## 1 Introduction  
Charged defects in non-metals, such as semiconductors and insulators, play a central role in determining electronic and electrical behavior of materials for semiconductor and battery applications [1]. Point defects are also very relevant to actinide materials such as UO\({}_{2}\) under extreme conditions of temperature, pressure, and irradiation, as they play a major role in accommodating stoichiometry variations [2]. In addition, defects induce lattice distortions in the perfect crystal, leading to changes in volume which can, at least in principle, be measured experimentally [3; 4] and compared to theoretical calculations [4; 5; 6; 7; 8; 9]. These lattice distortions are particularly important in determining the interaction of defects with external strains, dislocations, and grain boundaries [10; 11], and play a central role in the evolution of microstructures in materials. The defect formation volume is an intrinsic property of the defect; theory for computing the volume change due to point defects gained attention in the early 1980s with a focus on point defects in ionic crystals [5; 6]. It is important to note that experimental measurements of the volume change associated with point defects yield the defect-averaged response of a mass and charge balanced system [3; 4; 5; 6]; as a result there is little understanding of the volume change or lattice distortion associated with individual charged defects. Investigations of the formation volume of charged point defects in UO\({}_{2}\)[9], Si [12; 13; 14; 15], ZnO [16], In\({}_{2}\)O\({}_{3}\)[17], CeO\({}_{2}\)[18], SrTiO\({}_{3}\)[4; 19] indicate that the defect formation volume is a strong function of the charge state of the defect, and as electrons are added (or removed) to the system the formation volume increases (or decreases) linearly with charge. Defect formation and relaxation volumes are closely related concepts. The defect relaxation volume is simply the change in volume of a computational cell due to relaxation of the crystal lattice around the defect site. By contrast, the defect formation volume is calculated as the change in the volume associated with the formation reaction for the defect; it, therefore, accounts for the reference volume associated with the reservoir to or from which atom(s) are moved in order to create the defect in the simulation. We describe defect relaxation and formation volume and the differences between the two in more detail in Section 2.2. The theory of relaxation volume and lattice relaxation around the defect site can be used to obtain the so-called defect elastic dipole tensor [20; 21], which relates the atomic structure of the point defect with its elastic distortion of the lattice. The elastic dipole tensor [7; 20; 22], \(G_{ij}\), captures the symmetry of the point defect and is given as:  
\[G_{ij}=C_{ijkl}\Delta\Omega_{kl}^{r} \tag{1}\]  
where \(C_{ijkl}\) is the elastic constant tensor, and \(\Delta\Omega_{kl}^{r}\) is the relaxation volume tensor of the defect [23]. The elastic dipole tensor describes the change in formation energy of the defect, to first order, in the presence of external strains [10]. Therefore, computation of the defect relaxation volume tensor quantifies the lattice distortion around the defect and thus the elastic interaction with other defects and strain fields [11; 24]. An accurate description of the dipole tensor is critical for developing models of transport in complex microstructures. Real materials contain dislocations, grain boundaries, and other defect structures that can be described by elastic strain fields. The dipole tensor provides a means for relating changes in energy of a given point defect as its position varies in that strain field.