The Phase-Field Crystal (PFC) model describes atoms in a lattice through a continuous periodic density field and investigates diffusive time scales. In the amplitude expansion of the PFC model (APFC), a coarse-grained description of this density is obtained by focusing on its complex amplitudes and, in turn, on their dynamics. These amplitudes vary on length scales larger than the atomic spacing but still retain details of the crystal lattice [1]. After outlining the basics of this approach, numerical simulations based on the APFC model and the finite element method are shown to reproduce defects structures in two and three dimensions for different crystal symmetries [2]. Continuous deformation fields derived from the complex amplitudes are shown to characterize dislocations. Connections with macroscopic elasticity theory are then discussed [2, 3]. Finally, representative applications are shown, such as the prediction of defect motion in binary systems with defects induced by Cottrell Atmospheres [4]. These findings assess the APFC model as a powerful coarse-graining of the PFC model. More importantly, the description of crystal structures through amplitudes emerges as a natural framework connecting atomic-scale lattice deformations and continuum elasticity.

[1] M. Salvalaglio and K. R. Elder. Coarse-grained modeling of crystals by the amplitude expansion of the phase-field crystal model: an overview. arXiv:2202.04196. [2] M. Salvalaglio et al, Closing the gap between atomic-scale lattice deformations and continuum elasticity, npj Comput. Mater. 5, 48 (2019). [3] M. Salvalaglio, et al, A coarse-grained phase-field crystal model of plastic motion, J. Mech. Phys. Solids 137, 103856 (2020). [4] M. Salvalaglio, et al. Mesoscale Defect Motion in Binary Systems: Effects of Compositional Strain and Cottrell Atmospheres, Phys. Rev. Lett. 126, 185502 (2021).