11institutetext: National Centre for Nuclear Research, ul. Andrzeja Sołtana 7,
05-400 Otwock, Poland 22institutetext: Department of Physics, The Florida
State University, Tallahassee, Florida 32306, U.S.A. 33institutetext: Heavy
Ion Laboratory, University of Warsaw, ul. Pasteura 5a, 02-093 Warsaw, Poland

# Strong coupling effects in near-barrier heavy-ion elastic scattering

N. Keeley 11 K.W. Kemper 2233 K. Rusek 33

(Received: date / Revised version: date)

###### Abstract

Accurate elastic scattering angular distribution data measured at bombarding
energies just above the Coulomb barrier have shapes that can markedly differ
from or be the same as the expected classical Fresnel scattering pattern
depending on the structure of the projectile, the target or both. Examples are
given such as 18O+184W and 16O+148,152Sm where the expected rise above
Rutherford scattering due to Coulomb-nuclear interference is damped by
coupling to the target excited states, and the extreme case of 11Li
scattering, where coupling to the 9Li + $n$ \+ $n$ continuum leads to an
elastic scattering shape that cannot be reproduced by any standard optical
model parameter set. An early indication that the projectile structure can
modify the elastic scattering angular distribution was the large vector
analyzing powers observed in polarised 6Li scattering. The recent availability
of high quality 6He, 11Li and 11Be data provides further examples of the
influence that coupling effects can have on elastic scattering. Conditions for
strong projectile-target coupling effects are presented with special emphasis
on the importance of the beam-target charge combination being large enough to
bring about the strong coupling effects. Several measurements are proposed
that can lead to further understanding of strong coupling effects by both
inelastic excitation and nucleon transfer on near-barrier elastic scattering.
A final note on the anomalous nature of 8B elastic scattering is presented as
it possesses a more or less normal Fresnel scattering shape whereas one would
a priori not expect this due to the very low breakup threshold of 8B. The
special nature of 11Li is presented as it is predicted that no matter how far
above the Coulomb barrier the elastic scattering is measured, its shape will
not appear as Fresnel like whereas the elastic scattering of all other loosely
bound nuclei studied to date should eventually do so as the incident energy is
increased, making both 8B and 11Li truly “exotic”.

##  1 Introduction

Angular distributions of the differential cross section for heavy-ion elastic
scattering at incident energies close to the Coulomb barrier, when plotted as
a ratio to the Rutherford scattering cross section, typically exhibit the
characteristic form often referred to as a Fresnel scattering pattern, viz.
one or more minor oscillations about the Rutherford value at small angles
followed by a larger peak before an essentially exponential fall-off as a
function of scattering angle. The large peak is due to interference between
scattering from the Coulomb and nuclear potentials and is often referred to as
the Coulomb-nuclear interference peak or the Coulomb rainbow peak. As the
incident energy is reduced towards the Coulomb barrier the small-angle
oscillatory structure and the Coulomb-nuclear interference peak move to larger
angles and become less pronounced until they disappear completely; Fig. 1
illustrates typical behaviour (optical model calculations are plotted rather
than actual data for the sake of clarity).

Figure 1: Elastic scattering angular distributions for the 16O + 208Pb system
at several near-barrier incident energies calculated using the optical model.
Optical potential parameters were taken from a fit to the data of Ref. Rud01 .
The nominal Coulomb barrier for this system corresponds to an incident energy
of about 83 MeV.

However, some systems are found to have angular distributions that differ
markedly from the norm in that where the Coulomb-nuclear interference peak
should be there is instead a large reduction of the elastic scattering cross
section compared to the Rutherford value. Figure 2 graphically illustrates
this phenomenon: Fig. 2 (a) shows a normal near-barrier heavy-ion elastic
scattering angular distribution, that for 95 MeV 16O incident on a 208Pb
target Rud01 , which contrasts with that in Fig. 2 (b) for 90 MeV 18O incident
on a 184W target Tho77 .

Figure 2: (a) Elastic scattering angular distribution for 95 MeV 16O incident
on a 208Pb target. The circles represent the data of Ref. Rud01  (re-
normalised to the Rutherford value at forward angles on this plot) while the
solid curve represents an optical model fit. (b) Elastic scattering angular
distribution for 90 MeV 18O incident on a 184W target. The circles represent
the data of Ref. Tho77  while the solid curve denotes the result of a coupled
channels calculation including Coulomb and nuclear coupling to the first
$2^{+}$ excited state of 184W. The dashed curve denotes the result of a no
coupling optical model calculation using the same optical potential as in the
coupled channels calculation.

The observed depletion of the elastic scattering cross section in the 18O +
184W system is due to strong coupling, in this case strong quadrupole Coulomb
coupling to the first $2^{+}$ excited state of 184W Tho77 . Similar effects
have also been observed due to strong quadrupole Coulomb coupling in the
projectile in the 20Ne + 208Pb system Gro78 . It is these strong coupling
effects on the elastic scattering angular distribution that form the subject
of this review. Such a review is timely since with the increasing quality of
radioactive beams rather precise measurements of angular distributions of the
elastic scattering of exotic nuclei are now feasible and it is possible to
probe the projectile structure through such scattering. Such measurements have
already been made for several weakly-bound exotic nuclei and the resulting
angular distributions are found to exhibit all the characteristics of strong
coupling effects, in these cases due to strong coupling to the projectile
continuum.

Note that in both figures we have plotted the cross section on a linear scale
rather than the more usual logarithmic one. In this we follow the
recommendation of Satchler Sat75 ; Bal75  who pointed out that a semi-
logarithmic plot can obscure the details of the nuclear-Coulomb interference
region, the part of the angular distribution most sensitive to the nuclear
interaction: “A particularly sensitive part of the angular distribution is the
oscillatory region just before the exponential fall below the Rutherford cross
section. (The details of this region may be overlooked if, as is often done,
the ratio-to-Rutherford cross section is shown on a semi-logarithmic plot. It
is more revealing to use a linear plot.)” Bal75 .

We begin with a survey of elastic scattering measurements for systems
involving stable nuclei that exhibit strong coupling effects from both
inelastic and transfer couplings. This survey aims to be representative rather
than exhaustive, in that we give examples of the types of system that exhibit
strong coupling effects. We then review measurements of the elastic scattering
of exotic nuclei that also exhibit strong coupling effects before summarising
the necessary conditions for the existence of these effects. Finally we make
some suggestions for future measurements along with the accuracy required to
explore further this phenomenon.

##  2 Strong coupling effects with stable beams

###  2.1 Systems with effects due to target coupling

The first published data to show strong coupling effects on the elastic
scattering were for the 18O and 12C \+ 184W systems at incident energies of 90
and 70 MeV respectively Tho77 . As can be seen in Fig. 3 the effect is less
marked for the 12C + 184W data, due to the smaller charge product for this
projectile, although the 12C incident energy is somewhat higher with respect
to the Coulomb barrier than for the 18O + 184W data and this will also play a
rôle in reducing the effect.

Figure 3: Elastic scattering data for the 18O + 184W and 12C + 184W systems at
90 and 70 MeV respectively. The solid curves denote the results of optical
model (18O + 208Pb) or coupled channels (18O \+ 184W and 12C + 184W)
calculations. Taken from Ref. Tho77 . Reprinted Fig. 1 with permission from
Tho77 . ⓒ 1977, The American Physical Society

Also shown in Fig. 3 is the measured angular distribution for the elastic
scattering of a 90 MeV 18O beam from a 208Pb target which exhibits the usual
Fresnel shape, providing further proof that the strong coupling effect is due
to the 184W target and not the projectile. Incidentally, a comparison of Fig.
2 and Fig. 3 well illustrates the value of presenting near-barrier heavy-ion
elastic scattering angular distributions on a linear scale.

These data were taken with the use of a QDDD magnetic spectrometer in order to
obtain the resolution necessary to separate inelastic scattering to the 111
keV $2_{1}^{+}$ first excited state of 184W from the elastic scattering. As
Thorn et al. Tho77  were careful to point out, previous measurements of the
“elastic scattering” of heavy ions from heavy deformed targets obtained the
usual Fresnel scattering angular distributions because they could not resolve
inelastic scattering to the ground state rotational bands. Thorn et al. also
showed, by explicit coupled channels calculations, that the elastic scattering
data can be well described by coupling to the 111 keV $2_{1}^{+}$ state of
184W, the effect on the elastic scattering being almost entirely due to the
Coulomb coupling (it should be noted that these elastic scattering data cannot
be described by conventional optical model potentials).

Angular distributions for the elastic scattering of 72 MeV 16O ions from 152Sm
Kim79 ; Sto81  and, to a lesser extent from 148Sm Sto81  show similar
behaviour, see Fig. 4.

Figure 4: Elastic scattering data for the 16O + 148,152Sm systems at incident
energies of 72 MeV. The data are taken from Ref. Sto81 . The solid curves
denote the result of coupled channels calculations including
$0^{+}\longleftrightarrow 2^{+}\longleftrightarrow 4^{+}$ rotational model
coupling.

Again, the main source of the effects on the elastic scattering angular
distribution is Coulomb excitation of the first $2^{+}$ excited states of the
targets, although as Kim points out Kim79 , nuclear couplings are important
for a detailed reproduction of both the elastic and inelastic scattering
angular distributions. Data were also obtained at lower energies for the 16O +
148,150,152Sm systems by Talon et al. Tal81 . These data too show significant
deviations from Rutherford scattering but the effect on the shape of the
angular distribution is not so dramatic as for the higher energy data. Details
of the experimental procedure for the 72 MeV 16O + 148,152Sm measurements are
not given in Refs. Kim79 ; Sto81  but the lower energy measurements of Talon
et al. Tal81  employed a QDDD magnetic spectrometer.

Figure 4 provides an unambiguous demonstration of the link between the
importance of the coupling effect and the strength of the $B(E2)$ and the
excitation energy of the $2^{+}_{1}$ state since the charge products and, to a
very good approximation, the Coulomb barriers are identical for the 16O +
148Sm and 152Sm systems. While both these Sm isotopes are deformed, 152Sm has
a $B(E2;0^{+}_{1}\rightarrow 2^{+}_{1})$ nearly five times that of 148Sm plus
a much lower excitation energy. This is confirmed by the 72.3 MeV 16O + 144Sm
elastic scattering data of Abriola et al. Abr89  which show a classic Fresnel
scattering angular distribution, 144Sm being a weakly-coupled spherical
nucleus with a relatively high-lying first excited state.

The above examples demonstrate that, under the right conditions, strong
coupling to low-lying excited states of the target can give rise to near-
barrier elastic scattering angular distributions that deviate significantly
from the usual Fresnel pattern. We shall explore fully what these conditions
are in a later section. For the present, suffice it to say that all systems
where the effect has been observed and has been ascribed to target couplings
involve heavy deformed target nuclei.

###  2.2 Systems with effects due to projectile coupling

Elastic scattering angular distributions for the 20Ne + 208Pb system have been
measured for incident 20Ne energies of 161.2 MeV Bal75  and 131 MeV Gro78 .
The 161.2 MeV data at first sight appear to show the usual Fresnel scattering
pattern and an acceptable description of these data could be obtained with
standard optical potentials, with the exception of the angular region between
$34^{\circ}$ and $50^{\circ}$ Bal75 . The 131 MeV data, however, unambiguously
show the strong coupling effect, the shape of the angular distribution being
similar to that for the 16O + 148Sm system (cf. Figs. 4 and 5).

Figure 5: Elastic scattering data for the 20Ne + 208Pb system at incident
energies of 131 MeV (a) and 161.2 MeV (b). Data are taken from Refs. Gro78
and Bal75  respectively. The solid curves denote the results of coupled
channels calculations including coupling to the $2^{+}_{1}$ state of 20Ne. The
dashed curves denote the results of no coupling optical model calculations
using the same optical potentials as in the appropriate coupled channels
calculations.

Both data sets were obtained using position sensitive silicon detectors
mounted on moveable arms. Coupled channels calculations reveal that the dip in
the elastic scattering angular distribution just before the main Coulomb-
nuclear interference peak at 161.2 MeV is a residual effect due to the strong
coupling to the 20Ne $2^{+}_{1}$ state, see Fig. 5 (b). These data therefore
clearly show that the strong coupling effect diminishes as the incident energy
is increased.

Elastic scattering angular distributions have also been measured for the 24Mg
+ 208Pb and 26Mg + 208Pb systems at incident energies of 145 MeV Eck81  and
200 MeV Hen89  and 200 MeV Hen91  respectively. The 145 MeV 24Mg + 208Pb data
were taken using an Enge split pole magnetic spectrometer but both sets of 200
MeV data employed standard position sensitive silicon detectors. The 200 MeV
data have elastic scattering angular distributions of similar shape to that
for the 161.2 MeV 20Ne + 208Pb data, i.e. there is only a residual strong
coupling effect, this time clearly visible as a distinct dip in the cross
section just before the main Coulomb-nuclear interference peak. The effect is
smaller for 26Mg, as might be expected due to its weaker collectivity compared
to 24Mg. The 145 MeV 24Mg + 208Pb data clearly show the strong coupling
effect, see Fig. 6.

Figure 6: Elastic scattering data for the 24Mg + 208Pb system at an incident
energy of 145 MeV. The solid curve denotes an optical model fit including a
long-range polarisation potential due to Coulomb excitation of the $2^{+}_{1}$
of 24Mg while the dashed curve indicates the result of an optical model
calculation with the same potential but without the Coulomb polarisation
potential. Taken from Ref. Eck81 . Reprinted Fig. 4 with permission from Eck81
. ⓒ 1981, The American Physical Society

Our final example of a system with strong coupling effects due to the
projectile is 28Si + 208Pb. Elastic scattering data for this system are
available at incident energies of 162 MeV Eck81 , 166 MeV Voj87 , 209.8 MeV
Chr84  and 225 MeV Kol84 . The influence of strong coupling is apparent for
all these data sets although the effect becomes weaker as the incident energy
is increased. The measurements of Refs. Eck81 ; Voj87 ; Kol84  were made using
Enge split pole spectrometers while that of Ref. Chr84  employed a QDDD
spectrometer.

For all these systems it is strong Coulomb coupling to the $2^{+}_{1}$ excited
state of the projectile that is predominantly responsible for the strong
coupling effect on the elastic scattering, similarly to what was found for the
systems with strongly coupled targets. This raises the intriguing question as
to what the angular distribution for a system with a strongly deformed
projectile and a strongly deformed target would look like.

###  2.3 Systems with effects due to projectile and target coupling

As an example of a system with strong coupling in both projectile and target
we cite 20Ne + 148Nd. Data for the elastic and inelastic scattering in this
system were measured using a QDDD magnetic spectrometer at an incident energy
of 116 MeV Jia91 . The shape of the angular distribution is similar to those
for systems with strong coupling in either the target alone or the projectile
alone, thus the presence of strong coupling in both target and projectile does
not seem to affect the qualitative nature of its effect on the elastic
scattering, see Fig. 7.

Figure 7: Elastic scattering data for the 20Ne + 148Nd system at an incident
energy of 116 MeV. The data are taken from Ref. Jia91 . The solid curve
denotes the result of a coupled channels calculation including both 20Ne and
148Nd $0^{+}_{1}\rightarrow 2^{+}_{1}$ coupling (but no mutual excitation),
the dotted curve the result of a calculation including the 148Nd
$0^{+}_{1}\rightarrow 2^{+}_{1}$ coupling only, the dot-dashed curve a
calculation including the 20Ne $0^{+}_{1}\rightarrow 2^{+}_{1}$ coupling only
and the dashed curve the no coupling calculation.

In spite of a large cross section for mutual excitation of the 20Ne and 148Nd
$2^{+}_{1}$ states it was found that coupling to mutual excitation had little
influence on the elastic scattering angular distribution Jia91 . The effect
seems to be due to the individual couplings to the separate $2^{+}_{1}$
states, although the calculations plotted in Fig. 7 suggest that the combined
effect of projectile and target couplings is greater than the sum of its
parts.

###  2.4 Strong coupling effects with polarised projectiles

An example of a different type of strong coupling effect is provided by the
analysing powers for elastic scattering of polarised heavy ion beams.
Analysing powers measured in experiments with polarised beams were found to be
even more sensitive to the coupling effects than the cross sections. In the
1980s many experiments with polarised 6,7Li and 23Na beams were performed at
the Max Planck Institute for Nuclear Physics in Heidelberg, Germany fick .
Later on these studies were moved to the Nuclear Structure Facility of the
Daresbury Laboratory, U.K., and in the second part of the 1990s to the Florida
State University, U.S.A. The very first experiments with vector polarised 6Li
beams scattered from nickel targets already revealed large values of the
vector analysing powers (iT11) that could not be explained by a “static” spin-
orbit interaction derived from the deuteron-target or the nucleon-nucleon
spin-orbit potentials by means of folding methods. The analysing powers
predicted by optical model calculations with spin-orbit interactions
calculated in this way were much too small to explain the experimental data.

Figure 8: Elastic scattering angular distributions for polarised 6Li + 58Ni
compared to optical model and coupled channels calculations. The analysing
powers are almost completely generated by the coupling to the 3+ resonant
state of 6Li. Taken from Ref. nishioka , Fig. 5.

The explanation of this “spin crisis” came from microscopic coupled channels
calculations that included couplings to the 3+ resonant first excited state of
6Li nishioka . In Fig. 8 the 6Li + 58Ni elastic scattering data (angular
distributions of the differential cross section and the vector analysing
powers) taken from rusek  are compared with the model predictions. The short-
dashed curves represent the results of an optical model calculation with the
spin-orbit potential derived from the deuteron-target interaction by means of
the Watanabe-type cluster-folding method. This calculation underpredicted the
measured differential cross section and generated values of the analysing
power that are very close to zero. Inclusion of couplings to the resonant
first excited state of the projectile in coupled channels calculations
produced much larger values of the analysing powers, close to experiment
(solid curves). The inclusion of further couplings to the resonances at 4.31
and 5.65 MeV excitation energy improved the description of the data still
further (dashed curves). Further improvement would require couplings to the
non-resonant $\alpha$ \+ $d$ continuum.

Figure 9: Comparison of 60 MeV 6Li + 26Mg elastic scattering data with the
results of coupled channels calculations. The dotted curves denote the result
of an optical model calculation including a cluster-folded spin-orbit
potential, the dashed curves denote the result of a coupled channels
calculation including couplings to all three 6Li resonant states and the solid
curves denote the result of a coupled discretised continuum channels
calculation including couplings to the non-resonant $\alpha$ \+ $d$ continuum.
Note that the analysing power arises entirely from channel couplings. Taken
from Ref. rusek1 . Reprinted Fig. 3 with permission from rusek1 . ⓒ 1994, The
American Physical Society

This is illustrated in Fig. 9. The experimental data for 60 MeV polarised 6Li
+ 26Mg elastic scattering were obtained at the Nuclear Structure Facility of
the Daresbury Laboratory rusek1 . As in the previous case, optical model
calculations with the cluster-folded spin-orbit potential included generated
negligible values of the vector analysing power (dotted curves). Couplings to
all three resonances of 6Li improved the description of the differential cross
section angular distribution and produced sizable values of the analysing
power (dashed curves). Finally, inclusion of couplings to the non-resonant
continuum resulted in a good reproduction of both sets of data (solid curves).
One should mention that these calculations were free of any adjustable
parameters.

Finally, in Ref. Van89  an extensive coupled channels analysis of 6Li + 12C
and 6Li + 16O elastic and inelastic scattering data found that for these light
target systems the elastic scattering vector analysing power could only be
satisfactorily explained by explicitly including couplings to the $L=2$
resonances of 6Li. However, it was found that the static spin-orbit potential
did have an influence on the final result for the calculated analysing power
for these systems via a complicated interference with the dynamic coupling
effects.

###  2.5 Strong coupling effects without strong couplings

The title of this final sub-section dealing with strong coupling effects on
elastic scattering involving stable heavy-ion beams would appear at first
sight to be something of a paradox: how is it possible to have a “strong
coupling” effect on the elastic scattering in the absence of strong couplings?
However, this phenomenon does occur.

Figure 10: Elastic scattering data for the 58Ni + 64Ni system at incident
energies of 203.8 MeV (left) and 219.2 MeV (right). The solid curves denote
coupled channels calculations including coupling to the $2^{+}_{1}$ states of
both nuclei while the dashed curves are optical model calculations. Taken from
Ref. Rui92 , Fig. 4.

Neither 58Ni nor 64Ni are considered to exhibit strong coupling, their
relatively high-lying $2^{+}_{1}$ levels ($E_{\mathrm{x}}=1.45$ MeV for 58Ni
and 1.35 MeV for 64Ni) being rather weakly-coupled vibrational states. The
elastic scattering data of Ref. Rui92  for the 58Ni + 64Ni system at two near-
barrier energies nevertheless show the characteristic “strong coupling” form
rather than the usual Fresnel scattering pattern, see Fig. 10. The apparent
paradox may be resolved by considering the large charge product for this
system which transforms the weak coupling of the Ni $2^{+}_{1}$ levels into a
strong Coulomb coupling.

Similar effects should be exhibited by the near-barrier elastic scattering of
most weakly-coupled medium mass vibrational nuclei from a sufficiently heavy
target. This indeed seems to be the case, see e.g. the 40Ar + 208Pb elastic
scattering data of Ref. Arl80  and the 58Ni + 208Pb data of Ref. Bec87 .
However, a possible exception is the 40Ca \+ 208Pb system. A quasi-elastic
scattering angular distribution for this system has been reported for an
incident 40Ca energy of 236 MeV Szi04 . While it is not explicitly stated in
Ref. Szi04  exactly which channels are included in the definition of “quasi-
elastic” the angular distribution does exhibit the shape associated with
strong coupling effects, from which we may infer that the elastic scattering
angular distribution will also do so. However, coupled channels calculations
suggest that, due to the doubly-magic nature of 40Ca and its consequent
structure properties, the effect is not produced by inelastic couplings.
Rather, it appears that this system may be a case where strong transfer
couplings produce a similar effect on the elastic scattering. Calculations
presented in Ref. Szi04  seem to bear this out and conventional coupled
reaction channels calculations including the 208Pb(40Ca,41Ca)207Pb single
neutron pickup coupling confirm this, see Fig. 11.

Figure 11: Calculated elastic scattering angular distributions for the 40Ca +
208Pb system at an incident energy of 236 MeV for the no coupling (dashed
curve) and coupled reaction channels (solid curve) cases. The coupled reaction
channels calculation includes couplings to the 208Pb(40Ca,41Ca)207Pb single
neutron pickup reaction only. The points denote the quasi-elastic scattering
data of Ref. Szi04 .

##  3 Strong coupling effects with radioactive beams

The elastic scattering of certain light, weakly-bound radioactive nuclei from
heavy targets has also been found to exhibit the non-Fresnel pattern angular
distributions characteristic of strong coupling effects. For these nuclei it
is strong coupling to the low-lying continuum that is responsible. A general
review of available data for light radioactive beams was given in Ref. Kee09
so that we shall only give representative examples here.

A particularly good example of the effect of strong coupling to the low-lying
continuum on the near-barrier elastic scattering of a weakly-bound radioactive
nucleus is the 6He + 208Pb system. Figure 12 (a) shows the 27 MeV 6He + 208Pb
elastic scattering angular distribution of Kakuee et al. Kak03 .

Figure 12: (a) Elastic scattering angular distribution for the 6He + 208Pb
system at an incident energy of 27 MeV. The solid curve denotes a coupled
discretised continuum channels calculation while the dashed curve denotes the
corresponding no coupling optical model calculation. The dotted curve denotes
the result of a coupled discretised continuum channels calculation with
nuclear coupling only (but including diagonal Coulomb potentials). The data
are taken from Ref. Kak03 . (b) The solid curve denotes a coupled discretised
continuum channels calculation of the 6He + 208Pb elastic scattering at an
incident 6He energy of 45 MeV while the dashed curve denotes the corresponding
no coupling optical model calculation. The dotted curve denotes the result of
a coupled discretised continuum channels calculation with nuclear coupling
only (but including diagonal Coulomb potentials).

Coupled discretised continuum channels calculations have confirmed that the
effect is dominated by Coulomb dipole coupling to the low-lying $\alpha+n+n$
continuum. However, since this is a strong Coulomb coupling effect the charge
product of the projectile-target system must be sufficiently large for the
effect to be observed; the 6He + 64Zn data of Refs. Dip03 ; Dip04  exhibit the
classic Fresnel shape. This question was examined further in Ref. Kuc09  where
a series of coupled discretised continuum channels calculations suggested that
a target with an atomic number greater than about 80 was required in order
unambiguously to see the strong coupling effect. Figure 12 (b) underlines that
careful selection of the incident energy is also required if the strong
coupling effect is to be observed; if the energy is too high with respect to
the nominal Coulomb barrier then even for a 208Pb target the 6He elastic
scattering is expected to present a more or less conventional Fresnel type
shape (45 MeV represents an incident energy approximately 2.5 times that
equivalent to the nominal Coulomb barrier for the 6He + 208Pb system).

Preliminary data for the elastic scattering of 8He from a 208Pb target at an
incident energy of 22 MeV have recently been published Mar13 . They show a
similar shape to the 6He + 208Pb data at the same incident energy Aco11 ,
although given the higher breakup thresholds of 8He, combined with a
considerably larger spectroscopic factor for single-neutron removal, coupling
to neutron stripping reactions may turn out to have a more prominent rôle in
8He elastic scattering than in 6He where its effect is most important at large
angles, see e.g. Ref. Kee08 .

The elastic scattering of 11Li from 208Pb at near- and sub-barrier energies
has recently been measured and shows spectacular departures from the Fresnel
shape Cub12 . This too has been shown to be due mostly to strong Coulomb
dipole coupling to the low-lying continuum. The coupling effect is much
stronger than for 6He, in part due to the extra charge on the 11Li nucleus but
mostly due to the much stronger Coulomb dipole polarisability of 11Li Kee13 .
Figure 13 plots the data for 29.8 MeV 11Li + 208Pb elastic scattering compared
to a coupled discretised continuum channels calculation. The great importance
of Coulomb couplings may be seen by comparing the solid and dotted curves in
the figure, nuclear coupling only becoming important for angles greater than
about $80^{\circ}$.

Figure 13: Elastic scattering angular distribution for the 11Li + 208Pb system
at an incident energy of 29.8 MeV. The solid and dashed curves denote the
results of coupled discretised continuum channels and no coupling calculations
respectively. The dotted curve shows the result of a coupled discretised
continuum channels calculation with nuclear coupling only (but including
diagonal Coulomb potentials). The data are taken from Ref. Cub12 . Adapted
from Ref. Kee13 , Fig. 1.

Quasi-elastic scattering data for the 11Be + 64Zn system have been measured at
an incident energy of 28.7 MeV DiP10 . The data are quasi-elastic since
inelastic scattering to the low-lying bound first excited state of 11Be could
not be separated from the elastic scattering, although for most practical
purposes they may be regarded as pure elastic. For this system, and unlike the
case for the 6He + 64Zn elastic scattering Dip03 ; Dip04 , the angular
distribution has the characteristic shape associated with strong coupling
effects. The persistence of strong coupling effects for elastic scattering
from a medium mass target could partly be due to the extra charge on the 11Be
nucleus compared to 6He. However, coupled discretised continuum channels
calculations found that, unlike all the other systems mentioned in this
review, strong nuclear couplings—in this case to the low-lying 10Be + $n$
continuum—appeared to have the most important influence on the 11Be elastic
scattering Kee10 , see Fig. 14.

Figure 14: Quasi-elastic scattering angular distribution for the 11Be + 64Zn
system at an incident energy of 28.7 MeV. The solid curve denotes the result
of the full coupled discretised continuum channels calculation including both
Coulomb and nuclear couplings, while the dot-dashed curve denotes the result
of a similar calculation but with nuclear couplings only while retaining the
diagonal Coulomb potentials. The dashed curve is the result of a calculation
including coupling (both Coulomb and nuclear) to the 320 keV $1/2^{-}$ first
excited state of 11Be only. Taken from Ref. Kee10 . Reprinted Fig. 3 with
permission from Kee10 . ⓒ 2010, The American Physical Society

The exact importance of the nuclear couplings is to some extent model
dependent, see Ref. DiP12 . However, it is clear that the nuclear couplings
are far more important than for 6He or 11Li (or, indeed for the strongly-
coupled systems involving stable nuclei) and are vital in reproducing the
shape of the angular distribution. The calculations of Refs. Kee10  and DiP12
both neglect the possibility of excitation of the 10Be core. Extensions of the
standard coupled discretised continuum channels technique have been developed
to include core excitation effects Sum06 ; Sum14 ; Die14  and have recently
been applied to these data Die14 . It was found that the standard coupled
discretised continuum channels calculations of Ref. DiP12  gave similar
results for the quasi-elastic scattering to those of the more sophisticated
model including core excitation Die14  although the latter was in better
agreement with the inclusive breakup cross section data.

We end this brief survey of strong coupling effects for systems involving
radioactive beams with another paradox. In this case it is a nucleus that
would be expected a priori to show such effects in its elastic scattering but
which apparently does not, 8B. The threshold against 8B $\rightarrow$ 7Be +
$p$ breakup is just 137.5 keV so that the cross section for this process
should be large and one would naively expect commensurate effects on the
elastic scattering due to coupling to the continuum. However, coupled
discretised continuum channels calculations suggest that the influence on the
elastic scattering of coupling to the 7Be + $p$ continuum is small, even for a
208Pb target Kee09 , see Fig. 15.

Figure 15: Elastic scattering angular distribution for the 8B + 208Pb system
at an incident energy of 60 MeV. The full curve denotes the result of a
coupled discretised continuum channels calculation while the dashed curve
represents the result of the no coupling optical model calculation. Adapted
from Fig. 25 of Ref. Kee09  (see Ref. Kee09  for details of the calculation).

While the model of 8B employed in these calculations is simplified to the
extent that excitations of the 7Be core are ignored, the 170.3 MeV 8B + NatPb
elastic scattering data of Yang et al. Yan13  suggest that it is sufficiently
realistic. Coupling effects are expected to be minimal at this
energy—approximately three times the Coulomb barrier—and indeed the measured
angular distribution shows a classic Fresnel scattering pattern Yan13 .
However, calculations for near-barrier energies suggest that any suppression
of the nuclear-Coulomb interference peak for 8B + 208Pb elastic scattering
will be due largely to “static” effects rather than coupling to the continuum
Kee09 . This is borne out by similar calculations Lub09  for the near-barrier
8B + 58Ni data of Ref. Agu09 . Unfortunately 8B is a difficult beam to
produce, particularly at lower energies, so that the available near-barrier
data do not have a sufficient number of points to confirm this experimentally
Agu09 .

Part of the explanation for this behaviour may come from the fact that in 8B
the “valence” particle is a proton rather than a neutron, thus allowing subtle
interference effects involving the Coulomb interaction with the core in
addition to the presence of a Coulomb barrier tending to suppress the large
radius tail of the valence particle wave function. Indeed, Kumar and
Bonaccorso Kum11  have shown that with regard to the nuclear breakup the
proton in 8B behaves like a more tightly bound neutron. Later work showed the
importance of interference effects for breakup observables Kum12 .
Nevertheless, the charge on the proton does not seem to account entirely for
the lack of coupling effect on the near-barrier 8B elastic scattering, since
coupling to breakup has a much larger effect for 6Li elastic scattering at
similar energies with respect to the Coulomb barrier in spite of the very much
larger breakup threshold (and consequently much smaller breakup cross section)
for this projectile compared to 8B, see for example Ref. Kee10a .

##  4 Conditions governing strong coupling effects

###  4.1 General considerations

In this section we explore the conditions necessary for strong coupling
effects on near-barrier elastic scattering to manifest themselves. In the
previous two sections we have seen how systems with strong coupling exhibit
near-barrier elastic scattering angular distributions of a characteristic
shape that, at first sight, appears to have nothing in common with the usual
Fresnel-type elastic scattering pattern. However, like the conventional
Fresnel pattern the strong coupling angular distribution shape does evolve as
a function of bombarding energy, see for example the 28Si + 208Pb data of
Refs. Eck81 ; Voj87 ; Chr84 ; Kol84 . In order to illustrate this more clearly
we present in Fig. 16 a series of coupled channels calculations for the 18O +
184W system at incident energies ranging from somewhat below the nominal
Coulomb barrier ($V_{\mathrm{B}}=70.8$ MeV, equivalent to a bombarding energy
of about 77.7 MeV) to approximately $1.5\times V_{\mathrm{B}}$.

Figure 16: Coupled channels calculations for the 18O + 184W system at several
sub- and near-barrier bombarding energies. The input parameters are identical
with those of the coupled channels calculations in Ref. Tho77 . The nominal
Coulomb barrier for this system corresponds to a bombarding energy of about
77.7 MeV.

The calculations use the same optical potential and coupling parameters as
those of Ref. Tho77  since in this figure we are investigating the qualitative
evolution of the shape of the angular distribution as a function of bombarding
energy rather than attempting to make quantitatively accurate predictions of
the elastic scattering at a particular energy.

A comparison with Fig. 1 will show that, despite the superficially completely
different shapes of the elastic scattering angular distributions for the
strongly-coupled system, what we find in Fig. 16 is in fact a standard
Fresnel-type scattering pattern superimposed on a strong Coulomb coupling
angular distribution shape. As the bombarding energy is increased from 75 MeV
we see an evolution from a completely structureless angular distribution to
the emergence of a strong Coulomb-nuclear interference peak as the real
nuclear potential becomes of more importance, exactly as in the classic
Fresnel scattering pattern. The difference in the strongly-coupled system is
that now, due to depletion of the elastic scattering by the strong Coulomb
excitation, the Coulomb-nuclear interference maximum is peaked at much less
than the Rutherford value, except for energies considerably above the Coulomb
barrier.

###  4.2 Strong coupling equivalent to a long-range DPP

Thus far we have explained strong coupling effects in terms of coupled
channels calculations. However, on publication of the 18O + 184W data of Thorn
et al. Tho77  it was almost immediately demonstrated that they could also be
explained in terms of a long-range dynamic polarisation potential (DPP)
induced by the Coulomb coupling which, when added to a conventional optical
model potential, provided a good description of the data. This DPP could
easily be calculated from a single parameter, the $B(E\lambda)$ value, making
use of some approximations Lov77 ; Lov77a ; Bal78 ; Bal79 . The formalism of
Baltz et al. Bal78 ; Bal79  is more sophisticated than that of Love et al.
Lov77 ; Lov77a  although both give similar results. There is now a
considerable literature on the calculation using various approximations of
Coulomb excitation DPPs and their application to elastic scattering data, most
recently including radioactive beam data.

Love et al. and Baltz et al. assume the sudden approximation, which leads to a
predominantly imaginary DPP (the real parts of their DPPs are in fact
negligible) and most subsequent work on Coulomb DPPs has made similar
assumptions. However, if the adiabatic approximation is appropriate, i.e. if
the collision time $\gg$ the period of internal motion, a purely real,
attractive DPP results Sat83 . In reality, the DPP will be complex and the
real and imaginary parts of the DPPs extracted from coupled discretised
continuum channels calculations for light weakly-bound nuclei are usually
comparable in size, see e.g. Ref. Mac09 . The form of the real part of such
DPPs at large radii matches very well the adiabatic model expression for the
(purely real) DPP which is directly related to the polarisability of the
weakly-bound nucleus. The dipole polarisability of light weakly-bound nuclei
may thus be obtained from fits to the real part of the DPP extracted from
appropriate coupled discretised continuum channels calculations, see for
example Refs. Kee13  and Par11 . Starting from analytic expressions for the
imaginary polarisation potential for Coulomb dipole excitation, the
corresponding real polarisation potential has been obtained through a
dispersion relation involving the Coulomb dipole excitation energy And94 ,
thus yielding an analytic expression for a complex Coulomb dipole DPP. This
formalism was applied to the 11Li + 208Pb elastic scattering at 50 MeV,
producing a prediction for the angular distribution in remarkable qualitative
agreement with the recent measurements at somewhat lower incident 11Li
energies Cub12 .

While coupled channels calculations are now routine, at least for stable
nuclei where the number of states involved is small, the expressions for the
Coulomb DPPs do give valuable insight into the conditions necessary for strong
coupling effects to be maximal. For example, Love et al. Lov77a , referring to
their expression for the Coulomb DPP state that it: “…provides an explicit
guide as to when coupling to a particular inelastic channel may be important.
Clearly, the effects should be most important in those cases in which the
target and/or projectile has a large $B(E\lambda)$ and the projectile and/or
the target possesses a large Z.”

###  4.3 The conditions under which strong coupling effects are expected

This leads us conveniently into a discussion of the necessary conditions under
which strong coupling effects on the near-barrier elastic scattering are
expected to appear. Frahn and Hill Fra78  have given an excellent summary of
how the strong coupling influence depends on various parameters which we may
conveniently quote here: “…based solely on the long-ranged nature of the
polarization potential …some general predictions could be made:

  1. i)

Dynamic polarization causes an overall reduction of the differential cross
section which is small at small angles and becomes gradually stronger with
increasing $\theta$.

  2. ii)

The polarization probability increases rapidly with increasing deformation of
the target and/or projectile.

  3. iii)

For a given ratio of the energy to the Coulomb barrier energy, the
polarization increases strongly with the mass numbers of projectile and
target.

  4. iv)

For a given projectile-target system the polarization effect diminishes
rapidly with increasing energy.”

to which we might also add the excitation energy of the strongly coupled
state; even a strongly coupled state will not produce the characteristic
strong coupling elastic scattering angular distribution if its excitation
energy is high. This is, of course, a somewhat subjective statement since
whether or not the strong coupling effect is present also depends on the
incident energy. However, since such effects are usually limited to incident
energies relatively close to the Coulomb barrier states with excitation
energies of approximately 5-10 MeV would not be expected to have much
influence on the elastic scattering in the relevant energy regime regardless
of their coupling strength. A proper investigation of the interplay of these
two linked properties—excitation energy of the coupled state(s) and projectile
incident energy—is beyond the scope of the present review but would form an
interesting subject for further study.

Summarising the above information, we would expect strong coupling effects to
be present in the near-barrier elastic scattering under the following
circumstances:

  1. 1.

Systems where the target is a well-deformed medium to heavy mass nucleus with
a low-lying (a few hundred keV or less) first excited state. In this instance
the projectile may be a relatively light, more or less inert nucleus such as
12C or 16O or similar, the only requirement being that the charge product is
sufficiently large since the Coulomb coupling effect is dominant in such
systems. If the target is sufficiently heavy, i.e. it carries a large enough
charge, even $\alpha$ particle elastic scattering should show the strong
coupling effect under these circumstances, see for example Refs. Lov77a ;
Rus09 .

  2. 2.

Systems where the projectile is a well-deformed light nucleus with a strongly
coupled ground state rotational band and the target is a heavy inert nucleus
(208Pb or similar). Again, since the coupling effect is Coulomb dominated
exactly how heavy the target nucleus needs to be for the effect to be plain
will depend on the projectile charge. In this case the first excited state of
the deformed projectile need not be particularly low-lying (1-2 MeV is
sufficient).

  3. 3.

Systems where the projectile is a well-deformed light nucleus and the target a
well-deformed medium to heavy mass nucleus. It appears that coupling to mutual
excitation of the target and projectile has little or no influence on the
elastic scattering in these systems although the overall effect of individual
coupling to the projectile and target excited states does seem to be greater
than the sum of its parts.

  4. 4.

Systems where both the projectile and target are of medium mass or greater. In
this case the nuclei involved may have only relatively weak couplings but the
system will still exhibit strong coupling effects on the elastic scattering
due to the large charge product. Any medium mass (or greater) nucleus incident
on, say, a 208Pb target should give rise to the characteristic strong coupling
shape for its near barrier elastic scattering. Doubly-magic medium mass nuclei
may prove exceptions, cf. 40Ca.

  5. 5.

Systems with weakly-bound radioactive beams incident on heavy targets such as
197Au and 208Pb. There are two caveats to this statement: a weak binding
energy does not automatically guarantee the existence of the strong coupling
effect, as the case of 8B proves and a heavy target may not always be
necessary to see the effect, cf. 11Be and 11Li.

  6. 6.

Well angular momentum and Q value matched transfer reactions can give similar
strong coupling effects to those due to strong inelastic couplings. Such
couplings may turn out to be important for exotic nuclei.

In all these cases it should also be emphasised that the optimum effect will
be for energies somewhat greater than the Coulomb barrier, about 20–30 %
greater than $V_{\mathrm{B}}$.

Finally, to illustrate further the importance of excitation energy on the
strong coupling effect at a given incident energy close to the Coulomb barrier
we present in Fig. 17 test coupled channels calculations for the $\alpha$ \+
238U system at a bombarding energy of 24.7 MeV. The calculations are similar
to those of Rusek Rus09  except that for our purposes we have only included
coupling to the $2^{+}_{1}$ state of 238U (omission of coupling to the other
members of the ground state band has little effect on the result beyond
damping out the small scale oscillations about the Rutherford value at small
angles).

Figure 17: Coupled channels calculations for the $\alpha$ \+ 238U system at a
bombarding energy of 24.7 MeV. The various curves correspond to a no coupling
optical model calculation (dotted curve) and coupled channels calculations
with the excitation energy of the 238U $2^{+}_{1}$ state multiplied by factors
of 1, 10, 50 and 100.

The actual excitation energy of the $2^{+}_{1}$ state, 45 keV, was multiplied
by factors of 10, 50 and 100 in order to investigate the influence of
excitation energy on the strong coupling effect. It is found that the
excitation energy has to be multiplied by a factor of about 30, i.e. an
excitation energy of 1.35 MeV, before the characteristic strong coupling shape
has more or less disappeared in favour of the usual Fresnel-type elastic
scattering pattern. Although there are still significant coupling effects for
$E_{\mathrm{x}}\times 50$ and 100 they are of such a nature that they could be
subsumed into the optical model fitting procedure. These calculations
demonstrate that even a strongly coupled state will not give rise to the
characteristic strong coupling elastic scattering angular distribution shape
if its excitation energy is too high, although exactly what constitutes “too
high” may also depend on other factors such as the projectile incident energy.

##  5 Suggested future experiments

In this section we propose systems that would be interesting candidates for
further study of the strong coupling effect in the light of our conclusions
concerning the conditions under which such effects should be most important.
Suggestions are made for experiments with both stable and radioactive beams
and an indication of the required precision of the measurements is given. We
have limited ourselves to what we hope are practical measurements, either
already or in the near future. All coupled channels calculations in the
following subsections were performed using the code FRESCO Tho88 .

###  5.1 Heavy beams on a heavy target

Due to what we might term the magnifying effect of the large charge product,
the elastic scattering of a heavy nucleus from a similarly heavy target ought,
in principle, to provide a good example of the “strong coupling effect without
strong coupling.” As an example, we suggest the 120Sn + 208Pb system.
Measurement of the pure elastic scattering would require a detection energy
resolution of about 1 MeV in order to resolve inelastic scattering to the
first excited state of 120Sn ($E_{\mathrm{x}}=1.17$ MeV). An incident 120Sn
energy of 900 MeV should be suitable, thus a detection energy resolution of
about 0.1 % would be required.

We present the result of a coupled channels calculation for a 900 MeV 120Sn
beam incident on a 208Pb target in Fig. 18 (a).

Figure 18: (a) Coupled channels calculation for the 120Sn + 208Pb system at a
bombarding energy of 900 MeV. The calculation including Coulomb and nuclear
couplings to the $2^{+}_{1}$ state of 120Sn and the $3^{-}_{1}$ state of 208Pb
is denoted by the solid curve while the dashed curve denotes the result of the
no coupling optical model calculation. (b) Coupled channels calculation for
the 132Sn + 208Pb system at a bombarding energy of 900 MeV. The calculation
including Coulomb and nuclear couplings to the $2^{+}_{1}$ state of 132Sn and
the $3^{-}_{1}$ state of 208Pb is denoted by the solid curve while the dashed
curve denotes the result of the no coupling optical model calculation.

The calculation does not aim at a quantitatively accurate prediction of the
angular distribution but it should be reasonably realistic. The optical
potential was calculated according to the prescription of Broglia and Winther
Bro81  with the imaginary part taken to be the same as the real part but with
a depth divided by a factor of 4. The $B(E2)$ and $B(E3)$ values were taken
from Raman et al. Ram01  and Kibédi and Spear Kib02  respectively. The nuclear
deformation lengths were derived from the $B(E\lambda)$ values assuming the
collective model and radii of $1.2\times\mathrm{A}^{1/3}$ fm. A precision of
$\pm 5$ % on each point of the angular distribution would be more than
sufficient not only to demonstrate the effect but also show the details of the
structure, provided that a sufficient number of points could be measured. Such
a measurement should therefore be feasible over a reasonable period of time
given the availability of a suitable 120Sn beam.

By way of contrast, in Fig. 18 (b) we present the result of a coupled channels
calculation for the radioactive 132Sn beam incident on a 208Pb target, again
at an incident energy of 900 MeV. The 132Sn nucleus is doubly-magic and its
first excited state, a $2^{+}$ level, is consequently situated at the rather
high energy of 4.04 MeV. We would therefore anticipate a much reduced strong
coupling effect. The calculation suggests that this will indeed be the case,
at least as far as inelastic excitations are concerned; in fact, the residual
“strong coupling” effect is almost entirely due to coupling to the 208Pb
$3^{-}_{1}$ state. The coupled channels calculation took the $B(E2)$ value for
132Sn from Pritychenko et al. Pri13 , the nuclear deformation length being
derived in a similar fashion to those in the 120Sn + 208Pb calculation. The
optical potential was again calculated using the prescription of Broglia and
Winther Bro81 . In this case a detection energy resolution of about 0.25 %
would be required, since it would be necessary to resolve the $3^{-}_{1}$
state of 208Pb ($E_{\mathrm{x}}=2.61$ MeV) in order to measure pure elastic
scattering. A precision of $\pm 5$ % on each point of the angular distribution
would be more than adequate to demonstrate the predicted difference between
the 120Sn + 208Pb and 132Sn + 208Pb elastic scattering. Such a measurement
should therefore be a feasible radioactive beam experiment.

###  5.2 Deformed beams on a heavy deformed target

We have already seen in subsection 2.3 that systems involving a deformed light
nucleus incident on a deformed heavy target lead to interesting results with
effects due to strong coupling in both the projectile and the target. It would
be of interest to extend these studies to a more strongly coupled target such
as 184W for instance in order to check whether what we might term the “greater
than the sum of its parts effect” is universal. Suitable beams for this
purpose would be 20Ne and 24Mg or 28Si. Sufficient energy resolution to
measure pure elastic scattering would require the use of a spectrometer to
detect the scattered beam particles but this ought to be possible since a
detection energy resolution of about 100 keV would be necessary to separate
the first excited state of 184W.

In Fig. 19 we plot the results of coupled channels calculations for the 20Ne +
184W system at a bombarding energy of 120 MeV, about 23 % higher than the
nominal Coulomb barrier for this system.

Figure 19: Coupled channels calculations for the 20Ne + 184W system at a
bombarding energy of 120 MeV. The dotted curve denotes the no coupling optical
model calculation, the dashed curve a coupled channels calculation including
coupling to the 20Ne $2^{+}_{1}$ state only, the dot-dashed curve a coupled
channels calculation including coupling to the 184W $2^{+}_{1}$ state only and
the solid curve the full coupled channels calculation including coupling to
both target and projectile $2^{+}_{1}$ states (but not mutual excitation).

The calculations again employed the Broglia-Winther prescription for the
optical model parameters, the imaginary potential depth being set at a quarter
of the real depth. The $B(E2)$ values were taken from Ref. Ram01  and the
nuclear deformation lengths were taken from Refs. Gro78  and Tho77  for 20Ne
and 184W respectively. In this system, unlike the 20Ne + 148Nd data of Jia91 ,
the coupling effect is dominated by the target coupling (to be expected since
the coupling effect is much stronger in 184W). However, the calculations also
suggest that the overall coupling effect is not “greater than the sum of its
parts” in this case, thus making it an interesting system for further
experimental investigation.

As a second example we take the 24Mg + 184W system at an incident 24Mg energy
of 145 MeV, again about 23 % greater than the nominal Coulomb barrier for this
system. Coupled channels calculations similar to those just described for the
20Ne + 184W system are presented in Fig. 20.

Figure 20: Coupled channels calculations for the 24Mg + 184W system at a
bombarding energy of 145 MeV. The dotted curve denotes the no coupling optical
model calculation, the dashed curve a coupled channels calculation including
coupling to the 24Mg $2^{+}_{1}$ state only, the dot-dashed curve a coupled
channels calculation including coupling to the 184W $2^{+}_{1}$ state only and
the solid curve the full coupled channels calculation including coupling to
both target and projectile $2^{+}_{1}$ states (but not mutual excitation).

The $B(E2)$ value for 24Mg was taken from Ref. Ram01  and the corresponding
nuclear deformation length was taken from Ref. Eck81 . The coupling effect is
again dominated by the target coupling and seems to show similar behaviour to
20Ne + 184W with regard to the cumulative effect of coupling to target and
projectile states.

It is interesting to note that in both these cases where the target coupling
effect is predicted to dominate there appears to be a destructive interference
effect between projectile and target coupling influences. This would certainly
bear further investigation and good elastic scattering data (at the $\pm$ 5 %
level) for these systems are desirable. Both experiments should be feasible
with currently available beams provided access to a spectrometer is possible
in order to obtain the required detection energy resolution of about 100 keV.

###  5.3 Light weakly-bound radioactive beam on a heavy target

While experiments to measure the near-barrier elastic scattering of light
weakly-bound radioactive beams from heavy targets have already been performed
for many projectiles of interest, as outlined in section 3, there remain a few
systems which should prove interesting and which have not yet been studied.
One such system should be 15C + 208Pb. The 15C nucleus is relatively weakly
bound, its threshold against 15C $\rightarrow$ 14C + $n$ breakup being 1.218
MeV. However, perhaps the most important aspect of its structure is that it is
a single neutron halo nucleus with the valence neutron in a $2s_{1/2}$
orbital, thus raising the intriguing possibility of strong coupling effects
due to neutron stripping because of the long tail on the wave function of the
valence neutron. Coupled reaction channels calculations have already
demonstrated that the 208Pb(15C,14C)209Pb single neutron stripping reaction
should have a significant coupling effect on sub-barrier 15C + 208Pb elastic
scattering Kee07 .

In Fig. 21 we present a coupled reaction channels calculation showing the
effect of coupling to the single neutron stripping reaction on the 15C + 208Pb
elastic scattering angular distribution for an incident 15C energy of 65 MeV.

Figure 21: Coupled reaction channels calculation for the 15C + 208Pb system at
a bombarding energy of 65 MeV. The dashed curve denotes the no coupling
optical model calculation while the solid curve denotes the coupled reaction
channels calculation including coupling to the 208Pb(15C,14C)209Pb single
neutron stripping.

The entrance channel 15C + 208Pb optical potential was obtained by Watanabe-
type cluster folding of 14C + 208Pb and $n$ \+ 208Pb optical potentials over
the internal wave function of the 15C ground state. The 15C wave function was
calculated assuming a Woods-Saxon well of “standard” geometry:
$R_{0}=1.25\times{\mathrm{A}}^{1/3}$ fm, $a=0.65$ fm, the well depth being
adjusted to give the correct single neutron binding energy, and a spin-orbit
component of the same geometry with a fixed well depth of 6 MeV. No attempt
was made to tune these parameters. The 14C + 208Pb potential parameters were
taken from a fit to the 12C + 208Pb data of Ref. San01  nearest to the
appropriate energy using the potential geometry of Ref. Bal75  and adjusting
the real and imaginary depths. The neutron potential was the central part of
pot I of Ref. Ann85 . The exit channel 14C + 209Pb potential parameters were
taken from a similar fit to the appropriate data of Ref. San01  to those used
to generate the entrance channel potential. The
$\left<{}^{15}\mathrm{C}|^{14}\mathrm{C}+n\right>$ and
$\left<{}^{209}\mathrm{Pb}|^{208}\mathrm{Pb}+n\right>$ overlaps and
spectroscopic factors were as in Ref. Kee07 .

Figure 21 shows that the single neutron stripping alone would be expected to
give the characteristic “strong coupling” shape to the elastic scattering
angular distribution in this system. However, the weak binding of 15C implies
that breakup coupling to the continuum could produce a similar effect. In Fig.
22 (a) we present the result of a coupled discretised continuum channels
calculation for 15C + 208Pb at an incident energy of 65 MeV.

Figure 22: (a) Coupled discretised continuum channels calculations for the 15C
+ 208Pb system at a bombarding energy of 65 MeV. The dashed curve denotes the
no coupling optical model calculation while the solid curve denotes the
coupled discretised continuum channels calculation including coupling to the
14C + $n$ continuum. The dotted curve denotes a coupled discretised continuum
channels calculation with nuclear couplings only. (b) Similar calculations for
an incident 15C energy of 190 MeV, equivalent to approximately three times the
nominal Coulomb barrier for the 15C + 208Pb system. Note the different angular
ranges.

The calculations are based on the same Watanabe-type folding procedure as was
used to calculate the entrance channel potential in the coupled reaction
channels calculation. Couplings to the $L$ = 0, 1, 2, 3 and 4 14C + $n$
continuum as well as the 0.74 MeV $5/2^{+}$ bound first excited state of 15C
were included with couplings up to multipolarity $\lambda=4$. The continuum
was divided into bins in momentum ($k$) space of width $\Delta k=0.1$ fm-1 up
to a maximum value $k_{\mathrm{max}}=0.6$ fm-1. The calculated angular
distribution exhibits the strong coupling shape typical of other weakly-bound
radioactive beams scattered from heavy targets. Also plotted on Fig. 22 (a),
as the dotted curve, is the result of a coupled discretised continuum channels
calculation with nuclear coupling potentials only (diagonal Coulomb potentials
were retained). It will be noted that the nuclear couplings are important for
15C, suggesting that it is more similar to 11Be than to 6He or 11Li where the
Coulomb couplings are dominant. However, the trade off between nuclear and
Coulomb couplings is to some extent energy dependent, and test calculations
for 11Li + 208Pb for an incident energy of 40 MeV yield a relative importance
for nuclear and Coulomb coupling similar to that for 65 MeV 15C + 208Pb shown
in Fig. 22 (a), suggesting that 6He may be more exceptional than at first
thought in that Coulomb breakup couplings appear to be dominant over the whole
range of near-barrier energies, see Fig. 12 and Fig. 1 (c) of Ref. Kee13 , for
example. More data (and more sophisticated calculations) are clearly desirable
to investigate further this fascinating question.

The near barrier 15C + 208Pb elastic scattering is therefore expected to
exhibit strong coupling effects both on account of couplings to the continuum
and strong single neutron stripping couplings. In this respect it should be
somewhat similar to the recent data for near-barrier 8He + 208Pb elastic
scattering Mar13 . However, Fig. 22 (b) underlines the necessity of performing
the measurements at energies close to the Coulomb barrier. Figure 22 (b)
presents the result of a similar coupled discretised continuum channels
calculation to that shown in Fig. 22 (a) but for an incident 15C energy of 190
MeV, approximately three times the nominal Coulomb barrier for this system.
The continuum was extended up to a maximum value $k_{\mathrm{max}}=1.2$ fm-1,
the bins for $k$ values above 0.6 fm-1 being of width $\Delta k=0.2$ fm-1. The
predicted elastic scattering angular distribution is now seen to be of an
essentially standard Fresnel type pattern with a slight residual strong
Coulomb coupling effect just before the main Coulomb-nuclear interference
peak, similar to that of the 161.2 MeV 20Ne + 208Pb data of Fig. 5 (b).

###  5.4 Heavier radioactive beams on a heavy target

Many heavier non weakly-bound radioactive nuclei are either known to be or are
predicted to be more strongly deformed than the corresponding isotopes in the
valley of stability. As such they should also be interesting candidates for
strong coupling effects on their elastic scattering from heavy targets. As
examples we take 30Ne and 32Mg, both with lower lying and more strongly
coupled first excited states than their already strongly coupled stable
counterparts 20Ne and 24Mg. In Fig. 23 we present the result of a coupled
channels calculation for the elastic scattering of 30Ne from a 208Pb target at
an incident 30Ne energy of 131 MeV.

Figure 23: Coupled channels calculation for the 30Ne + 208Pb system at an
incident 30Ne energy of 131 MeV. The solid curve denotes the result of a
coupled channels calculation including coupling to the $2^{+}_{1}$ state of
30Ne only while the dashed curve denotes a no coupling optical model
calculation using the same optical potential as in the coupled channels
calculation.

The optical model potential parameters are the same as those used in the 131
MeV 20Ne + 208Pb calculation shown in Fig. 5 and the $B(E2)$ value was taken
from Ref. Pri13 . The nuclear deformation length was derived from the $B(E2)$
assuming the collective model expression and a charge radius of
$1.2\times{\mathrm{A}}^{1/3}$ fm; this value was then multiplied by a factor
of 0.75 before insertion in the coupled channels calculation since nuclear
deformation lengths derived from fits to inelastic scattering data tend to be
somewhat smaller than the values derived from the corresponding $B(E\lambda)$.
As a comparison of Figs. 5 and 23 will show, the coupling effect for 30Ne is
considerably larger than that for 20Ne at the same energy. While Fig. 23
should not be taken as a quantitatively reliable prediction—there may be
additional effects due to strong transfer couplings and differences in the
underlying optical potential parameters for 30Ne compared to 20Ne—it does
demonstrate that 30Ne would be an interesting candidate for study when beams
becomes available.

In Fig. 24 we present the results of a similar calculation for the 32Mg +
208Pb system at an incident 32Mg energy of 200 MeV.

Figure 24: Coupled channels calculation for the 32Mg + 208Pb system at an
incident 32Mg energy of 200 MeV. The solid curve denotes the result of a
coupled channels calculation including coupling to the $2^{+}_{1}$ state of
32Mg only while the dashed curve denotes a no coupling optical model
calculation using the same optical potential as in the coupled channels
calculation.

The optical model potential parameters were taken from a coupled channels fit
to the 200 MeV 24Mg + 208Pb elastic scattering data of Ref. Hen89  which
included coupling to the $2^{+}_{1}$ state of 24Mg only. The 32Mg $B(E2)$ was
again taken from Ref. Pri13  and the nuclear deformation length derived in a
similar way to the 30Ne + 208Pb calculations. The coupling effect is
considerably larger than for the 24Mg + 208Pb system at the same energy
although the same caveats apply to the quantitative reliability of the
prediction as for the 30Ne + 208Pb case.

##  6 Discussion

We have shown in this review that strong coupling effects on near-barrier
heavy ion elastic scattering have a long history, going back nearly forty
years. The important point to draw from the large body of work of this type
conducted with stable beams is that the elastic scattering can, under the
right conditions, be sensitive to the nuclear structure of the colliding
nuclei, projectile or target or both. The advent of beams of weakly-bound
light radioactive nuclei at suitable energies (and of suitable quality) has
led to a revival of interest in this topic, see for example Ref. Dia14  for a
recent article.

In addition to its intrinsic interest from the point of view of studies of the
reaction mechanism, measurements of the elastic scattering for systems where
the “strong coupling effect” is apparent may have other uses. The sensitivity
of the elastic scattering angular distribution to the details of the nuclear
structure may be exploited as a means of extracting nuclear structure
information, in particular the $B(E\lambda)$ value for the strongly coupled
state. This was already suggested in the context of the application of an
analytical form of the long-ranged Coulomb DPP by Baltz et al., Bal78 :
“Indeed, in the situation where the long-range potential arises dominantly
from a single state, sub-Coulomb elastic scattering analyzed in terms of our
analytical expression might provide an alternative method of determining the
experimental $B(E2)$ to that single state.” Bal78 .

While we would not advocate such measurements as a replacement for the
traditional methods such as Coulomb excitation, for suitable radioactive beams
a measurement of the near-barrier elastic scattering from an appropriate
target might provide a valuable adjunct. The method has much to recommend it,
since in the case of the 18O \+ 184W system the coupling effect on the elastic
scattering was almost exclusively due to the Coulomb coupling to the
$2^{+}_{1}$ state of 184W and was thus sensitive to the $B(E2)$ value for
excitation of this state; the other members of the rotational band seemed to
have little or no influence. The same was found for the 20Ne \+ 208Pb elastic
scattering where the 20Ne $2^{+}_{1}$ was the relevant state; coupling to the
$4^{+}_{1}$ state had little or no effect on the elastic scattering (unlike
the inelastic scattering to the 20Ne $2^{+}_{1}$). However, practical
considerations linked with the likely quality of medium to heavy mass
radioactive beams will severely limit the utility of the method, since it is
essential that pure elastic scattering is measured, thus requiring good
detection energy resolution.

That the near-barrier elastic scattering angular distribution can, under the
right circumstances, clearly show differences due to the details of the
nuclear structure of the interacting nuclei is demonstrated by Fig. 25.

Figure 25: Elastic scattering angular distributions for 131 MeV 20Ne + 208Pb
and 132 MeV 22Ne + 208Pb. The solid curves denote coupled channels fits
including couplings to the $0^{+}_{1}$, $2^{+}_{1}$ and $4^{+}_{1}$ states of
20Ne and 22Ne, see Ref. Gro84  for details. Taken from Ref. Gro84 . Reprinted
Fig. 1 with permission from Gro84 . ⓒ 1984, The American Physical Society

The main influence on the elastic scattering is due to coupling to the
$2^{+}_{1}$ state of the projectile. In 20Ne the $2^{+}_{1}$ state has an
excitation energy of 1.63 MeV and the $B(E2;0^{+}_{1}\rightarrow
2^{+}_{1})=0.034$ e2b2 Ram01 , whereas in 22Ne the $2^{+}_{1}$ excitation
energy is 1.27 MeV and the $B(E2;0^{+}_{1}\rightarrow 2^{+}_{1})=0.023$ e2b2
Ram01 . This difference between the two isotopes translates to an effect of
order 10 % in the measured elastic scattering angular distributions in the
region of the Coulomb-nuclear interference peak and as Fig. 25 shows, the
precision required clearly to demonstrate the effect is well within the
capabilities of measurements with stable beams. However, effects of this order
should also be visible in measurements with radioactive beams of the quality
now becoming available and which will be available at the second generation
radioactive beam facilities when these come on line (a precision of between
2-3 % in measurements of elastic scattering angular distributions has already
been achieved with some currently available radioactive beams, notably 6He).

With the continuing development of existing radioactive ion beam facilities
including advanced detector system updates, the future for these studies is
bright. Further exploration of possible strong coupling effects due to
transfer reactions should be particularly propitious. However, we emphasise
that care is needed in the selection of beam and target combinations and the
choice of incident energy; it is clear from the data and calculations
presented in this review that a beam energy close to the nominal Coulomb
barrier for the system under study is essential if strong coupling effects on
the elastic scattering are to be observed unambiguously. The one exception to
this rule seems to be 11Li, where the coupling to the continuum is so strong
that it should persist to energies well in excess of the Coulomb barrier, at
least for heavy targets. On the other hand, in 8B we have the intriguing case
of a nucleus which ought to be exotic—it has the lowest known threshold
against breakup and the predicted breakup cross sections are very large—and
yet is not, at least not in the conventional sense, since coupling to the 7Be
+ $p$ continuum has an almost negligible effect in coupled discretised
continuum channels calculations. Why this should be so remains “a riddle,
wrapped in a mystery, inside an enigma” Chu39 . One might almost say that 8B
is exotic because it is not exotic …Nevertheless, it is clear that much work
remains to be done in this fascinating field, proving that elastic scattering
measurements are neither trivial nor uninteresting.

## Acknowledgements

The authors would like to thank Dr. D.D. Caussyn for help with scanning
figures for this article. KWK acknowledges partial support from the Florida
State University Robert O. Lawton Fund. Figures 8 and 10 are ⓒ 1984 and 1992
respectively, reprinted with permission from Elsevier.

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a factor of a few within the AGN lifetime. In Fig. D.4, we show the mass
growth of different ECOs without capping accretion rates at the Eddington
limit. Here we see that if the embed accretion rate is not capped at
Eddington, then ECOs will swiftly grow into IMBHs. This can pose substantial
self-consistency problems for our (and likely other) AGN disk models, as a
large population of IMBHs will throttle the gas supply to the central SMBH by
opening gaps in the gas and also accreting most of the gas onto themselves.
This is a major consideration in our choice to cap ECO growth at Eddington.
Note that in this experiment, we have chosen the conditions that produced the
least mass growth in Fig. 11 (high initial BH spins, and $10^{7}M_{\odot}$
SMBH).

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