(a) Derive linear density expressions for FCC [100] and [111] directions in terms of the atomic radius \(R\). (b) Compute and compare linear density values for these same two directions for copper (Cu).

The atomic radius is crucial for understanding a variety of physical properties of materials, especially when exploring crystal lattice structures such as those found in metals. In the case of FCC crystals, the atomic radius, denoted as \(R\), is half the distance between the centers of two adjacent atoms.

When we consider the arrangement of atoms in an FCC structure, we find that the unit cell is constructed of atoms at each corner and at the center of each face. Since the atoms are in contact along the face diagonal, the atomic radius can be correlated directly to the unit cell edge length. As calculated in our exercise, the relationship is given by \(a = 2\sqrt{2}R\), where \(a\) is the edge length of the cubic unit cell.

The importance of the atomic radius becomes evident when determining properties such as linear density, which essentially measures how tightly atoms are packed along a certain direction in a crystal. The atomic radius thus serves as a fundamental parameter for calculating the spacing between atoms and subsequently their arrangement in different crystallographic directions.

Crystallographic directions are vectors that show orientation in a crystal lattice; they are denoted in square brackets, such as [100] or [111]. These directions are key for understanding properties like linear density because they exemplify the directional dependence of atomic arrangement within a crystal.

In our FCC example, the [100] direction is a straight line through the center of one face of the cube to the opposite face parallel to one of the axes of the cube. In contrast, the [111] direction is a diagonal that passes through the body of the cube, connecting opposite corners.

For the FCC structure, the [111] direction is densely packed as it contains a sequence of atoms in a straight line—crucial for understanding why its linear density is higher compared to the [100] direction. These crystallographic directions not only influence mechanical properties like strength and ductility but also help in predicting how materials will behave when subjected to external forces or treatments.

The unit cell edge length, represented as \(a\), is a primary dimension that defines the size of the smallest repeating unit of the crystal structure. In an FCC crystal, the edge length is intimately tied to the atomic radius via the equation \(a = 2\sqrt{2}R\). This relationship is derived from the geometric arrangement of atoms within the FCC unit cell, where each face-centered atom is in contact with the corner atoms.

The edge length is a determining factor for calculating the volume of the unit cell, packing factor, and as shown in the example exercise, the linear density in different crystallographic directions. Understanding this parameter is pivotal as it helps to bridge atomic scale features with macroscopic properties of materials, which is fundamental for various applications in material science and engineering. Notably, the unit cell edge length is a direct link to the atomic scale, which emphasizes the importance of atomic radius once again in the broader context of crystallography and material properties.