Crystal structures II Spacefilling and crystal stability 1. Space filling of crystal structures (Analytical task) By assuming hard spheres touching each other, calculate the space filling of the • simple cubic (sc) structure, • face centered cubic (fcc) structure, • hexagonal closed packed (hcp) structure with an ideal ratio c a = q 8 3 . 2. Stability of a crystal lattice (Tool task) In the following task, we determine the lowest energy configuration of a crystal. The potential energy is here modelled by a modified Lennard-Jones potential V =4( σ r ) 12 - ( σ r ) 6 + A r σ 2 + B r σ + C, (1) with =0.0104 eV and σ =3.4 ˚ A. The polynomial A ( r σ ) 2 + B r σ + C is used to bring this potential smoothly to zero for a given potential cutoff radius r cut . The potential is intended to model the interaction between Argon atoms and was implemented in the code MiniMol, a minimal Molecular Statics / Molecular Dynamics tool. A detailed description of how to handle MiniMol can be found on our webpage in the tool description“HandsOn MiniMol”. Prepare rectangular simulation cells for Argon in the • face centered cubic (fcc) structure, • body centered cubic (bcc) structure, • hexagonal closed packed (hcp) structure with an ideal c a = q 8 3 ratio. Vary the lattice constant a from of 3.0 ˚ A to 6.0 ˚ A and plot the energy per atom given by MiniMol with respect to a. Which is, according to this model, the most stable crystal structure of Argon? Determine the lattice constant and the cohesive energy of this model. Please, turn page -→ 1