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# Multi-Messenger Constraint on the Hubble Constant $H_{0}$
with Tidal Disruption Events

Thomas Hong Tsun Wong  [  Department of Physics, University of California, San
Diego, California, 92092, USA

###### Abstract

Tidal disruption events (TDEs), apart from producing luminous electromagnetic
(EM) flares, can generate potentially detectable gravitational wave (GW) burst
signals by future space-borne GW detectors. In this Letter, we propose a
methodology to constrain the Hubble constant $H_{0}$ by incorporating the TDE
parameters measured by EM observations (e.g., stellar mass, black hole (BH)
mass and spin, and other orbital parameters) into the observed TDE GW
waveforms. We argue that an accurate knowledge of the BH spin could help
constrain the orbital inclination angle, hence alleviating the well-known
distance-inclination degeneracy in GW waveform fitting. For individual TDEs,
the precise redshift measurement of the host galaxies along with the
luminosity distance $D_{\rm L}$ constrained by EM and GW signals would give a
self-contained measurement of $H_{0}$ via Hubble’s law, completely independent
of any specific cosmological models.

††preprint: APS/123-QED

##  I Introduction

The method of utilizing the emissions of gravitational wave (GW) by compact
object mergers, known as the “standard sirens” (well-defined sources emitting
at some known frequencies), to measure $H_{0}$ was long proposed [1]. From
Hubble’s law:

$v_{\rm H}=cz=H_{0}D_{\rm L}~{},$ (1)

the detected GW waveform provides constraints on $D_{\rm L}$ while the bright
electromagnetic (EM) counterpart measures the redshift, known as the “bright
siren” (an EM-observable “standard siren”). It was only until recently has
this technique been implemented in the binary neutron star merger event
GW170817 [2]. The uncertainty in $H_{0}$ measurement is, however, dominated by
the degeneracy between $D_{\rm L}$ and the inclination angle $\eta$, defined
here as the angle between the orbital angular momentum and the line of sight
111Note that this definition differs by $90^{\circ}$ as compared with the
conventional description, of the binary system from the GW template-waveform
fitting [2, 4], as seen for a small-angle approximation:

$h_{\rm GW}\propto\frac{\cos\eta}{D_{\rm L}}~{},$ (2)

where $h_{\rm GW}$ is the detected GW strain amplitude. By incorporating the
multi-messenger information of the event, the viewing-angle-dependent features
of various EM emission models (e.g., gamma-ray burst and kilonova) are
exploited to arbitrate the distance-inclination degeneracy, providing a
tighter constraint on $D_{\rm L}$, and subsequently $H_{0}$ ([4] and
references therein).

One would naturally question whether compact object mergers remain the sole
astrophysical sources to measure $H_{0}$ in a multi-messenger approach. As
long as massive objects revolve around each other, GW emissions are
guaranteed, therefore tidal disruption of stars by massive black holes (BHs)
would present themselves as viable candidates due to the fact that immense EM
radiation is released during the transient event [5, 6, 7, 8]. When a star
approaches the galactic central supermassive black hole (SMBH) at a
sufficiently close distance, the tidal radius $r_{\rm T}\approx\left(M_{\rm
BH}/m_{\star}\right)^{1/3}r_{\star}$, where $M_{\rm BH}$, $m_{\star}$,
$r_{\star}$ are the BH mass, stellar mass, and stellar radius, respectively,
the star is then torn apart given that the tidal field of the hole exceeds the
star’s self-gravity [9]. The disrupted stellar material would be stretched
into a debris stream, approximately half of it will gradually dissipate
orbital energy into EM radiation, and eventually circularize into an accretion
disk. Optical, near-UV, X-ray all-sky surveys have detected up to a hundred or
so events, and one to two orders of magnitude more are expected in the coming
decade [10, 11].

Tidal disruption events (TDEs) could only generate GW bursts as the star is
often disrupted within an orbital timescale, i.e. the intact star does not
survive an entire orbit to produce a full period of GW waveform [5]. An open
comprehensive living catalog of TDE GW waveforms has been built to explore a
wide range of parameters [8]. Given their relatively long orbital timescale
prior to disruption ($\sim 10^{2-4}\,{\rm s}$), the characteristic GW burst
frequency is approximately in the range of $0.1-10\,{\rm mHz}$, which
corresponds to the designed sensitivities of the upcoming space-borne GW
detectors [12, 13, 14, 15]. But in fact, most TDE GW signals are incapable of
generating a large enough signal-to-noise ratio to trigger a detection for
_LISA_ [12] but would lie well within the detection limit of post-_LISA_
detectors [14, 15, 7, 16]. The TDE GW observed rate by _LISA_ is predicted to
remain half a dozen or so for the entire four-year mission [7] as the
characteristic strains of the events are weak given typical TDE parameters,
which are estimated as [5, 8]:

$\begin{split}h_{\rm GW}&\sim 10^{-22}\left(\frac{D_{\rm L}}{20\,{\rm
Mpc}}\right)^{-1}\times\\\ &~{}~{}~{}~{}\,\beta\left(\frac{r_{\star}}{{\rm
R}_{\odot}}\right)^{-1}\left(\frac{m_{\star}}{{\rm
M}_{\odot}}\right)^{4/3}\left(\frac{M_{\rm BH}}{10^{6}\,{\rm
M}_{\odot}}\right)^{2/3}~{},\end{split}$ (3)

where $r_{\rm p}$ is the pericenter radius and $\beta=r_{\rm p}/r_{\rm T}$ is
the penetration parameter 222As a simple approximation, $\beta\geq 1$ usually
refers to the complete disruption of the star, but [21] has shown that the
cutoff between survival and disrupted sometimes lies above unity. (quantifying
how deeply the orbit penetrates into the BH gravitational potential well).

The pessimistic observed rate by _LISA_ inspires us to explore the possibility
of using a very limited number of events to independently measure $H_{0}$. We
hereby propose using TDE EM observations to constrain as many parameters as
possible prior to GW waveform fitting, resulting in a remarkably improved GW
constraint on the luminosity distance $D_{\rm L}$. This work focuses on
gathering the cumulative modeling effort in obtaining TDE parameters,
exploring their intercorrelations, and most importantly, proposing the _first_
methodology to constrain $H_{0}$ via TDEs. Prior to detecting GW signals,
using simulated waveforms to run the following analysis in an attempt to
constrain $H_{0}$ would inevitably lead to a circular argument, therefore this
letter serves as a primal investigation on the foundational idea of exploiting
multi-messenger signals of TDEs to measure $H_{0}$.

In Section II, we illustrate the recent progress in constraining all waveform-
dependent TDE parameters by EM observations with physical modeling, laying the
foundational work to further propose the novel approach to alleviate the
distance-inclination degeneracy using the constrained BH spin parameters, the
methodology is then presented with an estimation on which parameter(s) would
most dominate the $H_{0}$ uncertainty. In Section III, we discuss possible
ways to further improve the precision of $H_{0}$ determination.

##  II Multi-Messenger Constraints on $D_{\rm L}$

Given the ability to localize TDEs with the current multi-wavelength surveys,
the redshifts of the host galaxies can be comfortably measured with high
certainty [10, 11]. In order to estimate an accurate Hubble constant $H_{0}$,
the problem lies in constraining the luminosity distance $D_{\rm L}$ from
their GW counterparts.

Unfortunately, in order to accurately match the observed waveforms with the
templates, one could not rely solely on the approximate peak amplitude in
Eq.(3) which depends mainly on three parameters. Provided the number of
waveform-dependent parameters, there is only hope to constrain $D_{\rm L}$,
and hence $H_{0}$, if most, if not all, these TDE parameters could be
constrained to some extent with EM observations. We hereby show the parameter
inter-dependencies (summarized in Fig.2) and how they are expected to yield a
constrained $H_{0}$.

###  II.1 EM Constraints on TDE Parameters

####  II.1.1 Three main parameters: $M_{\rm BH}$, $m_{\star}$, $\beta$

TDE-host black hole mass is one of the easiest to infer since there are a few
$M_{\rm BH}$-galaxy relations available [18, 19, 20] and simulations/models
specifically for TDE-specific scenarios [5, 21, 22, 23, 24, 25]. The common
TDE EM-GW detection rate is highly limited [7], a more accurate constraint on
the $M_{\rm BH}$ of the few detectable events could perhaps be computed in a
case-by-case TDE-model-dependent manner instead of using the global galaxy
relations, even if the latter has a slightly better constraint than the
former.

Focusing on the disruption of main-sequence (MS) stars, the mass-radius
relation is given by [26], such that all $r_{\star}$-dependence turns into
$m_{\star}$ accordingly. MCMC method that fits both the peak luminosity and
the color temperature of the observed TDEs could constrain $m_{\star}$ down to
a $\sim$few % uncertainty (but sometimes a lot higher) [23, 25]. It is indeed
a challenging task to check whether the mass constraint is accurate given
there is yet to exist another independent EM-measurement, additional GW
waveform information could complement the deficiency based on the approximate
duration $\tau$/frequency $f$ of the burst [5, 8]:

$f\sim\frac{1}{\tau}\approx 10^{-4}\,{\rm
Hz}\times\beta^{3/2}\left(\frac{m_{\star}}{{\rm
M}_{\odot}}\right)^{1/2}\left(\frac{r_{\star}}{{\rm
R}_{\odot}}\right)^{-3/2}~{}.$ (4)

Orbital parameters remain the most challenging-to-constrain variables as the
TDE observables do not depend as sensitively as they do on the masses. The
$\beta$ dependency on TDE light curves is investigated with hydrodynamical
simulations [21], resulting in co-dependency on both masses. A relatively
weaker constraint on $\beta$ is through analyzing the probability distribution
among the EM-observed TDE population [27]. Both works provided corresponding
analytical formulae. For the high signal-to-noise TDE detections by _LISA_ , a
lower bound on $\beta$ (e.g., $\beta_{\rm min}\gtrsim 10$ for $M_{\rm
BH}=10^{6}\,{\rm M}_{\odot}$) can be imposed [7]. Given the rough EM-
dependence, it could potentially be slightly more beneficial to further
constrain $\beta$ from the TDE GW waveform as the overall shape of the
polarizations and the duration of the GW burst change when $\beta$ is varied
[8]. The EM constraint can simply be used as the prior knowledge with a large
uncertainty between $\beta_{\rm min}\gtrsim\beta\geq\beta_{\rm max}$, where
$\beta_{\rm max}$ happens at $r_{\rm p}=r_{\rm Sch}$, i.e., all stellar
materials are swallowed at pericenter passage thus EM signal detection is
unlikely.

It is important to note that these three parameters are not completely
independent, e.g., a less massive, i.e., more compact, star could not be
tidally disrupted by a very massive BH, but will instead be swallowed whole,
i.e. $r_{\rm T}\leq r_{\rm Sch}$, where $r_{\rm Sch}$ is the Schwarzschild
radius of the hole. Some regions in the parameter space are thus automatically
ruled out for any EM-observable event.

####  II.1.2 Degeneracy-breaking parameters:
Black hole spin and inclination angle: $a_{\rm BH}$, $\theta$, $\eta$

Upon first glance, the spin properties of the host BH might seem like a sub-
dominant factor, but the way they correlate with the orbital inclination angle
$\eta$ could ultimately lead to a probable alleviation of the known distance-
inclination degeneracy.

Figure 1: Schematic illustration of how a typical $\eta$ orbit would evolve
into an accretion disk of specific orientation given a BH spin offset. When
the (large) BH spin orientation sufficiently differs from the orbital angular
momentum of the stellar orbit, the disrupted stream debris would evolve away
from the initial orbital plane, resulting in a stream collision somewhere else
(red explosion) [21]. The position of the intersection could constrain the
final orientation of the circularized accretion disk, where then the viewing-
angle dependent model may be applied [28]. The $z$-axis is set to be the line
of sight (los).

The dimensionless spin magnitude and the spin orientation are defined by
$0\leq a_{\rm BH}\equiv cJ/GM_{\rm BH}^{2}\leq 1$ and the angle between the
spin vector and the stellar orbital angular momentum,
$0^{\circ}\,{\rm(prograde)}\leq\theta\leq 180^{\circ}\,{\rm(retrograde)}$,
respectively. Spin parameters could be constrained with the observed light
curve at peak accretion [29] (most sensitive for high $\beta$) or with X-ray
reverberation technique when the accretion disk is formed [30]. If an observed
TDE has a rapidly spinning host BH, the launching of a relativistic jet via
the Blandford-Znajek mechanism [31] would be advantageous in further
constraining the BH spin parameters [32].

The parameter $\eta$ has never been investigated with EM observables as the
incoming orbit of the star has (nearly) no influence on the
multiwavelength/multi-epoch signatures as we could not see lights at the exact
disruption phase. However, when GW is included as part of the multi-messenger
analysis, $\eta$ is of utmost importance.

The first-ever analysis on incorporating viewing-angle-dependent models was
used for the case of binary neutron star merger GW170817 [4], where physical
models of gamma-ray burst and kilonova are used to constrain the system
inclination by modeling their light curves and spectra-photometry. We propose
that a similar approach could be implemented with the work of [28, 33], in
which the TDE spectral features depend mainly on the viewing angle 333[33]
found the dependency between the $L_{\rm optical}/L_{\rm X-ray}$ ratio and the
BH accretion rate $\dot{M}_{\rm acc}(t)$ along a specific viewing angle.
Despite the additional layer of complexity, $\dot{M}_{\rm acc}(t)$ is,
however, not challenging to be deduced from observation [52].. Assuming that
the X-ray and optical/UV emissions originate from the inner part of the
accretion disk and the disk luminosity reprocessed by the expanding outflow,
respectively 444The origin of optical emission is still currently under
debate: some believe it is from the shocks created during the stream self-
intersection [51, 48], while some think it comes from the reprocessing layer
surrounding the disk [52, 28]., when viewing the TDE system edge-on, the
intrinsic X-ray emission from the disk will be reprocessed by the optically
thick outflow, optical/UV luminosity would dominate the observed spectrum;
when viewing the TDE system face-on, we would then expect to look into the
optically thin funnel and see a stronger X-ray luminosity from the exposed
inner disk. Given that both optical and X-ray luminosities are measured for
many TDE candidates, their ratios $L_{\rm optical}/L_{\rm X-ray}$ could
potentially indicate the approximate inclination angle of the geometrically
thick outer accretion disk [28, 33], i.e., increase in viewing angle of the
disk generally augments the optical-to-X-ray luminosity ratio.

To link the orientations of the disk and the stellar orbital plane, for BH
spins that are aligned with the orbital plane’s normal ($\theta=0$), we could
safely assume that the stellar materials would on average remain on its
incoming orbital plane due to symmetry, such that the luminosity ratio could
directly be used to infer $\eta$. For cases where the direction of the
moderate/high BH spin is significantly offset from the orbital plane’s normal
(incoming stellar orbits are often randomly oriented), relativistic precession
for close encounters (high $\beta$) would induce deflections of the debris
stream out of the initial orbital plane, leading to a certain period when the
TDE flare is not observable [36]. When the stream eventually self-intersects
and dissipates its orbital energy during circularization, it is unlikely that
the circularized disk will lie on the original orbital plane [37]. Hence, in
order to utilize this analysis, both BH spin parameters should _not_ be
neglected, their relations are illustrated by the stream evolution in Fig.1
and indicated in Fig.2. When the spin parameters are constrained by the
modeling discussed in Section BH spin, the simulation [36] could then be
implemented to predict how the initial orbit plane would undergo multiple
windings and end up on the final accretion disk plane where the stream
intersection finally happens, hence yielding the orbital inclination angle
$\eta$ through forward modeling.

The spin parameters have to be constrained by EM observations as they are the
input parameters for determining the orbital inclination angle $\eta$, meaning
that if $a_{\rm BH}$ and $\theta$ are fitted through GW waveform, the
distance-inclination degeneracy would still remain.

####  II.1.3 Other orbital parameters: $e$, $\phi$

From the GW waveform simulation [8], the strain amplitude dependence of the
orbital eccentricity $e$ is approximately an order of magnitude smaller than
$\beta$ and $\theta$. Albeit the mild sensitivity in $e$, implying the
insignificant contribution to the uncertainty of $D_{\rm L}$, EM constraints
are still possible. The common assumption is that most TDE stars have a
roughly parabolic ($e\approx 1$) flyby orbit [38]. Hyperbolic orbits
($e\gtrsim 1$) are automatically not taken into consideration since the
stellar materials are unbound after the disruption and would not produce a
detectable EM signal; and near-circular orbits are highly unlikely based on
loss cone dynamics [39]. The eccentricity is then essentially constrained to
some values in between given by the fallback rate [40].

There exists one orbital orientation parameter $\phi$, which is the angle
between the stellar pericenter axis and the projection of the line of sight
onto the orbital plane [8], being the least dependent of all. This angle can
hardly be constrained by EM observations nor TDE models and shall not possess
any prior in the waveform fitting. (All angles discussed $\theta$, $\eta$, and
$\phi$ are defined identically to [8].)

###  II.2 Luminosity Distance $D_{\rm L}$ and $H_{0}$ Estimate

####  II.2.1 Key Methodology

Figure 2: Graphical model describing how the relations amongst TDE parameters
and how EM and GW observations are combined to yield $H_{0}$ from a single
TDE. The main TDE parameters ($M_{\rm BH}$, $m_{\star}$, $\beta$) are
indicated by blue circles. Dashed arrows illustrate how parameters are
obtained via other parameters, which involve some simulations/models and
additional EM observables (e.g., light curves and spectra). EM-constrained
parameters combining with the GW waveform fitting process would give the PDF
of $D_{\rm L}$, then with the observed host galaxy redshift, the PDF of
$H_{0}$ (likelihood) is found. The $H_{0}$ likelihood and cosmologically
determined prior then give rise to the posterior (the final determination of
$H_{0}$ by a single TDE). The filled boxes indicate the outputs of each
procedure.

Using suitable TDE models to fit the corresponding observables discussed in
Section II.1, all seven EM-constrained parameters ($M_{\rm BH}$, $m_{\star}$,
$\beta$, $a_{\rm BH}$, $\theta$, $\eta$, $e$) would have their corresponding
probability density functions (PDFs) obtained through simulations and model-
fitting. As for $D_{\rm L}$ and $\phi$, they are expected to freely vary
during the waveform fitting, and no prior assumption should be made on $D_{\rm
L}$ to prevent any bias. With the knowledge to constrain the inclination angle
$\eta$ as an input parameter, the distance-inclination degeneracy is expected
to be relieved to a very large extent.

All nine variables and their corresponding uncertainties would be used to
generate a huge catalog of GW waveforms [8], centering at the constrained
parameter values to avoid exploring the large parameter space. Note that some
parameter values are prohibited ($r_{\rm T}\leq r_{\rm Sch}$), e.g., large
$M_{\rm BH}$ for disruption of small $m_{\star}$, high $\beta$ for specific
$M_{\rm BH}$ and $m_{\star}$. The theoretically generated waveforms would then
be fitted with that of the observed in a similar manner as in [41], yielding a
PDF of $D_{\rm L}$. The PDF of $D_{\rm L}$ would translate directly to the PDF
of $H_{0}$ using Eq.1. Setting this PDF as the likelihood in Bayesian
formalism [2] and the $H_{0}$ from other cosmological studies [42, 43] as the
prior, the posterior $H_{0}$ is expected to peak near the prior value while
eliminating the other $H_{0}$ peaks derived from $D_{\rm L}$. The graphical
description is shown in Fig.2.

####  II.2.2 Bottleneck in $H_{0}$ Measurement Uncertainty

Figure 3: TDE parameters that dominate the dependency on GW waveform amplitude
$|h_{\rm GW}|$. Amongst the seven EM-constrained parameters, varying these
three parameters: $M_{\rm BH}$ (solid), $m_{\star}$ (dotted), and $\beta$
(dashed), would fluctuate $h_{\rm GW}$ to a large extent. All waveforms are
centered at $t_{\rm burst}$, the time when the peak amplitude is reached. The
other parameters are chosen as follows: $a_{\rm BH}=0$, $\theta=0$, $e=1$,
$\eta=0$, $D_{\rm L}=20\,{\rm Mpc}$. Plotted from simulation results [8].

As long as the parameters are entangled in such a complicated manner (Fig.2),
what we could do is, from an order-of-magnitude point of view, estimate which
parameter(s) would predominantly contribute to the uncertainty $\sigma_{D_{\rm
L}}$.

Fig.3 shows the main TDE parameters that impact the GW burst amplitudes (the
exact waveform of $h_{\rm GW}$ here is less important than those of LIGO
events as TDEs often only generate single bursts), while the rest either
become important only in extreme scenarios (high $\beta$ or near-maximal BH
spin) or are always subdominant. These three parameters, coincidentally, are
often the input parameters in most simulations (as seen from the number of
arrows pointing out of them in Fig.2), thus their uncertainties are cumulative
and are projected onto the rest. Amongst them, we believe the penetration
parameter $\beta$ ought to contribute the largest uncertainty of $H_{0}$. Even
though $M_{\rm BH}$ is used thrice to determine other parameters (while twice
by $\beta$), $M_{\rm BH}$ is typically better constrained than $\beta$ [22,
23, 27], unless for very massive BHs where the detectable $\beta$ range can be
as narrow as order of unity. $\sigma_{H_{0}}$ is found to be dominated by the
distance-inclination degeneracy [2, 4], implying that $\sigma_{\eta}$ should
dominate over the rest. Given that $\beta$ is used to determine $\eta$,
$\sigma_{\beta}$ should in turn dominate. It is understandable as the
magnitude of the off-plane precession sensitively depends on how close the
stellar debris orbits around the spinning BH [36]. $\sigma_{M_{\rm BH}}$ would
then be the next dominating uncertainty.

The few hundred TDE GW waveforms in the presently enlarging library [8] have a
resolution too low in the 9-dimensional parameter space to yield a reasonable
fitting. This should immediately raise the question: What is the approximate
number of waveforms required to result in a reasonable fit, which then
translates into a reasonable $H_{0}$ precision? If a uniform search in
parameter space is implemented, the number of waveforms generated would
skyrocket as the number of parameters increase. Having established that each
parameter affects the waveform to different extents, it is only sensible to
vary densely on the parameters of dominant contributions, such as $M_{\rm
BH}$, $\beta$, and $\eta$. Adaptive resolution on which parameters to explore
should precede uniformly increasing the total number of waveforms across all
parameter spaces in the catalog. Ultimately, the goal of the multi-messenger
analysis is to better constrain $D_{\rm L}$, not finding the best-fit TDE
parameters.

##  III Discussion

As predicted by [5, 7, 8, 16], even the optimistic TDE GW detection rate by
_LISA_ is expected to remain a few for the entire duration of the mission. It
is therefore of utmost importance that the analyses of the few limited multi-
messenger TDE observations could be maximized, stressing the power of this
methodology to _independently measure $H_{0}$ with a handful of events_. To
strengthen the constraining power of TDE parameters as a whole, the GW burst
signal during disruption could trigger the immediate follow-up EM observations
such that light from the pre-peak epoch can be captured. When _DECIGO_ and the
other next-generation spaceborne GW detectors are eventually in operation, the
expected thousands to millions of TDE detections might in turn place EM
observation as the bottleneck of the multimessenger era, but by then a
statistically significant measurement of $H_{0}$ from TDEs should already be
obtained. If the uncertainty on $D_{L}$, hence $H_{0}$, could be reduced even
by some small portion, after incorporating EM constraints with this proposed
methodology, this would then conclusively demonstrate the functionality of TDE
multi-messenger $H_{0}$ measurement, while placing the development of TDE
modeling at the bottleneck of the analysis.

For typical cases, $\sigma_{\beta}$ would be dominant, still, there are
certain possible ways to further constrain $\beta$: by brute force, we would
benefit from a GW TDE triggering of pre-peak high-cadence EM observation [21];
more $\beta$-sensitive observable could be found with improved modeling; or
exploiting the potentially huge detectable TDE population by post-_LISA_
interferometers, then the $\beta$-distributions [27] could directly constrain
$H_{0}$ and not the individual $\beta$ in each event.

Given the modeling complication and intertwining relations among parameters,
measuring $H_{0}$ with TDEs is clearly a non-trivial task and would likely
require a collaborative effort in the field. All in all, it is manifest that
the proliferating EM and GW detections of TDEs and more comprehensive TDE
simulations in the next decade should lead to both precise and accurate
measurements of the Hubble constant in addition to the standard siren
approach.

###### Acknowledgements.

I thank S.K. Li, Paul C.W. Lai, Lars L. Thomsen, and George M. Fuller for
useful comments and discussions.

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# Fusion at near-barrier energies within quantum diffusion approach

V.V.Sargsyan1, G.G.Adamian1, N.V.Antonenko1, W. Scheid2, and H.Q.Zhang3 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia
2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
3China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China

###### Abstract

Within the quantum diffusion approach the role of neutron transfer in the
fusion (capture) reactions with toughly and weakly bound nuclei is discussed.
The breakup process is analyzed. New methods for the study of the breakup
probability are suggested.

###### pacs:

25.70.Jj, 24.10.-i, 24.60.-k
Key words: sub-barrier capture, neutron transfer, quantum diffusion approach

##  I Introduction

The nuclear deformation and neutron-transfer process have been identified as
playing a major role in the magnitude of the sub-barrier fusion (capture)
cross sections Gomes . There are a several experimental evidences which
confirm the importance of nuclear deformation on the fusion. The influence of
nuclear deformation is straightforward. If the target nucleus is prolate in
the ground state, the Coulomb field on its tips is lower than on its sides,
that then increases the capture or fusion probability at energies below the
barrier corresponding to the spherical nuclei. The role of neutron transfer
reactions is less clear. The importance of neutron transfer with positive
$Q$-values on nuclear fusion (capture) originates from the fact that neutrons
are insensitive to the Coulomb barrier and therefore they can start being
transferred at larger separations before the projectile is captured by target-
nucleus. Therefore, it is generally thought that the sub-barrier fusion cross
section will increase because of the neutron transfer.

The fusion (capture) dynamics induced by loosely bound radioactive ion beams
is currently being extensively studied. However, the long-standing question
whether fusion (capture) is enhanced or suppressed with these beams has not
yet been answered unambiguously. The study of the fusion reactions involving
nuclei at the drip-lines has led to contradictory results.

##  II Quantum diffusion approach for capture

In the quantum diffusion approach EPJSub ; EPJSub02 ; EPJSub1 ; EPJSub2 ;
EPJSub3  the capture of the projectile by the target-nucleus is described with
a single relevant collective variable: the relative distance between the
colliding nuclei. This approach takes into consideration the fluctuation and
dissipation effects in collisions of heavy ions which model the coupling of
the relative motion with various channels (for example, the non-collective
single-particle excitations, low-lying collective dynamical modes of the
target and projectile). The nuclear static deformation effects are taken into
account through the dependence of the nucleus-nucleus potential on the
deformations and mutual orientations of the colliding nuclei. We have to
mention that many quantum-mechanical and non-Markovian effects accompanying
the passage through the potential barrier are taken into consideration in our
formalism EPJSub ; EPJSub1 .

The capture cross section is a sum of partial capture cross sections EPJSub ;
EPJSub1

$\displaystyle\sigma_{cap}(E_{\rm c.m.})$ $\displaystyle=$
$\displaystyle\sum_{J}\sigma_{\rm cap}(E_{\rm c.m.},J)=$ (1) $\displaystyle=$
$\displaystyle\pi\lambdabar^{2}\sum_{J}(2J+1)\int_{0}^{\pi/2}d\theta_{1}\sin(\theta_{1})\int_{0}^{\pi/2}d\theta_{2}\sin(\theta_{2})P_{\rm
cap}(E_{\rm c.m.},J,\theta_{1},\theta_{2}),$

where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie
wavelength, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$
is the nucleon mass), and the summation is over the possible values of angular
momentum $J$ at a given bombarding energy $E_{\rm c.m.}$. Knowing the
potential of the interacting nuclei for each orientation with the angles
$\theta_{i}(i=1,2)$, one can obtain the partial capture probability $P_{\rm
cap}$ which is defined by the passing probability of the potential barrier in
the relative distance $R$ coordinate at a given $J$. The value of $P_{\rm
cap}$ is obtained by integrating the propagator $G$ from the initial state
$(R_{0},P_{0})$ at time $t=0$ to the final state $(R,P)$ at time $t$ ($P$ is a
momentum):

$\displaystyle P_{\rm cap}=\lim_{t\to\infty}\int_{-\infty}^{r_{\rm
in}}dR\int_{-\infty}^{\infty}dP\
G(R,P,t|R_{0},P_{0},0)=\lim_{t\to\infty}\frac{1}{2}{\rm
erfc}\left[\frac{-r_{\rm
in}+\overline{R(t)}}{{\sqrt{\Sigma_{RR}(t)}}}\right].$ (2)

The second line in (2) is obtained by using the propagator
$G=\pi^{-1}|\det{\bf\Sigma}^{-1}|^{1/2}\exp(-{\bf q}^{T}{\bf\Sigma}^{-1}{\bf
q})$ (${\bf q}^{T}=[q_{R},q_{P}]$, $q_{R}(t)=R-\overline{R(t)}$,
$q_{P}(t)=P-\overline{P(t)}$, $\overline{R(t=0)}=R_{0}$,
$\overline{P(t=0)}=P_{0}$,
$\Sigma_{kk^{\prime}}(t)=2\overline{q_{k}(t)q_{k^{\prime}}(t)}$,
$\Sigma_{kk^{\prime}}(t=0)=0$, $k,k^{\prime}=R,P$) calculated for an inverted
oscillator which approximates the nucleus-nucleus potential $V$ in the
variable $R$. The frequency $\omega$ of this oscillator with an internal
turning point $r_{\rm in}$ is defined from the condition of equality of the
classical actions of approximated and realistic potential barriers of the same
hight at given $J$. This approximation is well justified for the reactions and
energy range, which are here considered.

We assume that the sub-barrier capture mainly depends on the optimal one-
neutron ($Q_{1n}>Q_{2n}$) or two-neutron ($Q_{2n}>Q_{1n}$) transfer with the
positive $Q$-value. Our assumption is that, just before the projectile is
captured by the target-nucleus (just before the crossing of the Coulomb
barrier) which is a slow process, the transfer occurs and can lead to the
population of the first excited collective state in the recipient nucleus
SSzilner  (the donor nucleus remains in the ground state). So, the motion to
the $N/Z$ equilibrium starts in the system before the capture because it is
energetically favorable in the dinuclear system in the vicinity of the Coulomb
barrier. For the reactions under consideration, the average change of mass
asymmetry is connected to the one- or two-neutron transfer ($1n$\- or
$2n$-transfer). Since after the transfer the mass numbers, the isotopic
composition and the deformation parameters of the interacting nuclei, and,
correspondingly, the height $V_{b}=V(R_{b})$ and shape of the Coulomb barrier
are changed, one can expect an enhancement or suppression of the capture. If
after the neutron transfer the deformations of interacting nuclei increase
(decrease), the capture probability increases (decreases). When the isotopic
dependence of the nucleus-nucleus potential is weak and after the transfer the
deformations of interacting nuclei do not change, there is no effect of the
neutron transfer on the capture. In comparison with Ref. Dasso , we assume
that the negative transfer $Q-$values do not play visible role in the capture
process. Our scenario was verified in the description of many reactions
EPJSub1 ; EPJSub2 ; EPJSub3 .

##  III Results of calculations

Because the capture cross section is equal to the complete fusion cross
section for the reactions treated, the quantum diffusion approach for the
capture is applied to study the complete fusion. All calculated results are
obtained with the same set of parameters as in Ref. EPJSub . Realistic
friction coefficient in the relative distance coordinate $\hbar\lambda$=2 MeV
is used. Its value is close to that calculated within the mean-field
approaches obzor . For the nuclear part of the nucleus-nucleus potential, the
double-folding formalism with the Skyrme-type density-dependent effective
nucleon-nucleon interaction is used EPJSub ; EPJSub1 . The parameters of the
nucleus-nucleus interaction potential $V(R)$ are adjusted to describe the
experimental data at energies above the Coulomb barrier corresponding to
spherical nuclei. The absolute values of the experimental quadrupole
deformation parameters $\beta_{2}$ of even-even deformed nuclei in the ground
state and of the first excited collective states of nuclei are taken from Ref.
Ram . For the nuclei deformed in the ground state, the $\beta_{2}$ in the
first excited collective state is similar to the $\beta_{2}$ in the ground
state. For the quadruple deformation parameter of an odd nucleus, we choose
the maximal value from the deformation parameters of neighboring even-even
nuclei (for example,
$\beta_{2}$(231Th)=$\beta_{2}$(233Th)=$\beta_{2}$(232Th)=0.261). For the
double magic and neighboring nuclei, we take $\beta_{2}=0$ in the ground
state. Since there are uncertainties in the definition of the values of
$\beta_{2}$ in light-mass nuclei, one can extract the ground-state quadrupole
deformation parameters of these nuclei from a comparison of the calculated
capture cross sections with the existing experimental data. By describing the
reactions 12C+208Pb, 18O+208Pb, 32,36S+90Zr, 34S+168Er, 36S+90,96Zr, 58Ni +
58Ni, and 64Ni + 58Ni, where there are no neutron transfer channels with
positive $Q$-values, we extract the ground-state quadrupole deformation
parameters $\beta_{2}$=-0.3, 0.1, 0.312, 0.1, 0, 0.05, and 0.087, for the
nuclei 12C, 18O, 32S, 34S, 36S, 58Ni, and 64Ni, respectively, which are used
in our calculations.

###  III.1 Role of neutron transfer in capture process at sub-barrier energies

After the neutron transfer in the reaction 40Ca($\beta_{2}=0$) +
96Zr($\beta_{2}=0.08$)$\to^{42}$Ca($\beta_{2}=0.247$) + 94Zr($\beta_{2}=0.09$)
(Fig. 1) or 40Ca($\beta_{2}=0$) +
124Sn($\beta_{2}=0.095$)$\to^{42}$Ca($\beta_{2}=0.247$) +
122Sn($\beta_{2}=0.1$) (Fig. 1) the deformation of the nuclei increases and
the mass asymmetry of the system decreases, and, thus, the value of the
Coulomb barrier decreases and the capture cross section becomes larger (Fig.
1). In Fig. 2, we observe the same behavior in the reactions
58Ni($\beta_{2}=0.05$) + 132Sn($\beta_{2}=0$)$\to^{60}$Ni($\beta_{2}=0.207$) +
130Sn($\beta_{2}=0$) ($Q_{2n}=7.8$ MeV), 58Ni($\beta_{2}=0.05$) +
130Te($\beta_{2}=0$)$\to^{60}$Ni($\beta_{2}=0.207$) + 128Te($\beta_{2}=0$)
($Q_{2n}=5.9$ MeV), 64Ni($\beta_{2}=0.087$) +
132Sn($\beta_{2}=0$)$\to^{66}$Ni($\beta_{2}=0.158$) + 130Sn($\beta_{2}=0$)
($Q_{2n}=2.5$ MeV), and 64Ni($\beta_{2}=0.087$) +
130Te($\beta_{2}=0$)$\to^{66}$Ni($\beta_{2}=0.158$) + 128Te($\beta_{2}=0$)
($Q_{2n}=0.55$ MeV). One can see a good agreement between the calculated
results and the experimental data Liang ; TimmersCa40Zr96 ;
Stefanini40ca116124sn . So, the observed capture enhancement at sub-barrier
energies in the reactions mentioned above is related to the two-neutron
transfer channel. One can see that at energies above and near the Coulomb
barrier the cross sections with and without two-neutron transfer are almost
similar. Since the two-neutron transfer causes a larger change of the
deformations of the nuclei in the reactions 58Ni + 132Sn,130Te than in the
reactions 64Ni + 132Sn,130Te, at sub-barrier energies the capture enhancement
in the reactions with 58Ni is larger than in the reactions with 64Ni (Fig. 2).

Figure 1:  The calculated capture cross sections versus $E_{\rm c.m.}$ for the
indicated reactions 40Ca + 96Zr (solid line), 40Ca + 90Zr (dashed line), and
48Ca + 124Sn (solid line). For the reactions 40Ca + 96Zr,124Sn, the calculated
capture cross sections without the neutron transfer process are shown by
dotted lines. The experimental data (symbols) are from Refs. TimmersCa40Zr96 ;
Stefanini40ca116124sn .  Figure 2:  The same as in Fig. 1, for the reactions
58,64Ni + 132Sn (solid lines) and 58,64Ni + 130Te (dashed lines). The
experimental data (symbols) are from Refs. Liang ; Liang0 . For the reactions
58,64Ni + 132Sn (dotted lines) and 58,64Ni + 130Te (dash-dotted lines), the
calculated capture cross sections without the neutron transfer are shown.
Figure 3:  (Color online) The calculated reduced capture cross sections versus
$(E_{\rm c.m.}-V_{b})/(\hbar\omega_{b})$ in the reactions 40Ca+124Sn (solid
line), 48Ca+124Sn (dashed line), 48Ca+124Sn (dotted line), and 48Ca+132Sn
(dash-dotted line).  Figure 4:  (Color online) The calculated capture cross
sections versus $E_{\rm c.m.}$ for the reactions 40Ca+124Sn (solid line) and
48Ca+124Sn (dashed line). The experimental data for the reactions 40Ca+124Sn
(solid squares) and 48Ca+124Sn (open squares) are from Ref. Kol . In the
calculations the barriers were adjusted to the experimental values.  Figure 5:
(Color online) The calculated capture cross sections versus $E_{\rm c.m.}$ for
the reactions 40Ca+132Sn (solid line) and 48Ca+132Sn (dashed line). The
experimental data for the reactions 40Ca+132Sn (solid squares) and 48Ca+132Sn
(open squares) are from Ref. Kol . In the calculations the barriers were
adjusted to the experimental values.

One can make unambiguous statements regarding the neutron transfer process
with a positive $Q$-value when the colliding nuclei are double magic or semi-
magic. In this case one can disregard the deformation and orientation effects
before the neutron transfer. To eliminate the influence of the nucleus-nucleus
potential on the capture (fusion) cross section and to make conclusions about
the role of deformation of colliding nuclei and the nucleon transfer between
interacting nuclei in the capture (fusion) cross section, a reduction
procedure is useful Gomes2 . It consists of the following transformations:

$E_{\rm c.m.}\rightarrow x=\dfrac{E_{\rm
c.m.}-V_{b}}{\hbar\omega_{b}},\qquad\sigma_{cap}\rightarrow\sigma_{cap}^{red}=\dfrac{2E_{\rm
c.m.}}{\hbar\omega_{b}R_{b}^{2}}\sigma_{cap},$

where $\sigma_{cap}=\sigma_{cap}(E_{\rm c.m.})$ is the capture cross section
at bombarding energy $E_{\rm c.m.}$. The frequency
$\omega_{b}=\sqrt{V^{{}^{\prime\prime}}(R_{b})/\mu}$ is related with the
second derivative $V^{{}^{\prime\prime}}(R_{b})$ of the total nucleus-nucleus
potential $V(R)$ (the Coulomb + nuclear parts) at the barrier position
$R_{b}$. With these replacements we compared the reduced calculated capture
(fusion) cross sections $\sigma_{cap}^{red}$ for the reactions
40,48Ca+124,132Sn (Fig. 3). The choice of the projectile-target combination is
crucial, and for the systems studied one can make unambiguous statements
regarding the neutron transfer process with a positive $Q$-value when the
interacting nuclei are double magic or semi-magic spherical nuclei. In this
case one can disregard the strong direct nuclear deformation effects. In Fig.
3, one can see that the reduced capture cross sections in the reactions
40Ca+124,132Sn with the positive $Q_{2n}$-values strongly deviate from those
in the reactions 48Ca+124,132Sn, where the neutron transfers are suppressed
because of the negative $Q$-values. After two-neutron transfer in the
reactions
40Ca($\beta_{2}=0$)+124Sn($\beta_{2}=0.1$)$\to^{42}$Ca($\beta_{2}=0.25$)+122Sn($\beta_{2}=0.1$)
($Q_{2n}$=5.4 MeV) and
40Ca($\beta_{2}=0$)+132Sn($\beta_{2}=0$)$\to^{42}$Ca($\beta_{2}=0.25$)+130Sn($\beta_{2}=0$)
($Q_{2n}$=7.3 MeV) the deformation of the light nucleus increases and the mass
asymmetry of the system decreases and, thus, the value of the Coulomb barrier
decreases and the capture cross section becomes larger (Fig. 3). So, because
of the transfer effect the systems 40Ca+124,132Sn show large sub-barrier
enhancements with respect to the systems 48Ca+124,132Sn. We observe that the
$\sigma_{cap}^{red}$ in the 40Ca+124Sn (48Ca+124Sn) reaction are larger than
those in the 40Ca+132Sn (48Ca+132Sn) reaction. The reason of that is the
nonzero quadrupole deformation of the heavy nucleus 124Sn. It should be
stressed that there are almost no difference between $\sigma_{cap}^{red}$ in
the reactions 40,48Ca+124,132Sn at energies above the Coulomb barrier.

In Figs. 4 and 5 one can see a good agreement between the calculated results
and the experimental data in the reactions 40,48Ca+124,132Sn. This means that
the observed capture enhancements in the reactions 40Ca+124,132Sn at sub-
barrier energies are related to the two-neutron transfer effect. Note that the
slope of the excitation function strongly depends on the deformations of the
interacting nuclei and, respectively, on the neutron transfer effect.

To describe the reactions 40,48Ca+132Sn and 48Ca+124,132Sn (Figs. 4 and 5), we
extracted the values of the corresponding Coulomb barrier $V_{b}$ for the
spherical nuclei. There are differences between the calculated and extracted
$V_{b}$. From the direct calculations of the nucleus-nucleus potentials (with
the same set of parameters), we obtained
$V_{b}$(40Ca+124Sn)-$V_{b}$(48Ca+124Sn)=2.3 MeV,
$V_{b}$(40Ca+132Sn)-$V_{b}$(48Ca+132Sn)=2.2 MeV,
$V_{b}$(40Ca+124Sn)-$V_{b}$(40Ca+132Sn)=1.3 MeV, and
$V_{b}$(48Ca+124Sn)-$V_{b}$(48Ca+132Sn)=1.2 MeV. From the extractions, we got
$V_{b}$(40Ca+124Sn)-$V_{b}$(48Ca+124Sn)=1.1 MeV,
$V_{b}$(40Ca+132Sn)-$V_{b}$(48Ca+132Sn)=1.0 MeV,
$V_{b}$(40Ca+124Sn)-$V_{b}$(40Ca+132Sn)=-0.3 MeV, and
$V_{b}$(48Ca+124Sn)-$V_{b}$(48Ca+132Sn)=-0.4 MeV, which seem to be
unrealistically small. However, these differences of $V_{b}$ do not influence
the slopes of the excitation functions but only lead to the shifting of the
energy scale. With realistic isospin trend of $V_{b}$
$\sigma_{cap}$(40Ca+124Sn)$<\sigma_{cap}$(48Ca+124Sn) and
$\sigma_{cap}$(40Ca+132Sn)$<\sigma_{cap}$(48Ca+132Sn) at energies above the
corresponding Coulomb barriers.

Figure 6:  (Color online) The same as in Fig. 1, for the indicated reactions
60Ni + 100Mo,150Nd (solid lines), and 64Ni + 100Mo,150Nd (dashed lines). For
the reactions 60Ni + 100Mo and 60Ni + 150Nd, the calculated capture cross
sections without the neutron transfer are shown by dotted lines. The
experimental data for the reactions 60Ni + 100Mo (closed squares) and 64Ni +
100Mo (open squares) are from Ref. Scarlassara .

One can find reactions with a positive $Q$-values of the two-neutron transfer
where the transfer weakly influences or even suppresses the capture process.
This happens if after the transfer the deformations of the nuclei do not
change much or even decrease. For instance, in the reactions
60Ni($\beta_{2}\approx 0.1$) +
100Mo($\beta_{2}=0.231$)$\to^{62}$Ni($\beta_{2}=0.198$) +
98Mo($\beta_{2}=0.168$) ($Q_{2n}=4.2$ MeV), 64Ni($\beta_{2}\approx 0.087$) +
100Mo($\beta_{2}=0.231$)$\to^{66}$Ni($\beta_{2}=0.158$) +
98Mo($\beta_{2}=0.168$) ($Q_{2n}=0.94$ MeV), and 60Ni($\beta_{2}\approx 0.1$)
+ 150Nd($\beta_{2}=0.285$)$\to^{62}$Ni($\beta_{2}=0.198$) +
148Nd($\beta_{2}=0.204$) ($Q_{2n}=6$ MeV) we expect a weak dependence of the
capture cross section on the neutron transfer (Fig. 6). There is the
experimental evidence Scarlassara  of such an effect for the 60Ni + 100Mo
reaction. So, the two-neutron transfer channel with large positive
$Q_{2n}$-value weakly influences the fusion (capture) cross section. The
reduced capture cross sections in the reactions 60Ni + 100Mo,150Nd are close
to each other in contrast to those in the reactions 58,64Ni + 132Sn,130Te. The
60Ni + 150Nd reaction has even a small suppression due to the neutron
transfer.

Figure 7:  The calculated capture cross sections vs $E_{\rm c.m.}$ for the
reactions 32S+108,110Pd (dashed lines) and 36S+106,108Pd (solid lines) (a,b).
For the 32S+110Pd reaction (a), the calculated capture cross section without
the neutron transfer process is shown by a dotted line. For the reactions
32S+110Pd, the experimental data from Pengo  and Stefanini3236s110pd  are
marked by open squares and stars, respectively.  Figure 8:  The same as in
Fig. 7, for the reactions 32S+104,106Pd (dashed lines) and 36S+104,110Pd
(solid lines) (a,b). The dotted lines correspond to the reactions
32S+104,106Pd when the neutron transfer is disregarded. The experimental data
(symbols) are from Ref. Pengo .  Figure 9:  The calculated capture cross
sections vs $E_{\rm c.m.}$ for the reactions 32S+100,102,104Ru (dashed lines)
(a,b,c) and 36S+100,102,104Ru (solid lines) (a,b). The dotted lines correspond
to the reactions 32S+102,104Ru (a,c) when the neutron transfer is disregarded.
The experimental data (symbols) are from Ref. Pengo .

Figures 7-9 show the capture excitation function for the reactions
32,36S+Pd,Ru as a function of the bombarding energy. One can see a relatively
good agreement between the calculated results and the experimental data Pengo
. The $Q_{2n}$-values for the $2n$-transfer processes are positive (negative)
for all reactions with 32S (36S). At energies above and near the Coulomb
barrier the cross sections with and without two-neutron transfer are almost
similar. After the $2n$-transfer (before the capture) in the reactions
32S($\beta_{2}=0.312$)+110Pd($\beta_{2}=0.257$)$\to^{34}$S($\beta_{2}=0.252$)+108Pd($\beta_{2}=0.243$),
32S($\beta_{2}=0.312$)+108Pd($\beta_{2}=0.243$)$\to^{34}$S($\beta_{2}=0.252$)+106Pd($\beta_{2}=0.229$),
32S($\beta_{2}=0.312$)+106Pd($\beta_{2}=0.229$)$\to^{34}$S($\beta_{2}=0.252$)+104Pd($\beta_{2}=0.209$),
32S($\beta_{2}=0.312$)+104Pd($\beta_{2}=0.209$)$\to^{34}$S($\beta_{2}=0.252$)+102Pd($\beta_{2}=0.196$),
or
32S($\beta_{2}=0.312$)+104Ru($\beta_{2}=0.271$)$\to^{34}$S($\beta_{2}=0.252$)+102Ru($\beta_{2}=0.24$),
32S($\beta_{2}=0.312$)+102Ru($\beta_{2}=0.24$)$\to^{34}$S($\beta_{2}=0.252$)+100Ru($\beta_{2}=0.215$),
32S($\beta_{2}=0.312$)+100Ru($\beta_{2}=0.215$)$\to^{34}$S($\beta_{2}=0.252$)+98Ru($\beta_{2}=0.195$)
the deformations of the nuclei decrease and the values of the corresponding
Coulomb barriers increase. As a result, the transfer suppresses the capture
process in these reactions at the sub-barrier energies. The suppression
becomes stronger with decreasing energy (Figs. 7-9). As seen in Fig. 7, the
capture cross sections calculated without two-neutron transfer are larger than
those calculated with two-neutron transfer in the case of the 32S+110Pd
reaction. The enhancement of the sub-barrier fusion for the reactions with 32S
with respect to the reactions with 36S is related to a larger deformation of
34S in comparison with 36S. We observe the same behavior in the reactions
32,36S+94,96,98,100Mo.

Figure 10:  (Color online) The calculated (solid line) capture cross sections
vs $E_{\rm c.m.}$ for the reactions 16O+76Ge and 18O+74Ge (the curves
coincide). For the 18O+74Ge reaction, the calculated capture cross sections
without neutron transfer are shown by dotted line. The experimental data for
the reactions 16O+76Ge (open circles) and 18O+74Ge (open squares) are from
Ref. Jia . The experimental data for the 16O+76Ge reaction (solid circles) are
from Ref. 16OAGe .  Figure 11:  The calculated capture cross sections vs
$E_{\rm c.m.}$ for the reactions 18S+112,118,124Sn (solid, dashed and dotted
lines, respectively) (a) and 32S+112,116,120Sn (solid, dashed and dotted
lines, respectively) (b). The experimental data (symbols) are from Ref. AOASn
; Tripathi .

Figures 10 and 11 show the excitation functions for the reactions
18O+74Ge,112,118,124Sn and 32S+112,116Sn. For the 32S-induced reactions,
$Q_{2n}>0$. For the projectile 18O there is a large range of positive
$Q_{2n}$-values, for example, varying from 1.4 MeV for 18O+124Sn up to 5.5 MeV
for 18O+112Sn. The agreement between the calculated results and the
experimental data Jia ; AOASn  is rather good. As seen in Fig. 11, the cross
sections increase systematically with the target mass number and run nearly
similarly down to the lowest energy treated. In the reactions
32S($\beta_{2}=0.312$)+112Sn($\beta_{2}=0.123$)$\to^{34}$S($\beta_{2}=0.252$)+110Sn($\beta_{2}=0.122$),
32S($\beta_{2}=0.312$)+116Sn($\beta_{2}=0.112$)$\to^{34}$S($\beta_{2}=0.252$)+114Sn($\beta_{2}=0.121$),
18O($\beta_{2}=0.1$) + 74Ge($\beta_{2}=0.283$)$\to^{16}$O($\beta_{2}=0$) +
76Ge($\beta_{2}=0.262$),
18O($\beta_{2}=0.1$)+112Sn($\beta_{2}=0.123$)$\to^{16}$O($\beta_{2}=0$)+114Sn($\beta_{2}=0.121$),
18O($\beta_{2}=0.1$)+118Sn($\beta_{2}=0.111$)$\to^{16}$O($\beta_{2}=0$)+120Sn($\beta_{2}=0.104$),
and
18O($\beta_{2}=0.1$)+124Sn($\beta_{2}=0.095$)$\to^{16}$O($\beta_{2}=0$)+126Sn($\beta_{2}=0.09$)
the $2n$-transfer suppresses the capture process (Figs. 10 and 11). The sub-
barrier capture cross sections for the systems 18O+ASn studied here do not
show any strong dependence on the mass number of the target isotope. Our
results show that cross sections for reactions 16O+76Ge (16O+114,120,126Sn)
[$Q_{2n}<0$] and 18O+74Ge (18O+112,118,124Sn) are very similar (Fig. 10). Just
the same behavior was observed in the recent experiments 16,18O+76,74Ge Jia .

###  III.2 Neutron transfer in reactions with weakly bound nuclei

After the neutron transfer in the reactions
13C+232Th($\beta_{2}=0.261$)$\to^{14}$C($\beta_{2}=-0.36$)+231Th($\beta_{2}=0.261$)
($Q_{1n}=1.74$ MeV),
15C+232Th($\beta_{2}=0.261$)$\to^{14}$C($\beta_{2}=-0.36$)+233Th($\beta_{2}=0.261$)
($Q_{1n}=3.57$ MeV) the deformations of the target or projectile nuclei in
these reactions and in the 14C+232Th($\beta_{2}=0.261$) ($Q_{1n,2n}<0$)
reaction are the same. In Fig. 12 the calculated cross sections slightly
increase with the mass number of C, and are nearly parallel down to the lowest
energy treated. There is a relatively good agreement between the calculated
results EPJSub3  and the experimental data Alcorta ; CTh  for the reactions
12,13,14C+232Th, but the experimental enhancement of the cross section in the
15C+232Th reaction at sub-barrier energies cannot be explained with our and
other Alcorta  models. Because we take into account the neutron transfer
(15C$\to^{14}$C), one can suppose that this discrepancy is attributed to the
influence of the breakup channel Gomes  which is not considered in our model.
However, it is unclear why the breakup process influences only two
experimental points at lowest energies. Different deviations of these points
in energy from the calculated curve in Fig. 12 create doubt in an influence of
the breakup on the kinetic energy. So, additional experimental and theoretical
investigations are desirable.

Figure 12:  (Color online) The calculated (lines) and experimental (symbols)
capture cross sections vs $E_{\rm c.m.}$ for the reactions 12C+232Th (dash-
dotted line, solid triangles), 13C+232Th (dotted line, open triangles),
14C+232Th (solid line, open squares), and 15C+232Th (dashed line, solid
squares). The experimental data are from Refs. Alcorta ; CTh .  Figure 13:
The calculated (lines) and experimental (symbols) capture cross sections vs
$E_{\rm c.m.}$ for the reactions 12C+208Pb (dash-dotted line), 13C+208Pb
(dotted line), 14C+208Pb (solid line), and 15C+208Pb (dashed line). The
experimental data (solid squares) for the 12C+208Pb reaction are from Ref.
12C208Pb .

The question is whether the fusion of nuclei involving weakly bound neutrons
is enhanced or suppressed at low energies. This question can been addressed to
the systems 12-15C+208Pb Alamanos3 . After the neutron transfer in the
reactions
13C+208Pb($\beta_{2}=0$)$\to^{14}$C($\beta_{2}=-0.36$)+207Pb($\beta_{2}=0$)
($Q_{1n}=1.74$ MeV),
15C+208Pb($\beta_{2}=0$)$\to^{14}$C($\beta_{2}=-0.36$)+209Pb($\beta_{2}=0.055$)
($Q_{1n}=3.57$ MeV) the deformations of the light nuclei are the same as in
the 14C+208Pb($\beta_{2}=0$) ($Q_{1n,2n}<0$) reaction. The heavy nuclei are
almost spherical. This means that the slopes of the excitation functions are
almost the same (Fig. 13). As in the case of the 15C+232Th reaction, we do not
expect enhancement of the capture cross section in the 15C+208Pb reaction
owing to the neutron transfer. The same effect was observed in Ref. Alamanos3
. The study of the reactions 15C+208Pb,232Th at sub-barrier energies provides
a good test for the verification of the effect of weakly bound nuclei on
fusion and capture because it reveals the role of other effects besides
neutron transfer.

Figure 14:  The calculated (solid line) and experimental (symbols) capture
cross sections vs $E_{\rm c.m.}$ for the reaction 9Li+70Zn. The experimental
data are from Ref. Vino .

By assuming that the $2n$-transfer process takes place and the break-up
channels are closed, one can predict almost the same capture cross sections
for the reaction with large positive $Q_{2n}$ value 6He+206Pb (9Li+68Zn) and
for the complemented reaction 4He+208Pb (7Li+70Zn). Indeed, after the transfer
in the reactions 6He+206Pb$\to^{4}$He($\beta_{2}=0$)+208Pb($\beta_{2}=0.055$)
($Q_{2n}=13.13$ MeV), 9Li+86Zn$\to^{7}$Li($\beta_{2}\approx
0.4$)+70Zn($\beta_{2}=0.248$) ($Q_{2n}=9.60$ MeV) they become equivalent to
the reactions 4He+208Pb and 7Li+70Zn. Therefore, the slopes of the excitation
functions in the reactions with 6He (9Li) and 4He (7Li) should be similar.
This conclusion supports the experimental data of Ref. Wolski , where the
authors concluded that the fusion enhancement in the 6He+206Pb reaction (with
respect to the 4He+208Pb reaction) is rather small or absent.

By assuming that the $2n$-transfer process occurs, we calculated the capture
cross sections for the 9Li+70Zn reaction (Fig. 14). The agreement with the
experimental data of Ref. Vino  is quite satisfactory. At lowest energies, the
calculated cross section is by factor of $\sim 5$ less than the experimental
value. The experimental data are well reproduced by the model Bala  where two-
neutron transfer from the 70Zn leads to 11Li halo structure and molecular bond
between the nuclei in contact enhances the fusion cross section. Note that
two-neutron transfer 9Li+70Zn$\to^{7}$Li+72Zn with $Q_{2n}=8.6$ MeV is much
energetically favorable than the two-neutron transfer
9Li+70Zn$\to^{11}$Li+68Zn with $Q_{2n}=-15.4$ MeV. These observations deserve
further experimental and theoretical investigations including the breakup
channel.

###  III.3 Breakup probabilities

The difference between the calculated capture cross section
$\sigma_{cap}^{th}$ in the absence of breakup and the experimental complete
fusion cross section $\sigma_{fus}^{exp}$ can be ascribed to the breakup
effect with the probability EPJSub4

$\displaystyle P_{\rm BU}=1-\sigma_{fus}^{exp}/\sigma_{c}^{th}.$ (3)

If at some energy $\sigma_{fus}^{exp}>\sigma_{cap}^{th}$, the values of
$\sigma_{cap}^{th}$ was normalized so to have $P_{\rm BU}\geq 0$ at any
energy. Note that $\sigma_{fus}^{exp}=\sigma_{fus}^{noBU}+\sigma_{fus}^{BU}$
contains the contribution from two processes: the direct fusion of the
projectile with the target ($\sigma_{fus}^{noBU}$), and the breakup of the
projectile followed by the fusion of the two projectile fragments with the
target ($\sigma_{fus}^{BU}$). A more adequate estimate of the breakup
probability would then be:

$\displaystyle P_{\rm BU}=1-\sigma_{fus}^{noBU}/\sigma_{cap}^{th},$ (4)

which leads to larger values of $P_{\rm BU}$ than the expression employed by
us. However, the ratio between $\sigma_{fus}^{noBU}$ and $\sigma_{fus}^{BU}$
cannot be measured experimentally but can be estimated with the approach
suggested in Ref. Maximka . The parameters of the potential are taken to fit
the height of the Coulomb barrier obtained in our calculations. The parameters
of the breakup function Maximka  are set to describe the value of
$\sigma_{fus}^{exp}$. As shown in Ref. Maximka  and in our calculations, in
the 8Be+208Pb reaction the fraction of $\sigma_{fus}^{BU}$ in
$\sigma_{fus}^{exp}$ does not exceed few percents at $E_{\rm c.m.}-V_{b}<$4
MeV. This fraction rapidly increases and reaches about 12–20%, depending on
the reaction, at $E_{\rm c.m.}-V_{b}\approx$10 MeV. Because we are mainly
interested in the energies near and below the barrier, the estimated
$\sigma_{fus}^{BU}$ does not exceed 20% of $\sigma_{fus}^{exp}$ at $E_{\rm
c.m.}-V_{b}<$10 MeV. The results for $P_{\rm BU}$ are presented, taking
$\sigma_{fus}^{noBU}$ into account in Eq. (4).

Figure 15:  (Color online) The dependence of the extracted breakup probability
$P_{BU}$ vs $E_{c.m.}-V_{b}$ for the indicated reactions with 9Be-projectiles
in %. Formula (4) was used.  Figure 16:  (Color online) The same as in Fig.
15, but for the indicated reactions with 6,7,9Li-projectiles.

As seen in Figs. 15 and 16, at energies above the Coulomb barriers the values
of $P_{\rm BU}$ vary from 0 to 84%. In the reactions 9Be+144Sm,208Pb,209Bi the
value of $P_{\rm BU}$ increases with charge number of the target at $E_{\rm
c.m.}-V_{b}>3$ MeV. This was also noted in Ref. PRSGomes5 . However, the
reactions 9Be+89Y,124Sn are out of this systematics. In the reactions
6Li+144Sm,198Pt,209Bi the value of $P_{\rm BU}$ decreases with increasing
charge number of the target at $E_{\rm c.m.}-V_{b}>3$ MeV. While in the
reactions 9Be+89Y,144Sm,208Pb,209Bi the value of $P_{\rm BU}$ has a minimum at
$E_{\rm c.m.}-V_{b}\approx 0$ and a maximum at $E_{\rm
c.m.}-V_{b}\approx-(1-3)$ MeV, in the 9Be+124Sn reaction the value of $P_{\rm
BU}$ steadily decreases with energy. In the reactions 6Li+144Sm,198Pt,209Bi,
7Li+208Pb,209Bi, and 9Li+208Pb there is maximum of $P_{\rm BU}$ at $E_{\rm
c.m.}-V_{b}\approx-(0-1)$ MeV. However, in the reactions 6Li+208Pb and
7Li+165Ho $P_{\rm BU}$ has a minima $E_{\rm c.m.}-V_{b}\approx 2$ MeV and no
maxima at $E_{\rm c.m.}-V_{b}\approx 0$. For 9Be, the breakup threshold is
slightly larger than for 6Li. Therefore, we cannot explain a larger breakup
probability at smaller $E_{\rm c.m.}-V_{b}$ in the case of 9Be.

##  IV Quasi-elastic and elastic backscattering - tools for search of breakup
process in reactions with weakly bound projectiles

The lack of a clear systematic behavior of the complete fusion suppression as
a function of the target charge requires new additional experimental and
theoretical studies. The quasi-elastic backscattering has been used Timmers ;
EPJSub4  as an alternative to investigate fusion (capture) barrier
distributions, since this process is complementary to fusion. Since the quasi-
elastic experiment is usually not as complex as the capture (fusion) and
breakup measurements, they are well suited to survey the breakup probability.
There is a direct relationship between the capture, the quasi-elastic
scattering and the breakup processes, since any loss from the quasi-elastic
and breakup channel contributes directly to capture (the conservation of the
total reaction flux):

$\displaystyle P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)+P_{BU}(E_{\rm
c.m.},J)=1,$ (5)

where $P_{qe}$ is the reflection quasi-elastic probability, $P_{BU}$ is the
breakup probability, and $P_{cap}$ is the capture probability. The quasi-
elastic scattering ($P_{qe}$) is the sum of all direct reactions, which
include elastic ($P_{el}$), inelastic ($P_{in}$), and a few nucleon transfer
($P_{tr}$) processes. In Eq. (5) we neglect the deep inelastic collision
process, since we are concerned with low energies. Equation (5) can be
rewritten as

$\displaystyle\frac{P_{qe}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}+\frac{P_{cap}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}=P_{qe}^{noBU}(E_{\rm c.m.},J)+P_{cap}^{noBU}(E_{\rm c.m.},J)=1,$ (6)

where

$P_{qe}^{noBU}(E_{\rm c.m.},J)=\frac{P_{qe}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}$

and

$P_{cap}^{noBU}(E_{\rm c.m.},J)=\frac{P_{cap}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}$

are the quasi-elastic and capture probabilities, respectively, in the absence
of the breakup process. From these expressions we obtain the useful formulas

$\displaystyle\frac{P_{qe}(E_{\rm c.m.},J)}{P_{cap}(E_{\rm
c.m.},J)}=\frac{P_{qe}^{noBU}(E_{\rm c.m.},J)}{P_{cap}^{noBU}(E_{\rm
c.m.},J)}=\frac{P_{qe}^{noBU}(E_{\rm c.m.},J)}{1-P_{qe}^{noBU}(E_{\rm
c.m.},J)}=a.$ (7)

Using Eqs. (5) and (7), we obtain the relationship between breakup and quasi-
elastic processes:

$\displaystyle P_{BU}(E_{\rm c.m.},J)=1-P_{qe}(E_{\rm
c.m.},J)[1+1/a]=1-\frac{P_{qe}(E_{\rm c.m.},J)}{P_{qe}^{noBU}(E_{\rm
c.m.},J)}.$ (8)

The reflection quasi-elastic probability $P_{qe}(E_{\rm
c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$ for bombarding energy $E_{\rm c.m.}$ and
angular momentum $J=0$ is given by the ratio of the quasi-elastic differential
cross section $\sigma_{qe}$ and Rutherford differential cross section
$\sigma_{Ru}$ at 180 degrees Timmers . Employing Eq. (8) and the experimental
quasi-elastic backscattering data with toughly and weakly bound isotopes-
projectiles and the same compound nucleus, one can extract the breakup
probability of the exotic nucleus. For example, using Eq. (8) at backward
angle, the experimental $P_{qe}^{noBU}$[4He+AX] of the 4He+AX reaction with
toughly bound nuclei (without breakup), and $P_{qe}$[6He+A-2X] of the 6He+A-2X
reaction with weakly bound projectile (with breakup), and taking into
consideration $V_{b}$(4He+AX)$\approx V_{b}$(6He+A-2X) for the very asymmetric
systems, one can extract the breakup probability of the 6He:

$\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm
c.m.},J=0)[^{6}He+^{A-2}{\rm X}]}{P_{qe}^{noBU}(E_{\rm
c.m.},J=0)[^{4}He+^{A}{\rm X}]}.$ (9)

Comparing the experimental quasi-elastic backscattering cross sections in the
presence and absence of breakup data in the reaction pairs 6He+68Zn and
4He+70Zn, 6He+122Sn and 4He+124Sn, 6He+236U and 4He+238U, 8He+204Pb and
4He+208Pb, 8Li+207Pb and 7Li+208Pb, 7Be+207Pb and 10Be+204Pb, 9Be+208Pb and
10Be+207Pb, 11Be+206Pb and 10Be+207Pb, 8B+208Pb and 10B+206Pb, 8B+207Pb and
11B+204Pb, 9B+208Pb and 11B+206Pb, 15C+204Pb and 12C+207Pb, 15C+206Pb and
13C+208Pb, 15C+207Pb and 14C+208Pb, 17F+206Pb and 19F+208Pb, leading to the
same corresponding compound nuclei, one can analyze the role of the breakup
channels in the reactions with the light weakly bound projectiles 6,8He, 8Li,
7,9,11Be, 8,9B, 15C, and 17F at near and above the barrier energies. On other
side, the experimental uncertainties could be probably smaller when the same
target-nucleus AX is used in the reactions with weakly and toughly bound
isotopes. Then, one can extract the breakup probability of the 6He [$\Delta
E=V_{b}(^{4}{\rm He}+^{A}{\rm X})-V_{b}(^{6}{\rm He}+^{A}{\rm X})]$:

$\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm
c.m.},J=0)[^{6}{\rm He}+^{A}{\rm X}]}{P_{qe}^{noBU}(E_{\rm c.m.}+\Delta
E,J=0)[^{4}{\rm He}+^{A}{\rm X}]}.$ (10)

For the very asymmetric systems, one can neglect $\Delta E$.

Using the conservation of the total reaction flux, analogously one can find
the following expression

$\displaystyle P_{BU}(E_{\rm c.m.},J)=1-\frac{P_{el}(E_{\rm
c.m.},J)}{P_{el}^{noBU}(E_{\rm c.m.},J)},$ (11)

which relates the breakup and elastic scattering processes.
$P_{el}^{noBU}(E_{\rm c.m.},J)$ is the elastic scattering probability in the
absence of the breakup process. So, one can extract the breakup probability of
the 6He at the backward angle:

$\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el}(E_{\rm
c.m.},J=0)[^{6}{\rm He}+^{A-2}{\rm X}]}{P_{el}^{noBU}(E_{\rm
c.m.},J=0)[^{4}{\rm He}+^{A}{\rm X}]}$ (12)

or

$\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el}(E_{\rm
c.m.},J=0)[^{6}{\rm He}+^{A}{\rm X}]}{P_{el}^{noBU}(E_{\rm c.m.}+\Delta
E,J=0)[^{4}{\rm He}+^{A}{\rm X}]}.$ (13)

One concludes that the quasi-elastic or elastic backscattering technique could
be a very important tool in breakup research. We propose to extract the
breakup probability directly from the quasi-elastic or elastic backscattering
probabilities of systems mentioned above.

##  V Summary

The quantum diffusion approach was applied to study the role of the neutron
transfer with positive $Q$-value in the capture reactions at sub-, near- and
above-barrier energies. We demonstrated a good agreement of the theoretical
calculations with the experimental data. We found, that the change of the
magnitude of the capture cross section after the neutron transfer occurs due
to the change of the deformations of nuclei. The effect of the neutron
transfer is an indirect effect of the quadrupole deformation. When after the
neutron transfer the deformations of nuclei do not change or slightly
decrease, the neutron transfer weakly influences or suppresses the capture
cross section. Good examples for this effect are the capture reactions 60Ni +
100Mo,150Nd, 18O + 64Ni,112,114,116,118,120,122,124Sn,204,206Pb, and
32S+96Zr,94,96,98,100Mo,100,102,104Ru,104,106,108,110Pd,112,114,116,118,120,122,124Sn.
at sub-barrier energies. Thus, the general point of view that the sub-barrier
capture (fusion) cross section strongly increases because of the neutron
transfer with a positive $Q$-values has to be revised.

The neutron transfer effect can lead to a weak influence of halo-nuclei on the
capture. Comparing the capture cross sections calculated without the breakup
effect and experimental complete fusion cross sections, the breakup was
analyzed in reactions with weakly bound projectiles. A trend of a systematic
behavior for the complete fusion suppression as a function of the target
charge and bombarding energy is not achieved. The quasi-elastic or elastic
backscattering was suggested to be an useful tool to study the behavior of the
breakup probability.

We thank P.R.S. Gomes and A. Lépina-Szily for fruitful discussions and
suggestions. This work was supported by DFG, NSFC, RFBR, and JINR grants. The
IN2P3(France)-JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are
gratefully acknowledged.

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