Deformation twinning is one of the major deformation mechanisms in crystals, which plays an important role in determining the mechanical properties of metals and alloys. One of the important issues to understand twinning mechanisms is the determination of the deformation path. However, due to a lack of theoretical tools, a fundamental relationship between symmetry breaking and the deformation path has not been established in materials science, which conceals the physical origin of deformation twinning. Utilizing a graph approach for deformation pathways, we show that twinning modes in hexagonal close-packed (hcp) titanium and zirconium are dictated by both the symmetry of hcp and the symmetry breaking associated with the bcc to hcp transformation. Our work not only opens another avenue to investigate the symmetry and symmetry breaking in hcp crystals, but also provides insight into the physical origin of crystalline defects.

中文翻译：

---

断裂对称性决定的孪生路径：六角形密堆积的钛和锆中变形孪生的再探讨

变形孪生是晶体的主要变形机制之一，它在决定金属和合金的机械性能中起着重要的作用。理解孪生机制的重要问题之一是变形路径的​​确定。然而，由于缺乏理论工具，材料科学中尚未建立对称破坏与变形路径之间的基本关系，这掩盖了变形孪生的物理起源。利用图形方法的变形路径，我们表明六方密堆积（hcp）钛和锆中的孪生模式是由hcp的对称性和与bcc到hcp转变相关的对称破坏所决定的。