(a) Compare planar densities (Section 3.11 and Problem 3.53 ) for the \((100),(110),\) and (111) planes for FCC. (b) Compare planar densities (Problem 3.54 ) for the \((100),(110),\) and (111) planes for \(\mathrm{BCC}\)

Understanding the Face-Centered Cubic (FCC) crystal structure is essential for studying materials science. FCC is one of the most common arrangements for metallic elements. In an FCC crystal, atoms are arranged in a way that each face of the cubic unit cell has an atom at its center, along with atoms at each of the eight corners. This results in a highly dense and closely packed structure.

During the calculation of planar densities for different planes in FCC, such as \(100\), \(110\), and \(111\), we closely examine how these atoms are distributed across various planes. The contribution of shared atoms is accounted for by dividing the atoms by the number of unit cells that share them. For instance, corner atoms are shared by four unit cells, hence they contribute a quarter of an atom each to a plane. This intricate placement of atoms in the FCC structure leads to different planar densities for each plane.

The Body-Centered Cubic (BCC) crystal structure is another pivotal pattern utilized by metals. Here, atoms are placed at the eight corners of a cube and one atom at the very center of the cube. Unlike FCC, where the faces are centered, BCC's additional atom is situated in the body of the cube. This slight difference alters the arrangement’s bulk properties significantly.

When examining various planes such as \(100\), \(110\), and \(111\) in BCC, the central atom plays a crucial role in determining the planar density. For planes that cut through the central atom, the count includes the complete atom, while the corner atoms are still shared and hence counted as fractions. This unique positioning impacts the atomic packing and hence affects the planar density for each specific plane.

Atomic packing factor (APF) is a measure of the efficiency with which atoms are packed in a crystal structure. It is defined as the ratio of the volume occupied by atoms in a unit cell to the total volume of the unit cell. In structures like FCC and BCC, the APF gives us insight into the material's density and can influence properties like strength and ductility.

The APF for the FCC crystal structure is about 0.74, which is the highest possible for spherical atoms, indicating a highly efficient packing. BCC has a lower APF of approximately 0.68. These values help us understand why FCC metals are typically more malleable than BCC metals, as the more densely packed atoms can slide past each other more easily.

The lattice parameter, often denoted as 'a', is the physical dimension that defines the size of the unit cell in crystal structures. It is a critical factor in calculating planar densities, as seen in the exercise for FCC and BCC structures. The lattice parameter determines the distance between atoms and the spatial arrangement within the crystal.

For any crystallographic calculation, including the derivation of planar densities, the value of 'a' underpins the entire computation process. For example, the area of the certain planes is a function of the lattice parameter: the area of \(100\) plane is calculated as \(a^2\), while the area of \(111\) plane is derived using the square root of 3 over 4 times \(a^2\). This constant, 'a', is integral to understanding the microscopic world of crystal structures and their properties.