# 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.

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**ABSTRACT:**Progress in breakup reaction theories, like the distorted wave Born approximation, the continuum discretized coupled channels method and the dynamical eikonal approximation, is brought into focus. The need to calculate exclusive reaction observables and the utility of benchmark tests as arbitrators of theoretical models are discussed. KeywordsBreakup theories-post and prior forms-distorted wave Born approximation-continuum discretized coupled channels-dynamical eikonalPramana 01/2010; 75(1):127-136. · 0.56 Impact Factor - SourceAvailable from: Stefan Typel[Show abstract] [Hide abstract]

**ABSTRACT:**Essential properties of halo nuclei can be described in terms of a few low-energy constants. For neutron halo nuclei, analytical results can be found for wave functions and electromagnetic transition matrix-elements in simple but well-adapted models. These wave functions can be used to study nuclear reactions; an especially simple and instructive example is Coulomb excitation. A systematic expansion in terms of small parameters can be given. We present scaling laws for excitation amplitudes and cross sections. The results can be used to analyze experiments like ${}^{11}$Be Coulomb excitation. They also serve as benchmark tests for more involved reaction theories.European Physical Journal A 09/2008; · 2.04 Impact Factor

Page 1

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

1 Introduction

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 [5]. 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-

trophysics [6].

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) [9] (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 [10].

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 [11] (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.

Page 2

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 Formalism

2.1 Finite range distorted wave born approximation -

Method 1

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

d3σ

dEbdΩbdΩc

=

2π

¯ hva

?

lµ

1

(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 [12]. The reduced amplitude

βlµin post form finite range distorted wave Born approx-

imation is given by

βlµ=?exp(γkc− αK)|Vbc|Φlµ

× ?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. [9]). 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(−)

b

stituted with appropriate Coulomb distorted waves. For

pure Coulomb breakup, of course, the interaction between

a?

(2)

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

[14].

A more detailed description of the formalism can be

found in Refs. [9,10].

2.2 First order multipole Coulomb dissociation

-Method 2

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

dσE1

dEbc

=dB(E1)

dEbc

nE1. (3)

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

[7]. 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

dB(E1)/dEbc∝

√SnE3/2

(Ebc+ Sn)4.

bc

(4)

This distribution has a maximum for Ebc = 3/5Sn [7].

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

Page 3

R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..3

mean square radius, r, of the single-particle state, namely

[7]:

B(E1) = (Z(1)

effe)23

4π?r2? (5)

and more in general for any multipolarity, λ, (Eq. 65 in

Ref.[8])

B(Eλ) = (Z(λ)

effe)22λ + 1

4π

?r2λ? (6)

where the effective charge is defined as

Z(λ)

eff= Zc

?Ab

A

?λ

+ Zb

?−Ac

A

?λ

, (7)

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:

?r2λ? =

(2λ)!

(2a)2λ

(8)

and the reduced transition probability becomes

B(Eλ) = (Z(λ)

effe)2(2λ + 1)!

4π(2a)2λ= (Z(λ)

effe)2(2λ + 1)!

4π

?

¯ h2

8µSn

?λ

(9)

having used the relation a2= 2µSn/¯ h2, with µ being the

reduced mass of the fragment – fragment system in the

final channel.

Therefore in the present case by using Eq. (9), one

obtains

σE1∝ B(E1) ∝

1

Sn. (10)

that relates the total dipole cross-section with the one-

neutron separation energy.

3 Results

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

configuration.

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)

ProjectileSn

Vdepth

(MeV)

67.33

69.00

70.99

72.53

73.89

(MeV)

0.100

0.250

0.504

0.750

1.000

11Be

19C

0.160

0.350

0.530

0.750

1.000

47.43

48.77

49.77

50.82

51.86

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

[7], 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-

Page 4

4R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..

00.2 0.4

Erel (MeV)

0.6

0.81

0

1000

2000

3000

4000

5000

6000

dσ/dErel [mb/MeV]

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.

0 0.2

Separation energy Sn (MeV)

0.4

0.6

0.81

0

0.2

0.4

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 [17] suggest a value of 0.58 ± 0.09 MeV based on

indirect measurements [2,18]. In Ref. [2], 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. [18], 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

measurements.

0

0.5

1

1.5

2

Erel (MeV)

0

2000

4000

6000

8000

dσ/dErel [mb/MeV]

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

020

θat (deg.)

40

100

101

102

103

104

105

dσ/dθat (mb/rad)

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.

11Be on

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.

Page 5

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 [19].

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

transition.

0 0.20.4 0.60.81

0

200

400

600

0 0.20.4 0.60.81

0

400

800

1200

00.2 0.4 0.60.81

X

0

500

1000

1500

dσ/dX (mb)

30 MeV/u

72 MeV/u

400 MeV/u

0 0.20.40.60.81

0

800

1600

2400

p1/2→ s1/2

s1/2→ p3/2

s1/2→ p1/2

p1/2→ d3/2

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

00.20.4 0.60.81

0

0.5

1

1.5

2

0 0.2 0.40.6 0.81

0

1

2

3

30 MeV/u

72 MeV/u

400 MeV/u

0 0.20.40.60.81

X

0

0.5

1

1.5

2

dσ/dX (mb)

0 0.2 0.40.60.81

0

1

2

3

p1/2→ p3/2

p1/2→ f5/2

s1/2→ d5/2

s1/2→ d3/2

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

Page 6

6R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..

(I)

2s1/2

even

odd

parities

E1E2

(II)

2s1/2

even

parity

E2

E1

deep

1p1/2

odd

parity

shallow

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

see text.

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

continuum states.

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 [21]. 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

for.

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

0 200

Separation energy (Sn keV)

400

600

800 1000

0

2000

4000

6000

σ (mb)

0

5000

10000

15000

20000

FRDWBA

σ ~ 1/Sn

Total E1

11Be

19C

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

theory.

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

Page 7

R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..7

10-1

1500

100

101

102

103

104

105

σ (mb)

00.2 0.4

Sn (MeV)

0.6

0.81

0

500

1000

σE1/σE2

E1

E2

(a)

(b)

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-

tion II).

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

heavier/charged clusters.

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