{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:24:00Z","timestamp":1760059440917,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T00:00:00Z","timestamp":1749772800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["U2267207","HQ24042","KF24014","202401001561"],"award-info":[{"award-number":["U2267207","HQ24042","KF24014","202401001561"]}]},{"name":"Doctoral Research Fund of Hunan Institute of Technology","award":["U2267207","HQ24042","KF24014","202401001561"],"award-info":[{"award-number":["U2267207","HQ24042","KF24014","202401001561"]}]},{"name":"Provincial Applied Specialty Subject Open Fund for Safety Science and Engineering Subject","award":["U2267207","HQ24042","KF24014","202401001561"],"award-info":[{"award-number":["U2267207","HQ24042","KF24014","202401001561"]}]},{"name":"Hunan Provincial Department of Education Teaching Reform Key Project Fund","award":["U2267207","HQ24042","KF24014","202401001561"],"award-info":[{"award-number":["U2267207","HQ24042","KF24014","202401001561"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Lewis\u2019 theory of multiple scattering has been modeled as a random walk on a unit sphere for calculating the multiple scattering angular distribution of charged particles, which is more intuitive and mathematically simpler. This formalism can lead to the Goudsmit\u2013Saunderson theory and the Lewis theory of multiple scattering angular distribution, thus providing an easier-to-understand framework to unify both the Goudsmit\u2013Saunderson and the Lewis theories. This new random walk method eliminates the need for integro-differential expansions in Lewis theory and is faster at calculating multiple scattering angular distributions, reducing the required Legendre series terms by 80% at small step (path) length (&lt;20) and providing much greater calculation efficiency. Crucially, the random walk formalism explicitly preserves spherical symmetry by treating angular deflections as steps on a unit sphere, enabling the efficient sampling of scattering events while maintaining accuracy. Further, a robust algorithm for numerically calculating multiple scattering angular distributions of electrons based on the Goudsmit\u2013Saunderson and Lewis theories has been developed. Partial wave elastic scattering differential cross-sections, generated with the program ELSEPA, have been used in the calculations. A two-point Gauss\u2013Legendre quadrature method is used to calculate the Legendre coefficients (multiple scattering moments).<\/jats:p>","DOI":"10.3390\/sym17060945","type":"journal-article","created":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T04:20:23Z","timestamp":1750134023000},"page":"945","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Symmetry-Driven Modeling of Electron Multiple Scattering: A Random Walk Approach on the Unit Sphere"],"prefix":"10.3390","volume":"17","author":[{"given":"Zining","family":"Ni","sequence":"first","affiliation":[{"name":"School of Safety and Management Engineering, Hunan Institute of Technology, Hengyang 421002, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9035-6526","authenticated-orcid":false,"given":"Muhammad Abdul","family":"Wasaye","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Hunan Institute of Technology, Hengyang 421001, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6469-6172","authenticated-orcid":false,"given":"Irfan","family":"Siddique","sequence":"additional","affiliation":[{"name":"School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China"}]},{"given":"Zhiqiang","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Safety and Management Engineering, Hunan Institute of Technology, Hengyang 421002, China"}]},{"given":"Mohsin","family":"Saeed","sequence":"additional","affiliation":[{"name":"Department of Physics, University of the Punjab, Lahore 54590, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,13]]},"reference":[{"key":"ref_1","first-page":"135","article-title":"Monte Carlo Calculations of the Penetration and Diffusion of Fast Charged Particles","volume":"Volume 1","author":"Alder","year":"1963","journal-title":"Methods in Computational Physics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1103\/PhysRev.58.292","article-title":"Multiple Scattering of Fast Electrons and Alpha-Particles, and \u201cCurvature\u201d of Cloud Tracks Due to Scattering","volume":"58","author":"Williams","year":"1940","journal-title":"Phys. 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