Physics 463Winter 2012HW # 02 Solutions1. The Wigner-Seitz cell is a special kind of primitive cell where the lattice point is in the “center” of the cell.Each point in the boundary is the minimum lies equidistant between one or more other lattice points thanthe original “central” lattice point.The construction is more illustrative.Draw a line between the centrallattice point and other nearby lattice points. Draw a plane that perpendicularly bisects this line segment. Thesmallest volume enclosed is the Wigner-Seitz cell.The area of the Wigner-Seitz cell is|a1×a2|provided thata1anda2are primitive vectors. Some conventionalcells are not primitive cells. In fact, due to tessellation, the area of any cell divided by the number of latticepoints will yield the area of the Wigner-Seitz cell.Oblique,A=a1a2|sinφ|Rectangular,A=a1a2Centered Rectangular,A=12a1a2Hexagonal,A=|sin(120◦)|a1a2=√32a21Square,A=a211
2. Coordination number is the number of nearest neighbors. This value is the same for any lattice point due to thedefinition of a lattice. The packing fraction is determined by finding the largest spheres at each lattice pointwithout intersection (each sphere must have the same radius due to lattice symmetry). The packing fractionis simply the ratio of the volume inside the spheres to the total volume.