Core excitation in the elastic scattering and breakup of $^{11}$Be on
  protons
Core excitation in the elastic scattering and breakup of 11Be on protons
N. C. Summers1, 2, 3 and F. M. Nunes3, 4
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996
Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854
National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824
(Dated: October 25, 2018)
The elastic scattering and breakup of 11Be from a proton target at intermediate energies is studied.
We explore the role of core excitation in the reaction mechanism. Comparison with the data suggests
that there is still missing physics in the description.
PACS numbers: 24.10.Eq, 25.40.Cm, 25.60.Gc, 27.20.+n
I. INTRODUCTION
Nuclear reactions offer the most diverse methods to
study nuclei at the limit of stability. Understanding re-
action mechanisms in nuclear processes involving nuclei
near the driplines is of great importance, particularly at
this time, when there is such a high demand for accuracy
on the structure information to be extracted from the
data. Reaction and structure models are undoubtedly
entangled, therefore improving reaction models often im-
plies incorporating more detailed structure models in the
description [1].
It is generally accepted that, in reactions with loosely
bound nuclei, the coupling to the continuum needs to be
considered. Continuum effects are very much enhanced
in breakup but can also have imprints on other reac-
tion channels, for example elastic and inelastic scatter-
ing. One framework that explicitly includes continuum
effects is the Continuum Discretized Coupled Channel
(CDCC) method [2]. A large amount of work has been
devoted to the analysis of experiments within this frame-
work [3, 4, 5, 6, 7] and in general results are very good.
The eXtended Continuum Discretized Coupled Chan-
nel (XCDCC) method [8, 9] was recently developed. It
brings together a coupled channel description of the pro-
jectile with a coupled channel model of the reaction, en-
abling the description of interference between the multi-
channel components of the projectile as well as dynam-
ical excitation of the core within the projectile, during
the reaction [9]. The model has been applied to the
breakup of 11Be→10Be+n and 17C→16C+n on 9Be at
≈ 60 MeV/nucleon [8, 9]. The projectiles are described
within a two-body core + n multi-channel model, where
the core can be in the ground state but also in an ex-
cited state. This model produces breakup cross sections
to specific final states of the core, given a coupled chan-
nel Hamiltonian for the projectile. Results presented in
Ref. [8, 9] show that core excitation effects in the total
cross section to the ground state of the core are small,
but become very large when considering the total popula-
tion of the core’s excited state. Other differences can be
seen in angular and energy distributions but at present
no such data is available. The effect of core excitation
needs to be studied in other regimes, for very light and
very heavy targets, as well as a variety of energies.
In this work we concentrate on the proton target. In
this case the process is nuclear driven, and recoil effects
are very important. Several reactions of 11Be on protons
have been measured in a number of facilities, namely
elastic scattering at 50 MeV/nucleon [10], quasi-elastic
and breakup at 64 MeV/nucleon [11] as well as transfer
at 35 MeV/nucleon [12].
11Be proton elastic scattering at 49.3 MeV/nucleon
was performed in GANIL [10], at the same time as the
elastic scattering of the core 10Be at 59.4 MeV/nucleon.
Even though the outcoming 11Be measurements corre-
spond to quasi-elastic, these are essentially elastic as
the contribution from the first excited state is negligible.
Standard optical potentials (either density folding as in
JLM [13] or global optical potentials coming from elas-
tic fits as in CH89 [14]) could reproduce the 10Be elastic
reasonably well, requiring small renormalizations of the
real and imaginary parts of the interaction (λV = 0.9 and
λW = 1.1 for CH89) [10] . Larger renormalizations were
required in order to reproduce the distribution of 11Be
(λV = 0.7 or λW = 1.3 for CH89) [10].
It is clear from Ref. [10] that the global optical poten-
tial overestimates the elastic cross section for 11Be. In
Ref. [15] the elastic scattering of 11Be on 12C was success-
fully described using a 10Be+n two-body model, incor-
porating breakup effects. As the 10Be-target interaction
was fixed by the 10Be elastic scattering data, the large
modification in the 11Be+12C elastic data was described,
without renormalization, purely through breakup effects.
Due to the loosely bound nature of the last neutron in
11Be this loss of flux from the elastic channel can be at-
tributed to breakup into 10Be+n. It is thus possible that
the large renormalizations for 11Be scattering on protons
[10] are also due to breakup effects.
In Ref. [11], elastic data is only described after large
renormalizations of both the real and imaginary part of
the 10Be+p interaction (λV = 0.75 and λW = 1.8), much
larger than those used in Ref. [10]. These same renormal-
izations can no longer describe the 10Be+p elastic data
from Ref. [10], and are inconsistent with few-body reac-
tion theory. We will re-examine the elastic scattering of
http://arxiv.org/abs/0704.1586v1
10/11Be+p to see if one can consistently describe both
sets of data using the same interaction for 10Be+p, by
including continuum and core excitation.
In addition to elastic and inelastic measurements,
breakup data from NSCL exist at 63.7 MeV/nucleon [11].
This breakup data is integrated into two wide energy
bins due to statistics. The lower energy bin covers the
1.78 MeV resonance and a reasonable angular distribu-
tion is obtained, which underpredicts the cross section
[11]. The higher energy bin covers resonances that are
thought to be built on excited core states. The cal-
culations presented in Ref. [11] failed to reproduce the
shape of this higher energy bin, and the authors suggest
that the source of the disagreement may be due to an
active core during the reaction. Now that it is possi-
ble to include core excitation in the reaction mechanism
[9] we will re-examine the breakup data using a consis-
tent 10Be+p interaction, and including systematically the
coupling to the 2+ state in 10Be.
Transfer reactions have also been performed with the
11Be beam, at 35.3 MeV/u in GANIL [12] with the aim
of extracting spectroscopic factors for the ground state.
While the reaction mechanisms proved to be more com-
plicated than the 1-step DWBA theory, results for (p,d)
show evidence for a significant core excited component.
The inverse reaction, 10Be(d,p)11Be, has also been stud-
ied in GANIL [16], the main interest being the resonance
structure of 11Be. This illustrates how transfer is being
used beyond the standard application of spectroscopy of
bound states, underlining the need to better understand
the transfer mechanism and its coupling to the contin-
All these different data offer a good testing ground for
theory. A comprehensive theoretical study [17] focusing
on 10Be(d,p) show inconsistencies of the extracted spec-
troscopic factors for data at different energies. Optical
potential uncertainties and core excitation effects could
be at the heart of the problem.
In this work we perform calculations including elastic,
inelastic, and breakup channels of 11Be on protons at in-
termediate energies. We explore explicitly the effect of
the inclusion of core excitation in the reaction mecha-
nism. Comparison to elastic and breakup data will be
presented here. The analysis of the inelastic channel is
presented in [18] and we leave a detailed study of the
transfer channel for a future publication. In section II
we provide the details of the calculations. In section III
we present the results: first for the elastic channel (III A),
then for the breakup (III B). Finally, in section IV, we
draw our conclusions and provide an outlook into the
future.
II. DETAILS OF THE CALCULATIONS
The calculations for breakup of loosely bound systems
on protons have a rather different convergence require-
ment as compared to the breakup on heavier systems.
energy V RV aV W RW aW
40 60.84 1.000 0.7 23.16 0.600 0.6
60 31.64 1.145 0.69 8.78 1.134 0.69
TABLE I: 10Be-proton Woods-Saxon potential parameters.
All energies are in MeV and lengths in fm.
The model space needs to span large excitation energies,
while the radial dependence can be reduced significantly.
For the CDCC calculations at 40 MeV/nucleon, the
continuum was discretized upto 35 MeV, with 10 bins
upto 10 MeV for s-, p-, and d-waves, and 8 bins from
10–35 MeV. We include 12 bins from 0–35 MeV for all
other partial waves up to lmax = 4. The same binning
scheme was used for the XCDCC calculations, except
that the higher bin density upto 10 MeV was only used
for channels with outgoing ground state core components.
Partial waves up to lmax = 4 were used for the coupled
channels projectile states.
For the 60 MeV/nucleon CDCC calculations, a slightly
different binning scheme was adopted to match the ex-
perimental energy bin integrations. From 0–0.5 MeV, 2
bins were used; over the observed energy bins from 0.5–
3.0 and 3.0–5.5 MeV, 3 bins were used in each case; and
from 5.5–30 MeV, 6 bins. For the XCDCC calculations
where the outgoing channel had excited core states, only
1 bin was used from 0–0.5 MeV, 1 bin for each observed
energy range, and 5 bins above.
The radial integrals for the bins were calculated upto
40 fm in steps of 0.1 fm. The radial equations in the
CDCC method were calculated for 30 partial waves with
the lower radial cutoff for the integrals set to 4 fm inside
the point Coulomb radius, and matched to the asymp-
totic Coulomb functions at 150 fm.
The 11Be bound state potential parameters are taken
from Ref. [19], using the Be12-pure interaction for the
CDCC calculations and the Be12-b for the XCDCC cal-
culations. The 10Be-proton interaction is fitted to the
proton elastic data available at the two energies. A good
fit could be obtained from a renormalized CH89 interac-
tion [14]. The parameters are given in Table I. For the
cases including 10Be excitation, the OM potentials were
deformed with the same β2 deformations as used in the
11Be bound state. The coupling matrix elements to the
excited state in 10Be assume a rotational model with the
deformation fitted to the experimental B(E2) strength
[20]. The deformation length is in good agreement with
that obtained from inelastic scattering ot the 2+ state in
10Be [21], and the optical potential used here reproduces
the angular distribution of the inelastic scattering well.
0 20 40 60 80
 (deg)
Be OM
Be CF
Be CDCC
Be XCDCC
FIG. 1: (Color online) 10/11Be elastic scattering on a proton
target at ∼40 MeV/nucleon, using an optical model fit to the
10Be elastic and various 11Be reaction models for the 11Be
data. The experimental data are from GANIL [22].
0 20 40 60
 (deg)
Be OM
Be CDCC
Be XCDCC
FIG. 2: (Color online) 10/11Be elastic scattering on a proton
target at ∼60 MeV/nucleon, using an optical model fit to the
10Be elastic and various 11Be reaction models for the 11Be
data. The experimental data are from Ref. [10] (10Be) and
Ref. [11] (11Be).
III. RESULTS
A. Elastic channel
The elastic scattering is the first test on the reaction
model. In Fig. 1 we show the 10Be and 11Be elastic data
and theoretical calculations at ∼40 MeV/nucleon. The
optical model for the 10Be (dashed/black line) is fitted to
the 10Be elastic data (open circles). The cluster folding
model (dotted/red line) folds the 10Be+p and n-p interac-
tions over the 11Be ground state wave function to produce
the 11Be+p potential. Also shown in Fig. 1 is the effect of
the 11Be continuum within CDCC (dot-dashed/blue line)
and core excitation within XCDCC (solid line). Even
though there is significant improvement over the simple
optical model when including breakup, results still over-
estimate the 11Be elastic cross section at larger angles,
and no improvement is found by including excited core
contributions.
Calculations were repeated for a higher energy, around
60 MeV/nucleon, where both 10Be and 11Be elastic data
exist. Once again, when the 10Be data is fitted with an
optical model, and the 11Be elastic is described within the
CDCC approach, the cross section is over-estimated (see
Fig. 2). Note that the data at this energy does not span
a large angular range, but it is evident that the pattern
of over-predicting the 11Be cross section remains.
Other reaction calculations have been performed in an
effort to describe this data [23], which consisted of a
transfer to the continuum approach in which the breakup
continuum was described using the deuteron basis. This
also failed to describe the data when the 10Be potential
was fixed to the elastic data. The same pattern of over-
predicting the 11Be elastic was also seen at a lower energy
of ∼40 MeV/nucleon [23].
As pointed out earlier, in [10, 11] large renormaliza-
tion factors were needed to reproduce the elastic cross
section. By including more relevant reaction channels,
one might account for a part of the renormalization re-
quired, corresponding to flux that is being removed from
the elastic channel. This suggests that there are still
channels coupled to the elastic that have not been con-
sidered. Preliminary calculations including the deuteron
transfer channel along with the breakup in the 11Be basis
show improvement at small angles, but the disagreement
still remains at large angles. Due to large non-orthonality
corrections, CDCC calculations including the deuteron
transfer coupling turn out to be numerically challenging.
They will be discussed in a later publication.
B. Breakup: comparison with data at ∼60
MeV/nucleon
Breakup data was also obtained at 63.7 MeV/nucleon
[11], summed into two energy bins. The first covers the
energy range 0.5–3.0 MeV, which spans the 1.78 MeV res-
onance, predominantly a d-wave neutron coupled to the
ground state of the core. The second energy bin is over
the energy range 3.0–5.5 MeV, which spans a resonance
at 3.89 MeV, thought to be predominantly an s-wave
neutron coupled to a 10Be(2+) core [11]. In Ref. [11],
CDCC results were presented which underestimated the
cross section for the lower energy bin, but did not repro-
duce the higher energy bin. It was suggested that since
the higher energy bin spanned a resonance with a pos-
sible excited core component, the disagreement could be
due to the spectator core approximation in the standard
CDCC theory. Since XCDCC can handle excited core
components, this data is re-examined.
The breakup angular distribution data and the asso-
ciated theory prediction for the lower energy bin (0.5–
3.0 MeV) and the higher energy bin (3.0–5.5 MeV) are
presented in Figs. 3 and 4 (the equivalent of Figures 3b
and 3c of Ref. [11]). Firstly, the CDCC calculations of
Ref. [11] were redone, with a higher CDCC bin density.
0 20 40
 (deg)
XCDCC
Be+n)p @ 63.7 MeV/nucleon
0.5-3.0 MeV
FIG. 3: 11Be breakup at 60 MeV/nucleon with the relative
energy between breakup fragments in the range 0.5–3.0 MeV.
Experimental data are from Ref. [11].
0 20 40
 (deg)
XCDCC
) x10
Be+n)p @ 63.7 MeV/nucleon
3.0-5.5 MeV
FIG. 4: (Color online) 11Be breakup at 60 MeV/nucleon with
the relative energy between breakup fragments in the range
3.0–5.5 MeV. Experimental data are from Ref. [11].
We find that converged CDCC calculations, for the lower
energy bin, do in fact agree well with the data (dashed
line in Fig. 3), contrary to what is presented in Ref. [11].
Results do not change significantly when core excitation
is included with XCDCC (solid line in Fig. 3), as could be
expected expected due to the resonance structure in this
energy region. One can conclude that a single-particle
description for the first d5/2 resonance is adequate.
The data for the higher energy bin is not well described
within the single particle CDCC model (dashed line in
Fig. 4). To see if this discrepancy can be explained by ex-
cited core contributions, we include the excited 10Be(2+)
components in the reaction mechanism, within XCDCC
(solid line). As shown in Fig. 4, core excitation lowers the
cross section but does not significantly change the shape
of the distribution. It becomes clear that core excitation
does not help to reproduce the shape of the higher en-
ergy angular distribution. The main reason for this is
that for the 11Be coupled channel model of [20], most of
the breakup ends up in the 10Be ground state. Fig. 4
also shows the breakup cross section to the 0+ and 2+
states of 10Be (red/dotted line and the dot-dashed/blue
line respectively). We see that whereas the ground state
distribution has the original shape of the CDCC calcu-
lation, the shape of the distribution to the excited state
reproduces the data (to illustrate this fact, we show the
breakup cross section to 10Be(2+) multiplied by 10 by
the dashed/blue line). The reason for this maybe that
the large number of resonances in this region are not re-
produced well by our particle-rotor model for 11Be. The
only resonance that appears in this model is the 3/2+, for
which the width is not narrow enough to attract signifi-
cant cross section. Some suggest that more exotic struc-
tures are responsible for resonances in this region [24].
Without exotic resonances built on excited core compo-
nents in our structure model, the breakup cross section
is still dominated by ground state 10Be fragments.
IV. CONCLUDING REMARKS
A consistent analysis of reactions involving the halo
nucleus 11Be on protons, at two intermediate energies
(∼40 and ∼60 MeV/nucleon) are performed and com-
pared with data. An optical model approach, based on
a cluster folding potential constructed from the 10Be+p
potential fitted to the appropriate elastic data, is un-
able to describe the 11Be elastic data. The inclusion of
breakup effects improve the description, but theoretical
predictions still overestimate the elastic cross section at
larger scattering angles. The inclusion of core excitation
does not affect the elastic distribution significantly. Note
however that these results include no artificial renormal-
ization of the optical potential. Elastic scattering ex-
periments with radioactive beams at large facilities have
repeatedly been undermined. The fact that the best re-
action models are still unable to fully describe the mecha-
nisms for the 11Be case, shows the need for a more varied
and better elastic scattering experimental program.
In this work we also study the breakup channel explic-
itly. Core excitation in the description of the continuum,
within XCDCC, produces a slight modification of the dis-
tribution. These breakup calculations are compared to
the data at 63.7 MeV/nucleon, for two energy bins 0.5–
3.0 MeV and 3.0–5.5 MeV. For the lower energy bin, the
shape of the angular distribution is well reproduced by
the models. The same cannot be said for the higher en-
ergy bin.
The XCDCC calculations predict breakup states to
specific states of the core 10Be. This level of detail is
still not available in the data, but it could be helpful in-
formation, even at an integrated level, to identify possible
causes for the remaining disagreement with the data.
Another important point is related to the basis used
to describe the breakup states. As discussed in Ref. [25],
within CDCC, one can describe the three body final state
continuum 10Be+n+p in the 11Be continuum basis or in
the deuteron continuum basis. In this work we used the
11Be basis. Work in Ref. [25] shows that in practice the
two choices do not provide the same result. Efforts are
underway to tackle this problem within a Faddeev frame-
work [26]. These results may have important implications
to the theory-experiment mismatch.
Acknowledgements
We thank the high performance computing center
(HPCC) at MSU for the use of their facilities. This
work is supported by NSCL, Michigan State University,
the National Science Foundation through grant PHY-
0555893, by NNSA through US DOE Cooperative Agree-
ment DEFC03-03NA00143 at Rutgers University, and by
the US DOE under contract No. DE-FG02-96ER40983
at the University of Tennessee.
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