Role of Higher Multipole Excitations in the Electromagnetic Dissociation of One Neutron Halo Nuclei
ABSTRACT We investigate the role of higher multipole excitations in the electromagnetic dissociation of one-neutron halo nuclei within two different theoretical models -- a finite range distorted wave Born approximation and another in a more analytical method with a finite range potential. We also show, within a simple picture, how the presence of a weakly bound state affects the breakup cross section.
- [show abstract] [hide abstract]
ABSTRACT: A distorted-wave Born-approximation (DWBA) formalism is developed to describe the ''elastic'' and ''inelastic'' breakup of the projectile in the heavy-ion-induced reactions. Using a ''local momentum approximation'' and the surface localization property of these reactions expressions for the DWBA transition amplitude are derived. One of the attractive features of the present formulation is the fact that the radial integrals involved in it are the same as those contained in the transition amplitude evaluated with a zero range approximation. The validity of the approximations used in these derivations has been tested and it is shown that these approximations are well fulfilled.Annals of Physics - ANN PHYS N Y. 01/1985; 163(2):265-287.
- [show abstract] [hide abstract]
ABSTRACT: We have measured the excitation energy spectrum of the 10Be+n system in the Coulomb dissociation of the one-neutron-halo nucleus 11Be on a Pb target at 72A MeV. We have determined the dipole strength distribution dB (E1) /dEx which shows a strong peak at about 800 keV excitation energy. The spectral shape is consistent with a direct breakup model. The Coulomb post accelaration effect has been observed also to support the direct breakup mechanism. The E1 dominance of the Coulomb dissociation is confirmed by observed angular distributions of 10Be in the 11Be rest frame.Physics Letters B 01/1994; · 4.57 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Some cross section transformation problems are treated which occur in coincidence measurements on nuclear reactions with three fragments in the exit channel. Expressions are given for the transformation from the coordinates of two fragments to the coordinates of their relative and centre-of-mass motion. The influence of constraints, for instance on the reaction Q-value, is taken into account.Nuclear Instruments and Methods in Physics Research. 01/1982; 200:361-368.
arXiv:0712.3330v1 [nucl-th] 20 Dec 2007
EPJ manuscript No.
(will be inserted by the editor)
Role of Higher Multipole Excitations in the Electromagnetic
Dissociation of One Neutron Halo Nuclei
R. Chatterjee1,2, L. Fortunato1and A. Vitturi1
1Dipartimento di Fisica and INFN, Universit` a di Padova, via F. Marzolo 8, I-35131, Padova, Italy
2Theory Group, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata 700064, India
Received: date / Revised version: date
Abstract. We investigate the role of higher multipole excitations in the electromagnetic dissociation of
one-neutron halo nuclei within two different theoretical models – a finite range distorted wave Born ap-
proximation and another in a more analytical method with a finite range potential. We also show, within
a simple picture, how the presence of a weakly bound state affects the breakup cross section.
PACS. 24.10.-i Nuclear reaction models and methods – 24.50.+g Direct reactions
The electromagnetic or Coulomb dissociation method is
a well established method to study the properties of a
wide variety of nuclei – from stable to weakly bound drip
line ones. That the electromagnetic interaction is quantifi-
able has led to the development of several analytical and
semi-analytical reaction models in nuclear physics ranging
from the semi-classical to the purely quantal. Theoretical
studies in Coulomb breakup reactions have also received
a boost, in recent years, with increasing interest in nuclei
far from the valley of stability, where one encounters iso-
topes which are often extremely unstable (especially those
closer to the driplines) and have structure and properties
different from stable ones. Investigating these nuclei also
opens up the interesting prospect of testing the limits of
theoretical models across the nuclear chart.
Coulomb dissociation has a rich history of being used
as a probe to investigate projectile structure information.
For instance, it would place constraints on their electric
dipole response [1,2,3,4]. It has also been used as an use-
ful indirect method in nuclear astrophysics. One can relate
the Coulomb breakup cross section to the corresponding
photodisintegration cross section and in turn relate it to
the inverse radiative capture cross section . One is thus
able to simulate and measure reaction cross sections, in
the laboratory on earth, of stellar reactions which goes
on at extremely low energies. This method thus provides
an ideal theoretical laboratory to study the physics of
breakup reactions as a tool for nuclear structure and as-
In this paper we primarily investigate the role of higher
multipole excitations as a function of neutron separation
energy in the electromagnetic dissociation of one neutron
halo nuclei with direct reactions and with two different
theoretical methods. The first method is based on the
post form finite range distorted wave Born approxima-
tion (FRDWBA)  (henceforth referred to as Method 1,
see section 2.1). In this model the electromagnetic interac-
tion between the core and the target nucleus is included to
all orders and the breakup contributions from the entire
non-resonant continuum corresponding to all the multi-
poles and the relative orbital angular momenta between
the valence nucleon and the core fragment are included.
Full ground state wave function of the projectile, of any
angular momentum configuration, enters as an input to
the theory. This method has also been referred to as the
Coulomb wave Born approximation in the literature .
The results obtained within this method are compared
with those obtained from a standard first order Coulomb
dissociation theory based on multipole expansion under
the aegis of the Alder-Winther theory  (henceforth in-
dicated with Method 2, see section 2.2). The multipole
strengths entering in this second approach can be calcu-
lated in a simple single-particle picture for one-neutron
halo nuclei as a function of different one-neutron separa-
tion energies (artificially varied from very weakly bound to
more stable systems). For the single-particle potential de-
scribing the interaction between the valence(halo) neutron
and the core, we will either use a Woods-Saxon potential,
which has to be numerically solved, or a square well po-
tential, leading to more analytical expressions [7,8] for the
B(Eλ) distributions. Our aim is to see how far the pre-
dictions from these two models tally with each other.
The comparison of the two methods allows us to ad-
dress the relative importance of dipole and quadrupole
breakup contributions in Coulomb dissociation. We shall
show the sensitivity of the reaction observables to the de-
tails of nuclear structure: in particular, in the case of11Be,
the low-lying continuum is strongly affected by the pres-
ence of a weakly-bound 1p1/2excited state.
2 R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..
The paper is organised in the following way. A brief
summary of the FRDWBA formalism and the relevant
analytical quantal results with the finite range potential
are given in section 2. Our results, with a description of
the structure model, relevant reaction cross sections and
the effect of having a weakly bound state near the particle
emission threshold are discussed in section 3. Summary
and conclusions of our work are in section 4.
2.1 Finite range distorted wave born approximation -
We consider the reaction a + t → b + c + t, where the
projectile a breaks up into fragments b (charged) and c
(uncharged) in the Coulomb field of a target t. The triple
differential cross section for the reaction is given by
(2l + 1)|βlµ|2
ρ(Eb,Ωb,Ωc) . (1)
Here vais the a–t relative velocity in the entrance chan-
nel and ρ(Eb,Ωb,Ωc) the phase space factor appropriate
to the three-body final state . The reduced amplitude
βlµin post form finite range distorted wave Born approx-
imation is given by
× ?χ(−)(kb)χ(−)(δkc)|χ(+)(ka)? ,
where, kb, kcare Jacobi wave vectors of fragments b and c,
respectively in the final channel of the reaction, kais the
wave vector of projectile a in the initial channel and Vbc
is the interaction between b and c. Φlµ
wave function of the projectile with relative orbital angu-
lar momentum state l and projection µ. In the above, K
is an effective local momentum associated with the core-
target relative system, whose direction has been taken to
be the same as the direction of the asymptotic momentum
kb[13,9]. α,δ and γ in Eq. 2, are mass factors relevant to
the Jacobi coordinates of the three body system (see Fig.
1 of Ref. ). χ(−)’s are the distorted waves for relative
motions of b and c with respect to t and the center of mass
(c.m.) of the b−t system, respectively, with ingoing wave
boundary condition and χ(+)(ka) is the distorted wave for
the scattering of the c.m. of projectile a with respect to
the target with outgoing wave boundary condition.
Physically, the first term in Eq. (2) contains the struc-
ture information about the projectile through the ground
state wave function Φlµ
a, and is known as the vertex func-
tion, while the second term is associated only with the
dynamics of the reaction. The charged projectile a and
the fragment b interacts with the target by a point Cou-
lomb interaction and hence χ(−)
stituted with appropriate Coulomb distorted waves. For
pure Coulomb breakup, of course, the interaction between
ais the ground state
(kb) and χ(+)(ka) are sub-
the target and uncharged fragment c is zero and hence
χ(−)(δkc) is replaced by a plane wave. This will allow the
second term of Eq. (2), the dynamical part, to be evalu-
ated analytically in terms of the bremsstrahlung integral
A more detailed description of the formalism can be
found in Refs. [9,10].
2.2 First order multipole Coulomb dissociation
Alternative to the previous method, more standard ap-
proaches are based on the multipole expansion of the Cou-
lomb field. In the time-dependent Alder-Winther formal-
ism, adapted for continuum states, the excitation proba-
bility for a given impact parameter and bombarding en-
ergy is proportional to the dB(Eλ)/dEbcdistribution. In
the case of high bombarding energies the kinematic part
of the dipole cross-section can be interpreted in terms of
equivalent photon number, nE1, leading to
If one further assumes that nE1 is weakly dependent on
the fragment – fragment relative energy after breakup,
then the breakup cross-section σE1is directly proportional
to the total B(E1).
Within this line of reasoning all the information on the
structure is contained in the matrix element of the elec-
tromagnetic operator. In a single-particle description of
the dissociation of a halo nucleus, the transition can be
attributed to the promotion of the valence neutron from
a bound to a continuum state. The B(Eλ) is obtained by
evaluating the matrix element involving the initial bound
and final continuum wave functions defined in the projec-
tile mean-field potential.
For the sake of pushing the mathematical treatment as
far as possible one can choose a finite square well potential
. In the limit of small binding energy, the bound and
unbound states may be written in terms of their asymp-
totic form as first order spherical Henkel functions and
spherical Bessel functions (of an appropriate order) re-
spectively. Since the energy is small, the largest part of
the contribution to the dipole strength will come from the
outer region. This fact allows one to derive simple expres-
sions for the strength distribution, namely dB(E1)/dEbc,
in terms of the binding energy, Sn, and the relative con-
tinuum energy, Erel= Ebc. As an explicit example, if one
considers electric transitions from a weakly bound s-orbit
to the p-continuum, the dipole strength is given by
This distribution has a maximum for Ebc = 3/5Sn .
This prediction will be discussed later in the paper.
As found in Ref. [7,8] in this extreme single-particle
picture the total integrated B(E1) is connected to the
R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..3
mean square radius, r, of the single-particle state, namely
B(E1) = (Z(1)
and more in general for any multipolarity, λ, (Eq. 65 in
B(Eλ) = (Z(λ)
effe)22λ + 1
where the effective charge is defined as
with Zi and Ai (i = b,c) being the charges and atomic
numbers of breakup fragments, respectively and A being
the atomic number of the projectile.
If the potential is a square well the expectation value
of the r2λoperator in the ground state s wave function is:
and the reduced transition probability becomes
B(Eλ) = (Z(λ)
effe)2(2λ + 1)!
effe)2(2λ + 1)!
having used the relation a2= 2µSn/¯ h2, with µ being the
reduced mass of the fragment – fragment system in the
Therefore in the present case by using Eq. (9), one
σE1∝ B(E1) ∝
that relates the total dipole cross-section with the one-
neutron separation energy.
3.1 Ground state structure of11Be and19C
To describe the structure of one-neutron halo nuclei11Be
and19C, we use single particle wave functions for the va-
lence neutron constructed by assuming the neutron-core
interaction to be of Woods-Saxon type. For the purpose
of better understanding the properties of weakly bound
nuclei, we artificially vary the depth of the potential in
order to obtain a set of different binding energies (among
which also the experimental one). In this way we simulate
situations ranging from nuclei at the valley of stability
to the neutron drip line for the same angular momentum
For the ground state of11Be, we have considered the
following configuration : a s – wave valence neutron cou-
pled to 0+ 10Be core, namely [10Be(0+) ⊗ 1s1/2ν] with a
one-neutron separation energy, Sn. A similar method is
adopted for19C : a18C (0+) core coupled to a neutron in
the 2s orbital. The radius and diffuseness parameters of
the Wood-Saxon well for each case have been taken to be
1.15 fm and 0.5 fm, respectively. The list of parameters
for the two isotopes is given in table 1.
Table 1. Potential parameters. The single particle wave func-
tion, is constructed by assuming the valence neutron-core inter-
action to be of Woods-Saxon type whose depth is adjusted to
reproduce the corresponding value of the binding energy with
fixed values of the radius and diffuseness parameters (taken to
be 1.15 fm and 0.5 fm, respectively)
3.2 Relative energy spectra and angular distributions
within the FRDWBA
In this subsection we shall calculate within the FRDWBA
(method 1) the relative energy spectra in the breakup of
11Be and19C on a heavy target for different binding ener-
gies of the projectile by varying the one neutron separation
energy, for the same angular momentum configuration of
the system. We shall also present the angular distribution
of the projectile c.m. with respect to the target in the
breakup of11Be on208Pb at four different beam energies.
In Fig. 1, we show the relative energy spectra for the
breakup of11Be on a208Pb target at 72 MeV/u incident
beam energy, for different one-neutron separation energies:
Sn= 100 keV (solid), 200 keV (dotted), 504 keV (dashed),
750 keV (dot-dashed) and 1000 keV (plus signs). Among
them the actual experimental value is 504 keV. Note that
for Sn= 100 and 200 keV the results are scaled by multi-
plying them with 0.1 and 0.5, respectively.
It is to be noted that the bound state neutron was in
the s-state (initial channel) and the transition to all pos-
sible states in the final channel has been taken care of by
the post form of the transition matrix (Eq. 2). To compare
our results with those predicted from the analytical model
, we plot the peak of the relative energy spectra as a
function of one-neutron separation energy in the breakup
of11Be on a208Pb target at 72 MeV/nucleon in Fig. 2.
It is seen that the slope is approximately 0.4 which
is pretty close to the simple estimate obtained from dis-
tribution 4. The discrepancy can be attributed to the ap-
4R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..
Sn = 100 keV (scaled X 0.1)
= 200 keV (scaled X 0.5)
= 504 keV
= 750 keV
= 1000 keV
11Be + 208Pb →
10Be + n + 208Pb
Ebeam = 72 MeV/u
Fig. 1. Relative energy spectra in the breakup of11Be on a
208Pb target at 72 MeV/u incident beam energy, for different
one-neutron separation energies. For Sn = 100 and 200 keV
the results are scaled by 0.1 and 0.5, respectively.
Separation energy Sn (MeV)
Peak Position (MeV)
Fig. 2. Peak of the relative energy spectra as a function of
(fictitious) one-neutron separation energies in the breakup of
11Be on a208Pb target at 72 MeV/nucleon.
proximate description of the wave functions implicit in the
derivation of the formula.
Fig. 2 could also serve a different purpose. Since the
dependence of the peak of the relative energy spectra as a
function of the one-neutron separation energy is approx-
imately linear, it could also be used to get an heuristic
idea about the separation energy of a system from the
experimental relative energy spectra. As an example, in
Fig. 3 we show the relative energy spectra in the breakup
of19C on a208Pb target at 67 MeV/nucleon, for a set of
different binding energies of19C : Sn= 160 keV (solid),
350 keV (dotted), and 530 keV (dashed). In fact, there is
still an open debate over the precise experimental deter-
mination of one neutron separation energy of19C. While
direct mass measurements suggest a rather low value of
0.16 ± 0.11 MeV [15,16], recent compilations from Audi
et. al  suggest a value of 0.58 ± 0.09 MeV based on
indirect measurements [2,18]. In Ref. , Coulomb disso-
ciation of19C on a208Pb target at 67 MeV/nucleon and
subsequent analyses of relative energy and angular distri-
butions yielded a value Sn = 0.53 ± 0.13 MeV for19C,
while in Ref. , breakup of19C on a9Be target at 60
MeV/nucleon suggested two values, Sn= 0.8 ± 0.3 MeV
and Sn= 0.65 ± 0.15 MeV, based on parallel momentum
Sn = 160 keV (scaled X 0.33)
Sn = 350 keV
Sn = 530 keV
19C + 208Pb →
18C + n + 208Pb
Ebeam = 67 MeV/u
Fig. 3. Relative energy spectra in the breakup of
208Pb target at 67 MeV/u incident beam energy, for different
(fictitious) one-neutron separation energies. For Sn = 160 keV
the results are scaled 0.33.
19C on a
Ea = 72 MeV/u
= 10 MeV/u
= 30 MeV/u
= 20 MeV/u
11Be + 208Pb →
10Be + n + 208Pb
Fig. 4. Angular distribution of the projectile c.m. with respect
to the target for the breakup of
incident beam energies.
208Pb at different
Thus by accurately measuring the strength function
in the breakup of19C and comparing it with our method
one should be able to determine the most probable range
for this binding energy. However, one should also keep in
mind, that not only the peak position but the whole shape
of the curve is characteristic of the given binding energy
(see Eq. 3).
In Fig. 4, we show the angular distribution of the pro-
jectile c.m. with respect to the target in the breakup of
11Be on a208Pb target at four different beam energies
varying from low energy 10 MeV/u to 72 MeV/u. We see
that the distributions are more forward peaked as one in-
creases the incident beam energy.
R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..5
3.3 Effect of a weakly bound state near threshold
In this subsection we investigate yet another feature of
the breakup dynamics of neutron rich halo nuclei – that
of the relative importance of the E1 and E2 transitions
to the continuum as a function of the various fictitious
binding energies of the projectile. However, in doing so
we would also like to study the effect of having a weakly
bound state near the particle emission threshold, as is the
case in11Be, whose first excited state (p1/2) is bound by
only 0.18 MeV.
In an ideal case one should look for a unique poten-
tial which might be able to yield the s1/2 (ground state)
and the p1/2 (first excited state) in11Be at the right posi-
tion. However this straightforward prescription fails in the
case of11Be, where one faces an inversion with respect to
the normal ordering of states and thus in a single-particle
picture we are forced to choose two quite distinct poten-
tials for the ground and first excited states. In addition to
the ’deep potential’ used for the groundstate, we need a
’shallow potential’ of depth 37.38 MeV to obtain the p1/2
bound by 0.18 MeV in11Be. Of course one should keep
in mind that we are simulating with one-body potentials
what in reality might be a more complicated situation (eg.
a deformed core).
Next comes the question of constructing the contin-
uum states. The ground state wave function is very sen-
sitive to the choice of the binding energy as we have seen
in the previous sections while calculating several reaction
observables. However, the prescription to treat the contin-
uum states is of paramount importance. We thus consider
two options here (see Fig. 5) for the potential generating
the continuum wave functions – (I) one in which all the
continuum states are generated with the deep potential,
thereby ensuring that the ground state and the contin-
uum state are generated by the same potential prescrip-
tion and (II) another in which the even and odd parity
continuum states are generated by the deep and shallow
potentials, respectively. Naturally, in these options the E2
transitions would remain the same as the even parity con-
tinuum states are generated by the deep potential in both
cases. What distinguishes these options is the fact that for
the E1 transitions, the relevant continuum states are gen-
erated by deep and shallow potentials for options (I) and
(II), respectively. Furthermore in option (II) the presence
of the bound p state absorbs part of the E1 strength .
We shall now calculate the electric dipole and qua-
drupole angular distribution (dσ/dθat) of the c.m. of the
projectile in its breakup on a heavy target for different
beam energies, within the framework of the Alder-Winther
theory of Coulomb dissociation [11,20]. Since the grazing
angle (θgr) is a function of the beam velocity, instead of
directly plotting (dσ/dθat) as a function of the projectile
c.m. scattering angle (θat), for each beam energy, it would
be prudent to rescale the angular distribution by measur-
ing the angles in units of the grazing angle. Therefore we
plot dσ/dX as a function of X = θat/θgr. Indeed the area
under such a curve would again give the total one-neutron
removal cross section for the concerned multipolarity and
0 0.20.4 0.60.81
0 0.20.4 0.60.81
00.2 0.4 0.60.81
Fig. 6. Scaled angular distributions for different beam ener-
gies, from the ground and the excited states of
allowed E1 transitions to the continuum are shown in the top
and lower panels, respectively, in grazing angle units. For more
details see text.
11Be for all
0 0.2 0.40.6 0.81
0 0.2 0.40.60.81
Fig. 7. Scaled angular distributions for different beam ener-
gies, from the ground and the excited states of
allowed E2 transitions to the continuum are shown in the top
and lower panels, respectively, in grazing angle units. For more
details see text.
11Be for all
In Fig. 6, we show the angular distributions of the
center of mass of the projectile with respect to the tar-
get for E1 in the breakup of11Be on a208Pb target at
three different beam energies 30, 72 and 400 MeV/u as
a function of X for breakup from both the ground and
first excited states of11Be. For beam energies 30, 72 and
400 MeV/u, θgr= 8.12,3.62 and 0.995 degs., respectively.
The top panel shows the allowed E1 transitions from the
s1/2 (ground state) to all possible continuum states, while
the bottom panel shows it from the p1/2 (first excited
state) to all possible continuum states.
In Fig. 7, we show the angular distributions of the
center of mass of11Be in its breakup on a208Pb target
for E2 transitions at three different beam energies 30, 72
6R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..
Fig. 5. A schematic picture of potential options to generate the continuum states. In option (I) the ground state (which happens
to be a s – state) generates both the even and odd parity continuum states. In option (II) the even and odd parity continuum
states are eigenfunctions of potentials generating the ground (s) and excited (p) states of11Be, respectively. For more details
and 400 MeV/u as a function of X for breakup from both
the ground and first excited states. The top panel shows
the allowed E2 transitions from the s1/2 (ground state)
to all possible continuum states, while the bottom panel
shows it from the p1/2 (first excited state) to all possible
In both these calculations the even parity continuum
states are generated from the potential which also sets the
s1/2 ground state (at 0.5 MeV) and the odd parity con-
tinuum states are generated within the shallow potential
that also gives the p1/2 excited state at 0.18 MeV. As ex-
pected, below the grazing angle we see a domination of E1
transitions over the E2. It is interesting to note that the
direct breakup contribution from the first excited state is
of the same order of magnitude or even greater as that
from the ground state. However, in a reaction process the
breakup via the excited state is a two-step process and
therefore its cross section would be small in comparison
with the direct breakup from the ground state . The in-
verse process, namely the capture process via the excited
state may be of importance for astrophysical considera-
tions. A more thorough investigation of the relative role
of the two step E1 via a weakly bound excited state with
respect to a one step E2 from the ground state is called
3.4 Relative importance of dipole and quadrupole
contributions to break-up cross-section
In this subsection we investigate the relative importance
of dipole and quadrupole contributions to break-up cross-
section. This is forst done by comparing the total break-
up cross section obtained within the FRDWBA, which in-
cludes the contribution of all multipolarities, with the pure
Separation energy (Sn keV)
σ ~ 1/Sn
Fig. 8. Total one-neutron removal cross section as a function
of different one-neutron separation energies in the breakup of
11Be on208Pb at 72 MeV/nucleon (top panel) and19C on208Pb
at 67 MeV/nucleon (bottom panel). The solid lines show the
FRDWBA values, while the dashed lines show the 1/Sn curve.
In the top panel filled circles show the total E1 contribution,
which is almost indistinguishable from the FRDWBA values.
dipole cross-section obtained within the Alder-Winther
As an example in Fig. 8, we show the total one-neutron
removal cross section in the breakup of11Be on a208Pb
target at 72 MeV/nucleon and19C on208Pb at 67 MeV/nucleon,
in the top and bottom panels, respectively, as a func-
tion of different one-neutron separation energies. FRD-
WBA results are shown in solid lines while the filled cir-
cles show the total E1 contribution calculated with the
Alder-Winther theory with potential option – I (in the top
R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..7
Fig. 9. (a) Total and (b) relative importance of dipole (solid)
and quadrupole (dashed) breakup cross sections on Pb at 72
MeV/nucleon, as a function of one neutron separation energies
of11Be. The solid lines in both figures are those in which even
parity continuum states are generated from the s1/2and the
odd parity continuum states are generated from the p1/2 –
potential option II. The dotted lines show those in which a
unique single particle potential is used to generate both the
ground and the continuum states – potential option I.
panel). Since these two results are almost coincident, one
can infer that the effect of higher multipoles is negligible.
In other words, moving away from the valley of stability
towards the drip lines does not alter the predominance of
dipole dissociation in the breakup process. Interesting to
note that at higher binding energy, the hyperbolic 1/Sn
behaviour (dashed line in Fig. 8), predicted by the simple
analytical model (section 2.2) fits quite well, while at low
binding energy we see some deviation from the 1/Sncurve
(see Refs. [7,8]).
We now turn our attention to the details of the total
one-neutron removal cross section with and without tak-
ing into account the effect of the p1/2 excited state, within
the Alder-Winther theory. In Fig. 9, the dipole and qua-
drupole breakup cross sections (upper panel) on Pb at 72
MeV/nucleon, and their relative importance (lower panel)
are shown as a function of various fictitious one-neutron
separation energies of11Be. The solid black lines in both
figures are those in which even parity continuum states
are generated from the deep potential and the odd par-
ity continuum states are generated from the shallow one
(option II above). The dotted lines show those in which
a unique single particle potential is used to generate both
the ground and the continuum states (option I above). It is
interesting to note that, although the absolute cross sec-
tions are strongly dependent on the binding energy, the
relative importance of E1/E2 transitions shows a much
weaker behaviour in both options (and especially in op-
4 Summary and Conclusions
In this paper, we have compared two different theoretical
models of breakup reactions by calculating several reac-
tion observables like relative energy spectra, angular dis-
tributions and breakup cross sections, taking the neutron
separation energy as a parameter. Thus we have theoreti-
cally simulated situations ranging from weakly bound iso-
topes to stable ones for the same angular momentum con-
figuration of the system.
We have first calculated the relative energy spectra in
Coulomb induced breakup processes for various projectiles
and beam energies as a function of the neutron separation
energy, within the framework of the post form FRDWBA.
In this model the electromagnetic interaction between the
core and the target nucleus is included to all orders and
the breakup contributions from the entire continuum cor-
responding to all the multipoles and the relative orbital
angular momenta between the valence nucleon and the
core fragment are included in the theory.
We have studied the relative importance of dipole and
quadrupole breakup contributions in Coulomb dissocia-
tion under the framework of the Alder-Winther theory.
We have constructed a single particle toy model for the
structure of the halo nucleus11Be, with separate Woods-
Saxon potentials describing the bound states of the sys-
tem, namely s1/2 (ground state) and p1/2 (first excited
state). To the unique continuum generated for each of
these configurations, we calculated Coulomb breakup cross
sections on Pb by varying again the one neutron separa-
tion energy of11Be. We have found that the relative im-
portance of the E1/E2 transition remains fairly constant.
Finally the investigation of the one-neutron breakup
cross section as a function of separation energy obtained
by comparing the results of two theories revealed that, as
one goes away from the valley of stability towards the drip
lines (where one would encounter predominantly weakly
bound isotopes) higher multipoles, other than the dipole,
do not play any significant role in the breakup process.
That the two results were almost identical also opens up
an interesting opportunity. In the calculation of astrophys-
ical S - factor via the Coulomb dissociation method it is
crucially important that only a particular multipolarity
is almost solely responsible for the breakup process. A
word of caution is in order here: these results have been
obtained for a weakly-bound neutron halo and in princi-
ple one might expect a different behaviour for the case of
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