\documentclass[a4paper]{jpconf}
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\newcommand{\kfi}{|\phi \>}
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\oeq
\oP_R(N)=\frac{1}{2\pi}\int_0^{2\pi} \stb \d \tet \stf e^{i\tet(\oN_R-N)},
\label{eq:projector}
\ceq
where
\oeq
\oN_R = \sum_{s} \sdf \int \stb \d \vr \stf \oad(\vr s) \sdf \oa(\vr s)
\sdf \Theta(x)
\label{eq:NG}
\ceq
counts the number of particles in the $x>0$ region ($\Theta(x)=1$ if $x>0$ and 0 elsewhere).
Isospin is omitted to simplify the notation.
The projector defined in Eq.~(\ref{eq:projector}) can be used to compute the probability to find $N$ nucleons in $x>0$ in the final state $\kfi$,
\oeq
\left|\oP_R(N)\kfi\right|^2=\<\phi|\oP_R(N) |\phi\>=\frac{1}{2\pi}\int_0^{2\pi} \stb \d \tet \stf e^{-i\tet{N}}\bfi\phi_R(\tet)\>,
\label{eq:proba}
\ceq
where $|\phi_R(\tet)\>=e^{{i\tet\oN_R}}\kfi$.
Note that $|\phi_R(\tet)\>$ is an independent particle state and, then,
the last term in Eq.~(\ref{eq:proba}) is the determinant of the matrix of the occupied single particle state overlaps~\cite{sim10b}:
\oeq
\bfi\phi_R(\tet)\>=\det (F)
\ceq
with
\oeq
F_{ij}= \sum_{s} \int \stb\d \vr \sdf{\az_i^s}^*(\vr) {\az_j^{s}}(\vr) e^{i\tet\Theta(x)}.
\ceq
The integral in Eq.~(\ref{eq:proba}) is discretized using $\tet_n=2\pi{n}/M$ with the integer $n=1\cdots{M}$.
Choosing $M=300$ ensures numerical convergence for the $^{16}$O+$^{208}$Pb system.
Fig.~\ref{fig:proba} shows the resulting transfer probabilities at $E_{c.m.}=74.44$~MeV (left) and at $E_{c.m.}=65$~MeV (right).
In agreement with the results presented in section~\ref{sec:fusion}, the most probable channel is $Z=6$ and $N=8$ at the barrier. However, lowering the energy reduces the transfer probabilities and the main channel is $Z=N=8$ well below the barrier, corresponding to inelastic and elastic channels.
To compare with experimental data on the $^{16}$O+$^{208}$Pb reaction, we plot in Fig.~\ref{fig:proba_TDHF+exp} the transfer probabilities from TDHF as a function of the distance of closest approach $R_{min}$ between the collision partners~\cite{cor09}, assuming a Rutherford trajectory~\cite{bro91}:
\oeq
R_{min}={Z_1Z_2e^2}[1+\mbox{cosec}(\theta_{c.m.}/2)]/{2E_{c.m.}}
\label{eq:R_min}
\ceq
where $\theta_{c.m.}$ is the center of mass scattering angle.
Recent data from Ref.~\cite{eve11} are shown in Fig.~\ref{fig:proba_TDHF+exp} for sub-barrier one- and two-proton transfer channels.
We see that TDHF overestimates the one-proton transfer probabilities and underestimates the strength of the two-proton transfer channel.
This discrepancy is interpreted as an effect of pairing interactions~\cite{sim10b,eve11}.
Indeed, paired nucleons are expected to be transferred as a pair, increasing (reps. decreasing) the two-nucleon (single-nucleon) transfer probability. Note, however, that the sum of one- and two-proton transfer probabilities is well reproduced by TDHF~\cite{sim10b}.
For $R_{min}>13$~fm, however, the TDHF calculations reproduces reasonably well the sum of one and two-proton transfer channels. For $R_{min}<13$~fm, sub-barrier fusion, not included in the TDHF calculations, reduces transfer probabilities~\cite{sim10b,eve11}.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{proba_TDHF+exp.eps}
\caption{Proton number probability
as function of the distance of closest approach in the small outgoing fragment of the $^{16}$O+$^{208}$Pb reaction. TDHF results are shown with lines. Experimental data (open symbols) are taken from Ref.~\cite{eve11}.}
\label{fig:proba_TDHF+exp}
\end{center}
\end{figure}
These studies emphasize the role of pairing interactions in heavy-ion collisions.
The recent inclusion of pairing interactions in 3-dimensional microscopic codes~\cite{ass09,eba10,ste11} gives hope in our ability to describe such data more realistically in a near future.
\section{Fusion and quasi-fission in heavy systems\label{sec:QF}}
To a reasonably good approximation, the fusion barrier for light and intermediate mass systems is determined by the frozen barrier.
This approximation fails, however, for heavy systems with typical charge products $Z_1 Z_2\geq 1600$ which are known to exhibit fusion hindrance~\cite{gag84}.
Above this threshold, an extra-push energy is usually needed for the system to fuse~\cite{swi82}.
In fact, at the energy of the frozen barrier, heavy systems are more likely to encounter quasi-fission, i.e., a re-separation in two fragments after a possible mass exchange.
We now investigate the reaction mechanism in heavy systems with possible fusion hindrance.
We first illustrate the fusion hindrance with TDHF calculations of fusion thresholds.
Then, we investigate the quasi-fission process.
\subsection{TDHF calculations of fusion hindrance}
Let us first consider the $^{90}$Zr+$^{124}$Sn system which has a charge product $Z_1Z_2=2000$, and, then,
is expected to exhibit a fusion hindrance.
Indeed, the proximity model~\cite{blo77} predicts a barrier for this system $V^{prox.}\simeq215$~MeV, while TDHF calculations predicts
that the system encounters a fast re-separation at this energy, as shown in Fig.~\ref{fig:ZrSn}~\cite{ave09}.
In the same figure, we observe a long contact time at $E_{c.m.}=240$~MeV, which is interpreted as a capture trajectory leading to fusion.
The additional energy needed to fuse is then $\sim15$~MeV, in excellent agreement with the extra-push model ~\cite{swi82} which predicts $E^{X-push}\simeq14.8$~MeV.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Sn_Zr.eps}
\caption{Distance between the centers of mass of the fragments as a function of time in head-on
$^{90}$Zr+$^{124}$Sn collisions for different center of mass energies. }
\label{fig:ZrSn}
\end{center}
\end{figure}
TDHF calculations of fusion hindrance have been performed for several systems, as shown in Fig.~\ref{fig:xpush}, where the TDHF fusion thresholds are compared with the interaction barriers predicted by the proximity model~\cite{blo77} and with the extra-push model~\cite{swi82}.
We observe an increase of the fusion threshold with TDHF as compared to the proximity model which assumes frozen reactants.
This indicates that dynamical effects are playing an important role in the reaction by hindering fusion.
The order of magnitude of the additional energy needed to fuse is similar to the one predicted with the extra-push model.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{heavy.eps}
\caption{TDHF fusion thresholds for several heavy systems are compared with the proximity barrier~\cite{blo77} and with results from the extra-push model~\cite{swi82}.
}
\label{fig:xpush}
\end{center}
\end{figure}
\subsection{The quasi-fission process\label{sec:QFTDHF}}
The quasi-fission mechanism is a fast re-separation of the fragments, with usually a partial equilibration of their mass. Typical quasi-fission times are of the order of few zepto-seconds~\cite{tok85,rie11,sim12b}.
Quasi-fission becomes dominant in heavy systems and is mostly responsible for the fusion hindrance discussed in the previous section.
As an example, Fig.~\ref{fig:ZrSn_dens} shows the density profiles for the $^{90}$Zr+$^{124}$Sn head-on collision at $E_{c.m.}=235$~MeV.
We observe that the two fragments are in contact during $\sim5$~zs.
A dinuclear system is then formed before re-separation in two fission-like fragments.
\begin{figure}
\begin{center}
\includegraphics[width=15cm]{Sn_Zr_dens.eps}
\caption{Density profile in the $^{90}$Zr+$^{124}$Sn head-on collision at $E_{c.m.}=235$~MeV. From Ref.~\cite{ave09}.}
\label{fig:ZrSn_dens}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{densL0.eps}
\caption{Snapshots of the TDHF isodensity at half the saturation density in the $^{40}$Ca+$^{238}$U system for different initial orientations and $E_{c.m.}$.}
\label{fig:densQF}
\end{center}
\end{figure}
Extensive calculations are ongoing on the $^{40}$Ca+$^{238}$U system to compare with recent measurements performed at the Australian National University~\cite{wak12b}.
Examples of density evolutions are shown in Fig.~\ref{fig:densQF} for two different initial conditions.
In both cases, a quasi-fission process is obtained.
In particular, we observe an important multi-nucleon transfer from the heavy fragment toward the light one.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{mass.eps}
\hspace{0.5cm}
\includegraphics[width=10cm]{time.eps}
\caption{TDHF calculations of the mass of the fragments (top) and of the quasi-fission time (bottom) in $^{40}$Ca+$^{238}$U central collisions as a function of the center of mass energy (divided by the proximity barrier~\cite{blo77}). For quasi-fission times larger than 23~zs, only a lower limit is given. Two different orientations of the $^{238}$U are considered (see inset). }
\label{fig:QFtime}
\end{center}
\end{figure}
Fig.~\ref{fig:QFtime} presents final fragment masses (top) and quasi-fission times (bottom) for two different orientations of the $^{238}$U.
Comparing these two figures, we observe that the mass equilibration (i.e., the formation of two fragments with symmetric masses) is not complete and varies with the life-time of the dinuclear system, i.e., the longer the contact, the larger the mass transfer.
We also see that all the calculations with the $^{238}$U deformation axis aligned with the collision axis lead to a quasi-fission with partial mass equilibration and quasi-fission times smaller than 10~zs.
Shell effects may affect the final outcome of the reaction by favouring the production of fragments in the $^{208}$Pb region.
In particular, this orientation never leads to fusion, while the other orientation produces long contact times above the barrier which may be associated to fusion.
Calculations of non-central $^{40}$Ca+$^{238}$U collisions are ongoing in order to compare with experimental data.
\section{Actinide collisions \label{sec:actinides}}
Collisions of actinides form, during few zs, the heaviest nuclear systems available on Earth.
In one hand, such systems are interesting to study the stability of the QED vacuum under strong electric fields~\cite{rei81,ack08,gol09}.
In the other hand, they might be used to form neutron-rich heavy and super-heavy elements via multi-nucleon transfer reactions~\cite{zag06,zag12,ked10}.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{density.eps}
\caption{Snapshots of the isodensity at half the saturation density in $^{238}$U+$^{238}$U central collisions at $E_{c.m.}=900$~MeV from TDHF calculations. Snapshots are given at times $t=0$, 1.5, 2.7, and $4.2$~zs from top to bottom. From Ref.~\cite{gol09}.}
\label{fig:density}
\end{center}
\end{figure}
Actinide collisions have been studied with the TDHF approach~\cite{cus80,gol09,ked10,sim11b}.
Fig.~\ref{fig:density} shows density evolutions in $^{238}$U+$^{238}$U central collisions at $E_{c.m.}=900$~MeV for different initial conditions.
We see that the initial orientation of the nuclei plays a crucial role on the reaction mechanism~\cite{gol09}, with the production of a third fragment in the tip-tip collision (left), or net mass transfer in the tip-side configuration (middle).
In the latter case, $\sim6$~protons and $\sim11$~neutrons, in average, are transferred from the right to the left nucleus, corresponding to the formation of a $^{255}$Cf primary fragment.
Similar calculations have been performed on the $^{232}$Th+$^{250}$Cf system~\cite{ked10}.
An example of density evolution is shown in the right panel of Fig.~\ref{fig:dist}.
In this case, we observe a net transfer of nucleons from the tip of the $^{232}$Th to the side of the $^{250}$Cf, corresponding to an inverse quasi-fission process, i.e., the exit channel is more mass-asymmetric than the entrance channel. Indeed, in this case, a $^{265}$Lr fragment is formed in the exit channel.
It is worth mentioning that these calculations predict inverse quasi-fission for this specific orientation only, i.e., when the tip of the lighter actinide is in contact with the side of the heavier one. Indeed, the other orientations induce ''standard'' quasi-fission~\cite{ked10}.
Note also that another inverse quasi-fission mechanism is predicted in actinide collisions due to shell effects in the $^{208}$Pb region~\cite{vol78,zag06,zag12}.
In the previous example, the $^{265}$Lr heavy fragment indicates the average $N$ and $Z$ of a distribution.
The fluctuations and correlations of these distributions have been computed with the Balian-V\'en\'eroni prescription~\cite{bal84} using the \textsc{tdhf3d} code~\cite{sim11,sim11b}.
Fig.~\ref{fig:dist}(left) shows the resulting probabilities assuming Gaussian distributions of the form
\oeq
P(z,n)=\(2\pi\sigma_N\sigma_Z\sqrt{1-\rho^2}\)^{-1} \exp\[ -\frac{1}{1-\rho^2} \( \frac{n^2}{\sigma_N^2} + \frac{z^2}{\sigma_Z^2} -\frac{2\rho nz}{\sigma_N\sigma_Z} \) \],
\label{eq:Gauss}
\ceq
where $\sigma_{N,Z}$ are the standard deviations and $0\le\rho<1$ quantifies the correlations between the $N$ and $Z$ distributions~\cite{sim12b}.
We see that many new $\beta-$stable and neutron-rich heavy nuclei could be produced if their excitation energy is low enough to allow their survival against fission.
This inverse quasi-fission process needs further investigations,
in particular to predict realistic production cross-sections.
\begin{figure*}
\begin{center}
\includegraphics[width=15cm]{dist.eps}
\caption{(right) Snapshots of the isodensity at half the saturation density in $^{232}$Th+$^{250}$Cf central collisions at $E_{c.m.}=916$~MeV. (left) Gaussian distributions of $N$ and $Z$ with widths and correlations computed with the BV prescription (linear color scale). The solid line represents the predicted $\beta-$stability line.}
\label{fig:dist}
\end{center}
\end{figure*}
\section{Conclusions and perspectives}
The TDHF theory has been applied to the study of heavy ion collisions at energies around the Coulomb barrier.
Its time-dependent nature allows to investigate dynamical effects responsible for the modification of the fusion thresholds.
In particular, the coupling between the relative motion and proton transfer in $^{16}$O+$^{208}$ decreases the barrier by reducing the Coulomb repulsion.
Particle transfer probabilities are predicted using a particle-number projection technique.
A comparison with experimental data shows the importance of pairing correlations on transfer.
The latter could be studied with new time-dependent Hartree-Fock-Bogoliubov codes.
In particular, one could answer the question on the origin of these pairing correlations: Are they present in the ground-states or are they generated dynamically during the collision?
For heavy and (quasi-)symmetric systems, a fusion hindrance is observed due to the quasi-fission process.
The possibility to study the quasi-fission mechanism with a fully microscopic quantum approach such as the TDHF theory is promising.
It will help to understand the strong fusion hindrance in quasi-symmetric heavy systems and may provide a guidance for new fusion experiments with exotic beams.
Actinide collisions have been investigated both within the TDHF approach and with the Ballian-V\'en\'eroni prescription.
A new inverse quasi-fission mechanism associated to specific orientations was found.
This mechanism might help to produce new neutron-rich heavy nuclei.
A systematic investigation of this effect is mandatory to help the design of future experimental equipments dedicated to the study of fragments produced in actinide collisions.
\section*{Acknowledgements}
D. Hinde, M. Dasgupta and M. Evers are warmly thanked for useful discussions.
A.~W. is grateful to CEA/Saclay, IRFU/SPhN, where part of this work has been performed.
The TDHF calculations were performed on the NCI National Facility
in Canberra, supported by the Commonwealth Government.
Support from ARC Discovery grants DP06644077 and DP110102858 is acknowledged.
\section*{References}
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\bibliography{biblio}
\end{document}