Do Halo Nuclei Follow Rutherford Elastic Scattering at Energies Below the Barrier?
The Case of
11
Li
M. Cubero,
1,2
J. P. Ferna
´
ndez-Garcı
´
a,
3,4
M. Rodrı
´
guez-Gallardo,
3
L. Acosta,
5
M. Alcorta,
1
M. A. G. Alvarez,
3,4
M. J. G. Borge,
6,
*
L. Buchmann,
7
C. A. Diget,
8
H. Al Falou,
9
B. R. Fulton,
8
H. O. U. Fynbo,
10
D. Galaviz,
11
J. Go
´
mez-Camacho,
3,4
R. Kanungo,
9
J. A. Lay,
3
M. Madurga,
6
I. Martel,
5
A. M. Moro,
3
I. Mukha,
4
T. Nilsson,
12
A. M. Sa
´
nchez-Benı
´
tez,
5
A. Shotter,
7,13
O. Tengblad,
6
and P. Walden
7
1
Instituto de Estructura de la Materia CSIC, E28006 Madrid, Spain
2
CICANUM, Universidad de Costa Rica UCR, Apartado 2060 San Jose
´
, Costa Rica
3
Departamento de FAMN, Universidad de Sevilla, 41080 Seville, Spain
4
Centro Nacional de Aceleradores, Universidad de Sevilla/Junta de Andalucı
´
a/CSIC, 41092 Seville, Spain
5
Departamento de Fı
´
sica Aplicada, Universidad de Huelva, 21071 Huelva, Spain
6
Instituto de Estructura de la Materia CSIC, E28006 Madrid, Spain
7
TRIUMF, V6T2A3 Vancouver, British Columbia, Canada
8
Department of Physics, University of York, YO 10 5DD Heslington, York, United Kingdom
9
Department of Astronomy and Physics, Saint Mary’s University, Halifax B3H3C3, Nova Scotia, Canada
10
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus, Denmark
11
CFNUL, Universidade de Lisboa, 1649-003 Lisbon, Portugal
12
Fundamental Physics, Chalmers University of Technology, 41296 Go
¨
teborg, Sweden
13
School of Physics and Astronomy, University of Edinburgh, EH9 3JZ, Edinburgh, United Kingdom
(Received 7 October 2012; published 26 December 2012)
The first measurement of the elastic scattering of the halo nucleus
11
Li and its core
9
Li on
208
Pb at
energies near the Coulomb barrier is presented. The
11
Li þ
208
Pb elastic scattering shows a strong
reduction with respect to the Rutherford cross section, even at energies well below the barrier and down to
very small scattering angles. This drastic change of the elastic differential cross section observed in
11
Li þ
208
Pb is the consequence of the halo structure of
11
Li, as it is not observed in the elastic scattering of its
core
9
Li at the same energies. Four-body continuum-discretized coupled-channels calculations, based on a
three-body model of the
11
Li projectile, are found to explain the measured angular distributions and
confirm that the observed reduction is mainly due to the strong Coulomb coupling to the dipole states in
the low-lying continuum of
11
Li. These calculations suggest the presence of a low-lying dipole resonance
in
11
Li close to the breakup threshold.
DOI: 10.1103/PhysRevLett.109.262701 PACS numbers: 25.60.Bx, 25.60.Gc, 25.60.t
One century ago, Rutherford [1] inferred the structure of
the atom from the reaction data measured by Geiger and
Marsden [2]. Since then, nuclear structure properties have
often been deduced from nuclear reaction studies. With the
advent of the first accelerated radioactive beams, new nu-
clear structures were discovered, such as the existence of a
halo in some very loosely bound nuclei. In fact, 25 years
ago, Hansen and Jonson [3] interpreted the large interaction
cross section observed in
11
Li with light targets by Tanihata
et al. [4] as due to the high probability of the outermost
nucleons to be at large distances from the central core,
which they referred to as a halo structure. The halo structure
is a threshold phenomenon due to the low binding energy
of the last nucleons. Halo nuclei have several features in
common, such as a rather compact core, an extended neu-
tron distribution, and very few, if any, excited states.
The discovery of halo nuclei brought renewed interest in
the modeling of nuclear reactions. This peculiar structure
should affect the reaction properties at near-Coulomb bar-
rier energies. Due to their low binding energy, the descrip-
tion of reactions involving halo nuclei should incorporate
the coupling into the continuum. Current approaches to
reaction theory involve different approximations whose
validity needs to be checked when applied to halo nuclei.
The most neutron-rich bound lithium isotope,
11
Li,isa
fascinating case. Predicted to be unbound, it was identified
in 1966 [5] and it is the archetype of a Borromean halo
nucleus; i.e., the two different binary subsystems,
9
Li-n and
n-n, are unbound, whereas the three-body system is bound
by S
2n
¼ 369:15 0:65 keV [6]. The ground state density
distribution of
11
Li extends well beyond its core; i.e., the
rms matter radius for the
9
Li isotope is 2:44 0:06 fm [7],
while for
11
Li reported values are one fermi larger. No
bound excited states are known, but several resonances
at 1.1 MeV ( 0:5 MeV) and 2.5 MeV ( 1:5 MeV)
have been identified in different reaction studies with light
ions (see Ref. [8] for a recent review). Theoretical models
predict also different low-lying resonances with J
nn
¼ 0
þ
and 1
[9–13], but their existence and precise location have
not been clearly established experimentally.
Due to the loosely bound structure, the neutron halo is
easily polarizable in the strong electric field of a heavy
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target such as
208
Pb. In contrast to normal nuclei, where the
E1 response is dominated by the giant dipole resonance,
large soft electric dipole (E1) strength close to the breakup
threshold has been observed in halo nuclei. The strongest
E1 transition observed at low excitation energy was deter-
mined in an exclusive measurement of the Coulomb
dissociation of
11
Li at 770 MeV at RIKEN [14]. This large
E1 strength will induce a strong dynamic polarization
effect of the
11
Li projectile when scattered with a heavy
target, giving rise to long-range Coulomb couplings.
According to Ref. [15], this effect should manifest itself
at Coulomb barrier energies as a departure from the elastic
Rutherford scattering.
In order to disentangle the contribution of the loosely
bound structure of
11
Li from the reaction process, one
should know the behavior of the core,
9
Li, under the
same conditions. No data exist for the scattering of
11
Li
near the Coulomb barrier, and even for
9
Li the scattering
data are scarce. The cross section of
9
Li on the
208
Pb target
was measured at 86 MeV [16], but this energy is three
times the Coulomb barrier. Near the Coulomb barrier, the
fusion cross section of
9
Li þ
208
Pb was measured at center-
of-mass (c.m.) energies from 23.9 to 43.0 MeV [17].
However, the elastic cross section was not reported.
This Letter reports on the first measurement of the
scattering of the halo nucleus
11
Li on
208
Pb at incident
energies of 24.3 and 29.8 MeV, which are, respectively,
below and around the Coulomb barrier (V
b
28 MeV).
To characterize the behavior of the core, the
9
Li þ
208
Pb
scattering has also been measured with the same setup and
at the same c.m. energies of 23.1 and 28.3 MeV. The
11
Li
elastic scattering data show a stronger reduction of the
cross section than predicted in Ref. [15] for this reaction.
These data have been compared with four-body continuum-
discretized coupled-channels (CDCC) calculations, based
on a three-body description of the
11
Li nucleus, which takes
into account, in addition to the dipole Coulomb couplings,
other Coulomb multipoles and nuclear couplings to all
orders.
The experiment was performed in the postaccelerated
ISAC-II line at the TRIUMF facility (Vancouver, Canada).
A primary 500 MeV 100 A proton beam produced at
the TRIUMF cyclotron irradiated a Ta primary target. The
secondary beams of Li isotopes were transported to the
ISAC-II facility for postacceleration [18]. The average
intensity of the
11
Li beam, as detected in our monitor
detector located 280 mm downstream of the Pb target,
was 4300
11
Li=s.
The accelerated incoming beam impinged on a
208
Pb
target tilted 75
with respect to the beam direction. For the
9
Li study, two
208
Pb targets with different thicknesses
1.45 and 1:9mg=cm
2
were used. Only the 1:45 mg=cm
2
208
Pb target was bombarded by the
11
Li beam, in order to
minimize the loss of energy resolution due to straggling in
the target.
The experimental setup consisted of four telescopes,
T1-T4. Two of them, T1 and T2, were placed in the target
forward direction, each one consisting of a windowless
40 m thick double-sided silicon (16 16) strip detector
(DSSSD) [19] acting as a E detector and a 500 m thick
Si PAD as an E detector. Telescopes T3 and T4 were placed
in the target backward direction, and each one consisted of
a 20 m thick E single-sided silicon (16) strip detector
and a 60 m thick DSSSD (16 16) behind.
The segmentation of the detector system gives informa-
tion of 256 pixels per telescope either by matching front
and back strips of the DSSSD for telescopes 1 and 2 or
front and back detectors in telescopes 3 and 4. This con-
figuration permitted a high angular resolution (2
–3
,
depending upon the detector) with large angular coverage:
10
to 40
(T1), 30
to 60
(T2), 50
to 100
(T3), and 90
to 140
(T4)[20]. Due to the compact geometry used in
the setup, a refined determination of the angle subtended
by each pixel of T1 and T2 was done based on the fact
that the elastic scattering of
9
Li on
208
Pb at energies below
the barrier follows the Rutherford scattering formula.
For the
11
Li þ
208
Pb data, the method was applied to the
sum of elastic and breakup data. Further details on the
setup, data processing, and event selection can be found in
Refs. [20,21].
Elastic events were selected in the two-dimensional
plot of E versus E þ E energy spectra for each pixel.
A clear identification of the elastic peaks and fragments,
both in the
9
Li and in the
11
Li scattering data, was
achieved. Figure 1 illustrates the data obtained for
9
Li
and
11
Li beams scattered on the same 1:45 mg=cm
2
thick
208
Pb target and at equivalent c.m. energies. The two-
dimensional plot for the
11
Li scattered data, on the bottom
part of Fig. 1, shows the contribution of the elastic channel
and the
9
Li breakup data. The pixels contributing to the
selected angular ring of ð14 1Þ
are displayed on the left-
hand side of the figure. The relatively large contribution
of the breakup channel observed at this low energy and
forward angle is remarkable.
In Fig. 2, we show the two-dimensional plot corresp-
onding to
9
Li þ
208
Pb scattered data at a c.m. energy of
28.3 MeV for a pixel of telescope 4 centered at 128.2
.
It is remarkable that, at this near-threshold energy, one can
identify
8
Li events likely corresponding to the transfer
reaction
208
Pbð
9
Li;
8
LiÞ
209
Pb (Q ¼0:1 MeV) and the
possible contribution of
7
Li þ
210
Pb (Q ¼ 3 MeV) and
production from transfer or breakup, observed at very high
energies [22]. The contribution of these channels, when
present, has been removed in order to extract the elastic
scattering data. The differential cross section for each
detector was calculated by the sum of the counts in each
pixel divided by the sum of their solid angles correspond-
ing to a given angular ring.
The measured elastic angular distributions for
9
Li and
11
Li on
208
Pb are displayed in Fig. 3, relative to the
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Rutherford cross section. The upper and lower panels
correspond to E
c:m:
¼ 23:1 and 28.3 MeV, respectively.
For the lower energy (below the Coulomb barrier), the
9
Li data are very close to the Rutherford cross section,
whereas, for the higher energy, a Coulomb-nuclear inter-
ference maximum starts to be visible at
c:m:
75
,
accompanied by a smooth decrease beyond this angle.
This is reminiscent of the Fresnel-type pattern character-
istic of the scattering of ‘‘normal’’ nuclei near the Coulomb
barrier. On the other hand, the
11
Li data show a strong
reduction with respect to Rutherford, even at the energy
below the barrier. Moreover, the departure from Rutherford
scattering at both energies starts at very forward angles.
This behavior has been observed previously in the scatter-
ing of
6
He [23] and
11
Be [24], although the observed
reduction was not as striking as in the
11
Li case.
For a more quantitative understanding of this behavior,
we have compared the
11
Li þ
208
Pb data with four-body
CDCC calculations. The CDCC method [25,26] is a gen-
eralization of the coupled-channels formalism traditionally
used in the analysis of inelastic reactions populating ex-
cited states of the projectile or target. In CDCC, the model
space is extended so as to include the unbound states of the
projectile, thus allowing the evaluation of the projectile
breakup and its effect on the elastic scattering. Since the
positive energy states form a continuum, a discrete repre-
sentation in terms of a finite set of square-integrable func-
tions is commonly used. Due to the Borromean structure
of the
11
Li nucleus, we use here a recent extension of the
method appropriate for three-body projectiles (four-body
CDCC). This kind of calculation has successfully been
used to describe several
6
He induced reactions [27–29].
FIG. 2 (color online). Two-dimensional plot of E versus
E þ E for
9
Li cattering on a 1:9mg=cm
2 208
Pb target at
E
c:m:
¼ 28: 3 MeV for a pixel at 128.2
. Note the presence of
breakup channels already at this energy.
FIG. 3 (color online). Elastic differential cross section of
9
Li
and
11
Li on
208
Pb, plotted as a ratio to the Rutherford cross
section. In the upper part, it is shown for energies below the
barrier, E
c:m:
¼ 23:1 MeV, and in the bottom part for E
c:m:
¼
28:3 MeV. The optical model (OM) calculation for the
9
Li þ
208
Pb system is also shown in each panel. For details on the four-
body CDCC calculations, see the text.
FIG. 1 (color online). Two-dimensional plot of E versus
E þ E in T1 displayed for the pixels illustrated on the left
part of the figures. The scattered
9
Li þ
208
Pb data are shown in
the upper part for only one pixel centered at 13.9
. In the bottom
figure, the scattered
11
Li þ
208
Pb data are shown in the angular
sector of ð14 1Þ
. Contributions from the elastically scattered
11
Li and
9
Li reakup data are distinctly separated. The data
correspond to the same c.m. energy of 23.1 MeV.
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To describe the
11
Li nucleus, a three-body structure is
assumed with an inert and spinless
9
Li core surrounded
by two valence neutrons. The
11
Li states were obtained
solving the Schro
¨
dinger equation in hyperspherical coor-
dinates. We took the two-body interactions P4 from
Ref. [30]. In addition, an effective three-body force was
included and adjusted to reproduce the experimental two-
neutron separation energy. This gave a ground state wave
function with a rms of 3.30 fm, assuming a rms of 2.44 fm
for the
9
Li core [7]. Continuum states were grouped into
energy bins, up to a maximum energy of 5 MeV, following
the average method, recently extended to the three-body
continuum [29]. Continuum states with J
nn
¼ 0
þ
, 1
, 2
þ
,
and 3
were included.
The elastic differential cross section for this reaction is
very sensitive to the BðE1Þ strength close to the threshold
and, in particular, to the presence of a dipole resonance.
Since the precise location of this resonance has not yet
been firmly established, we have adjusted the position of
this resonance to the energy that best reproduces the elastic
and breakup cross sections. The latter will be presented
in a separate work [31]. This gives E
res
¼ 0:69 MeV (i.e.,
0.32 MeV above the breakup threshold). To place the
dipole resonance at the desired position, the three-body
force has been adjusted for the 1
continuum. This model
predicts a large E1 strength at low energies, even larger
than that extracted experimentally [14].
For the scattering calculations, the n þ
208
Pb and
9
Li þ
208
Pb interactions are also required. The former was taken
from the global parametrization of Koning and Delaroche
[32]. The real part of the
9
Li þ
208
Pb potential was gen-
erated microscopically, using the double-folding Sa
˜
o Paulo
potential [33], with the
9
Li matter density from Ref. [7] and
the
208
Pb density from a Hartree-Fock calculation. The
imaginary part was parametrized using a Woods-Saxon
potential for which the parameters (W
v
,r
i
,a
i
), along with
the normalization of the real part (N
r
), were adjusted in
order to reproduce the present
9
Li elastic data at E
c:m:
¼
28:3 MeV, giving rise to the parameters N
r
¼ 1:0, W
v
¼
46:5 MeV, r
i
¼ 1:33 fm, and a
i
¼ 0:47 fm. The optical
model calculations for
9
Li þ
208
Pb are compared with the
data in Fig. 3 (for further details about the procedure, see
Ref. [20]).
The CDCC coupled differential equations were solved
with the code
FRESCO [34], including both nuclear and
Coulomb couplings between the target and the projectile
to all orders. The four-body CDCC calculations are com-
pared with the
11
Li þ
208
Pb data in Fig. 3. To illustrate the
effect of the coupling to the breakup channels, we have
included also the calculation in which these couplings are
omitted (dotted line). This four-body CDCC calculation
without coupling to the continuum follows the
9
Li data up
to 90
but then decreases faster than the
9
Li data beyond
this angle, as illustrated in the lower part of Fig. 3. This
reduction is due to the absorptive effect arising from the
interaction of the halo neutrons with the target and also to
the extended size of the
11
Li ground state. This diminution,
however, is found to be insufficient to explain the
11
Li data.
Including the coupling to the continuum produces an addi-
tional reduction, and the resulting angular distribution is
found to describe very well the data at both energies. At
E
lab
¼ 24:3 MeV, the CDCC calculations underestimate
the data at the largest angles. Inclusion of continuum states
of even larger angular momenta (e.g., J
nn
¼ 4
þ
) could
increase the calculated elastic cross sections even further.
Our CDCC calculations indicate that the main interactions
responsible for the reduction of the elastic cross section
and the subsequent disappearance of the Coulomb-nuclear
interference peak are the dipole Coulomb couplings. This
special behavior of
11
Li has been associated with the effect
of Coulomb dipole polarizability [15]. The weakly bound
11
Li, in the strong Coulomb field of the target, gets dis-
torted and eventually breaks up. This reduces the elastic
cross sections similarly below (E
c:m:
¼ 23:1 MeV) and at
the barrier top (E
c:m:
¼ 28:3 MeV), where the absence of
the Coulomb-nuclear interference peak is extraordinary.
In summary, the first measurement of the elastic scatter-
ing of the halo nucleus
11
Li and its core
9
Li on
208
Pb at
energies below and on top of the Coulomb barrier is
presented. The
9
Li þ
208
Pb scattering data follow the
expected behavior for well-bound nuclei, obeying the
Rutherford formula below the barrier and exhibiting a
Fresnel-like diffraction pattern above the barrier. On the
other hand, the
11
Li þ
208
Pb elastic cross section departs
significantly from the standard behavior of a well-bound
nucleus such as its core
9
Li, showing a strong reduction
with respect to the Rutherford cross section both below and
around the Coulomb barrier. Four-body CDCC calcula-
tions, using a three-body model of
11
Li and including
Coulomb and nuclear couplings to all orders, reproduce
satisfactorily the experimental elastic angular distribu-
tions. The reduction of the cross section with respect to
the Rutherford scattering is attributed to the strong dipole
coupling between the ground and the continuum states in
11
Li. The presence of a low-lying dipole resonance, close
to the breakup threshold, is found to improve the agree-
ment with the data.
This work has been partially supported by Spanish
National Projects No. FPA2009-07387, No. FPA2009-
07653, No. FPA2009-08848, and No. FPA2010-22131-
C02-01, by the Consolider-Ingenio 2010 Program CPAN
(CSD2007-00042), and by the U.K. Science and
Technology Facilities Council through Grant No. EP/
D060575/1. M. C. acknowledges the support of the
CSIC-UCR and FCT Grant No. ISFL-2-275.
*mj.borge@csic.es
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week ending
28 DECEMBER 2012
262701-5
