Engineering Physics of High-Temperature Materials. Nirmal K. Sinha

efficiency in packing leads to the abundance of three main crystal structures. While these structures do not describe all materials, their prevalence makes them important to dive into and enables understanding of similar structures. Two of these structures are based on a simple cubic arrangement. The simple cubic structure consists of atoms placed on each corner of a cube. The simple cubic unit cell is thus a cube containing 1/8 of each corner atom for a total of one atom per unit cell. The distance between the corners of the cube is called the unit cell dimension, which is commonly denoted as the lattice constant, “a.” Assuming the highest packing efficiency in the simple cube and the model of an atom as incompressible spheres of radius “r” we have a = 2r. Three of the most common crystal structures in nature are the body‐centered cubic (BCC) structure, the face‐centered cubic (FCC) structure, and the hexagonal close‐packed (HCP) structure. As can be seen in Figure 2.1, the first two structures are based on the simple cube.

Figure 2.1 Basic crystal lattice structures: (a) BCC, (b) FCC, and (c) HCP crystals. The upper stick–ball representations demonstrate the basic atomic arrangement while the lower images represent the close packing. The side inset in (c) represents the top‐down view of the HCP structure with the shaded region demonstrating the primitive unit cell.

      The BCC structure takes the simple unit cell and increases the packing efficiency by having an additional atom at the center of the cube. Thus, there are two atoms per unit cell. The diagonal passing through the center of the BCC unit cell is called the close‐packed direction because atoms along the diagonal touch each other. The length of the diagonal is thus equal to 4r and . Common BCC materials include iron, chromium, and tungsten.

The FCC structure has an atom at each corner of the unit cell plus an atom at the center of each face of the unit cell. Thus, there are four atoms per unit cell. The closed‐packed directions of the FCC structure are the diagonals across each face of the unit cell, and this arrangement gives rise to the closest packing of spheres in a cube. When viewed from a 45° angle (i.e. viewing the (111) layers) and taking a slightly larger view, relative to the unit cell, the FCC structure can be seen to be made up of three types of planes. It is thus said to have an ABC stacking sequence (Figure 2.2). Common FCC materials include aluminum, gold, and nickel.

      The HCP structure, which is the last of the most common structures, also gives rise to the same packing efficiency as does the FCC structure and thus the closest packing of spheres in a cube. As shown in Figure 2.2, it can be seen to have an ABA stacking sequence. The HCP unit cell is not based on a simple cube, but rather, as its name implies, on a hexagonal structure with atoms at each corner of the hexagon. An additional atom is located at the center of each face of the hexagon and three more atoms are located in a plane between the hexagonal faces. The hexagonal unit cell contains within it three primitive unit cells (the smallest repeating unit). This primitive cell is represented as the shaded region in the top‐down view of the HCP unit cell in Figure 2.1c. The larger hexagonal unit cell is easier to visualize, and it contains six atoms per cell. Common HCP metallic materials include magnesium, zinc, and titanium. Snow and ice, which are among the most abundant, but nonmetallic, materials on Earth's surface, also belong to the hexagonal family system, though these materials do not have the close‐packed structure at normal pressures and temperatures on earth’s surface. Ice is discussed in greater detail in Section 2.6.

Figure 2.2 Stacking sequence in (a) FCC and (b) HCP crystals.

Variations of the three basic crystal structures – BCC, FCC, and HCP – give rise to a wide range of crystal structures. In addition, many solids have the ability to exist in more than one crystal structure. This is called polymorphism. Polymorphism often results due to exposure to different temperatures and pressures. For example, many metals will exist in a close‐packed structure (e.g. HCP) at low temperature, but will transform into the less dense BCC structure at higher temperatures.

Allotropy is a special subset of polymorphism. Allotropes are specifically polymorphs of the elements. Elemental carbon is one of the most common examples of allotropy due to the range of structures it can take. Allotropes of carbon include amorphous coal, graphite (hexagonal crystal system), diamond (FCC variation) and nanocarbon structures (such as buckminsterfullerenes and nanotubes). Comparing the properties of some common carbon allotropes demonstrates the great impact that crystal structure and overall microstructure can have on the properties of materials (Table 2.1).

2.2.2 Structure of Amorphous Solids

      Amorphous solids can best be described by what they are not. They are materials that exhibit elastic response like crystalline solids, but possess no crystal‐like long‐range order in the spatial arrangements of their atoms. Amorphous materials could be inorganic or organic and are also known as glasses and vitreous solids. Glasses are increasingly being used in wide‐ranging branches of science and technology. Bulk amorphous metals (aka metallic glasses) are increasingly being used as engineering materials owing to their extraordinary isotropic properties, including high strength, corrosion resistance, formability, and even biocompatibility (Khan et al. 2017; Jafary‐Zadeh et al. 2018). Molten minerals quenched to an amorphous state are used as analogs for magma (molten rocks) deep inside the surface of Earth (Davidovits 2005).

Table 2.1 Comparison of the properties of some basic carbon allotropes.

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	Coal	Graphite	Diamond	Buckminsterfullerene (C60)	Carbon nanotubes
Structure/hybridization	Amorphous (varies)	Trigonal planar (sp2)