ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 12, с. 2129-2135
ЯДРА
COMPARISON OF HALO OF nBe, 15C, AND 19C
© 2007 R. Kharab*, R. Kumar, P. Singh, H. C. Sharma1)
Department of Physics, Kurukshetra University, India Received November 24, 2006; in final form, February 28, 2007
We have compared the halo of11 Be,15C, and 19C nuclei by analyzing the one-neutron stripping reaction data on the Be target at 60, 54, and 57-MeV/A beam energies, respectively, within the framework of the eikonal approximation approach. The determination of effective range through the comparison of the total cross section data and prediction has revealed that the halo of19 C is well developed while that of15 C is the least and of11 Be lies in between these two. The longitudinal momentum distribution data also strengthen these observations.
PACS:25.60.Gc, 25.70.Mn, 25.60.Dz
1. INTRODUCTION
Following the pioneering work of Tanihata et al. [1] the breakup reactions of unstable nuclei lying in the close vicinity of neutron/proton drip line have confirmed the existence of a novel nuclear structure frequently refered to as nucleon halo. This peculiar nuclear structure consists of a nuclear core with normal nuclear density and diffused one or two valence nucleon(s). This threshold effect may be attributed to the tunneling out of outermost valence nucleon(s) to the classically forbidden region. Unexpectedly large matter radius, long tail in density distribution, small binding energy, and narrow longitudinal momentum distribution (LMD) of the constituent fragments in their ground state are some of the characteristic features of the halo nuclei. Because of these unique ground-state properties of halo nuclei, their breakup cross section on light or heavy target has been found as significantly enhanced.
In case of light target and at small impact parameter the breakup occurs predominantly by the nuclear interaction between the participants through two different mechanisms namely stripping and diffraction dissociation. In the stripping reaction one of the constituent fragments of the projectile is absorbed by the target while the other goes out [2]. The stripping reaction involving halo nuclei represents an efficient tool to ascertain various ground-state properties of these nuclei [3]. So far, the number of different theoretical models has been proposed to deal with the stripping reactions of halo nuclei [2, 4—7]. However,
''Department of Applied Sciences, Haryana College of Technology and Management, Kaithal, India. E-mail:kharabrajesh@rediffmail.com
at high energies the eikonal approximation approach has been considered as the most convenient one [8].
Experimentally, one-neutron (nBe, 19C) [9, 10], one-proton (8B) [11], and two-neutron (11Li) [12] halo structures have been well established. However, the nuclei with one valence neutron occupying l = = 0 orbital have attracted a significant attention. Besides being simple, the absence of centrifugal and Coulomb barrier strengthens their candidature for halo structure. Some of the light neutron-rich nuclei like 11 Be and 19C appear to have well-developed one-neutron halo structure while15C is good candidate for it. In the present work we have compared the halos of these nuclei by determining the effective range of the nuclear interaction between the valence neutron and core through comparison of experimental and theoretical stripping cross sections. The theoretical formalism is being described briefly in Section 2, while the results are discussed in Section 3. The conclusions are presented in Section 4.
2. THEORETICAL FORMALISM
The basic expression for the neutron absorption cross section, differential in the momentum of the core (kc) within the eikonal approximation approach, is given by [7]
da 11
dk
Mo
(2n)3 2Lo + 1 У d2bn[1 - \Sn(bn)\2] x
dre-ikc •rSc(bc )^Lo Mo (r)
x
x
Here, ^LqMq (r), the wave function for the relative motion of core and valence neutron, may be conveniently assumed to have the usual single-particle form ^LqMq (r) = Yl0Mo (r)RLo (r), where Rl0 is the radial part of the wave function and the YLqMq (r) denote usual spherical harmonics. The functions Sn(bn) and Sc(bc), the so-called profile functions, properly take into account the interaction of the valence neutron and the core with the target nucleus, respectively. The impact parameters bc and bn of the core and the valence neutron with respect to the target are related to the center-of-mass and relativemotion coordinates (R, r) of the fragments of the projectile through the following relation:
bc = Rj_ — rj_ and hn = Rj_ + r_|_
Ac + 1
A + 1
with Ac as the mass number of the core.
The use of integral / dkc exp[—lkc • (r — r')j = = (2^)35(r — r') to integrate Eq. (1) over kc yields the following expression for the total stripping cross section:
1 Y^ j d2bn[l - |S"ra(bra)|2] x (2)
a
2Lo + 1
Mo
X j Mo (r) \Sc(bc)\^LoMo (r).
Another important observable of the stripping reaction is the LMD and is obtained by integrating Eq. (1 ' over all transverse momenta kc^ of the core. It is written as
d(T 1 jd2bn[l-\Sn(bn)\2} x (3)
dkcz 4n
œ oo
xj r±dr±f (r±,bn) J dz J dz'eikcz(z-z') x
—oo —oo
x Rio (i +z2) Rl° (Vrl + z,2) Pl■
where PLo (r • r') is the Legendre polynomial and the function f (r±, bn) is given by:
2n
f(r±,bn) = ± J d(j) |5c(|bn — rj_|)|2. (4)
o
Equation (3) has been obtained by using
J dkc± exp[-fkc • (r - r')] = (2nfô(r± - r'±)
E^LoMo (r)YLoMo (r')
Mo
and
/x (2Lo + 1)
4n
Plo (i • r').
The main ingredients in the calculations of cross section and LMD are the profile functions and the radial part of the initial-state wave function of the projectile.
The simplest approximation for the profile function that is often used is the extreme strong-absorption black-disk (BD) model [13]:
Si(bi ) =
0, bi < Rt + Ri
1,bi > Rt + Ri
(5)
i = n,c.
In this model the expression for LMD may be simplified to the following form:
Rt+R n
da 1
dk7 = ~ 1
^ J bndbn J r±dr±f(r±,bn) x
o
Rt+Rc-b„
(6)
oo oo
J dz J dz'eik^z~z,) R*Lo + z2^j x
—oo —oo
xRLo (j/rl + z^PLo(r-r'),
and the corresponding total cross section may be rewritten as
Rt+Rn
) a = n J bndbn J r±dr±f (r±,bn) x (7)
Rt+Rc-bn
x J dzR*Lo (fiTï) RLo (fîT?) PLo{i • r),
—<x
where the function f (r±, bn) now becomes:
1 i (r* +bl-(Rc+Rtf ■ 1 - - COS ^ g—n
f (r±, bn) = { for r± < Rc + Rt + bn, 1, for r± > Rc + Rt + bn,
(8)
with Rt, Rc, and Rn being the radii of target, core, and the neutron, respectively. In the present study in order to check the dependence of the cross section and LMD to the nuclear boundary diffuseness, we have used the following diffused form of core profile function (Sc ):
Sc = 1-
1 + exp
bc - (Rt + Rc) A
(9)
with A being the diffuseness of the interaction potential. Here, bc = |bra — rj_| = \Jb^ + r\ — 2bnr± cos (f)
represents the impact parameter of the core with respect to the target.
x
P, fm-1
Fig. 1. Combination of finite-range parameter ¡3 and nuclear boundary diffuseness (A) which reproduce the total stripping cross section data of 11Be, 15C, and 19C nuclei.
For simplicity, we shall use the bound-state solution of a zero-range potential, i.e.,
R(r) = y/2a-
(10)
with a = e and n being the separation
energy of halo nucleon and reduced mass of the fragments, respectively, as the ground-state wave function of the projectile. It is quite justified for one-neutron halo nuclei in s state. To study the effect of the finite range of nuclear interaction between the fragments within the projectile, we have used the following form of ground-state wave function:
R(r) =
<2al3(a + l3) (e~ar -(/3 — a)2 r
(11)
It corresponds to the solution of Schrodinger equation in nonlocal separable potential introduced by Yamaguchi [14]. Here, the free parameter [ characterizes the finite range of nuclear interaction and is adjusted to reproduce experimental results. It may be mentioned here that when [ ^^ (i.e., range tends to zero), the Eq. (11) turns into Eq. (10), as expected.
3. RESULTS AND DISCUSSION
We have performed calculations of the stripping cross sections, integrated as well as differential in the longitudinal component of the momentum of outgoing core fragments, for 11 Be, 15C, and 19C nuclei on 9Be target. The major ingredients required are the radial part of the ground-state wave function of the projectile and the profile functions corresponding to the core—target and neutron—target interactions. Since the ground state of all the nuclei considered here is predominantly the s state, it is reasonable to describe these by solutions of Schro dinger equation in a nonlocal separable potential introduced by Ya-maguchi with free range parameter ([ ). The effects of the finite range of the nuclear forces between the constituents fragments of the projectile may be taken into account through the parameter [. In order to study the effect of the diffuseness of the nuclear surface, we have chosen the Fermi-distribution-type profile functions with free diffuseness parameter (A). The effects of finite range of nuclear forces and the nuclear boundary diffuseness on the stripping cross section have thus been investigated. The calculated cross section has been observed to be decreasing considerably with increasing diffuseness (A), while it increases with increasing finite range of nuclear
—ar
r
Effective range, fm
3.0
2.6
2.2
1.8
0
0.4
0.8
A, fm
Fig. 2. Variation of the effective range of nuclear interaction between the constituents fragments of the projectile with the nuclear boundary diffuseness.
d<5/dkcz, arb. units
2000
1000
5750
Yukawa wave function Yamaguchi wave function Data points
5850
5950
kcz, MeV/c
Fig. 3. The calculated longitudinal momentum distribution of18 C coming out from stripping reaction of19 C on Be target along with the data points at 57 MeV/A taken from [ 17]. Dashed and solid theoretical curves are overlapped completely.
0
d<3/dkcv arb. units
kcz, MeV/c
Fig. 4. The same as F
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