Analysis and Exploration of Kronig and Penney Potential Models in Crystal Lattice Structures
DOI:
https://doi.org/10.62292/10.62292/njp.v34i2.2025.254Keywords:
Kronig and Penney potential, Energy Levels, Periodic potential, Crystal latticeAbstract
This study delves into the intricate influence of crystal lattices on material properties, employing the Kronig and Penney potential model (KP) to understand wave functions and electronic band structures. Investigating classical and quantum mechanics, the research explores discrete atomic states through Schrödinger's equation, considering both finite and infinite potential scenarios. The results found that as (αa) increases, the energy within the +1 power scatter barrier experiences damping, reaching a constant defined by F(αa) = 6 × 〖10〗^(-5) (αa) + 0.9888. Electron confinement within the unit atom results in an infinite power scatter barrier, and energy (En) for any lattice is determined by E_n = 13.7 n^2. Temperature-induced variations signify increased energy, indicating free electron movement and a finite power scatter barrier. The electron's negligible velocity in the lattice plays a crucial role in determining amplitude. A decrease in ET corresponds to an increase in the velocity VT of the free electron. Extreme phonon energy in lattice structures is negligible due to its massless nature. The study notes increased energy in restricted electrons, causing a reduction in free electron energy due to an energy gap or barrier scatter. High temperatures are essential to reduce the energy band, facilitating electron transitions and resulting in an electron lifetime of 1.63 × 〖10〗^(-50) s. These insights deepen our understanding of crystal lattice dynamics, offering avenues for innovative applications in materials science and quantum physics.
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Angelo, Ziletti., Devinder, Kumar., Matthias, scheffler & Luca, M. Ghiringhelli (2017). Insightful classification of crystal structure using deep learning. Arxiv materials science. https://doi.org/arXiv:1709.02298v2
Benjamin, Friedlander (1984). Lattice structures for factorization of samples covariance matrix. pp. 163-169. https://doi.org/10.1063/1.1587011
Bootstrapping the Kronig-Penney model: 14 Dec 2022-Physical review (Physical review)-Vol. 106, Iss: 11. https://doi.org/10.1103/PhysRevD.106.116008
Cerutti, D. S., & Case, D. A. (2019). Molecular Dynamics Simulations of Macromolecular Crystals. Wiley interdisciplinary reviews. Computational molecular science, 9(4), e1402. https://doi.org/10.1002/wcms.1402
Eli, Alster., David, Montiel., Katsyo, Thornton's & Potter W. Voorhees (2017). Simulating complex crystal structures using the phase-field crystal model Physical reviews materials. Vol 1. Issue 6. Pp 060-801. DOI: https://doi.org/10.1103/PhysRev.96.218
Fischetti, M.V. & Vandenberghe, W.G. (2016). The Electronic Structure of Crystals: Computational Methods. In: Advanced Physics of Electron Transport in Semiconductors and Nanostructures. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01101-1_5
Graeme, J. Ackland., Nigel, B. Wilding & Alastair Bruce (1997). Evaluation of free energy differences between crystalline phases using the lattice switch montecurlo method: MRS proceedings (Cambridge University press). Vol 499 issue 1 pp 253-264. https://doi.org/10.1103/PhysRevLett.79.3002
Greiner, W. (2001). Quantum Mechanics; An Introduction. Fourth Edition. Springer, Berlin, Germany. pp. 181, 220 – 222. www.cambridge.org/core/books/introduction-to-quantum-mechanics/990799CA07A83FC5312402AF6860311E
Griffiths, D. J. (1995). Introduction to Quantum Mechanics. Prentice Hall, New Jersey. pp. 256 – 260. www.cambridge.org/core/books/introduction-to-quantum-mechanics/990799CA07A83FC5312402AF6860311E
Ibrahim, I. S., F. M. Peeters (1995). The magnetic Kronig–Penney model. American Journal of Physics (American Association of Physics Teachers)-Vol. 63, Iss: 2, pp 171-174. https://doi.org/10.1119/1.17977
Kakani, S.L and Kakani, S (2006). Modern Physics. Viva Books Private Limited, New Delhi. Pp 356-400. https://doi.org/urn:lcp:modernphysics0000kaka:lcpdf:768bf31c-97ca-44b5-ab5c-04e0b39bbf8f
Ken, A. Hawick (2012). Engineering internal domain specific languages software for lattice-based simulations: Journal of paralled and distributed computing and system. Doi: https://doi.org/10.2316/p.2012.790-043
Lan, lan., Zhi-Yuan, Lin & Kai-ming Ho. (2003). Lattice symmetry applied in transfer matrix methods for photon crystals. Journal of applied physics (American institute of physics). Vol 94. Issue 2. Pp 811-821.
Landau, L. D. and Lifshitz, E. M., (1991). Quantum Mechanics, Non-relativistic Theory, Volume 3 of Course of Theoretical Physics. Third edition, Pergamon Press, Oxford, England. pp. 119. https://www.scribd.com/document/410016083/Landau-L-D-Lifshitz-E-M-Volume-3-Quantum-Mechanics-non-relativistic-theory-3ed-Pergamon-1991-pdf
Li, J., Abramov, Y. A., & Doherty, M. F. (2017). New Tricks of the Trade for Crystal Structure Refinement. ACS central science, 3(7), 726–733. https://doi.org/10.1021/acscentsci.7b00130
Masatsugu, Suzuki (2015). Block theorem and energy band. https://bingweb.binghamton.edu/~suzuki/ModernPhysics/38_Bloch_theorem_and_energy_band.pdf
Matthew J. Blacker., Arpan, Bhattacharyya & Aritra, Banerjee (2022). Bootstrapping the Kronig-Penney model: https://doi.org/10.48550/arXiv.2209.09919
Newton, R. G., 2002. Quantum Physics: A Text for Graduate Student Springer-Verlag New York, Inc. pp. 181. ISBN 10: 0387954732 ISBN 13: 9780387954738. Publisher: Springer, 2002
Paul W. Betteridge, J. Roberto's Cairuther., Richard I. Cooper., Keith Prout & David J Watkins (2003). Crystals Version 12: software for guided crystal structure analysis. Journal of applied crystallography (international Union of crystallography). Vol 36. Issue 6. Pp 1487-1487. https://doi.org/10.1107/S0021889803021800
Ralph, Freese (2004). Automated lattice Drawing. Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, Pp 112-127, February 23-26, 2004, Proceedings https://link.springer.com/book/10.1007/b95548
Righini, R. (1985). Crystal structure and lattice dynamics calculations on CSe 2. Chemical Physics Letters, 115 (4–5), 381–386. https://doi.org/10.1016/0009-2614(85)85153-8
Ruho, Kondo (2021). Finding direct correlation function for desired two dimensional lattices with a phase field crystal: Physical reviews B. (American physical society). Vol 104. Issue 1. Pp. 014112. https://doi.org/10.1103/PhysRevB.104.014112
Salimen, Ana (2020). Kronig-Penney Model (Condensed Matter Assignment). https://www.scribd.com/document/449729125/kronig-penney-model-condensed-matter-assignmen t
Schiff, L. I., (1949): QUANTUM MECHANICS. McGraw-Hill Book Company, Inc. pp 169-172. https://www.scribd.com/document/318048394/Leonard-Schiff-Quantum-Mechanics
Terayama, K., Yamashita, T. & Oguchi, T. (2018). Fine-grained optimization method for crystal structure prediction. npj Comput Mater Vol 4. Issue 1. Pp 32. https://doi.org/10.1038/s41524-018-0090-y
Thaddeus, D. Ladd., J.R. Goldman., Fumiko Yamaguchi & Yoshuhisa, Yamamoto (2000). Decoherence in crystal lattice quantum computation. Applied physics A (springer-verlag). Vol 71. Issue 1. Pp 27-36
Urbano Oseguera. 01 Feb 1992- Classical Kronig–Penney model. American Journal of Physics (American Association of Physics Teachers)-Vol. 60, Iss: 2, pp 127-130. https://doi.org/10.1119/1.16929
