ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 2, с. 229-233
ЯДРА
CONTRIBUTION OF ELECTRIC QUADRUPOLE AND DIPOLE-QUADRUPOLE INTERFERENCE TERMS IN COULOMB BREAKUP OF 15C
© 2014 P. Singh1)*, S. Kharb2), M. Singh3)
Received January 14, 2013
The effects of electric quadrupole (E2) and dipole—quadrupole interference (E1—E2) terms in the Coulomb breakup of 15 C have been investigated within the framework of eikonal approximation. The sensitivity of Coulomb breakup cross section, differential in relative energy and Longitudinal Momentum Distribution (LMD)of core fragments, towards these terms have been examined. A very small (1% of E1) contribution of E2 transition has been predicted in integrated Coulomb breakup cross section. Further it is also found that the inclusion of E2 and E1 —E2 terms introduces a small asymmetry in the peak of relative energy spectrum and also increases the peak height of the spectrum. The contribution of dipole—quadrupole interference terms is clearly shown in LMD, as it introduces an asymmetry in the shape of LMD and enhances the matching between the data and predictions.
DOI: 10.7868/S004400271401019X
1. INTRODUCTION
In CNO (Carbon—Nitrogen—Oxygen), 14C(n, y)
151
C(p-)15N(n, y) )16O(n, y) O(n, a)14C
cycle the neutron capture reaction 14C(n, y)15 C attracts significant attention because of its considerable contribution in the synthesis of elements heavier than A > 20. Further, this reaction is also important in context of the ground-state configuration of 15C, which is final state achieved by 14C through neutron capture, as it has a moderate-sized neutron halo with low neutron binding energy [1—4]. Therefore, in order to predict the rate of reaction 14C(n, y)15 C the accurate information regarding the ground-state configuration of15 C and the precise measurement of reaction cross section are required. Preferably, the reaction cross sections are to be directly measured in the laboratory. But, despite of many efforts, the required environment like high temperature and density cannot be attained for performing such experiments in the laboratory. However, in recent years several indirect methods like elastic scattering; Coulomb excitation and dissociation; transfer reactions; nuclear knockout reactions; quasifree reactions; charge-exchange reactions, etc. have been
^Department of Physics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat, India.
2)Department of Applied Sciences, JMIT, Radaur, India.
3)Department of Applied Sciences, TERII, Kurukshetra, India. E-mail: panghal005@gmail.com
developed to extract cross sections relevant to as-trophysical processes [5]. The Coulomb dissociation process is being used frequently for obtaining the capture cross section in connection to astrophysical problems. But unfortunately it is not free from bias because of the presence of the contribution of electric quadrupole (e2) and dipole—quadrupole interference (E1 — E2) terms [6, 7]. In order to obtain conclusive observations regarding the astrophysical problems one has to estimate the exact contribution of E2 and E1— E2 interference terms in the Coulomb breakup process. Therefore, in the present work the effects of E2 and E1— E2 interference terms have been investigated on the major observables of Coulomb breakup reaction like, integrated Coulomb breakup cross section and cross section differential in relative energy and longitudinal momentum of the outgoing fragments.
2. THEORETICAL FORMALISM
Theoretically, in the energy range of interest the eikonal approximation is the most convenient model to describe the Coulomb breakup process. The transition amplitude for any electric multipolarity (Elm) in this approach is given as [8, 9]
0Y
lm
v^T+lf-
V c
l
J X
c
\v
X GElm - <fim(Q)M(Elm),
where Q is the momentum change in the scattering and is given as Q = 2k sin(6/2), with k and 6 as the incident momentum and the scattering angle, respectively, while q represents the relative motion momentum of the outgoing fragments after the breakup. The symbols 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. The hu gives the excitation energy of the projectile expressed as the sum of binding energy and relative energy of the outgoing fragments. Geiiu (f ) is the usual Winther and Alder functions. The non-dimensional functions, &m(Q), containing information regarding reaction mechanism are expressed in terms of Bessel functions as follows
= J Jm(QRx) Km (ç^^j Xdx.
1
The matrix element M (Elm) contains structural information about the projectile ground state and in the long-wavelength approximation it is defined as
M (Elm) =^2 Zk e Î (r)r{ Ylm(nk )fa(r) d3r, k=i,2 J
where 0i(0f) is the initial (final) state of the projectile. The initial state of the projectile is described by the usual single-particle shell-model wave function, i.e., (r) = RL(r)YLM(r), while the final state may be conveniently described by plane wave, i.e., 0f (r) = eiqr. These considerations along with usual properties of spherical harmonics modify the matrix element to the following form
M (Elm) = V^ZfVWTÏV2L + 1 x (2) x ^i-x^J==YXv(q) (LZ00|A0) (LlMm\\v) x
Xv
V2A + 1
x J r2drrl jX(qr)RL(r)
with
and
zff = Zi3l + (-1)1 Z232
ll
31(2) =
^2(1) m1 + m2
d4^ = \fc(Q, q)|2 dQ
d3q
(2^)3'
Substitution of Eq. (2) into Eq. (1) and the use of expression so obtained in Eq. (3) along with little mathematical exercise leads to the following expression
d2 _ z2a2
q2dq sin Odd y7^ ^ X (_1)(A2-Ai+ii-i2)/2(2Zi + l^(2l2 + l)v/2Ai + 1 X
x ^2A2 + 1
V2STÏ\cJ \cJ
x zfffzfffGhmGi2mKm(i)iLiiAi(q)ihi2\2(q) x
x (Ll\00|Ai0) (Ll200\\20) (AiA200|50) x
x (lil2m - m\S0) W(SlAL : l2Ai)Yso(q),
where the notation ILlA(q) represents the radial integral appearing in Eq. (2). The summation runs over all the quantum numbers except L and YS0(q) is the standard spherical harmonics without phase factor e%m. In case of 15C nuclei having a single valence neutron occupying zero-angular-momentum orbital the above equation has been simplified, by setting L = 0, to the following form
d2a
Zt2a2
q2dqsm9d0 ^/47r
(-
X (_l)(A2-Ai+ii-i2)/2(2Zi + + 1)^2Ai + 1 X
x V2A2 + 1
i ^yi-i x
V25 + 1 V c J \cJ
x ZiffZi2GhmGi2mKm(£)Ioi1 X1 (q)hi2X2 (q) x
x (0li00\Ai0) (OI200\A20) (AiA200\50) x
x (lil2m - m\S0) W(SI1A20 : hAi)Yso(q)■
Now the simple angular momentum algebra leads to the following explicit expressions (for detailed derivations readers should refer to [7, 10, 11]) for differential Coulomb dissociation cross section in longitudinal momentum distribution and relative energy of core fragments corresponding to electric dipole, quadrupole, and dipole—quadrupole interference terms
da Ei
dqz
4Z2(Zf ) a 2 2 s k
\qz \
3 y2 32
oii
(4)
(K2 - Ko2) {(1 + 2P2) - (1 - P2)Y2} +
Now the differential cross section for Coulomb excitation is expressed in terms of transition amplitude
as
2
+ -KoK^l - P2)y2
qdq,
(3)
da E2 dqz
ZKzffa2 10572/34
\
(5)
x
o
x
X
X
HŒPHAfl OH3HKA tom 77 № 2 2014
CONTRIBUTION OF ELECTRIC QUADRUPOLE AND DIPOLE-QUADRUPOLE
231
^ii2(7 - 10P2 + 3P4) + (Kl - Kl) x x (28 + 2OP2 + 57P4) + (7 + 5P2 - I2P4) x
x 72(2 — /32)2 ( '^KqKi — (Kf — Kq)
qdq,
d(JEl-E2
dqz
4Zi2Z1effZ2ffa2 /и
q i
Jt
572/33
x ^011 ^022
(K - Ko2)(2PI + ЗР3) +
+
2
-iloA'! - {Kl - K2)
x (Pi - P3)y2(2 - ß2)
qdq,
da e 1
dErel
4Z2(Zf)2a:2 2 2 ç A
372 ß2
011
(7)
(K2 - к2) {(1 + 2P2) - (1 - p2)y2} +
2
+ —KqКi(l - P2)72 ç
da e 2
22
f ZfiZfl^
dEKl-J 10572/?4 Ы 4 022 |
(8)
-?Ä"12(7 - 10P2 + 3P4) +
.Ç2
+ (K - К2)(28 + 20P2 + 57P4) + + (7 + 5P2 - I2P4)72(2 - ß2)2 x
x ( jKolû - (Kl - Kl)
2Erel
sine de,
daE1—E2 f izfzfzfa2 (uj
dErel
Iqz I
x I0111022
(Kl - K2)(2P1 + 3P3) +
+
-ÄoÄ'i - (Kl - Kl)
(P1 - P3)72(2 - ß2)
x sJ'2EKl ^ysmed,e,
da/dEiel, b/MeV 0.4
respectively. Here, K0, K1, P\, P2, and £ are the modified Bessel functions with order one and two,
3 4
Erel, MeV
Fig. 1. Relative energy spectrum of14C and n emerging from the Coulomb breakup of 15 C on Pb target at 68-MeV/u beam energy corresponding to electric dipole (El) (dotted curve), quadrupole (E2) (dashed curve), electric dipole—quadrupole interference terms (El—E2) (dash-dotted curve), and combination of all these (solid curve). Data points are taken from [12].
Legendre polynomial of order one and two with argument cos e, and adiabaticity parameter given as £ = (uR/jv). The total Coulomb breakup cross section may be obtained by integrating either Eqs. (4) and (5) over qz corresponding to electric dipole and quadrupole terms, respectively, or Eqs. (7) and (8) over Erej. Here, the contribution of Eqs. (6) and (9) may be ignored since these terms contribute nothing in the integrated cross section [6, 7].
3. RESULTS AND DISCUSSION
The major ingredient in our calculation is the radial part of the ground-state wave function of the projectile which strongly depends on the orbital occupancy of valence neutron and on the ground-state spin—parity. As far as ground-state of 15C is concerned during the present work the dominating spin couplingconfiguration 14C(0+) ® 2s1/2 with spectroscopic factor 0.91 has been considered [12], where the valence neutron occupies orbital and is attached to 14 C core with 1.128 MeV as the separation energy. The radial part of the wave function corresponding to this configuration is constructed by solving the corresponding Schrodinger equation using Wood—Saxon potential. The range and diffuseness parameters of the Wood—Saxon potential have been fixed at 2.34 and 0.60 fm, respectively, and depth (76 MeV) of the potential is adjusted to reproduce the ground-state separation energy of valence neutron. The wave
x
П
X
0
X
X
X
da/dpz, arb. units
pz, MeV/c
Fig. 2. Longitudinal momentum distribution of 14C emitted out from reaction 181 Ta(15 C, 15 C + n)
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