Final answer:
The question deals with sketching sampling lattices based on given basis vectors, illustrating Voronoi cells, and determining sampling density. These concepts are involved in understanding the arrangement of a crystalline solid and how the pattern repeats throughout space.
Step-by-step explanation:
The question at hand involves understanding the geometry and properties of two different sampling lattices, which are used to represent the arrangement of atoms or molecules in a crystal structure in two dimensions. Sampling lattices A and B are described by their basis vectors, respectively. Each basis vector pair defines a unique layout and hence a different unit cell. To visualize these, one would sketch the basis vectors on a two-dimensional graph, depicting the layout of the unit cells, and then use these to tile the entire plane to demonstrate how the lattice extends in space.
A Voronoi cell is a region around a lattice point that includes all points closer to that lattice point than to any other. These cells can be illustrated by constructing perpendicular bisectors between adjacent lattice points and defining the enclosed area. For each lattice defined, one would do this around a central lattice point to illustrate the boundaries of its Voronoi cell.
The sampling density of a lattice can be thought of as the number of sample points per unit area. It is inversely proportional to the area of the unit cell. Therefore, to calculate the sampling density, we would calculate the area of the unit cells defined by the basis vectors for each lattice, and then inversely relate this to determine the sampling density.
In the case of square and hexagonal lattices in two dimensions, we see a difference in packing and the number of lattice points that each can account for within the unit cell. This is crucial not only for the representation of solid crystals but also for determining properties such as density and coordination number. For example, in a simple cubic structure, each corner atom can be thought of as 'belonging' to eight different unit cells, thereby resulting in the unit cell itself containing only 1/8th of each atom at its corners.