Role of Higher Multipole Excitations in the Electromagnetic Dissociation of One Neutron Halo Nuclei

Article (PDF Available)inEuropean Physical Journal A 35(2) · January 2008with7 Reads
DOI: 10.1140/epja/i2007-10538-7 · Source: arXiv
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.
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. Chatterjee
1,2
, L. Fortunato
1
and A. Vitturi
1
1
Dipartimento di Fisica and INFN, Universit`a di Padova, via F. Marzolo 8, I-35131, 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
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 elec tromagnetic 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 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
non-resonant 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 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 sing le -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 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 low-lying continuum is strongly affected by the pres-
ence of a weakly-bound 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
1
(2l + 1)
|β
|
2
ρ(E
b
,
b
,
c
) .(1)
Here v
a
is the at relative velocity in the entrance chan-
nel and ρ(E
b
,
b
,
c
) the phase space factor appropriate
to the three-body final state [12]. The reduced amplitude
β
in post form finite range disto rted wave Bor n approx-
imation is given by
β
= hexp(γk
c
αK )|V
bc
|Φ
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
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
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 time-dependent Alder-Winther 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 oss-section can be interpreted in terms o f
equivalent photon number, n
E1
, leading to
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 single-particle 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 mean-field 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 s-orbit
to the p-continuum, 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 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
[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
<