Friedel's law guarantees an inversion-symmetric diffraction pattern for thin, light materials where a kinematic approximation or a single-scattering model holds. Typically, breaking Friedel symmetry is ascribed to multiple scattering events within thick, non-centrosymmetric crystals. However, two-dimensional (2D) materials such as a single monolayer of MoS2 can also violate Friedel's law, with unexpected contrast between conjugate Bragg peaks. We show analytically that retaining higher order terms in the power series expansion of the scattered wavefunction can describe the anomalous contrast between hkl and hkl¯peaks that occurs in 2D crystals with broken in-plane inversion symmetry. These higher-order terms describe multiple scattering paths starting from the same atom in an atomically thin material. Furthermore, 2D materials containing heavy elements, such as WS2, always act as strong phase objects, violating Friedel's law no matter how high the energy of the incident electron beam. Experimentally, this understanding can enhance diffraction-based techniques to provide rapid imaging of polarity, twin domains, in-plane rotations, or other polar textures in 2D materials.

中文翻译：

---

通过打破弗里德尔定律在二维材料中成像极性

弗里德尔定律保证了运动学近似或单散射模型适用的薄、轻材料的反演对称衍射图案。通常，破坏弗里德尔对称性归因于厚的非中心对称晶体内的多次散射事件。然而，二维 (2D) 材料，例如单层 MoS2 也可能违反弗里德尔定律，在共轭布拉格峰之间具有意想不到的对比。我们分析表明，在散射波函数的幂级数展开中保留高阶项可以描述 hkl 和 hkl¯peaks 之间的异常对比度，这种对比度发生在面内反演对称性破坏的二维晶体中。这些高阶项描述了从原子级薄材料中的同一原子开始的多个散射路径。此外，含有重元素的二维材料，如 WS2，总是作为强相物体，无论入射电子束的能量有多高，都违反弗里德尔定律。在实验上，这种理解可以增强基于衍射的技术，以提供二维材料中极性、双畴、面内旋转或其他极性纹理的快速成像。