научная статья по теме COULOMB BREAKUP OF NEUTRON-RICH ISOTOPES OF LIGHT NUCLEI Физика

Текст научной статьи на тему «COULOMB BREAKUP OF NEUTRON-RICH ISOTOPES OF LIGHT NUCLEI»

ЯДРА

COULOMB BREAKUP OF NEUTRON-RICH ISOTOPES

OF LIGHT NUCLEI

© 2008 P. Singh*, R. Kumar, R. Kharab**

Department of Applied Sciences, Technology Education and Research Institute, Kurukshetra, India; Department of Physics, Kurukshetra University, India Received October 4, 2007; in final form, March 17, 2008

The effects of higher order multipole transitions, in particular, E2 and E1-E2 interference, in the Coulomb dissociation of neutron-rich nuclei nBe, 14B, 15C, and 19C on Pb target at energies 72, 86, 550, and 77 A MeV, respectively, within the framework of the eikonal-approximation approach had been studied. The main steps involved in the derivation of the explicit expressions corresponding to dipole, quadrupole, and dipole—quadrupole-interference terms have been outlined. The calculations have revealed that the contribution of E2 transitions to the total cross section is finite but small, while that of E1-E2 interference is nil. Nevertheless, the E 1-E2 interference term introduces small distortions in the peak of the relative-energy spectrum. The calculated results have been compared with the corresponding data and the comparison favors a value of 0.530 MeV as the ground-state binding energy of19C.

PACS:25.60.Dz, 25.70.De, 25.70.Mn

1. INTRODUCTION

Now a day, it is a well established fact that there exist nuclei in the vicinity of neutron/proton drip lines having one or two very loosely bound valence nucleons extending too far out in space with respect to a dense charged core [1]. This peculiar nuclear structure is frequently referred to as nucleon "halo". Nucleon halo is basically a threshold effect resulting from the presence of a bound state close to the continuum and consists of a twofold structure composed of a core with normal nuclear density and a halo with diffuse valence nucleon(s) [2, 3]. The halo systems which involve a new form of nuclear matter are characterized by large interaction cross section, small binding energy, large matter radius, long tail in density distribution, and the narrow component in the momentum distribution of the constituent fragments [4—9]. Due to these unique properties, such systems need a stringent test regarding their structure.

Owing to the simplicity of reaction mechanism, the Coulomb dissociation reactions of halo nuclei on heavy targets provide a suitable channel to examine their structural properties. Basically, the process of Coulomb breakup, in the projectile frame of reference, is described as the absorption of a virtual photon by the projectile in the time-varying electromagnetic field of a heavy target. The Coulomb breakup cross

E-mail: panghal005@yahoo.co.in

E-mail: kharabrajesh@rediffmail.com

section is usually expressed in terms of the matrix element corresponding to electric dipole transition. Under suitable approximations, the matrix element represents the Fourier transform of the radial part of the ground-state wave function of the projectile. Thus, the ground-state wave function of projectile is conveniently obtained by inverse Fourier transform of the matrix element. However, the wave function so obtained may not be exact because of the presence of higher order multipole transitions, especially E2 and El-E2 interference, in the Coulomb breakup of the projectile.

Secondly, the Coulomb breakup of unstable nuclei attracts significant attention due to its emerging applications in determining the cross sections of radiative capture reactions of astrophysical interest at low energies [10]. But unfortunately, the information provided by the inverse cross section is not complete, because electromagnetic multipoles contribute with different strengths in capture as well as in Coulomb excitation reactions. The Coulomb excitation cross section has a nonnegligible contribution from the electric quadrupole (E2) transition, but the capture reaction at stellar temperature is completely dominated by the dipole transition. Thus, the capture cross section cannot be accurately determined without an independent measurement of the contribution from E2 transition.

In last two decades, a lot of unstable nuclei lying close to drip lines have been investigated intensively and one-neutron, two-neutron, and one-proton halo

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structures have been established. The one-neutron halo nuclei with zero orbital angular momentum in ground state have attracted a significant interest because of simplicity. Also, the absence of Coulomb as well as centrifugal barriers strengthens their candidature for halo structure. Keeping these facts in mind, we have systematically studied the Coulomb breakup of light neutron-rich isotopes, in particular, which are strong candidates for one-neutron halo structure. We have investigated the effects of E2 and El-E2-interference terms on the Coulomb breakup cross section, integrated as well as differential in the relative energy of the outgoing fragments, within the framework of the eikonal-approximation theory. The detailed description of the theoretical model is being given in Section 2. The calculated results are presented in Section 3, while Section 4 contains the conclusions drawn.

2. THEORETICAL FORMALISM

The Coulomb dissociation of halo nuclei has been investigated theoretically by several authors using different approaches [11 — 17]. However, at sufficiently high energy the eikonal approximation is the most convenient approach to describe the Coulomb breakup process. Under this approximation, when an energetic projectile having charge Zpe is projected onto a heavy target with charge Zte, then their relative motion trajectory is well approximated by a straight line and the transition amplitude for any electric multipolarity (Elm) is given by [18]

fC (Q, q)

TT

lm

ß7

l

x

c )

x G El

m I $m(Q)M (Elm),

here the standard notation for Bessel functions has been employed. In addition, hw = e + Eq = ^¡(rj2 +

+ q2) is the sum of the absolute value of the binding energy and the kinetic energy of the relative motion of the outgoing fragments.

M (Elm) is the matrix element for electric transition of multipolarity (lm), it contains structural information about projectile and under the long-wavelength approximation is given as

M (Elm) zj Î f (r)rj Ytm(n3 )&i(r)dr, (3)

j=1,2 J

with (fyf ) as the initial (final) state of the projectile. The initial or ground state of the projectile is described by the shell-model wave function (r) = = R(r)YLM(r), while the final state is considered as a plane wave, i.e., (r) = eiq'r. These assumptions for and along with the relation

/-wwu - /

(2h + l)(2l2 + 1) 4vr(2/3 + 1)

x (l1l2m1m2\l3m3) (I1I2001130)

lead to

M (Elm) = V^rZfy/2l + ly/2 L + 1 x (4)

E'

\v

V2XTT

Y\v (q) x

(1)

where Q is the momentum change in the scattering and is given by Q = 2k sin (9/2), with 9 and k as the scattering angle and the incident momentum of the center of mass of the projectile, respectively; q = ß1k1 — ß2k2 is the relative momentum of the outgoing fragments, k1 and k2 are the momenta of the corresponding clusters with masses m1 and m2, respectively, with ß1 = m2/(m1 + m2) and ß2 = = m1/(m1 + m2). The notations a, ß, 7, R, and GElm(c/v) represent the fine-structure constant, the velocity of projectile in units of the speed of light, the relativistic Lorentz factor, the interaction radius, and relativistic Winther—Alder functions, respectively. The nondimensional functions $m(Q), which contain information regarding reaction mechanism, are expressed in term of Bessel functions as follows:

x (Ll00\X0) (LlMm\\v) x

x J r2drrlj\(qr)RL(r),

where Zeff = [Z1pl1 + (-l) Z2pl2].

Now the differential cross section for Coulomb excitation is related to the transition amplitude through the following relation:

d'a = (5)

M

(2n)2k2'

Using Eq. (1) we may rewrite it as

7 2a2

d^ = ^TJk2R4 x (6)

ß 2 7 2

x L-limi(-i)m2hl2 X

M,l1 ,m1 l2,m2

$>m(Q) = J Jm(QRx)Km(^^jxdx-, (2) x M(El1m1)M*(El2m2)QdQ

x GEhmi (£) G*Ehm2 (£) <S>mi(QWm2(Q) x

dq

(2n)2k2'

x

1

X

0

COULOMB BREAKUP Inserting Eq. (4) into Eq. (6) and abbreviating the

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integral 2R2£2 f QdQ^*mi (Q)$ m-2 (Q) — fmim2 ), 0

with £ — uR/(jv) as the adiabaticity parameter, yield

d3a _ Z2a2

dq

8n2

^ L-1im1 (-i)m2 ¡1¡2 x (7)

M,U ,mi ,Л

l2,m2,\2,V2

I1+I2 — 2 /С\ * Í C\

x [—) GEiimi J GEhm2 J fmim2(0 x

x 4nZf1sZf2sL%li i-Л1 (-i)-Л2\11\21Y^X1V1 (q) x x YZ2u2(q) (LliMmi\XiVi) (Lli00\Xi0) x x (Ll2Mm2\X2V2) (Lh00\X20) x

x J r2drrl1 j\1 (qr)Rb(r) j r2drrl2 j\2 (qr)RL(r).

Integrating over ф, which is readily performed by using / et(v1—V2")фdф — 2n8(vi — u2), we obtain

d2 a

9 , • л ,л = zta2 X

q2dq sin Odd

^ Li(2m1+l1—Л1 ) x

(8)

M,l1,m1 , l2,m2 ,Л1,Л2,и

x (—i)(2m2 +l2—M)l2l2x—iX

f Ш \ l1+l2 —2

С J

X

x G¡imi Gi2m2 fmim2(0ZhZl2 X

Gl2m2 (x Гл1и(q)YT2„(q) (LliMmi\Xiv) (Lli00\Xi0) x x (Ll2Mm2\X2v) (Ll200\X20) (q)IU2Л2(q),

where the notation ILi\(q) represents the radial integration appearing in Eq. (4) and the Winther— Alder functions are redefined as GEim (c/v) = = il+mGElm (c/v). The relation

(9A)Ymv (M) = (2Ai + l)(2A2 + l)l1/2

E

4n(2S + 1)

x {X1X2w\Sms) (A1A2OOIS0) Ysms (q)

and symmetry properties of C.G. coefficients simplify Eq. (8) to the following form:

d2

a

Z?

a

q2dqsmed6 ^/47г

(9)

x y^ ( —i)(2L+S+l1 +Л2+m)( — 1)(Л2 — Л1 +l1—12)/2 x

x (2/1 + 1)(2¿2 + 1)v/2Ai + 1v/2A2 + 1 x

\/2S + 1 V с J \cJ

x ZiffziffGiimGi2mKm(^)ILii\i (q)ILi2x2 (q) x x (Lli00\ Ai0) (Ll200\ A2O) x x (AiA200\ 50) (lil2rn - m\ S0) x x W(SliA2L : l2Ai)Yso(q),

_ <x

where Km(() = 2R2i2 J QdQ$*m(Q,04>m(Q,0 and o

W(SliA2L : l2Ai) represents the Racah coefficient. The summation runs over all the quantum numbers except L and Y;50(q) is the standard spherical harmonics without phase factor e%m. In case of nuclei having a single valence neutron occupying orbital of zero angular momentum (L = 0), it is quit trivial, using some angular momentum algebra and properties of spherical harmonics, to obtain the following explicit expressions:

d2aE i

q2dq sin OdO

4Z2(Zf)2a

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