Lattice and magnetic degrees of freedom are strongly coupled in magnetic materials. We propose a consistent first-principles framework to explore the joint configurational space. For this, we define atomic spin moments from the projector-augmented wave formalism of density-functional theory and control them via Lagrangian constraints. We demonstrate our approach for vacancy formation and migration in collinear paramagnetic body-centered cubic iron, by implementing a relaxation scheme based on spin-space averaged forces (SSA relaxation). Based on these results we discuss the impact of the magnetic state on vacancy formation energies, migration barriers as well as relaxations.

中文翻译：

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在有效拉格朗日形式主义中由原子自旋约束计算得出的磁无序材料中缺陷周围的原子弛豫

晶格和磁自由度在磁性材料中紧密耦合。我们提出了一个一致的第一原理框架，以探索联合配置空间。为此，我们根据密度泛函理论的投影仪增强波形式主义定义原子自旋矩，并通过拉格朗日约束对其进行控制。通过实施基于自旋空间平均力（SSA弛豫）的弛豫方案，我们证明了在共线顺磁性体中心立方铁中空位形成和迁移的方法。基于这些结果，我们讨论了磁态对空位形成能，迁移势垒以及弛豫的影响。