to study the interplay between exotic
structures and reaction mechanisms.
Moreover the TDHF equation is strongly non linear
which is of great importance for reactions around the bar-
rier because it includes couplings between relative motion
and internal degrees of freedom of the collision partners.
Also TDHF provides a good description of collective mo-
tion and can even exhibit couplings between collective
modes [17]. In fact the TDHF theory is optimized for
the prediction of expectation values of one-body observ-
ables and gives their exact evolution in the extreme case
where the residual interaction vanishes. However, the
TDHF prediction of multipole moments in nuclear colli-
sion, for instance, may differ from the correct evolution
because of the omission of the residual interaction. An
improvement of the description would be given by the
inclusion of the effect of the residual interaction on the
dynamics, which would increase considerably the compu-
tational time and is beyond the scope of this paper.
The TDHF theory describes the evolution of the
one-body density matrix ρ(t) of matrix elements
〈rsq|ρ̂|r′s′q′〉 =
ϕ∗i (r
′s′q′)ϕi(rsq), where ϕi(rsq) =
〈rsq|i〉 denotes the component with a spin s and isospin
q of the occupied single particle wave-function ϕi. This
evolution is determined by a non linear Liouville-von
Neumann equation,
ρ− [h(ρ), ρ] = 0 (1)
where h(ρ) is the matrix associated to the self consistent
mean-field Hamiltonian. We have used the code built by
P. Bonche and coworkers [47] with an effective Skyrme
interaction [48] and SLy4d parameters [47]. In its actual
version, TDHF does not account for pairing interactions.
B. N/Z asymmetric reactions
As far as the dipole motion in the preequilibrium phase
is concerned, it is obvious that the main relevant asym-
metry responsible for such a motion is a difference in the
charge-to-mass ratio between the collision partners [6].
The associated experimental signature is an enhancement
of the γ-ray emission in the GDR energy region of the
compound system [22, 23, 24, 25, 26] which is attributed
to a dipole oscillation. Several informations about the
fusion path can be extracted from such a dipole oscilla-
tion and its corresponding γ-ray spectrum. For numerical
tractability we start our study of the fusion process with
a light system: 12Be+28S→40Ca. We first deduce the
γ-ray spectrum from the dipole motion. Then we study
the effects of the deformation of the compound system,
and of the impact parameter on this motion.
1. The preequilibrium GDR γ-ray spectrum
We first consider a central collision at an energy of
1 MeV/nucleon in the center of mass. The expectation
value of the dipole moment Q̂D is defined by
QD = 〈Q̂D〉 =
(Xp −Xn) (2)
whereXp =
〈x̂p〉
andXn =
〈x̂n〉
are the positions
of the proton and neutron centers of mass respectively.
The expectation value of the conjugated dipole moment
P̂D is then associated to the relative velocity between
protons and neutrons, and is defined by the relation
PD = 〈P̂D〉 =
(Pp − Pn) (3)
where Pp =
〈p̂p〉 and Pn =
〈p̂n〉 are the to-
tal proton and neutron moments respectively. These
definitions ensure the canonical commutation relation
Q̂D, P̂D
= i~.
The time evolutions of QD and PD are plotted in
Figs. 1-c and 1-b respectively. The trajectories in both
the (QD,t) and (PD,t) planes exhibit oscillations which
we attribute to the preequilibrium GDR. We also note
that PD(t) oscillates in phase quadrature with QD(t) and
that those oscillations are damped due to the one-body
dissipation. Consequently, the plot of PD as a function
of QD shown in Fig. 1-a is a spiral. The GDR period
extracted from these plots is around 107 fm/c, which
corresponds to an energy of ∼ 11.6 MeV.
FIG. 1: Time evolution of the expectation value of the dipole
moment, QD, and its conjugated moment, PD, in the reaction
12Be+28S→40Ca at an energy of 1 MeV/nucleon in the center
of mass and at zero impact parameter.
During the collision and before the equilibrium is
reached, a fast rearrangement of charges occurs within
the composite system [5], generating the γ-ray emission.
We extract the preequilibrium GDR γ-ray spectrum from
the Fourier transform of the acceleration of the charges
[9, 49]
(Eγ) =
|I(Eγ)|2
where α is the fine structure constant and
I(Eγ) =
The spectrum obtained from Eq. 4 is plotted in Fig. 2
(solid line). In order to have a spectrum without spu-
rious peaks coming from the finite integration time,
we multiply the quantity d
by a gaussian function
[50]. In addition, this function plays a role
of a filter in the time domain. This filter prevents the sig-
nal to be affected by the interaction between the nucleus
and the emitted nucleons which have been reflected on
the box [51]. We choose τ = 320 fm/c in our calculations.
This ensures the fact that the spectra are free of spuri-
ous effects coming from the echo. However this procedure
adds a width Γ ∼ ~
∼ 0.6 MeV. This is a drawback if
one is concerned with detailed spectroscopy. However, in
this paper, we are only interested by the gross properties
of the preequilibrium GDR in order to study the fusion
mechanisms. As we can see in Fig. 2, the preequilibrium
GDR energy is E
GDR = 11.64 MeV, which corresponds
FIG. 2: preequilibrium GDR γ-ray spectrum calculated in
the reaction 12Be+28S → 40Ca (solid line) at an energy of 1
MeV/nucleon in the center of mass and γ-ray spectrum of a
GDR built on the ground state of 40Ca (dotted line).
to the previous value deduced from the GDR oscillation
period.
The energy of the preequilibrium GDR is much lower
than the one of the GDR built on the spherical ground
state of the 40Ca. This situation will be now explored
into more details.
2. Deformation effect
To better characterize the preequilibrium GDR, it is
necessary to compare it with the usual GDR built upon
the CN ground state [20]. This GDR is generated by
applying an isovector dipole boost with a velocity kD on
the 40Ca HF ground state |ψ(t)〉 = exp
−ikDQ̂D
|HF 〉
yielding an oscillation of QD(t) and PD(t) in phase
quadrature as we can see in Fig. 3. The period of
the oscillation is around 80 fm/c which is lower than
in the fusion case and corresponds to a higher energy
(EGDR = 15.5 MeV) as it is shown in the associated
GDR γ-ray spectrum in Fig. 2 (dotted line). The lower
energy obtained for the fusing system reveals a strong
prolate deformation [5, 9, 10, 20]. The two mechanisms
(fusion reaction and dipole boost) are expected to gen-
erate a GDR with quite different dynamical properties.
This can be seen in the density plot projected in the re-
action plane shown in Fig. 4, which shows that in the
case of a fusion reaction, the CN relaxes its initial pro-
late elongation along the collision axis with a time which
is larger than the typical dipole oscillation period of the
FIG. 3: GDR built upon the HF ground state in 40Ca and
excited by an isovector dipole boost: evolution of the expec-
tation value of the associated dipole moment, QD, and its
conjugated moment, PD, as a function of time.
FIG. 4: Density plots projected on the reaction plane for dif-
ferent times in the case of the fusion reaction. Lines represent
isodensities.
Deformation effects can be studied all along the fusion
path [4, 20]. The quadrupole deformation parameter ǫ
is defined by a scaling of the axis from a spherical to a
deformed shape along the x-axis
Rx = R0(1 + α)
Ryz = R0(1− ǫ) (5)
where α is defined by the conservation of the volume of
the nucleus RxRyRz = R
0, which leads to
(2− ǫ)ǫ
(1 − ǫ)2
. (6)
If one neglects high order terms in ǫ, we get the usual
value α ≃ 2ǫ.
The deformation parameter is related to the expecta-
tion values of the monopole and quadrupole moments Q̂0
FIG. 5: Time evolution of the deformation, ǫ, in 40Ca
formed in the 12Be+28S fusion reaction at an energy of 1
MeV/nucleon in the center of mass. The time axis origin is
chosen when the maximum of the fusion barrier is reached.
The average preequilibrium deformation ǫp obtained from the
GDR energy (see Eq. 13) is represented by a dashed line.
and Q̂2 which are expressed by
Q0 = 〈Q̂0〉 =
dr ρ(r)r2 (7)
Q2 = 〈Q̂2〉 =
dr ρ(r)r2
. (8)
We can write Q2 as a function of Q0
Q2 = −
Q0 + 3
dr ρ(r)x2. (9)
Eqs. 5 and 7 lead to
dr ρ(r)x2 = (1 + α)2
Q0. (10)
Using Eqs. 6, 9, 10 and ǫ < 1, we get
ǫ(t) = 1−
2Q2(t)√
5Q0(t)
which, at first order in ǫ, becomes
ǫ(t) =
Q2(t)
5Q0(t)
. (12)
In Ref. [10] we used Eq. 12 to characterize the average
deformation. In Fig. 5 we present the time evolution of
the deformation, ǫ(t), obtained from the more general
FIG. 6: a) Energy of the preequilibrium GDR obtained from
the first oscillation of the dipole moment and b) the defor-
mation parameter, ǫ, obtained from Eq. 13 (dashed line) and
from Eq. 11 (solid line), as a function of the center of mass
energy.
expression of ε given in Eq. 11. We consider a 40Ca
formed in the 12Be+28S fusion reaction at an energy of 1
MeV/nucleon in the center of mass. The important point
here is that the deformation does not relax and strongly
affects the frequency of the oscillations. A lower energy is
expected for the longitudinal collective motion E
GDR in
the fused system as compared to the one simulated in a
spherical 40Ca [5, 7, 8, 9, 10, 20]. Following a macroscopic
model for the dipole oscillation, we expect the energy of
the GDR to evolve with the deformation along the x-axis
(collision axis) as
= (1 − ǫp)2 (13)
where ǫp is the average deformation during the preequi-
librium stage. The frequency of the GDR along the defor-
mation axis fulfills this relation with ǫp ≃ 0.13 in excel-
lent agreement with the observed deformation in Fig. 5.
We have also investigated the effect of the center of
mass energy ECM on the preequilibrium GDR energy
and on the deformation parameter (see Fig. 6). The GDR
energy exhibits small variations (less than 1 MeV) with
the center of mass energy (Fig. 6-a). For ECM < 40
MeV, the increase of E
with ECM is attributed to
the formation of a dinuclear system with a slow neck
dynamics at low energy [9]. The presence of the neck is
in fact expected to slow down the charge equilibration
process, and then to increase the GDR period.
For ECM > 40 MeV, Fig. 6-a a decrease of E
when ECM increases. As illustrated in Fig. 6-b, this is
associated to a larger quadrupole deformation when the
collision is more violent. Consequently, the higher the
center of mass energy, the more prolately deformed the
CN. In Fig. 6-b, the deformation is estimated from Eq. 13
(dashed line) and from Eq. 11 (solid line) at the first max-
imum after one oscillation of ǫ(t) (e.g. at t ∼ 225 fm/c
in the case of ECM = 1 MeV/u as we can see in Fig. 5).
We also observe in this energy domain a good agreement
between the deformations calculated with both methods.
This lowering of the GDR energy due to deformation is
not specific to nuclear physics. Indeed, an energy split-
ting of the isovector dipole mode has been observed in
fissioning atomic clusters due to a strong prolate defor-
mation of the fission phase [52]. In such systems, the use
of LASERs with the ”pulse and probe” technique is ex-
pected to give access to the deformation and also to the
fission time [53].
3. non central collisions
To better mimic the situation of a fusion reaction, we
extended our calculations to non-zero impact parameters.
In fact, a non central collision may excite collective ro-
tational states in the deformed preequilibrated CN. This
rotation may be coupled to the preequilibrium GDR [20].
In particular, the interplay of dipole vibration and defor-
mation can be affected by the rotation. In addition to
the center of mass coordinates with x along the beam
axis and y perpendicular to the reaction plane, we de-
fine a new coordinate system x′, y′, z′, where x′ is the
deformation axis, and y = y′ is the rotation axis (see
Fig. 7). In the head-on collision example studied previ-
ously, those two frames are the same. For symmetry rea-
sons, the dipole oscillation cannot occur along the z = z′
and y = y′ axis.
For non-central collisions, the oscillation is only for-
bidden along the y = y′ axis [5, 20]. In this case the
amplitude of the oscillation along x′ slightly decreases
with the impact parameter. This decrease becomes sig-
nificant at rather large impact parameters as we can see
in Fig. 8 where we have plotted the amplitude of the first
oscillation of the dipole moment along x′ (solid line) as
a function of the impact parameter. This decrease is ac-
companied by an oscillation of the dipole moment along
the z′ axis with a smaller amplitude which increases with
the impact parameter b. Both amplitudes are of the same
order when b ∼ 5 fm.
FIG. 7: Description of the two frames used in non central
collisions.
FIG. 8: Amplitude of the first oscillation of the dipole moment
along x′ (solid line) and along z′ (dashed line) as a function
of the impact parameter, b, in the 12Be+28S fusion reaction
at an energy of 1 MeV/nucleon in the center of mass.
The oscillation along the z′ axis results from a weak
symmetry breaking due to the rotation of the system
[10]. In order to demonstrate this, let us start with the
time-dependent Schrödinger equation in the laboratory
frame R: i~|ψ̇〉 = Ĥ |ψ〉. In the rotating frame R′, the
expression of the wave function is |ψ′〉 = R̂(α)|ψ〉 where
R̂(α) = e−iα(t)Ĵy is a rotation matrix, Ĵy is the generator
of the rotations around y and α(t) is the angle between
the two frames (see Fig. 7). We express the Schrödinger
FIG. 9: Time evolution of the total dipole moment for the
8Be+32S→40Ca reaction at an energy of 1 MeV/nucleon in
the center of mass. At time t = 0 fm/c, the distance between
the centers of mass of the nuclei is 92.8 fm. The arrow indi-
cates the time when the fusion barrier is reached. The dashed
line gives the result of the adiabatic model (cf. Eq. 15).
equation as −~α̇ĴyR̂−1|ψ′〉+i~R̂−1|ψ̇′〉 = ĤR̂−1|ψ′〉 and
we get [10]
i~|ψ̇′〉 =
R̂ĤR̂−1 + ~α̇Ĵy
|ψ′〉. (14)
Eq. 14 is the Schrödinger equation expressed in the
rotating frame R′ of the CN and Ĥ ′ = R̂ĤR̂−1+~α̇Ĵy is
the Hamiltonian expressed in this frame. The last term
induces a motion along the z′ axis from a dipole vibration
along x′. It is quantified by the dipole moment along z′
which is plotted as a dashed line in Fig. 8. This is a
clear manifestation of couplings between rotational and
vibrational motions in nuclei.
In this subsection we have shown that an N/Z asymme-
try in the entrance channel generates a dipole oscillation
during the preequilibrium phase of a fusion reaction. In
the next one we will see that, due to polarization effects,
such a motion also occurs in N/Z symmetric systems al-
though with a smaller amplitude.
C. N/Z symmetric reactions
We now examine the situation of a central collision in-
volving two N = Z nuclei using the example of 8Be+32S
at ECM = 1 MeV/nucleon (
8Be is bound with a strong
prolate deformation in Hartree-Fock calculations with
the SLy4d force). As we can see in Fig. 9, the amplitude
FIG. 10: Schematic representation of the isovector polariza-
tion due to Coulomb repulsion between protons that occurs
before fusion. The protons are represented by a solid line and
the neutrons by a dotted line. Xi is the position of the center
of mass of the nucleus i.
of the dipole oscillations is significantly reduced as com-
pared to the N/Z asymmetric case (cf. Fig. 1-c). In this
latter system (12Be+28S), the dipole oscillations are gen-
erated by the N/Z asymmetry, whereas in the 8Be+32S
reaction, they are only due to the mass asymmetry of
the two collision partners. Indeed, a mass asymmetry
induces a difference in the isovector polarization in the
collision partners. This polarization is due to Coulomb
repulsion between protons of the colliding nuclei before
the fusion starts [5].
To show it, let us use an adiabatic approach in which
we consider that the polarization of a nucleus at a dis-
tance X = X2 −X1 between the centers of mass is gen-
erated by the Coulomb field of its collision partner. Xi
is the position of the center of mass of the nucleus i.
The distance between the proton and neutron centers of
mass in nucleus i is supposed to be small as compared
to X (see Fig. 10). The equality between the external
Coulomb field and the restoring force between protons
and neutrons leads to a dipole moment in the nucleus i
QDi(t) ≃ (−1)i
NiZiZje
AiEGDR
imX(t)
where i 6= j = 1 (for 32S) or 2 (for 8Be). The GDR
energy is calculated in each collision partner from the
dipole response frequency following a small amplitude
dipole boost. We get EGDR = 23.0 MeV for
32S and
EGDRx = 17.2 MeV for
8Be along its deformation axis
which is chosen to be aligned with the collision axis. The
dipole moment in the total system becomes
QD(t) =
N1Z2 −N2Z1
X(t) +QD1(t) +QD2(t). (15)
The first term of the right hand side of Eq. 15 is usually
dominant for a N/Z asymmetric reaction [8]. However, it
vanishes for a N/Z symmetric one. In this case, one is left
with the sum of the dipole moments of the partners. This
simple adiabatic model (dashed line in Fig. 9) gives the
good trend of the total dipole moment up to the vicinity
of the contact point.
After the fusion starts, the dipole moment increases
and oscillates in the preequilibrium system. The adia-
batic model is too simple to describe this phenomenon.
In fact, due to the polarization, the nuclear interaction
acts first on neutrons and then is expected to modify
strongly the dipole moment at the initial stage of the
fusion [5].
The consequence of this polarization in a mass asym-
metric system is a dipole oscillation which can be in-
terpreted, as previously, in term of an excitation of a
preequilibrium GDR. However, the GDR excitation is
very small as compared to the N/Z asymmetric case. Of
course, for a mass and N/Z symmetric reaction no pree-
quilibrium GDR are allowed for symmetry reason [10].
As we will see in the next section, the special case of
an N/Z asymmetric and mass symmetric system exhibits
some interesting behaviors as far as the collective motions
are concerned.
D. mass asymmetry and isoscalar vibrations
In this subsection we study the couplings between the
isovector dipole motion and isoscalar vibrations in the
preequilibrium phase and their dependence on the mass
asymmetry in the entrance channel. The dipole motion
can be coupled to isoscalar vibrations through the non
linearity of the TDHF equation [10, 17, 19]. The presence
of such isoscalar vibrations in the preequilibrium system
depends on the structure of the colliding partners and
on their mass asymmetry. For instance a mass symmet-
ric system has a stronger quadrupole deformation at the
touching point than a mass asymmetric one. In such a
system a quadrupole vibration might appear.
Let us start this study with the time evolution of the
instantaneous dipole period [10] which is very sensitive to
couplings with isoscalar vibrations. We define this period
as being twice the time to describe half a revolution in the
spiral diagram representing the evolution of the system
in the (PD, QD) space. The resulting evolution is plotted
in Fig. 11 for two N/Z asymmetric central collisions:
• the mass asymmetric 12Be+28S reaction at ECM =
1 MeV/nucleon.
• the mass symmetric 20O + 20Mg reaction at
ECM = 1.6 MeV/nucleon.
The center of mass energy has been chosen to obtain
the same ECM/VB ratio for both reactions (VB is the
Coulomb barrier).
The mean values of the GDR period obtained for the
two reactions are different. For the mass symmetric re-
action, this value is ≃ 170 fm/c, whereas in the mass
asymmetric case it is ≃ 105 fm/c, in good agreement
with the one obtained from Fig. 1 (107 fm/c). This dif-
ference is attributed to a larger deformation of the CN
FIG. 11: Time evolution of the GDR period for 20O+20Mg
at 1.6 MeV/nucleon (solid line) and for 12Be+28S at
1 MeV/nucleon (dashed line). Both energies are in the center
of mass.
in the mass symmetric case which, in average, is ǫ ∼ 0.2
(from Eq. 12), as compared to the mass asymmetric sys-
tem (ǫ ∼ 0.13). Note that it is not appropriate to use
Eq. 13, to calculate the deformation from the observed
GDR energy frequency for 20O+20Mg since it is valid
only for small deformations.
The dipole moment time evolution for those two re-
actions (Figs. 1 and 12), shows that unlike 20O+20Mg,
the oscillations in the 12Be+28S system are dominated
by a single energy. This is consistent with the evolution
of the GDR period in Fig. 11 which is rather constant
in the mass asymmetric case whereas it exhibits strong
oscillations in the mass symmetric one. This anharmonic-
ity can also be seen in the GDR γ-ray spectrum of the
20O+20Mg reaction plotted in Fig. 13. Indeed, one can
clearly identify two peaks in this spectrum at 7.7 MeV
and 10.8 MeV.
To better understand what is the origin of the dif-
ferences between the two systems, we have calculated
the evolutions of the monopole Q0 and quadrupole Q2
moments defined by Eqs. 7 and 8 respectively. Those
evolutions are plotted in Fig. 14-a for 20O+20Mg. We
first note that Q2 is always positive, that is, the com-
pound system keeps a prolate deformation. In addition,
Q0 and Q2 exhibit strong oscillations with the same pe-
riod ∼ 165 fm/c. Therefore, we conclude that they have
the same origin which is interpreted as a vibration of
the density around a prolate shape [10]. This mode is
only excited in the mass symmetric channel: the evolu-
tions of Q0(t) and Q2(t) for the mass asymmetric reac-
tion (12Be+28S) at 1 MeV/nucleon in the center of mass
FIG. 12: Evolution of the expectation value of the dipole mo-
ment, QD, and its conjugated moment, PD, in the reactions
20O+20Mg→40Ca at an energy of 1.6 MeV/nucleon in the
center of mass.
FIG. 13: GDR γ-ray spectrum calculated in the
20O+20Mg→40Ca reaction at an energy of 1.6 MeV/nucleon
in the center of mass.
(thick lines in Fig. 14-b) do not show any significant oscil-
lation of these moments. Evolutions of Q0(t) and Q2(t)
at 1.6 MeV/nucleon in the center of mass are also plot-
ted (thin lines in Fig. 14-b). They do not exhibit any
significant oscillation neither. Therefore, the vibrations
observed in Fig. 14-a are not attributed to a difference
in the collision energy but to the mass asymmetry in the
entrance channel.
The monopole and quadrupole oscillations modify the
properties of the dipole mode in a time dependent way
[8, 20]. Let us consider a harmonic oscillator for the
dipole motion with a time dependent rigidity constant.
This is a way to simulate the non linearities of TDHF.
Indeed, the observed oscillation of the density modifies
the restoring force between protons and neutrons. This is
due to the fact that the density enters in the mean field
potential of the TDHF equation (Eq. 1). This restor-
ing force is lower along the deformation axis of a pro-
lately deformed nucleus than in the perpendicular axis.
Thus, variations of the density profile in the TDHF equa-
tion can be modeled by a corresponding variation of the
rigidity constant k(t). In such a model, the evolution
of the dipole moment is given by the differential equa-
tion Q̈D(t) + (k(t)/µ)QD(t) = 0 where µ =
the reduced mass of the system. We note ω0 the av-
erage pulsation related to the rigidity constant given by
k(t)/µ = ω20(1 + η cosωt), where ω is the pulsation of
the density oscillation deduced from Fig. 14-a and η is a
dimensionless constant which quantifies the coupling be-
tween the GDR and the other collective mode associated
to the density vibration. We thus have
Q̈D(t) + ω
0 [1 + η cosωt]QD(t) = 0. (16)
This equation is the so called Mathieu’s equation [10].
It is interesting to show how we can get this equation
from a more microscopic equation like the TDHF one
(Eq. 1) in a one dimensional framework. Following the
way of ref. [54], the Wigner transform of Eq. 1 for a local
self consistent potential V is
V f (17)
where f(x, p, t) =
ds exp(−ip.s/~) ρ(x+ s
, x− s
, t) is
the Wigner transform of the density matrix ρ(x1, x2, t) =
〈x1|ρ̂(t)|x2〉. The upper indices on the derivative opera-
tors in Eq. 17 stand for the function on which the oper-
ator acts. We have of course f = fp + fn where fp and
fn are the Wigner transforms of the proton and neutron
density matrices respectively.
We now apply the Wigner Function Moment (WFM)
method to get a closed system of dynamical equations for
the dipole and its conjugated moments. We calculate the
integrals on the phase space of Eq. 17 with the weights
xτ on the one hand, and pτ on the other hand (τ=1 for
protons and −1 for neutrons). The distance D between
proton and neutron centers of mass can be written as
dx dp x (fp − fn) and we get
(fp − fn)
dx dp x sin
V (fp − fn)
where the time dependence has been omitted for simplic-
ity. The right hand side term is the integral of multiple
p-derivatives of f so it vanishes because fp, fn and all
their p−derivatives vanish for |p| → ∞. With P being
the relative momentum between protons and neutrons
dp dx p (fp − fn) we get
. (18)
We now calculate the integral of Eq. 17 with the
weight pτ . Noting the matter density n(x, t) =
dp f(x, p, t) and the kinetic energy density A(x, t) =
dp p2 f(x, p, t) we have
(Ap −An) = −
(np − nn) .
Using A = 0 for |x| → ∞ we have
Ṗ = −
(np − nn) . (19)
Eqs. 18 and 19 are the system of dynamical equations
of motion we were looking for. It is important to stress
that this system of equations is obtained without approx-
imation for a local potential. To go further, we need an
explicit form of the potential. If we consider for instance
a harmonic oscillator V = kx2/2, we obtain the dipole
moment evolution equation: mD̈ = −k D with the solu-
tion D = D0 cosω0 t, where ω0 =
If a breathing mode occurs at a pulsation ω, then the
density n(x, t) oscillates with the pulsation ω: n(x, t) =
n0(x) [1 + λ(x) cosωt]. Since the potential is self consis-
tent, it also presents oscillations which are a function of
cosωt: V (x, t) ≡ V (x, cosωt). We assume for this poten-
tial the separable form V (x, t) = V0(x) (1 + F [cosωt]),
where V0(x) is the potential when no breathing mode
is excited. Using a harmonic picture for V0, that is,
V0(x) =
2, we get from Eqs. 18 and 19 the equa-
tion for the dipole moment QD =
Q̈D(t) + ω
0 (1 + F [cosωt])QD(t) = 0. (20)
We finally see that the Mathieu’s equation (Eq. 16) ap-
pears to be an approximation of Eq. 20 where only the
linear part of the function F(ξ) ≃ ηξ is conserved.
We have solved the Mathieu’s equation numerically
with a set of parameters suitable for our problem. The
pulsation of the density oscillation is extracted from
Fig. 14-a and we get ω ≃ 7.5 MeV/~. For the pulsation
of the GDR we choose the main peak at ωGDR ≃ 7.7
MeV/~ (see Fig. 12). It is related to the pulsation ω0
by the relation ω0 = rωGDR. The constants r and η are
tuned to reproduce approximatively the TDHF results
period. The parameter r is expected to be close to 1 but
not exactly 1 because of the presence of the oscillating
term which may slightly change the mean value of the
dipole pulsation. The solution of the Mathieu’s equation
oscillates with a time-dependent period which reproduces
the TDHF case quite well with r ≃ 1.1 and η ≃ 0.5 (see
Fig. 15).
FIG. 14: Evolution with time of the monopole (Q0, solid line)
and quadrupole (Q2, dashed line) moments in the reactions
20O+20Mg→40Ca at 1.6 MeV/nucleon (a) and for 12Be+28S
at 1 MeV/nucleon (thick lines) and 1.6 MeV/nucleon (thin
lines) (b). Both energies are in the center of mass.
FIG. 15: Time evolution of the GDR period calculated for the
reaction 20O+20Mg→ 40Ca at an energy of 1.6 MeV/nucleon
in the center of mass (solid line) and its modelization by the
Mathieu’s equation (dashed line).
FIG. 16: Evolution of the expectation value of the dipole
moment, QD, and its conjugated moment, PD, in the case of
the N/Z asymmetric reaction 40Ca+100Mo at a center of mass
energy of 0.83 MeV/nucleon.
In a recent paper [19], following the formalism devel-
oped in a study of non linear vibrations [17], we related η
to a matrix element of the residual interaction coupling
collective states.
As a consequence, the excitation of collective modes
such as the quadrupole and monopole vibrations is cou-
pled to the preequilibrium GDR. Such vibrations occur
only in the mass symmetric reaction we studied. The ef-
fects of this coupling are a reduction of the GDR energy
(estimated around 10 per cent in this case) and an addi-
tional spreading of the resonance line shape due to the
modulation of the dipole frequency.
E. comparison with experiments
As a test case, we have performed TDHF calcula-
tions of the reactions studied by Flibotte et al. [22].
In this paper, two systems have been investigated: an
N/Z asymmetric one (40Ca+100Mo) and an N/Z quasi-
symmetric one (36S+104Pd) at a center of mass energy
of 0.83 MeV/nucleon. These systems have been cho-
sen because they lead to the same composite system
(140Sm). The corresponding dipole evolutions obtained
from TDHF are plotted in Fig. 16 for the N/Z asymmet-
ric reaction and in Fig. 17 for the N/Z quasi-symmetric
one. A dipole oscillation is observed in both reactions
but with a stronger amplitude in the N/Z asymmetric
The preequilibrium GDR γ-ray spectra for those reac-
tions are calculated using Eq. 4 and plotted in Fig. 18-a.
FIG. 17: Evolution of the expectation value of the dipole
moment, QD, and its conjugated moment, PD, in the case of
the N/Z quasi-symmetric reaction 36S+104Pd at a center of
mass energy of 0.83 MeV/nucleon.
The area under the peak associated to the N/Z asymmet-
ric reaction (solid line) is considerably larger than the one
under the N/Z quasi-symmetric one (dashed line).
To estimate the importance of the preequilibrium γ-
ray emission with respect to the statistical decay and its
role on the fusion process, we have calculated the spec-
trum associated to the first chance statistical γ-ray decay.
It is obtained from the γ-ray emission probability in all
directions per energy unit assuming an equilibrated CN
[9, 55, 56]. Its expression is
E4γ e
E2γ − E2GDR
where m is the nucleon mass, ΓGDR and EGDR are the
width and the energy of the statistical GDR respectively,
and T is the temperature of the equilibrated CN. At first
order, the energy of the GDR does not depend on the
temperature [14]. We use the values EGDR = 15 MeV
and ΓGDR = 7 MeV. Following the same method as the
one employed in Ref. [9], we approximate the CN width
ΓCN with the total neutron width
ΓCN ≃ Γn =
2mr20A
T 2e−
T (22)
where Bn = 8.5 MeV is the neutron binding energy and
r0 = 1.2 fm. The temperature T is calculated from the
equation
where a ≃ 1/10 MeV−1 is the level density parameter
and E∗ = 71 MeV is the excitation energy. The resulting
spectrum is plotted in Fig. 18-a (dotted line).
We note that the N/Z asymmetric preequilibrium spec-
trum is comparable in intensity to the first step statistical
one. This fact has already been pointed out by Baran et
al. [9] who got a similar spectrum for the N/Z asymmet-
ric reaction with a semiclassical approach.
Another important conclusion which can be drawn
from Fig. 18-a is the lowering of the GDR γ-ray energy
for the non statistical part as compared to the statistical
one which is attributed to the deformation of the nucleus
(see sec. II B 2). This phenomenon is also reported by
Baran et al. [9]. In fact we get from Fig. 18-a a position
of the peak of about 7.5 MeV for the preequilibrium GDR
while Baran et al. obtained ∼9 MeV. On the experimen-
tal side, the γ-ray spectra are dominated by a statistical
background decreasing exponentially. In addition to this
background, the GDR creates a bump located around
the GDR energy (Fig. 1 of Ref. [22]). To get rid of the
statistical background, the authors of [22] linearized the
γ-ray spectra by dividing them by a theoretical statistical
background. The resulting spectra are plotted in Fig. 2
of Ref. [22]. This procedure is used by the authors to
determine the preequilibrium to statistical ratio for the
GDR component. However it cannot be used to deter-
mine the positions in energy of the peaks because the
division by an exponential background induces a shift in
energy which is different for both contributions (statisti-
cal and preequilibrium) if they are not centered around
the same energy, as expected from Fig. 18-a.
We modified the procedure as follows. First, we assume
that no preequilibrium γ-ray is emitted in the N/Z quasi-
symmetric reaction 36S+104Pd. We then subtract the
total γ-ray spectrum associated to the quasi-symmetric
reaction from the N/Z asymmetric one. These two spec-
tra are plotted in Fig. 1 of Ref. [22]. The result of this
subtraction is the preequilibrium component of the GDR
in the reaction 40Ca+100Mo, and is plotted in Fig. 18-b.
The error bars are both statistical and systematic due to
the graphical extraction of the data. Below 5 MeV the
systematic error is to high to get relevant data. Focusing
on the energy position of the preequilibrium component,
we note a good agreement between TDHF predictions
and experimental data.
To conclude, we extracted from existing data, for the
first time, an experimental observation of the lowering
of the preequilibrium GDR predicted by our TDHF cal-
culations. This analysis shows that the preequilibrium
GDR is, indeed, a powerful experimental tool to study
the fusion path. Another application of N/Z asymmetric
fusion reactions is proposed in the next section.
FIG. 18: a) preequilibrium GDR γ-ray spectrum calcu-
lated in the reactions 40Ca+100Mo (solid line) and 36S+104Pd
(dashed line). The dotted line represents the first chance sta-
tistical γ-ray decay spectrum. b) Experimental data resulting
from the subtraction of the γ-ray spectra obtained by Flibotte
et al. [22] in the reactions 40Ca+100Mo and 36S+104Pd.
III. FUSION/EVAPORATION CROSS
SECTIONS OF HEAVY NUCLEI
As mentioned in [9], the emission of a preequilibrium
GDR γ-ray decreases the excitation energy hence the ini-
tial temperature of the nucleus reaching the statistical
phase. The emission of preequilibrium particles, which
can be controlled in our example by the N/Z asymmetry,
is thus a new interesting cooling mechanism for the for-
mation of Heavy and Super Heavy Elements. For such
nuclei, the statistical fission considerably dominates the
neutron emission and the survival probability of the CN
becomes very small.
SHE must be populated at low excitation energy.
Firstly, because the smaller the excitation energy, the
smaller the fission probability. Secondly, because the
shell corrections decrease with excitation energy [57].
These corrections are responsible for the stability of the
transfermiums nuclei (Z > 100) in their ground state.
The quantum stabilization decreases quite rapidly with
excitation energy until the fission barrier vanishes. Those
two reasons are strong motivations to study the cooling
mechanisms involved in the preequilibrium phase of the
CN formation.
In the following, we expose one cooling mechanism re-
sponsible for the predicted enhancement of the survival
probability in the case of a N/Z asymmetric reaction. As
an illustration, we treat only the γ-emission part of the
preequilibrium GDR decay. Although it may play an im-
portant role, we do not treat the preequilibrium neutron
emission for two reasons:
• Only the direct neutron decay of giant resonances
can be assessed in TDHF. Then, we would be able
to describe only a small part of this neutron emis-
sion, the other parts being the sequential and sta-
tistical decays. Missing the sequential decay would
be a strong limitation of the description.
• We would need not only the number of emitted neu-
trons, but also their energy. Consequently, huge
spatial grid would have to be used in order to per-
form a spatial Fourier transform of the single par-
ticle wave functions, which is out of range of three
dimensional TDHF codes because of computational
limitations.
Let us define PE∗
init.
(E∗) the survival probability at an
excitation energy E∗ of a CN which started its statistical
decay at the energy E∗init. We also note P
surv and P
the final survival probabilities of the CN formed by N/Z
symmetric and asymmetric reactions respectively.
Fig. 19-a illustrates schematically the evolution of the
survival probability (x-axis) when the excitation energy
decreases (y-axis) in a case of an N/Z symmetry in the
entrance channel. In this case, no γ-ray emission is ex-
pected in the preequilibrium phase and the initial ex-
citation energy is always maximum E∗init = E
0 , where
E∗0 = Q + Ecm is the excitation energy when no pree-
quilibrium particles are emitted, Ecm is the center of
mass energy and Q = (M1 +M2 −MCN )c2. During the
statistical decay, the excitation energy decreases mainly
through neutron emission, but at the same time the
survival probability of the compound nucleus decreases
too. For instance, when the excitation energy reaches
E∗1 = E
0−EGDR, the survival probability P1 = PE∗0 (E
at this energy might be small. At the end of the decay,
when the excitation energy is zero, the survival probabil-
ity becomes PSsurv = PE∗0 (0) = P1PE
Fig. 19-b shows the same for an N/Z asymmetric re-
action. In this last case, the nucleus can emit a pree-
quilibrium GDR γ-ray with a probability Pγ . The nu-
clei which emit such a γ-ray begin the statistical decay
at a lower energy Einit = E
1 , whereas those which did
not emit a γ-ray still starts their decay at Einit = E
The probability for the latter case is 1 − Pγ . The sur-
vival probability at the end of the decay then reads
PAsurv = [(1− Pγ)P1 + Pγ ]PE∗1 (0). The ratio of the sur-
FIG. 19: Schematic representation of the CN population dur-
ing the statistical decay in the case of an N/Z symmetric
collision (a) and an N/Z asymmetric reaction (b).
vival probabilities between the N/Z symmetric and asym-
metric cases is
PAsurv
PSsurv
= 1 +
(1− P1) . (24)
We now use a simple model to get an estimate of this
quantity. It is clear that, to get a quantitative predictions
of survival probabilities, the studied mechanism has to
be included in more elaborated statistical models, which
is beyond the scope of this paper. The probability Pγ
can be calculated by integrating Eq. 4 over the energy
range. This can be done for example with a TDHF cal-
culation or using the classical electrodynamic formulae
from Ref. [49]. Following these formulae, we approximate
the probability to emit a preequilibrium GDR γ-ray per
interval of energy by
2e2QD(0)
3π(~c)3
E21 +
(E − E1)2 + ΓGDR
(E + E1)2 +
where E1 =
− ΓGDR2
is the “shifted” energy of
the damped harmonic motion and ΓGDR is the damping
width of the preequilibrium GDR. The initial value of the
dipole moment, QD(0), can be estimated from Eq. 15 at
the touching point and neglecting the polarization of the
collision partners [8]. We get
QD(0) ≃
R1 +R2
(Z1N2 − Z2N1)
where Ri is the radius of nucleus i.
To determine PE∗0 (E
1 ), we need to solve a system of
six equations: Eqs. 22, 23 and
= −Γn(t)
(Bn + T (t)) (25)
= −Γf(t)
P (t) (26)
Γf (t) =
~ω0ωs
Bf (t)
T (t) (27)
Bf (t) ≡ Bf [E∗(t)] = Bf (0)e−
Ed (28)
Eq. 25 gives the evolution of the excitation energy, assum-
ing as in [9] that the CN width can be identified to the
neutron width. This implies that we neglect the statisti-
cal gamma emission. This choice is justified by the fact
that the statistical neutron emission is much more prob-
able than the gamma emission in the excitation energy
domain of interest where the fission dominates, which is
above the neutron emission threshold Bn. Eq. 26 gives
the evolution of the survival probability against fission
P . Eq. 27 gives the evolution of the fission width. The
parameters ω0 and ωS are the oscillator frequencies of the
two parabolas approximating the potential V (x) in the
first minimum and at the saddle point respectively. The
variable x is related to the distance between the mass
centers of the nascent fission fragments (see [58]) and
β = 5× 1021s−1 is the reduced friction. Eq. 28 gives the
evolution of the fission barrier Bf . For SHE, this barrier
has only a quantum nature and vanishes at high excita-
tion energy. Ed ≃ 20 MeV is the shell damping energy
[58]. We consider that a CN with an excitation energy
between E∗1 and E
0 decays only by fission or neutron
emission.
We take here the example of the reac-
tion 124Xe+141Xe→265Hs∗ at the fusion barrier
(Ecm = Bfus), that is, an excitation energy E
0 = 54
MeV. With an energy and a width of the GDR of 13
MeV and 4 MeV respectively, the preequilibrium γ-ray
emission probability is Pγ ≃ 0.05. For the statistical
decay we take Bf [E
∗ = 0] ≃ 8.5 MeV, Bn = 6.5 MeV
and ω0 ≃ ωS ≃ 1 MeV/~. We also get a survival
probability PE∗0 (E
1 ) ≃ 0.01 which is small as compared
to Pγ . Following Eq. 24, the enhancement of the total
survival probability due solely to the N/Z asymmetry in
the entrance channel becomes PAsurv/P
surv ∼ 6.
To conclude, we see that such an effect may be useful
for the formation of Heavy and Super Heavy Elements.
Indeed, based on our conclusions, very asymmetric N/Z
collisions induced by radioactive ion beams that are com-
ing online in several laboratories, should allow the syn-
thesis SHE with a larger cross sections than are obtain-
able with beams of stable isotopes.
IV. CONCLUSION
In this paper we have performed TDHF calculations to
study in some details the properties of the preequilibrium
GDR that can be excited before the formation of a fully
equilibrated CN. We have shown that this probe can be
used to better understand the early stage of the fusion
path, and more precisely the charge equilibration. We
have clarified the role of the N/Z and/or mass asymme-
tries on the GDR excitation. The energy of the preequi-
librium GDR is expected to decrease with excitation en-
ergy, an effect attributed to a strong prolate shape asso-
ciated to the fused system. We presented the first experi-
mental indication of this shift in energy. The calculations
for an N/Z asymmetric collisions at non zero impact pa-
rameters have been performed and revealed couplings be-
tween the dipole oscillations and the CN rotation. Other
couplings between vibrational modes for mass symmetric
reactions have also been studied.
Finally we suggest that the use of N/Z asymmetric fu-
sion reactions is a good choice to synthesize Heavy and
Super Heavy Elements. In that case, the preequilibrium
GDR γ-ray emission cooling mechanism might be well
suited to reach the statistical phase with a low excita-
tion energy yielding a larger survival probability against
fission. The availability of radioactive beams with large
N/Z asymmetry and sufficient intensities for these kind of
studies will be extremely useful to check experimentally
our predictions in the near future.
Acknowledgments
This paper is dedicated to the memory of P. Bonche,
the author of the TDHF code we used. We are grateful
to M. Di Toro for a useful reading of the manuscript. We
also thank V. Baran, M. Colonna, D. Lacroix, D. Boilley
and J. P. Wieleczko for several fruitful discussions, and
P. Schuck for providing a pertinent reference.
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