199 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Neutron Halo Structure of Unstable Exotic 14 B and 18 N Nuclei Moath S. Hamad 1* and Ahmed N. Abdullah 2 1,2 Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq. *Corresponding Author. Received: 21 April 2025 Accepted: 24 July 2025 Published: 20 October 2025 doi.org/10.30526/38.4.4147 Abstract The ground-state properties like the nuclear densities and the root mean square (rms) radii for exotic neutron-rich nuclei 14 B and 18 N have been investigated using the Skyrme Hartree- Fock and Bear-Hodgson calculations to study the structure of these nuclei. The results of the evaluation are contrasted with the experimental data that is currently available. It found that a common feature of the neutron and matter densities for the above-selected exotic nuclei is the long tail behavior. We assumed that both 14 B and 18 N have a structure of the core nuclei 13 B and 17 N plus a valence neutron. It found that the structure of the valence one-neutron of 14 B and 18 N is a pure Configuration. The elastic charge form factors of the above selected exotic nuclei are evaluated using the plane wave Born approximation and compared with those of their stable isotope 10 B and 14 N. Keywords: Exotic nuclei, Elastic form factors, Bear-Hodgson, The Skyrme-Hartree-Fock. 1. Introduction The study on the structure of short-lived nuclei far from -stability has become a hot point in nuclear physics due to its exotic properties )1-4(. The resulting discovery of neutron halo in certain anomalous light neutron-rich nuclei was in 6 He, 11 Li, 11 Be, 14 Be, 19 C, etc )5-7(. The nuclear halo is a quantum effect that arises from the very weak binding of the valence nucleons and their occupation on the orbits with l = 0, 1 (low angular momentum), which allows the wave function of these valence nucleons to take on an extended radial dimension (8). The primary characteristic of the nuclear-matter density distribution in halo nuclei is the presence of a low-density tail at vast radial distances (9). The information about such nuclear structure can be extracted from the fragment momentum distribution of fragmentation reaction, total reaction cross section, Coulomb dissociation, and quadrupole moment (10, 11). Additional information on the nuclear structure can be obtained by the proton elastic scattering at intermediate energies. It gives insights into both the nuclear-matter density distribution and the nuclear-matter radius (12). This technique is well- established for the study of stable nuclei and may also be used to investigate unstable nuclei when applied to inverse kinetics with radiation beams. The methodology has been effectively utilized to examine the isotopic 12,14 Be (9) and 8,9,11 Li (13). The two-body (Core + n) and three-body (Core + 2n) models utilizing wave functions of various potentials have been employed to examine the ground state characteristics, including proton, neutron, and matter densities, of several halo nuclei, like as 22 N, 23 O, 14 Be, and 17 B, https://orcid.org/0009-0002-6219-1845 mailto:moathshihab99@gmail.com https://orcid.org/0000-0002-8633-1833 mailto:Ahmed.n@uobaghdad.edu.iq https://orcid.org/0009-0002-6219-1845 mailto:moathshihab99@gmail.com https://orcid.org/0000-0002-8633-1833 mailto:Ahmed.n@uobaghdad.edu.iq https://orcid.org/0009-0002-6219-1845 mailto:moathshihab99@gmail.com https://orcid.org/0000-0002-8633-1833 mailto:Ahmed.n@uobaghdad.edu.iq https://orcid.org/0009-0002-6219-1845 mailto:moathshihab99@gmail.com https://orcid.org/0000-0002-8633-1833 mailto:Ahmed.n@uobaghdad.edu.iq https://orcid.org/0009-0002-6219-1845 mailto:moathshihab99@gmail.com https://orcid.org/0000-0002-8633-1833 mailto:Ahmed.n@uobaghdad.edu.iq https://orcid.org/0009-0002-6219-1845 mailto:moathshihab99@gmail.com https://orcid.org/0000-0002-8633-1833 mailto:Ahmed.n@uobaghdad.edu.iq IHJPAS. 2025, 83(4) 200 etc (14-18). Also, the associated rms radii of these exotic nuclei have been studied by these models. The calculated results have been provided for the halo structure for the considered exotic nuclei. In this work, the characteristics of the ground state pertaining to exotic 14 B and 18 N nuclei, such as the proton [ ( )], neutron [ ( )] and matter [ ( )], the corresponding root mean square (rms) radii, and elastic charge form factors ( ( )), will be investigated using the Hartree-Fock (HF) and Bear-Hodgson (BH) calculations. We will examine the reaction cross- sections ( ) for that nucleus use the Kox formula (KF) and Glauber model (GM). 2. Materials and Methods The ( ) of exotic nuclei is (19): ( ) ( ) ( ) (1) where ( ) (core density) and ( ) (halo density) are expressed as (19): ( ) ∑ | ( )| (2) ( ) ∑ | ( )| (3) where ( ) and denote the occupation number of the orbit , respectively. The ( ) taken from Schrodinger equation of this solution of radial part and using BH potential (20): ( ) * ( ) ( ) + ( ) (4) Where and the ( ) given as (21): ( ) ( ) ( ) ⃗ ( ) (5) ( ) the central potential taken from following from (21): ( ) [ ( ) ] (6) ( ) is the spin orbit potential (21): ( ) [ ( ( ) ) ] (7) ( ) (for protons only) is Coulomb potential (21): ( ) { * + (8) and ( ) for neutrons. The is potential depth [in Equation 6] takes the BH form (22): (9) ( ) (10) The potential depth for neutrons ( ) and protons ( ) whereas the is constant. The Skyrme force are given by (23): ∑ ( ) ( ) ( )[ ( ) ⃗ ⃗ ( )] ( ) ⃗⃗ ⃗ ( ) ⃗ ( ) ( ⃗ ) ( ) ⃗ ( )( ) ⃗ (11) represents is space operator exchange, is the vector comprising the Pauli spin matrices ( ) denotes the delta function, ⃗ indicates is relative, and , , , , , , , , , and represent the parameters associated with the Skyrme force. The charge, as well as the densities of protons or neutrons with the range of the Skyrme HF methodology, are articulated by (24): IHJPAS. 2025, 83(4) 201 ( ) ∑ ( ) ( ) (12) where is the wave function of a single particle and denotes the probability of occupation for the state . The nucleus charge distributions [ ( )] can be obtained from the following folding relation (25): ( ) ∫ ( ) ( ) (13) where ( ) and are proton density. Where takes from following form of Gaussian (26): ( ) (√ ) ( ) (14) The core ( ), matter ( ), proton ( ) and neutron ( ) rms radii are obtained by (27): 〈 〉 [ ∫ ( ) ∫ ( ) ] (15) In PWBA, the elastic charge form factor ( ( )) is given by (28): ( ) ∫ ( ) ( ) (16) The use the GM (29) and KF (30) given, respectively, as: ∫[ ( )] (17) [ ( )] ( ) (18) 3. Results The ground-state characterizes nuclear densities, and the rms radii for exotic neutron-rich nuclei 14 B and 18 N have been investigated in the framework of the HF and BH calculations. The elastic ( ) of above selected exotic nuclei are evaluated using the PWBA. We use the KDEX Skyrme parameterization within HF calculations in this work. The values of the KDEX parameterization employed in our calculations are , , , , , , , , , (31). The densities of both core and tail (halo) parts in HF and BH calculations are described by radial wave functions for the HF and BH potentials, respectively. We assumed that both 14 B (J π , T= ,2) and 18 N (J π , T= ,2) have a structure of the core nuclei 13 B (J π , T= , ) and 17 N (J π , T= , ), with configurations {(1s1/2) 4 , (1p3/2) 7 , (1p1/2) 2 }and {(1s1/2) 4 , (1p3/2) 8 , (1p1/2) 3 , (1d5/2) 2 }, respectively plus valence one neutron . The valence neutron of both 14 B and 18 N is assumed to be in a pure 2s1/2 orbit. Table 1 displays the values of the BH parameters utilized in the present calculations for selected nuclei. The potential depth for neutrons ( ) and protons ( ) in core nuclei has been used the default of the NushellX@MSU program (32) has been used, where the for valence neutron and other parameters, provide the experimental of the last neutron as well as a matter (rms) radial for exotic nuclei. The parameter β has been fixed at a value of 0.51 )22(. Table 1.The BH parameters. Nuclei (MeV) (MeV) a0=aso fm r0=rso fm rc fm Core Halo 14 B 53.172 48.73 7.0 0.520 1.316 1.2 18 N 64.496 35.92 7.0 0.621 1.465 1.2 10 B 55.70 7.0 0.620 1.236 1.2 14 N 55.70 7.0 0.620 1.236 1.2 IHJPAS. 2025, 83(4) 202 The calculated for both protons and neutrons are presented in Table 2 together with the )33( of the valence neutron. Table 2.The calculated . Nuclei Proton Neutron ( ) )33( ( ) ( ) ( ) ( ) 14 B 1s1/2 67.78 -43.676 70.13 -48.289 ----- 1p3/2 54.89 -18.400 56.70 -21.947 ----- 1p1/2 ----- ----- 53.28 -15.223 ----- 2s1/2 ----- ----- 48.05 -0.97 -0.97 18 N 1s1/2 92.67 -70.285 96.09 -76.994 ----- 1p3/2 79.76 -44.947 82.63 -50.589 ----- 1p1/2 77.45 -40.447 80.38 -46.166 ----- 1d5/2 ----- ----- 70.17 -26.139 ----- 2s1/2 ----- ----- 35.92 -2.828 -2.828 The HF and BH calculations for the core ( ), matter ( ), proton ( ) and neutron ( ) rms radii (in fm) of selected halo nuclei are presented in Tables 3 and 4. For comparison purposes, the corresponding experiment rms radii (34-37) are also given in these tables. From these tables, we noted that the calculated results of our present study agree reasonably within the quoted error with the experimental results. Table 3. The calculated and radii and experimental ones. Nuclei (fm) (fm) HF BH Exp. (34) HF BH Exp. (34, 35) 14 B 2.42 2.44 2.46±0.12 2.89 2.77 2.77±0.04 18 N 2.57 2.57 2.49±0.15 2.87 2.80 2.80±0.04 Table 4. The calculated and rms radii and experimental ones. Nuclei (fm) (fm) HF BH Exp. (36) HF BH Exp. (37) 14 B 2.46 2.40 2.46±0.07 3.12 2.95 3.27±0.16 18 N 2.58 2.50 ----- 3.03 2.98 2.925±0.067 The calculated matter densities obtained by both HF (left part) and BH (right part) calculations for exotic 14 B and 18 N nuclei and core 13 B and 17 N nuclei, along with the tail (one-neutron halo) part, are shown in Figure 1. The blue, black, and dashed-red curves represent the core, tail part, and matter densities, respectively. The experimental matter densities for 14 B (38) and 18 N (39) are presented in this figure by a grey area for comparison. Figure 1a and b show the densities for 14 B, while Figure 1c and d correspond to those for 18 N. A common feature of the dashed-red curves, which can be shown in Figure 1, is the long tail behavior. Clearly, the dashed-red curves obtained with both HF and BH calculations lie within the experimental uncertainties and agree well with the experiment. Figure 2 a-d summarize the calculated proton (black curve) and neutron (blue curve) densities in 14 B (upper panel) and 18 N (lower panel) obtained by the HF and BH calculations. From these figures, it can be clearly seen that the neutron densities of 14 B and 18 N have a long tail with respect to the proton densities. This means that 14 B and 18 N are neutron-halo nuclei. Figure 3a-d compare the calculated results of matter densities for unstable 14 B and 18 N (dashed red distributions) and stable 10 B and 14 N (blue distributions) isotopes. From these figures, we can observe that, there is a difference in the behavior of the blue and dashed red distributions. This demonstrates a long tail in dashed red distributions and supports the halo structure of 14 B and 18 N nuclei. IHJPAS. 2025, 83(4) 203 Figure 1. The matter, core and halo density for halo nuclei 14 B and 18 N. Figure 2.The matter, proton and neutron density for halo nuclei 14 B and 18 N. IHJPAS. 2025, 83(4) 204 Figure 3. The matter for isotopes 10,14 B and 14,18 N. Figure 4a-d depicted the results of the longitudinal form factors (C0+C2) of 10,14 B (upper panel) and 14,18 N (bottom panel) isotopes calculated by HF and BH calculations. Therein, the blue and red curves refer to C0+C2 of unstable and stable isotopes, respectively. While the calculated results of C0 and C2 for unstable 14 B and 18 N nuclei are given by the dashed and black curves, respectively. For comparison the experimental ( ) for stable isotopes 10 B (40) and 14 N (41) are given by dotted symbols. The agreement is shown to be very well between the results of our calculations with the experimental data for 10 B and 14 N. First one can see from figures and the dashed red distributions decreases faster than the black distributions with increasing q. This is attributed to the charge density pulls out due to the addition of neutrons to 10 B and 14 N and thus the form factors decrease with increasing q. In this work, the of the exotic 14 B and 18 N nuclei on target 12 C are studied by the KF and GM with the OLA and summarized in Table 5 along with experimental results (42). The HO densities are used in GM calculations. It is demonstrated that the experimental data satisfactorily well by our calculations obtained by GM while the extracted by Kox formula closely agree with those experimental data within quoted error. IHJPAS. 2025, 83(4) 205 Figure 4. The longitudinal form factors of isotopes 10,14 B and 14,18 N. Table 5. The reaction cross sections of the exotic 14 B and 18 N nuclei on target 12 C. Halo nuclei Energy (MeV) (42) ( ) (mb) ( ) (mb) (42) KF GM 14 B 790 961 937 929±26 18 N 1020 1.077 1062 1046±8 The and moments for 10,14 B and 14,18 N isotopes are calculated in psd-model space using PSDMK interaction (43) and tabulated in Table 6 together with the experimental data (44). The effective charges of NuShellX@MSU code (NS) ( ) and free- nucleon g factors have been used to evaluate the and moments, respectively. In general, the theoretical and experimental results of and moments agree reasonably for all selected nucleus. Table 6. Calculated and experimental results of and moments. Nuclei Exp. (44) Exp. (44) 14 B 4.45 2.98±0.008 0.986 1.18±0.005 10 B 9.07 8.47±0.006 1.82 1.80±0.006 18 N 1.97 2.70±0.004 -0.107 -0.135±0.015 14 N 1.08 1.93±0.008 0.333 0.403±0.006 4. Conclusion The ground-state properties like the nuclear densities and the rms radii for exotic neutron- rich nuclei 14 B and 18 N have been investigated in the framework of the HF and BH calculations. This study draws the following conclusions: IHJPAS. 2025, 83(4) 206 The evaluated results are compared with available experimental data. It is found that both the HF and BH calculations are capable of providing theoretical predictions on the structure of exotic nuclei (considered in this study) and provide a satisfactory description of experimental data. The halo structure of the above exotic nuclei is emphasized through exhibiting the long tail performance in their calculated matter density distributions, where this performance is considered a distinctive feature of halo nuclei. It is found that the major difference between the calculated form factor of unstable nuclei ( 14 B, 18 N) and those of stable nuclei ( 10 B, 14 N) is attributed to the charge density pulls out due to the addition of neutrons to 10 B and 14 N, and thus the form factors decrease with increasing q. The calculated results of the reaction cross sections using the Kox formula and the Glauber model with an optical limit approximation are in good agreement with experimental data at high energy. The calculated results of the nuclear magnetic dipole and electric quadrupole moments using the Shell model calculations within the two-body effective interactions are in reasonable agreement with experimental data. Acknowledgment The researchers are very grateful to Prof. Dr. Alex Brown for his code (NuShellX@MSU code). Conflict of Interest The authors declare that they have no conflicts of interest. Funding The article was done depending on self-fund and no establishments supplied us. Ethical Clearance The work and calculations were accomplished using the NuShellX@MSU code References 1. 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