Quantifying and visualizing the microscopic degrees of freedom of grain boundaries in the Wigner–Seitz cell of the displacement-shift-complete lattice

We introduce a grain boundary (GB) translation vector, $t^{WS}$, to describe and quantify the domain of the microscopic degrees of freedom of GBs. It has long been recognized that for fixed macroscopic degrees of freedom of a GB there exists a large multiplicity of states characterized by different relative grain translations. More recently another degree of freedom, $\left[n\right]$, the number of GB atoms, has emerged and is now recognized as an equally important component of GB structural multiplicity. In this work, we show that all GB microstates can be uniquely characterized by their value of $t^{WS}$, which is located within the Wigner–Seitz (WS) cell of the Displacement-Shift-Complete lattice (DSCL) of the GB. The GB translation vector captures information about both the translation state and the number of GB atoms. We show that the density of GB microstates inside the WS cell of the DSCL is not uniform and can form clusters that correspond to different GB phases. The vectors connecting the centers of the clusters correspond to the Burgers vectors of GB phase junctions, which can be predicted without building the junctions. Using $t^{WS}$, we quantify GB excess shear and argue that it is defined up to a DSCL vector, which has implications for thermodynamic equilibrium conditions. Additionally, this work generalizes the definition of the number of GB atoms $\left[n\right]$ to asymmetric boundaries.