Pramana – J. Phys. (2020) 94:154 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-02011-x The neutron halo structure of 14B, 22N, 23Oand24F nuclei studied via the generalised Woods–Saxon potential AHMED N ABDULLAH Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq E-mail: [email protected] MS received 2 April 2020; revised 24 May 2020; accepted 6 July 2020 Abstract. The radial wave functions of the generalised Woods–Saxon (GWS) potential within the two-body model of (Core + n) have been used to study the ground-state density distributions of protons, neutrons and matter and the associated root mean square (rms) radii of neutron-rich 14B, 22N, 23O and 24F halo nuclei. The calculated results show that the radial wave functions of the generalised Woods–Saxon potential within the two-body model succeed in reproducing neutron halo in these exotic nuclei. Elastic electron scattering form factors for these nuclei are studied by combining the charge density distributions with the plane-wave Born approximation (PWBA). Keywords. Generalised Woods–Saxon potential; two-body model; one-neutron halo nuclei. PACS Nos 21.10.Gv; 25.30.Bf 1. Introduction of two-neutron 6He, 11Li, 12Be and 14Be halo nuclei using MSK7 parameter within the Skyrme–Hartree– The development of experiments using radioactive Fock (SHF) method. Tanaka et al [7] have used Glauber- nuclear beams allowed us to study nuclei far from the type calculation to measure the reaction cross-sections valley of stability, which led to the discovery of halo for 14Band8He on proton target. They have also phenomena [1,2]. Halo phenomenon is a quantum effect deduced the nucleon density distributions and the rms arises as a result of the combined impact of small binding proton, neutron and matter radii of these nuclei. Each energy of the valence particle(s) and their low angu- radius was consistent with some of the several theoreti- lar momentum (l = 0, 1) [3]. The experiments with cal values and also with some of the other experimental exotic (halo) nuclei are challenging because specific values. Abdullah [8] has used the SKxs25 parame- techniques are required due to the low intensity with ters within the SHF model to investigate the nuclear which the nuclei are produced and their short lifetime. structure of 8B, 17Ne, 23Al and 27P halo nuclei. It It is impossible to make short-lived nuclei as targets. was found that the SKxs25 parameters with this model Therefore, the experiments must be performed in inverse is suitable for describing the nuclear structure of the kinematics. In this case a beam of radioactive nuclei is above halo nuclei. Suzuki et al [9] studied the inter- 17 produced and it interacts with a stable target [4]. action cross-section (σI) for B on a carbon target The charge and nuclear matter distributions as well at 880 A MeV. The result for σI was 1118 ± 22 mb. as the root mean square (rms) radii of nuclear mat- They also studied the rms matter radii of 17Busing ter give important insights on nuclear wave functions the Glauber model based on the few-body reaction and nuclear potentials. The matter density distributions model and optical limit approximation. Their results and radii of the halo nuclei have been studied by mea- for these two different methods were 2.99 ± 0.09 and suring the reaction and interaction cross-sections. The 2.90 ± 0.06 fm, respectively. Tanaka et al [10]have relation between the nuclear density distribution and the measured the reaction cross-sections for 13,14,15Bon total reaction cross-section is studied using the Glauber Be, C, Al and proton targets using the Glauber model. model [5]. They have also deduced the proton, neutron and nucleon Abdullah [6] has investigated ground-state prop- density based on the Glauber-type calculation. The erties such as the binding energy per nucleon, the result suggests that there is a large tail in neutron den- ground-state densities and the corresponding rms radii sity distribution for 14B, which can be referred to as 0123456789().: V,-vol 154 Page 2 of 8 Pramana – J. Phys. (2020) 94:154 a one-neutron halo. The rms proton, neutron and matter where m is the reduced mass of the core and single 13,14,15 radii for B were also derived. Abdullah [11]has nucleon, εnj is the single-particle binding energy and used the two-body model of (Core+n) within the radial V (r) is the core potential which can be written as [13] wave functions of the cosh potential to investigate the ( ) = ( ) + ( ) L.S + ( ). ground-state features such as the proton, neutron and V r V0 r Vso r Vc r (5) matter densities, the rms nuclear proton, neutron, charge V (r) is the central potential which takes the form of the and mass radii of unstable neutron-rich 14B, 15C, 19C 0 GWS potential [14–16]: and 22N nuclei. The calculated results show that the two- body model with the radial wave functions of the cosh (r−R )/a V0 W0e 0 0 potential succeeds in reproducing neutron halo in these V0(r) =− − , (6) + (r−R0)/a0 ( − )/ 2 nuclei. 1 e 1 + e r R0 a0 In this study, we shall adopt the radial wave func- ( < < ) tions of the generalised Woods–Saxon (GWS) potential where V0 and W0 0 W0 150 MeV are the depths within the two-body model of (Core + n ) to study of the potential well. The GWS potential is reduced to the standard Woods–Saxon form for W0 = 0. the ground-state density distributions of protons, neu- ( ) trons and matter and the associated rms radii of Vso r is the spin-orbit potential expressed as [17]: neutron-rich 14B, 22N, 23Oand24F halo nuclei. Elas- tic electron scattering form factors for these nuclei ( ) = 1 d 1 Vso r Vso ( − )/ (7) will be studied by combining the charge density dis- r dr 1 + e r Rso aso tributions with the plane-wave Born approximation ( ) (PWBA). and Vc r is the Coulomb potential (for protons only) generated by a homogeneous charged sphere of radius Rc [18]: ⎧ 2. Theory ⎪ Ze2 ⎨ for r > Rc V (r) = r In the case of exotic (halo) nuclei, it is reasonable c ⎪ Ze2 3 r 2 (8) ⎩⎪ − for r ≤ R to parametrise the core and halo densities separately. 2 c Rc 2 2Rc Therefore, the ground-state matter density distribution for the halo nuclei can be written as [11] and Vc(r) = 0 for neutrons. The radii R0, Rso and Rc are usually expressed as [17] ρm(r) = ρc(r) + ρv(r), (1) 1/3 Ri = ri A , (9) where ρc(r) and ρv(r) are the core and valence (halo) densities, respectively and expressed as [12] where A is the mass number of the nucleus. The matter density of eq. (1) can be written as [17] 1 nj 2 ρc(r) = Xc Rnj (r) , (2) ρ (r) = ρ p(r) + ρn(r), 4π m (10) nj where ρ p(r) and ρn(r) are the proton and neutron den- 1 nj 2 ρv(r) = Xv Rnj (r) , (3) sities, respectively, written as [17] 4π p p p ρ (r) = ρ (r) + ρv (r) (11) where X n j refers to the number of protons or neutrons c c(v) ρn( ) = ρn( ) + ρn( ). r c r v r (12) in the sub-shell, nj and Rnj (r) are the radial wave functions of the GWS potential taken from the solu- ρc ( )[ρc( )] ρv ( ) [ρv( )] Here p r n r and p r n r are the core and tion to radial part of the Schrödinger equation using the valence proton [neutron] densities, respectively. GWS. From the proton density ρp(r) and the intrinsic charge The moving of valence neutron around the core can distribution f p of one proton, we can obtain the charge be described with the Schrödinger equation [13] distribution of the nucleus using the following folding 2 relation [19]: d Rnj (r) 2m + dr 2 2 ρ ( ) = ρ ( ) − , ch r p r f p r r dr (13) 2 ( + 1) × εnj − V (r) − Rnj (r) = 0, (4) 2m r 2 where f p takes the Gaussian form as follows [20]: Pramana – J. Phys. (2020) 94:154 Page 3 of 8 154 Table 1. Some properties for the selected nuclei. π Halo nucleus (J , T ) [22] Half-life τ1/2 [22] Core configuration Halo neutron configuration 14 − 4 7 2 B2, 212.5ms {(1s1/2) , (1p3/2) , (1p1/2) }2s1/2 22 − 4 8 3 6 N0, 423ms{(1s1/2) , (1p3/2) , (1p1/2) , (1d5/2) }2s1/2 23 + 4 8 4 6 O1/2 , 7/297ms{(1s1/2) , (1p3/2) , (1p1/2) , (1d5/2) }2s1/2 24 + 4 8 4 7 F3, 3 384 ms { (1s1/2) , (1p3/2) , (1p1/2) , (1d5/2) }2s1/2 Table 2. The GWS parameters employed in the present calculations. V0 = W0 (MeV) Proton Neutron Nuclei Core Valence Vso (MeV) a0 = aso (fm) r0 = rso (fm) rc (fm) 14B 68.127 64.891 62.540 6.0 0.512 1.385 1.439 22N 68.334 52.738 41.580 6.0 0.636 1.415 1.376 23O 58.216 50.787 36.760 6.0 0.723 1.498 1.371 24F 63.262 55.642 40.770 6.0 0.785 1.428 1.366 10B 59.758 59.758 6.0 0.517 1.341 1.477 15N 49.352 49.352 6.0 0.519 1.315 1.422 16O 49.587 49.587 6.0 0.516 1.311 1.413 19F 58.187 58.187 6.0 0.557 1.318 1.398 rms radii of one-neutron 14B, 22N, 23Oand24Fhalo 1 (− 2/ 2 ) ( )= r ap .