ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 3, с. 299-305
= ЯДРА ^^
EFFECT OF CORE DEFORMATION ON 17B HALO NUCLEUS
© 2014 Waleed S. Hwash1),2)*, Redzuwan Yahaya3), Shahidan Radiman3)
Received July 16,2013
This study aimed to investigate the two-neutron halo nucleus of 17B using the cluster model. The interaction between the two neutrons and deformed core 15 B was treated as a rigid rotor with a deformed Woods—Saxon potential and a spin—orbit interaction. The three-body description of the 17B nucleus was carried out using the Jacobi coordinates method. The three-body system of 17B was described with the hyperspherical harmonic method. The three-body energy-dependent on the deformation of the core was also studied. The binding energy, matter radius of 17B, and deformation of 15B were calculated by normalization and by using experimental data. Comparison of the results with experimental data demonstrated the deformation of the core, especially of the prolate shapes.
DOI: 10.7868/S004400271402010X
INTRODUCTION
With the advent of new technologies and facilities, numerous works have been conducted to achieve new insights in the study of nuclei far from the ^-stable line. Experimental works [1—3] on 11 Li, 14Be, and 17B light nuclei have shown abnormally large root-mean-square (RMS) radii, which are attributed to the halo effect [4]. The halo phenomenon is described as the threshold effect, which occurs in loosely bound systems where the particles are held in short-range potential wells. The more loosely the halo particles are confined, the more clearly the "halo stratosphere" is developed. The three-body halo candidates of choice are light drip-line nuclei with two neutrons orbiting the core, such as the 17B (=15B—n—n) nucleus. Since the first observation of 17B reported in 1973 [5], interest on this nucleus has continued to increase owing to its exotic properties (e.g., multi-neutron as well as beta-delayed emission [6], and an abnormally large matter radius [7, 8]). A large matter radius is an indication of a halo nucleus formed by numerous external neutrons, which surrounds a core that significantly increases the matter radius [9].
A number of studies have been conducted using the three-body models [10] and microscopic clus-
1)1 School of Applied Physics, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia.
2)Department of Physics, Faculty of Education for Pure Sciences, Anbar University, Iraq.
3)School of Applied Physics, Faculty of Science and Technology, Universiti Kebangsaan Malaysia Bang, Selangor, Malaysia.
E-mail: waleed973@yahoo.com
ter model [11] of 17B nucleus, and involves different 13B + di-neutron + di-neutron configurations. The generator coordinate method with 13B + di-neutron + di-neutron clusters is generally used to describe the 17B nucleus. The 13B wave functions are also constructed based on the 1p shell and the core excitations on the harmonic-oscillator model with all possible configurations. The parity and total angular momentum of17B projection have also been studied. The matter RMS radius and quadrupole moment of 17B have been found to be 2.81 fm and 2.74 e fm2, respectively, excluding the 15B—n—n configuration [11]. Ren and Xu [12] conducted research on the 17B ground state using a three-body model involving a 15B core and two outside neutrons. Though Ren and Xu explained the large RMS radius of 17B, it is still difficult to reliably calculate this based on a 15B + + n interaction. The shell-model spectrum of several boron isotopes has also been used by Warburton and Brown [13].
Several experiments on 17B have been performed [14, 15], which examined the mass and matter radii. The two-neutron separation energy (S2n) has been found to be 1.39 ± 0.14 MeV for 17B [ 14], whereas the matter radii of 17B are 4.10 ± 0.46 [8] and 3.0 ± 0.6 fm [16]. An abnormally large matter radius is interpreted as an indication of a neutron halo nuclei, specifically those in very neutron-rich and loosely bound nuclei of11 Li and 11 Be. The halo phenomenon is quite common for nuclei with large N/Z ratios that resulted in large differences in the Fermi energies of protons and neutrons. The weakening in the strong correlation of proton and neutron distribution is a characteristic of
a stable nucleus [17]. Thus, the halo or skin neutrons can be decoupled from the nuclear core to enable it to move independently from the rest of the nucleus. The electric quadrupole moment of the ground state is measured at Q = 38.6 ± 1.5 mb, which is very close to that of the N = 8 (e.g., the 13B isotope) but is in contrast to the results of the shell-model calculations. The shell-model calculation predicts an increase in the quadrupole moment corresponding to an increase in mass (assuming the conventional en = 0.5 value in the effective charge of the neutron). The deviation from the shell model can be ascribed to other factors caused by the decrease in the effective charge of the neutron, which results from the decoupling of the core and the valence neutrons [ 18].
In the present work, the 17B nucleus was investigated using the cluster model, with the assumption that the 17B nucleus consisted of the core 15B—n— n in the two-neutron halo nuclei. The three-body cluster model was solved using the hyperspherical harmonic method. The core was not inert but had some deformation dependence in this case, since the shells for neutrons and protons were not closed. As such, the extra neutrons and protons outside the closed shell in the core created some degree of deformation. Consequently, the deformation had an effect on the three-body energy in the two-neutron halo nuclei. The 17B nucleus was described using the Jacobi coordinates for the three-body problem, as formerly discussed for 14Be and 11 Li in [19] and [20], respectively.
THEORETICAL FRAMEWORK
The three-body system or two-neutron halo nuclei were defined in terms of the core and valence neutrons. The distance between each pair of particles was rjk, whereas the distance between the corresponding third particle and center of mass of the pair could be expressed in terms of the Jacobian coordinates x, y [19, 20], where
— \J Ajk^jk —
AjAk Ai + Ak
rjk,
y = \ A{jk)ir{jk)i
(Aj + Ak)Aj Ai + Aj + Ak
'(jk)i-
where x,y) contains the radial pattern, angular pattern, and spin of the remaining two particles relative to the core. The Jacobi coordinates (x,y) were transformed into the hyperspherical coordinates (hyperradius and hyperangle) and are defined as
P
2 , 2
x + y ,
d — arctan
(3)
The hyperspherical expansion of the three-body radial and angular wave functions is
Rn(p) —
P
5/2
n!
Po
(n + 5)
L5aiaä(z)ex (4)
The intrinsic Hamiltonian of the core determines a set of eigenstates 0core and eigenvalues ecore with
hcore (Ccore)0core(Ccore) = ^core^core (Ccore ). (1)
The total wavefunction of the system with regard to Jacobi coordinates is given as
*JM(x,y, £) = ^core(ecore)^(x,y), (2)
where z = p/p0 and
ly (Q) = NlkJy (sin Q)lx (cos Q)ly x (5) x ^+1/2'ly+1/2(cos2Q). The wavefunction of the valence neutrons is
(p,Q) = Rn (p)4x ly (Q). (6)
Thu^ ^(x,y)= (P'Q^ where Lilag(z) is the associated Laguerre polynomial of the order nlag =
= 0, 1, 2, ... and so on; and plx+1/2'ly+1/2(cos2Q) is the Jacobi polynomial. Equation (4) is the function of p owing to z dependence on p with z = p/p0, where values of p and p0 are explained in the next section. The n in the Eq. (4) is n = Ix + 1 and we
assumed that nlag = n. In addition, Nkxly is the normalization coefficient, and k is the hyperangular momentum quantum number k = lx + ly + 2n for n = 0, 1, 2, ...The wave function of the system comes from the internal wave function of each body in that system. The wave function of the relative motion of a neutron and 15B core is obtained by solving the Schrodinger equation in the spherical coordinates. More details about the hyperspherical harmonics formalism method are presented in the references [21, 22].
The system total Hamiltonian H is given by
H = T + h core(£) + VW O + (7)
+ ^core-n2(rcore-n2j + V^n-n(rn-n).
A two-body interaction between Vcore-n and Vn-n was applied to all pairs of interacting bodies with the potential taken as a deformed Wood—Saxon potential and as a spin—orbit interaction.
In the fixed-body frame, the radius of the deformed core can be expanded in the spherical harmonics. For simplicity, we retained only the quadrupole term, as seen in Eq. (10) below:
^core—n(rcore—n —
(8)
x
y
x
Table 1. Parameters of calculations
^X ly n K p, im po, im ro, im a, im aso, im Vo, MeV V;o,MeV
0 0 1 2 0.866 0.866 1.25 0.65 0.65 -74 -7.5
1 1 2 6 0.866 1.936 1.25 0.65 0.65 -74 -7.5
- - - - 1.936 - 1.25 0.65 - - -
2 2 3 10 0.866 2.958 1.25 0.65 - - -
2 2 3 10 1.936 - 1.25 - - - -
2 - - - 2.958 - 1.25 - - - -
-Vo
1 + exp ^
rC0K-n-R(e,4>)
+
, (01 , Vso + —^(2 Is)
m2c2
4rcore—n dr core—n
1 + exp
Vn—n(rn—n) -
e—n Rs
m2c2
(2ls)
V«
d
1 + exp
rn—n -R
4rn—n drn—n —i
(9)
with
R - Ro[1 + ß2V20(0,0)1,
R — Ro ' 1-25A(1or3e.
(10)
G-
-G
Y
T-configuration
Y-configuration
Fig. 1. Jacobi coordinates for three-body system.
for the average squared distance of nucleons for an A-body system from the position of the total center of mass is given by
1 A
A
E'
i=1
A
A
J2(ri - rcm)2,
(11)
(12)
i=i
The central potential and the spin—orbit potentials among n1 n2, n1—core, and n2—core were calculated using Eqs. (8) and (9), respectively. We assumed that the spin—orbit radius Rso = R and l represented the operator of orbital momentum between the core and a neutron, s represented the neutron spin of operator, and m = mn represented the mass of the ion. For practical calculation, we assumed that (h/mn)2 = = 2.0 fm2, f32 was the core quadrupole deformation, and Acore was the core mass number. The operator
where Ti is the position of i nucleon, rcm is center of mass and the RMS matter radius (rm)1/2 of the nucleus
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