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 oneneutron 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.
arXiv:0712.3330v1 [nuclth] 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. Chatterjee
1,2
, L. Fortunato
1
and A. Vitturi
1
1
Dipartimento di Fisica and INFN, Universit`a di Padova, via F. Marzolo 8, I35131, Padova, Italy
2
Theory 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
oneneutron 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 elec tromagnetic interaction is quantiﬁ
able has led to the development of several analytical and
semianalytical reaction models in nuclear physics ranging
from the semiclassical 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 e ncounters iso
topes which a re 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 a cross 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 o n their electric
dipole response [1,2,3,4]. It has a lso been used as an use
ful indirect method in nuclear astrophysics. One can relate
the Coulomb breakup cross sec tion to the corresponding
photodisintegration cros s 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 reac tio ns 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 r e actions a nd with two diﬀerent
theoretical methods. The ﬁrst method is based on the
post form ﬁnite r ange distorted wave Bor n 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
nonresonant continuum cor responding to all the multi
poles and the relative orbital angular momenta between
the valence nucleon and the core fra gment 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 a re compare d
with those obtained from a standard ﬁrst o rder Coulomb
dissociation theory base d on multipole expansion under
the aegis of the AlderWinther 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 singleparticle picture for oneneutron
halo nuclei as a function of diﬀerent oneneutron separa
tion energies (artiﬁcially varied from very weakly bound to
more stable systems). For the sing le particle potential de
scribing the interaction between the valence(halo) neutron
and the core, we will either use a WoodsSaxon potential,
which has to be numerically solved, or a square well po
tential, leading to more ana ly tical expressions [7,8] for the
B(Eλ) distributions. Our aim is to se e 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 of
11
Be,
the lowlying continuum is strongly aﬀected by the pres
ence of a weaklybound 1p
1/2
excited state.
2 R. Chatterjee, L. Fortunato, A. Vitturi: Higher Multipole Excitations in Electromagnetic Dissociation ..
The paper is organise d in the following way. A br ie f
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 str uctur e 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 s e c tion 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 bre aks up into fragments b (charged) and c
(uncharge d) in the Coulomb ﬁeld of a target t. The triple
diﬀerential cross section for the reaction is given by
d
3
σ
dE
b
dΩ
b
dΩ
c
=
2π
¯hv
a
X
lµ
1
(2l + 1)
β
lµ

2
ρ(E
b
, Ω
b
, Ω
c
) .(1)
Here v
a
is the a–t relative velocity in the entrance chan
nel and ρ(E
b
, Ω
b
, Ω
c
) the phase space factor appropriate
to the threebody ﬁnal state [12]. The reduced amplitude
β
lµ
in post form ﬁnite range disto rted wave Bor n approx
imation is given by
β
lµ
= hexp(γk
c
− αK )V
bc
Φ
lµ
a
i
× hχ
(−)
(k
b
)χ
(−)
(δk
c
)χ
(+)
(k
a
)i , (2)
where, k
b
, k
c
are J acobi wave vectors of fragments b and c,
respectively in the ﬁnal channel o f the reaction, k
a
is the
wave vector of projectile a in the initial channel and V
bc
is the interaction between b and c. Φ
lµ
a
is the ground state
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
k
b
[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
(+)
(k
a
) is the distorted wave for
the scattering of the c.m. of projectile a with respect to
the target with outgoing wave boundary c ondition.
Physically, the ﬁrst term in Eq. (2) contains the struc
ture information about the projectile through the ground
state wave function Φ
lµ
a
, and is known as the vertex func
tion, while the second term is associated only w ith the
dynamics of the reaction. The charged projectile a and
the fragment b interacts with the target by a point Cou
lomb interaction a nd hence χ
(−)
b
(k
b
) and χ
(+)
(k
a
) are sub
stituted with appropriate Coulomb distorted waves. For
pure Coulomb breakup, of course, the interaction between
the target and uncharged fragment c is zero and hence
χ
(−)
(δk
c
) 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 brems strahlung 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 multip ole expansion of the Cou
lomb ﬁeld. In the timedependent AlderWinther formal
ism, adapted for continuum s tates, the excitation proba
bility for a given impact parameter and bombarding en
ergy is proportional to the dB(Eλ)/dE
bc
distribution. In
the case of high bombarding energies the kinematic part
of the dipole cr osssection can be interpreted in terms o f
equivalent photon number, n
E1
, leading to
dσ
E1
dE
bc
=
dB(E1)
dE
bc
n
E1
. (3)
If one further assumes that n
E1
is weakly dependent on
the fragment – fragment relative energy after breakup,
then the breakup cross section σ
E1
is directly proportional
to the total B(E1).
Within this line of reasoning all the information on the
structure is contained in the ma trix element of the elec
tromagnetic operator. In a singleparticle description of
the dissociation of a halo nucleus, the transition can be
attributed to the promotion of the valence neutro n 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 meanﬁeld potential.
For the sake of pushing the mathematical treatment as
far as possible one can choose a ﬁnite s quare 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
sp e c tively. Since the energy is small, the largest part of
the contribution to the dipole stre ngth will come fr om the
outer region. This fact allows one to derive s imple expres
sions for the strength distribution, namely dB(E1)/dE
bc
,
in terms of the binding energy, S
n
, and the relative con
tinuum energy, E
rel
= E
bc
. As an explicit example, if one
considers electric transitions from a weakly bound sorbit
to the pcontinuum, the dipole strength is given by
dB(E1)/dE
bc
∝
√
S
n
E
3/2
bc
(E
bc
+ S
n
)
4
. (4)
This distribution has a ma ximum for E
bc
= 3/5S
n
[7].
This prediction will be dis c ussed later in the pa per.
As found in Ref. [7,8] in this extreme singleparticle
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 singleparticle state, namely
[7]:
B(E1) = (Z
(1)
eff
e)
2
3
4π
hr
2
i (5)
and more in general for any multipolarity, λ, (Eq. 65 in
Ref.[8])
B(Eλ) = (Z
(λ)
eff
e)
2
2λ + 1
4π
hr
2λ
i (6)
where the eﬀective charge is deﬁned as
Z
(λ)
eff
<