Single-particle energy states for a neutron and a proton are obtained by solving the time-independent Schrodinger equation for the mean-field Woods–Saxon potential along with the spin-orbit term. The wavefunctions are expanded as a linear combination of simple sine-wave basis states, which are eigenfunctions of the infinite spherical-well potential. The requisite algorithm based on matrix diagonalization is implemented in Free Open Source Software (FOSS) Scilab. Initial values for the simulation were taken from model parameters given in the book on Nuclear Structure by Bohr and Mottelson, which were then optimized to obtain the best convergence with the available experimental energy values of various nuclei with magic proton and neutron numbers. The level scheme, as well as the energy values for doubly magic nuclei 82 208Pb and 20 40Ca, which are obtained using our simulation, is presented in this paper. Finally, energy level diagrams for neutrons and protons with respect to mass number A were arrived at, based on those obtained for various magic nuclei. The evaluation method, which is based on the sine-wave basis, is akin to Fourier analysis. When done with the aid of FOSS Scilab, this technique becomes easily accessible to students at the under-graduate (UG) level and may be studied through small projects.

中文翻译：

---

基于正弦基的核壳模型仿真研究

中子和质子的单粒子能态是通过求解平均场 Woods-Saxon 势的与时间无关的薛定谔方程以及自旋轨道项来获得的。波函数被扩展为简单正弦波基态的线性组合，它们是无限球阱势的本征函数。基于矩阵对角化的必要算法在免费开源软件 (FOSS) Scilab 中实现。模拟的初始值取自 Bohr 和 Mottelson 在《核结构》一书中给出的模型参数，然后对其进行优化以获得与具有神奇质子和中子数的各种核的可用实验能量值的最佳收敛。双魔核 82 ​​208Pb 和 20 40Ca 的能级方案以及能量值，这是使用我们的模拟获得的，在本文中介绍。最后，根据对各种魔核获得的能级图，得出了中子和质子相对于质量数 A 的能级图。基于正弦波基础的评估方法类似于傅立叶分析。在 FOSS Scilab 的帮助下完成后，本科 (UG) 级别的学生可以轻松使用该技术，并且可以通过小项目进行研究。