标题： 带隙形成理论：对布拉格衍射模型的替代方案 显示英文标题

标题： Band gap formation theory: An alternative to the Bragg diffraction model

摘要： 能带，是固态物理中的一个关键概念，传统上通过电子波在晶体周期势场中的布拉格衍射来解释。 尽管被广泛接受，这一框架在二维系统中提出了基本问题，其中布拉格衍射——需要多方向波相互作用——简化为简单的干涉，因此无法解释能带的形成。 在本文中，我们引入了一种不依赖于布拉格反射的替代理论。 利用布洛赫波的薛定谔方程，我们将晶体晶格视为一组离散的观察点。 这种离散性引入了类似于信号处理中奈奎斯特频率的采样约束。 我们表明，当电子波数在周期势下变化而晶格间距保持不变时，能带自然地作为采样效应出现。 通过构建一个包含此效应的能量图，我们揭示出能带既源于波数的变化，也源于晶格作为离散采样器的作用，从而使能量曲线相对于奈奎斯特波数表现出平移和镜像对称性。 这种方法提供了能带形成的全新、物理基础的解释，并可以自然地扩展到更高维度。 显示英文摘要

摘要： The band gap, a key concept in solid-state physics, is traditionally explained by the Bragg diffraction of electron waves in the periodic potential of a crystal. Although widely accepted, this framework raises fundamental issues in one-dimensional systems, where Bragg diffraction-which requires multidirectional wave interactions-reduces to simple interference, thus failing to explain band gap formation. In this paper, we introduce an alternative theory that does not rely on Bragg reflection. Using the Schr\"{o}dinger equation for Bloch waves, we consider the crystal lattice as a discrete set of observation points. This discreteness introduces a sampling-like constraint analogous to the Nyquist frequency in signal processing. We show that when the electron wavenumber changes under a periodic potential while the lattice spacing remains fixed, a band gap naturally emerges as a sampling effect. By constructing an energy diagram that incorporates this effect, we reveal that the band gap originates from both the wavenumber change and the role of the lattice as discrete samplers, leading the energy curve to exhibit translational and mirror symmetries with respect to the Nyquist wavenumber. This approach provides a novel, physically grounded explanation of band gap formation and can be naturally extended to higher dimensions.