Analysis and Exploration of Kronig and Penney Potential Models in Crystal Lattice Structures

Abstract

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.