ЯДЕРНАЯ ФИЗИКА, 2015, том 78, № 3-4, с. 278-283

ЯДРА

ANALYSIS OF ORBITAL OCCUPANCY OF VALENCE NEUTRON IN 15C THROUGH COULOMB BREAKUP REACTIONS

©2015 P. Singh*

Department of Physics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, India

Received July 7, 2014

The Coulomb breakup reactions 208Pb(15C, 14C + n)208Pb and 181 Ta(15C, 14C + n)181Ta have been studied at 68 and 85 A MeV beam energies, respectively, within the framework of the eikonal approximation to investigate the orbital occupancy of valence neutron in the 15 C nucleus. The outcomes of the present work favor 0+ ® 2s1/2 as the core—neutron coupling for the ground-state structure with 0.91 as a spectroscopic factor.

DOI: 10.7868/S0044002715010171

1. INTRODUCTION

The peculiarities found in the structural composition of the nuclei lying in the close proximity of neutron/proton drip lines have attracted much attention of scientific community during last three decades. Consequently, remarkable work involving these nuclei has already been done both on theoretical as well as experimental front [1—23]. During these studies various structural aspects of these nuclei were investigated and a novel structure referred to as halo has been established for some of the very loosely bound nuclei. In the simplest picture the nucleon halo may be described as a two-fold structure having a core with normal nuclear density attached with diffused valence nucleon(s). Such structure may arise due to the tunneling of valence nucleon(s) to the classically forbidden region.

So far, various one- and two-nucleon halo structures have been observed around neutron/proton drip lines and the most famous among these are 11 Be,19C, 6He, 11 Li, 8B, 23Al, and 20Mg. However, the Carbon isotopic chain consisting of 15>17>19C remains the center of investigations because it offers more than one prominent candidates for halo structure. In the recent past, several experiments were performed involving 19C and it has been established that it is one-neutron halo system with [0+ ® 2s1/2]1/2+ as a predominance core—neutron spin coupling. Maddalena and Bazin [3, 7] suggest 3/2+ as spin—parity for 17C; as a result the valence neutron is expected to occupy d-orbital, which in turn results in a strong centrifugal barrier

E-mail: panghal005@gmail.com;pardeep.phy@dcrustm.

org

and eventually suppresses the possibility of halo structure in 17C. On the other hand, for 15C, Audi and Goss [8, 9] found 1/2+ as the ground state spin-parity. The possible ground-state configurations for 15 C corresponding to Jn = 1/2+ are 0+ ® 2s1/2 and 2+ ® 1d5/2. The latter represents a combination where the d-orbital valence neutron is attached with core in an excited state, which reduces the possibility of halo structure in 15C. The excited core states have been estimated by the measurements of photon emission in coincidence with momentum distribution of the projectile, the method for estimation of core excitation has been illustrated in detail in [10, 2428]. But 0+ ® 2s1/2 is the promising combination of core state and of valence neutron for having halo structure. The subsequent work carried out involving 15 C strengthens its candidature for halo structure and 14C(0+) ® 2s1/2 as the ground-state configuration with different values of spectroscopic factors has been proposed [10, 11]. Further, the possibility of occupation of d-orbital by valence neutron in 15C has also been considered in [17]. Hence, the ambiguous information regarding ground-state configuration of 15C needs further investigation. Therefore in the present work we studied the orbital occupancy of valence neutron in 15 C through 208Pb(15C, 14 C + + n)208Pb and 181 Ta(15C, 14C + n)181Ta Coulomb breakup reactions for which the interaction potential is well known. The theoretical formalism to describe the Coulomb breakup reaction is given in Section 2. In Section 3 the results of the present work are discussed in detail, while in Section 4 the conclusions are presented.

2. THEORETICAL FORMALISM

In the energy range considered here the eikonal approximation is the most suitable theoretical approach to describe the Coulomb breakup process, wherein the transition amplitude for any electric multipolarity (Elm) can be written as [12, 15, 20]

X (1)

Im

\cJ

x GElm $m(Q)M(Elm),

where the momentum change during the scattering is given by Q = 2k sin(6/2), with k and 6 as the incident momentum and the scattering angle, respectively. The relative motion momentum of the outgoing fragments after the breakup is represented by q. Zt, a, 3, y, and R stand for atomic number of the target, fine structure constant, velocity of the projectile in units of the speed of light, the relativistic Lorentz factor, and the interaction radius, respectively. GElm(c/v) are the usual Winther and Alder functions for electric excitation expressed in terms of associated Legendre polynomials, P™(c/v), and for m ^ 0 are written as[29]

G Elm ( - ) = Ï

l-\-m

y/Tßir ((l-m)\\l/2

l(2l + 1)!! \(l + m)!

2 A"1/2 ( (l + 1)(l + m)

-) -1

v

2l + 1

Pi

l1

l(l-m + 1) 21 + 1

m Pi-i

For m < 0 the Winther—Alder functions are obtained by

GeI-tu = ( —1 )mGElm •

The explicit expressions for Winther—Alder functions needed in this work are given below [29]

Ge 1-1 (-) = -ifSTT)'/^.

VW 3 v

The non-dimensional functions, &m(Q), contain information regarding reaction mechanism and are expressed in terms of Bessel functions

<x

$m(Q) = j Jm(QRx)Km (£x)xdx

with £ = uR/(jv) as the adiabaticity parameter. The matrix element M (Elm), containing structural information about projectile, may be formulated in the long-wavelength approximation and is written as

M (Elm) = zke [ f (r)rk Ylm(nk )fii(r)d3r. k=l,2 '

Here, fii is the initial state of the projectile and can be taken as single-particle shell-model wave function, i.e., fii(r) = RL(r)YLM(r), while the final, fif, state may be assumed as a plane wave, i.e., fif (r) = eiqr. These approximations along with the standard properties of spherical harmonics modify the matrix element to the following form

M (Elm) = V^ZfVWTlV2L + 1 x (2)

\v

V2A + 1

ÏAu(q){Ll00\X0) x

x {LlMm\\v) J r drrj\(qr)RL(r) 0

with

zeff = Zißl + (-i)lZ2ß2, ßi(2)

^2(1) m1 + m2

Now the differential cross section for Coulomb breakup is expressed in terms of transition amplitude

as

d3q

d a = \f (Q, q)\2 dû

(2n)3'

(3)

Substitution of Eqs. (1) and (2) in Eq. (3) and a trivial mathematical exercise lead to the following expression for the differential cross section:

d2

a

Z?

a

q2dq sin 9d9 ^/47r

(4)

x y^(_i)(2L+S+li +A2+m) (_i)(A2-Ax+li-l2)/2 x

x (2/1 + 1)(212 + 1)v/2Ai + 1v/2A2 + 1 X

x Gi1 mGi2mKm(i)lLi1\1 Ili2\2 x x (Ll\00|Ai0) (Ll200\\20) (AiA200|50) x x (li 12m _ m\S0) W(Sl^L : h\i)Yso(q), where the notation ILlx represents the radial integral

<x

J r2drrlj\(qr)RL(r) 0

appearing in Eq. (2). The summation runs over all the quantum numbers except L. YS0(q) is the

1

x

X

X

X

1

280

SINGH

standard spherical harmonics without phase factor eimSince the electric quadrupole term does not contribute significantly, only dipole term is considered here [12]. Equation (4) may be easily transformed to the following explicit expressions for calculating the relative energy spectrum and longitudinal momentum distribution (LMD) corresponding to the s orbital occupation and d-orbital occupation, respectively, [6, 12] for the case L = 0:

+ K2 - Ko2)}(2/2211 + 3/2213) + + 2P2 j-2 (Kf - Kq) + (^KoKi -- (K2 - K2)) 72} X

X I 2 ^211 + 9/211/213 - 6 /f13

qdq.

da El \

dErel / l=o

4 ZKzfYa*

-Ç -'011 x

372p2

(5)

(K2 - K02) {(1 + 2P2) - (1 - P2)Y2} +

2 n

+ -K0Kl(i-p2W

s

\j2£rel (^fzmOdB,

i da El \ V dqz ) L=0

4 Z?(Zf)W 372/32 * 'on

Here, K0, K1, P^ and P2 represent the modified Bessel functions of order one and two, and Legendre polynomial of order one and two.

3. RESULTS AND DISCUSSION

In numerical calculations the radial part of projectile ground-state wave function which depends on the orbital occupancy of valence neutron plays a major role. As mentioned earlier the valence neutron

15

(K2 - K02) {(1 + 2P2) - (1 - P2)Y2} +

2

+-KoK^l - P2)Y s

qdq.

For case L = 2:

/ da Ei \

V dETd J L=2

4Z2(Zff)2«2 2

2

75 y2 p 2

s2 X

5<^-K0Ki-(KÎ-K02)Jy2 +

+ (K2 - Ko2)}(2/2211 + 3/213) + + 2P2 j-2 (Kf - K$) + (jKoKi -

- K12 - K02

Y2 >x

x ( 2 ^211 + 9/211/213 - 6/|13 x ^2£rel (J^'sinM?,

da ei \ dfc /¿=2

4Z2(Zff)2a2 2

75 y2 p 2

2

5 < ( JKQK! - (K( - K20) ) 72 +

s /011 X (6) in 15C may occupy s-orbital or d-orbital corresponding to 0+ ® 2s1/2 and 2+ tg> 1d5/2 configuration. Here we have considered both cases and the wave functions corresponding to these configurations are constructed by solving the radial part of Schrodinger equation for Woods—Saxon potential. The geometrical parameters of the potential are tuned locally while its depth is adjusted to reproduce the effective binding energy (Sff) which is obtained by adding singleneutron separation energy and the excitation energy

(7) of the core state. Summary of parameters obtained for Woods—Saxon potential is given in the table.

Wave functions constructed using the parameters listed in the table are presented in Fig. 1 and the corresponding density distributions are given in Fig. 2. It is noticed from Figs. 1 and 2 that the wave function as well as density distribution for d-orbital valence neutron attached with the 2+ core state is much less extended in space in comparison to that of s-orbital neutron. The contraction of the wave function and rapid fall of density distribution may be attributed to the presence of centrifugal barrier for d-wave valence neutron. Moreover, the presence of 2+ core state increases the effective binding energy up to 8.228 MeV for 2+ ® 1d5/2 configuration and eventually confines the expansion of the corresponding wave function for the relative motion of core and valence neutron as it is shown in Fig. 1 by dotted curve. While the solid curve in Fig. 1 re

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