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Dynamic polarization potential due to
6
Li breakup on
12
C
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Pang, D. Y. and Mackintosh, R. S. (2011). Dynamic polarization p otential due to 6Li breakup on 12C. Physical
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2011 American Physical Society
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PHYSICAL REVIEW C 84, 064611 (2011)
Dynamic polarization potential due to
6
Li breakup on
12
C
D. Y. Pang
*
School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China
R. S. Mackintosh
†
Department of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, United Kingdom
(Received 28 September 2011; published 19 December 2011)
For
6
Li scattering from
12
C at five laboratory energies from 90 to 318 MeV, we study the dynamic polarization
potential, DPP, due to the breakup of the projectile. The breakup is evaluated using standard continuum discretized
coupled-channels formalism applied to a two-body cluster model of the projectile. The DPP is evaluated over a
wide radial range using both direct S-matrix-to-potential inversion and trivially equivalent local potential methods
which yield substantially and systematically different results. The radius at which the real DPP changes from
external repulsion to interior attraction varies s ystematically with energy. This should be experimentally testable
because, according to notch tests, this crossover radius is within a radial range to which elastic scattering should
be sensitive. The imaginary DPP has an emissive (generative) region at the lower energies; this may be associated
with counterintuitive properties of |S
L
|.
DOI: 10.1103/PhysRevC.84.064611 PACS number(s): 25.60.Gc, 24.50.+g, 24.10.Ht, 24.10.Eq
I. INTRODUCTION
We reopen an old question r elating to the interaction
potential between pairs of nuclei and the contribution to this
potential that is made by excitations of the interacting nuclei.
Such excitations contribute to the nucleus-nucleus interaction
in ways that are not naturally described by models in which the
local density assumption is implicit. Indeed, full understanding
of the origin of specific features of the dynamic polarization
potential (DPP) resulting from such excitations is still lacking;
this seems a rather basic gap in our understanding of nuclear
interactions. For example, just why does the specific process
to be discussed here result in a repulsive term in the nuclear
surface but an attractive term at smaller radii, and why (at
certain energies) does the imaginary part of the DPP exhibit
generative (emissive) radial regions?
It was found some 30 years ago that the real M3Y
folding model potential required [1,2] a factor of about 0.6in
order to fit
6
Li elastic-scattering angular distributions (ADs).
The explanation was found to lie in the breakup of the
projectile, and a good representation of the scattering was
found when the breakup was included using the continuum
discretized coupled-channels (CDCC) formalism [ 3–7]. Exact
inversion of the elastic-scattering S-matrix from such calcu-
lations [8] revealed explicitly the surface repulsion induced
by the breakup of the projectile, and approximate inversion
procedures revealed this too [5,6]; one of the themes of
this paper is to draw out differences between the results
of alternative inversion procedures. Later, S-matrix inversion
revealed generic features of the DPP [9,10], arising from the
breakup of deuterons as well as
6
Li, that were not confined to
the nuclear surface: breakup consistently generates repulsion
*
dypang@buaa.edu.cn
†
r.mackintosh@open.ac.uk
in the nuclear surface and attraction at smaller radii, with a
marked oscillatory pattern in the nuclear interior. There is also
further counterintuitive behavior discussed below.
In this work we study the DPP generated by the breakup of
6
Li scattering from
12
C at laboratory energies of 90, 123.5,
168.6, 210, and 318 MeV. In all calculations, a two-body
cluster model of
6
Li is used and no excitations of
12
Care
considered. Modest renormalization of the deuteron-
12
C and
4
He-
12
C interactions yields good fits to experimental
6
Li
ADs when breakup coupling is included. Many features of
the general approach followed here could be carried over to
more recent extensions of the CDCC formalism, for example,
Refs. [11–16].
Particular features of the work we describe are the
following.
(i) A comparison is made between potentials derived
from S-matrix inversion and the trivially equivalent
local potential TELP algorithm. The differences at all
energies are not small. Apart from the consequences
for nuclear scattering dynamics, this also raises general
questions about potential scattering. The real and
imaginary volume integrals J
R
and J
IM
, as defined
by Satchler [17], are well determined by inversion,
providing a concise measure of the DPP.
(ii) The DPPs that we present are not confined to the surface
region, and their overall properties, which we find to be
well-established by inversion, vary in a consistent and
systematic way with energy. Certain properties of the
DPP can be linked to the fact that, at the lower end
of the energy range studied, breakup coupling actually
increases |S
L
| over a range of L.
(iii) The sensitivity of elastic scattering to the potentials
within the nuclear overlap region is explored by means
of notch tests. In principle, elastic-scattering ADs
should be sensitive to the potential where the DPP
changes from repulsive to attractive.
064611-1
0556-2813/2011/84(6)/064611(9) ©2011 American Physical Society
D. Y. PANG AND R. S. MACKINTOSH PHYSICAL REVIEW C 84, 064611 (2011)
The potentials that we present are local and L-independent
representations of underlying nonlocal and L-dependent po-
tentials. As such, the behavior that we present has implications
for rather basic properties of nucleus-nucleus potentials that
cannot fully be accounted for within models based on the local
density approximation.
Section II specifies the details of the CDCC calculations and
the fits to the elastic scattering data. Section III explains
and compares the methods leading to the DPPs and presents
and compares the inverted potentials and the DPPs that are
found. Section IV presents the notch tests that establish the
radial sensitivity of the DPPs. Section V summarizes the results
and suggests further work.
II. CDCC CALCULATIONS
Angular distributions for
6
Li scattering elastically from
12
C at 90, 123.5, 168.6, 210, and 318 MeV were reported
in Refs. [18–21]. The experimental data for 90 MeV were
obtained from the nuclear database EXFOR/CSISRS [22], and
those for the other energies were obtained by digitizing from
Refs. [19–21].
A standard three-body CDCC model was used in our
analysis, based on an α + d cluster model of
6
Li in which
the deuteron spin was omitted and the ground state was
purely S wave. The α + d binding potential was taken to
be of Woods-Saxon form with parameters R = 1.9 fm and
a
0
= 0.65 fm [23–25]. The depth, 77.5 MeV, was adjusted
to give the correct binding energy of
6
Li and was fixed for
the calculation of the continuum states. Partial waves up
to L
max
= 1000 were solved for t he CDCC equations with
projectile-target separations out to 1000 fm. Both Coulomb
and nuclear breakup were included. The continuum bins were
calculated with cluster separations r 50 fm and the relative
orbital angular momenta between α and d were included
up to l = 2, higher l values having a small effect. The bin
states were constructed by discretizing continuum states up
to maximum α − d relative energies ε
max
= 35.3 MeV. The
continuum states were divided into 15 bins, which are equally
spaced in k space from k = 0uptok
max
= 1.5fm
−1
with steps
of 0.1 fm
−1
. The coupling potentials were constructed with
multipoles q 4. The CDCC calculations were performed
with
FRESCO [26] and convergence in all cases was verified by
calculations with an increased model space.
The real and imaginary parts of the α +
12
C and d +
12
C
potentials were obtained as follows: The starting point was
the pair of potentials interpolated from the energy dependence
of free α and d potentials [27], which were obtained by
fitting α +
12
C and d +
12
C elastic-scattering data from 10
to 100 MeV/nucleon using a single-folding model approach
with the JLMB model nucleon-nucleus potentials Ref. [28], as
described in Ref. [29]. The most appropriate context in which
to isolate and study the contribution of breakup to the
6
Li
optical potential is that in which the elastic-scattering AD is
fitted. Accordingly, and unlike many CDCC calculations, the
6
Li ADs were fitted by normalizing the d +
12
C and α +
12
C
real and imaginary potentials with the same factors, N
R
and
N
IM
, for the real and imaginary parts, respectively. These
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
0 20 40 60 80 100
σ/σ
Ruth
θ
c.m.
(deg)
90
123.5
168.6210318
x2500
x50
x0.02
x0.004
FIG. 1. Three-body CDCC calculations of
6
Li elastic scattering
from
12
C at incident energies between 90 and 318 MeV. The dots are
larger than the uniform error of 10% which was assumed for all data
points. The measured and calculated cross sections are offset by a
factor of 50 for clarity.
factors were therefore t he normalization factors for the
6
Li-
12
C
folded potential that would be responsible for scattering if the
breakup coupling were switched off. This folded potential
is the ‘bare” potential referred to in Sec. III below. The
experimental ADs were fitted by searching upon N
R
and N
IM
assuming uniform uncertainties of 10% for all data points. Be-
cause automatic searching with a converged CDCC calculation
is prohibitively time-consuming, the data were fitted by means
of a grid search. The fits to the AD data are depicted in Fig. 1.
The normalization factors N
R
and N
IM
, for each energy, are
listed in Table I. The DPPs themselves do not depend greatly
upon changes in the optical potentials that are represented by
TABLE I. Real and imaginary renormalization factors N
R
and
N
IM
for the α −
12
Candd −
12
C potentials. The last two columns
give volume integrals of the real and imaginary components
in MeV fm
3
.
E
lab
N
R
N
IM
J
R
J
IM
90.0 0.85 1.15 326.34 141.52
123.5 0.85 1.15 314.24 139.45
168.6 0.85 1.10 298.95 138.00
210.0 0.90 1.05 302.82 133.40
318.0 0.90 1.05 272.16 136.15
064611-2
DYNAMIC POLARIZATION POTENTIAL DUE TO
6
Li ... PHYSICAL REVIEW C 84, 064611 (2011)
0.6
0.7
0.8
0.9
1.0
1.1
1.2
2 3 4 5 6 7 8
U/U
bare
, real part
r (fm)
(a)
90 MeV
123.5 MeV
168.6 MeV
210 MeV
318 MeV
0.8
0.9
1.0
1.1
1.2
1.3
1.4
2 3 4 5 6 7 8
U/U
bare
, imaginary part
r (fm)
(b)
90 MeV
123.5 MeV
168.6 MeV
210 MeV
318 MeV
FIG. 2. Ratios between the full (bare + DPP) and the bare potentials for all five energies. Part (a) shows the ratio for the real part and part
(b) shows the ratio for the imaginary part.
these normalization factors. Table I also includes the volume
integrals J
R
and J
IM
of the real and imaginary components
of the (normalized) bare potentials. The volume integrals
are calculated following standard prescription [17], which
includes the factor (A
1
A
2
)
−1
and the sign convention that J
is positive for an attractive potential. Because only breakup
processes are included explicitly in this work, all other reaction
processes are represented through N
R
and N
IM
.
III. CALCULATION OF THE DPP
The formal DPP (see, e.g., Refs. [17,30]) is both nonlocal
and L dependent. Nevertheless, almost all phenomenological
or theoretical nucleus-nucleus potentials are local and L
independent, often involving some local equivalent repre-
sentation of exchange or other nonlocality. It is therefore
natural to calculate local and L-independent equivalents to
the underlying DPP, and that is reported here.
Calculation of the local equivalent of the formal DPP
arising from the complete set of nonelastic processes is
seldom attempted (but see, for example, Refs. [31,32] and
references therein.) However, there have been many studies
of DPPs due to specific coupled channels, early attempts
include Refs. [33–36]. In the present work we determine
the DPP that arises from one specific process, projectile
breakup, by calculating the potential that exactly reproduces
the S matrix, S
L
, when that process is included in a reaction
calculation, here a CDCC calculation. Subtracting the bare
interaction from such a potential directly yields the DPP arising
from the specific channels. Reference [37] gives an extensive
discussion of this coupled-channels-plus-inversion procedure
and its applications.
The local, L-independent potential that fits t he S matrix is
appropriate to making a connection between the local potential
of standard phenomenology and the processes that fall outside
local density models of the optical potential. Alternatives to
S-matrix inversion exist, for example, Ref. [38]; some are
compared in Ref. [39]; and Hussein et al. [40] discuss the
relevance of different types of inversion to different situations.
One such alternative to S-matrix inversion is the TELP [38],
which, with appropriate partial wave weighting [41], can be
output by the CC/CDCC code
FRESCO [26]. A TELP potential
is sometimes presented as the DPP due to specific processes.
We shall directly compare the DPP calculated from the
FRESCO
TELP with the DPP calculated by S-matrix inversion.
Various techniques for S-matrix-to-potential inversion are
surveyed in Ref. [42]; the results presented here exploit the
iterative-perturbative (IP) procedure [8,42,43] that can yield
precise inversions over a wide range of situations. This method
is the basis of the code
IMAGO [44], which can invert the
S-matrix for spin-0, spin-
1
2
and spin-1 projectiles.
TABLE II. For
6
Li scattering from
12
C at five laboratory energies, volume integrals J (in M eV fm
3
) of the two components of the
DPP induced by breakup of the projectile. The third column gives the change of rms radius of the real central component. The fourth
column gives the change in total reaction cross section and the fifth column gives the breakup cross section.
E
lab
(MeV) J
R
(MeV fm
3
) R
rms
(fm) J
IM
(MeV fm
3
) CS (mb) BU CS (mb)
90 −8.86 − 0.223 24.17 25.9 75.4
123.5 −6.71 −0.207 23.90 33.8 79.3
168.6 −6.39 −0.161 23.26 36.4 79.3
210 −6.21 −0.174 23.99 44.5 83.3
318 −9.60 −0.141 19.36 37.3 72.9
064611-3
D. Y. PANG AND R. S. MACKINTOSH PHYSICAL REVIEW C 84, 064611 (2011)
0.7
0.8
0.9
1.0
1.1
1.2
2 3 4 5 6 7 8
U/U
bare
, real part
r (fm)
(a)
90 MeV
123.5 MeV
168.6 MeV
210 MeV
318 MeV
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
2 3 4 5 6 7 8
U/U
bare
, imaginary part
r (fm)
(b)
90 MeV
123.5 MeV
168.6 MeV
210 MeV
318 MeV
FIG. 3. Ratios between the TELP full (bare + TELP DPP) and the bare potentials for all five energies. Part (a) shows the ratio for the real
part and part (b) shows the ratio for the imaginary part.
A. RESULTS OF S-MATRIX INVERSION
It has long been known that the real part of
6
Li potentials,
calculated using double-folding with interactions of the M3Y
type, must be reduced by around 40% to reproduce the ADs of
6
Li elastic scattering from nuclei [1]. This phenomenon was
later attributed to the breakup coupling effects in
6
Li scattering
[7]. We now present the local potentials that give the same ADs
as the CDCC calculations by inverting t he elastic-scattering
S-matrix from those calculations. Comparison of these with the
bare potentials will give a direct measure of the modification
of the single-channel potential by the coupling to breakup
channels.
The inverted potentials are most conveniently presented as
a ratio to the bare potential. In Fig. 2 we show the ratio of
U
inverted
= U
bare
+ U
DPP
over U
bare
for both real, Fig. 2(a), and
imaginary, Fig. 2(b), parts. For reference, we note that the
strong absorption radius (SAR) is around 6.15 fm at 90 MeV,
falling to 5.32 fm at 318 MeV. (Here we define the SAR as
the classical distance of closest approach for partial wave L
for which |S
L
|
2
= 0.5.) At the SAR, the depth of the real
part is reduced by about 35% at 90 MeV and by just 16 % at
318 MeV. Another view of the DPPs is given in the next section
where, in Figs. 4 to 8, they are compared with those calculated
using the approximate TELP procedure. We see there that the
TELP yields a much smaller reduction in the real potential
in the surface. In the far surface, and at the lowest energy,
these results are roughly consistent with the previously found
reduction factor of around 0.6 in the surface regions. But this
factor rises to about 0.84 at the highest incident energy and
at all energies the factor exceeds unity at smaller radii. The
question then arises as to whether the change from surface
repulsion to attraction further in, at about 4.2 fm for 90 MeV, is
of empirical significance. We return to this question in Sec. IV.
Table II presents the changes induced by breakup in the
real and imaginary volume integrals, as well as the change
in the rms radius of the real term. The quantities J
R
and
0.00
0.10
0.20
0.30
0.40
0.50
7 8 9 10 11 12
SMAT
TELP
-25
-20
-15
-10
-5
0
5
0 1 2 3 4 5 6
DPP real part (MeV)
r (fm)
90 MeV
(a)
-0.20
-0.10
0.00
7 8 9 10 11 12
SMAT
TELP
-12
-9
-6
-3
0
3
0 1 2 3 4 5 6
DPP imaginary part (MeV)
r (fm)
90 MeV
(b)
FIG. 4. Comparison of the S-matrix and TELP DPPs, (a) real and (b) imaginary parts, for 90 MeV.
064611-4