HW2_Solution - Physics 463 Winter 2012 HW 02 Solutions 1...

Physics 463
Winter 2012
HW # 02 Solutions
1. 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 than
the original “central” lattice point. The construction is more illustrative. Draw a line between the central
lattice point and other nearby lattice points. Draw a plane that perpendicularly bisects this line segment. The
smallest volume enclosed is the Wigner-Seitz cell.
The area of the Wigner-Seitz cell is | a
1
× a
2
| provided that a
1
and a
2
are primitive vectors. Some conventional
cells are not primitive cells. In fact, due to tessellation, the area of any cell divided by the number of lattice
points will yield the area of the Wigner-Seitz cell.
Oblique, A = a
1
a
2
| sin φ | Rectangular, A = a
1
a
2
Centered Rectangular, A =
1
2
a
1
a
2
Hexagonal, A = | sin(120
◦
) | a
1
a
2
=
√
3
2
a
2
1
Square, A = a
2
1
1

2. Coordination number is the number of nearest neighbors. This value is the same for any lattice point due to the
definition of a lattice. The packing fraction is determined by finding the largest spheres at each lattice point
without intersection (each sphere must have the same radius due to lattice symmetry). The packing fraction
is simply the ratio of the volume inside the spheres to the total volume.