For an FCC single crystal, would you expect the surface energy for a (100) plane to be greater or less than that for a (111) plane? Why? (Note: You may want to consult the solution to Problem \(3.54\) at the end of Chapter 3.)

Understanding the crystallographic planes within a Face-Centered Cubic (FCC) lattice is fundamental to grasping various material properties. In FCC structures, we consider two prominent crystallographic planes: the (100) and the (111) planes. These planes are defined by the orientation of the atoms within the lattice.

Visualize the (100) plane as a slice through the cube that includes all the atoms at the corners and the center of one face. This results in a square array of atoms, with each atom at these surface corners effectively shared by adjacent cells. On the other hand, the (111) plane slices through the cube diagonally, and by intersecting more densely packed atomic rows, it has a hexagonal close-packed arrangement. This increased density of atoms leads to more bonding interactions within the plane.

The varying atomic arrangements affect the material's surface energy, which is crucial for processes like crystal growth and surface reactions. For example, when determining which plane a crystal will grow or fracture along, understanding the surface energy associated with different crystallographic planes allows us to predict the behavior under various conditions.

The atomic arrangement in solids such as metals, semiconductors, and certain ceramics, where atoms are regularly spaced in a three-dimensional array, gives rise to crystalline structures. The FCC lattice is one such arrangement where each atom is positioned at the corners and the centers of the faces of the cube.

This pattern repeats itself throughout the entire crystal, creating a highly organized structure. The key to these arrangements lies in the unit cell, the smallest repeating unit in the crystal, which determines the physical and chemical properties of the material. For example, the density of a crystal depends on how closely packed the atoms are in the unit cell, and this packing can be different depending on the unit cell geometry.

Other common crystalline structures include Body-Centered Cubic (BCC) and Hexagonal Close-Packed (HCP); each offers different atomic arrangements and related properties. The type of unit cell and the way the atoms are positioned influence mechanical strength, ductility, and electronic properties of solids.

In crystalline structures, atoms are held together by bonds, which can be ionic, covalent, metallic, or a mix, depending on the material. The type and strength of bonding influence the material's properties substantially. In FCC crystals, atoms are bonded metallically, meaning they share a 'sea' of electrons that are not bound to any particular atom but move freely throughout the metal lattice.

This free electron movement grants metals their characteristic electrical conductivity and ductility. However, when analyzing surfaces, such as in the different planes of an FCC lattice, the arrangement of atoms drastically affects the number of bonds an atom forms. Atoms at the surface of a material typically have fewer neighbors and therefore fewer bonds. When bonds are broken, which happens at surfaces and interfaces, the material acquires surface energy.

The difference in bonding patterns between the (100) and (111) planes in an FCC structure directly correlates with their surface energy, affecting the stability and reactivity of these planes. Improved understanding of these bonding interactions helps material scientists tailor the surface characteristics of metals for specific applications such as catalysts or coatings.