2.2 Crystal Structure

As previously mentioned, materials are made of atoms and primarily based on the nature of atoms, different types of bonding can be found, as discussed in previous section. Besides different atoms and different atomic bonding, atom organization (atom packing) is another key factor influencing material properties. The properties of advanced materials are not so much affected by their overall chemical composition but rather by the specific arrangement of their constituents which we usually can't discriminate with our bare eye. The arrangement of the constituents of a material, i.e. the spatial distribution of elements, phases, orientations, and defects are subsumed under the term microstructure.

Atom arrangement in materials can be quite different, as shown in Figure 2.8. According to the space arrangement of atoms in solids, there are:

• Single crystalline: atoms are arranged periodically in space, possessing only one long-range regularity.

• Polycrystalline: atoms are arranged partially in periodic way, possessing many/several long-range regularity.

• Amorphous structure: atoms are randomly distributed in space, possessing short-range order but disordered in a long range.

• Quasicrystalline: having space order feature between crystalline and amorphous solids.

2.2.1 Unit Cells, Space Lattices and Lattice Point

Metallic and ceramic materials are usually crystalline. Also polymers can partially crystallize, a crystalline structure means in a physical sense, a strictly periodic arrangement of atoms. However, long before the atomistic structure of solids was known, crystals of minerals fascinated man and became a subject of scientific interest. The prominent feature of crystalline minerals is their external appearance with planar facets, which are characteristic of each mineral. According to the geometry of the facets, the crystallographers were able to group the crystals in terms of their shapes and symmetries into 32 classes (also referred to as point groups), which could be subdivided into seven crystal systems. These seven crystal systems can be defined by the macroscopic orientation of the crystal surfaces and their lines of intersection in an appropriately chosen crystal coordinate system. If there is no apparent symmetry, the crystal structure is called triclinic, and the angles between the crystal axes and their respective lengths are all different. The highest symmetry is represented by cubic crystals, where all crystal axes are equally long and are arranged under a mutual angle of 90°. The detail information of crystallography can be found in the textbook of solid state physics.

As mentioned, solids are made of atoms that are chemically bound to one another. Most solids, including all metals and most ceramics, have a crystalline structure. The strength and nature of the forces between the atoms of the crystal lattice and also the electron clouds around the nuclei determine the macroscopic properties of the solid, e.g., strength, elasticity and conduction of heat and charge. Taking two carbon materials, graphite and diamond, as an example, the importance of the crystalline structure for the determining of matter behavior can be clearly seen.

As shown in Figure 2.9, graphite belongs to hexagonal crystal system. It has a parallelly-layered, planar structure. In each layer, there are six-membered rings of carbon atoms. Within one layer, the carbon atoms are held together by strong covalent bonds. Between the layers, the atoms are held by weak van der Waals bonds. This is the reason for the weakness of graphite towards shearing forces and it can work as a lubricant. Contrary to graphite, diamond is made of tetrahedrally shaped and covalently bound carbon atoms, forming face-centered cubic crystal structure. Due to the strong covalent bonding, as well as the extremely rigid lattice, diamond is one of the hardest elements in nature.

When atoms are chemically bound to one another they have well-defined equilibrium separations that are determined by the condition that the total energy is minimized. Therefore, in a solid composed of many identical atoms, the minimum energy is obtained only when every atom is in an identical environment. This leads to a three-dimensional periodic arrangement that is known as the crystalline state. A metal would be representative of a crystalline solid (though not perfectly crystalline). The same is true for solids that are composed of more than one type of element. In this case, certain building blocks comprising a few atoms are the periodically repeated units.

Periodicity gives rise to a number of typical properties of solids. Periodicity also simplifies the theoretical understanding and the formal theory of solids enormously. Although a real solid never possesses exact three-dimensional periodicity, one assumes perfect periodicity as a model and deals with the defects in terms of a perturbation. Three-dimensional periodic arrangements of atoms or building blocks are realized in many different ways.

Therefore, in our study of crystalline structures, we will limit ourselves to the orderly arrangement of the atoms in their microscopic world. In this way, we represent atoms, ions, or molecules essentially as spheres of varying sizes occupying points at various distances from each other in space, hence the need for an axis system. Such a system is shown in Figure 2.10.

• Unit cell: The term unit cell is used to describe the basic building block or basic geometric arrangement of atoms in a crystal. You can compare a unit cell to a single brick in a brick wall. The box in Figure 2.10a is a unit cell. The angles between the principal planes are named α, β, and γ, where α is the angle measured in degrees between the x-y and x-z planes; angle β, between x-y and y-z planes; and angle γ, between the x-z and y-z planes. The sides of the box (unit cell), labeled a, b, and c, are the lattice parameters in the x, y, and z directions, respectively. These distances are also known as intercepts. To describe a particular axes or crystal system adequately, all six dimensions are needed, which are known as lattice constants.

• Space/crystal lattice and lattice points: If you repeat duplicating the unit cell in all three dimensions, you create a crystalline structure with a definite pattern (Figure 2.10b). This larger pattern of atoms in a single crystal is known as a space lattice or crystal lattice.

A space lattice is defined as a collection of mathematical points, arranged in such a way that each of the points is surrounded by precisely the same configuration, such that the constructing space is divided into small volumes of equal size, with atoms (ions or molecules) located at the intersections of these lines or between the various lines. The view will always be the same, independent of which point we chose as observation site. The points in space lattice are called lattice points. Every lattice point on one side of the observation site always has a corresponding lattice point in an identical position but situated on the opposite side.

• Primitive cell: We must keep in mind that the lines and points in a space lattice are only imaginary. The lattice concept is used to show the positions of atoms, molecules, or ions in relation to each other. We must also remember that the actual atoms in solids are located as close to each other as possible, thus attaining the lowest possible energy. Two atoms closest to each other would be represented by two spheres touching each other. The closer the atoms are, the denser the solid. In addition, we also must keep in mind that even if the lattice structure is basically simple, the crystal structure can be very complicated as the unit cell might consist of tens of thousands of atoms. This is the case in proteins and other organic system. If the unit cell only contains one atom/molecule, it is called primitive cell. The number of atoms or molecules per unit cell is given by:

where Ninterior, Nface, Ncorner is the number of atoms/molecules inside, on the face, on the corner, respectively, within a unit cell.

Metals are generally made of atoms with a close-packed structure, which means that the number of atoms per unit cell is greater than 1, the number of atoms in a simple crystal structure. The most common close packed crystal structures of metals are given in Table 2.1:

Organic materials are made of molecules holding together by van der Waals forces. Examples of their molecules per unit cell are given in Figure 2.11.

2.2.2 Crystal Systems

In this section, we will emphasize crystal systems composed of atoms and ions, but the structural particles of crystalline solids can be atoms, ions, or molecules. For example, solid methane, CH4, a molecular solid, has a face-centered cubic (FCC) structure, which means that there is a CH4 molecule at each corner and at the center of each face in its unit cell. The forces acting among these structural particles may be metallic bonds, interionic attractions, van der Waals forces, or covalent bonds.

Ⅰ. Cubic cells

The cubic system includes simple, body-centered and face-entered cubic cells.

The simple cubic unit cell (Figure 2.12a) consists of eight atoms located at each corner of the cube. Only polonium is crystallized in the simple cubic lattice. It must be remembered that if you represent these eight atoms with hard rubber balls and arrange them in accordance with this simple cubic unit cell, all eight atoms would be touching each other.

Another unit cell (Figure 2.12b) is known as the body-centered cubic (BCC). It is similar to the simple cubic unit cell, but contains an additional atom located in the center of the cube.

The third type of unit cell formed from the cubic axis system is the face-centered cubic (FCC). One atom at each corner and one in the center of each of the cube faces make up the complement of atoms. There is no atom at the center of the cube (Figure 2.12c).

For both the BCC and FCC structures, the same type of atoms must fill both the center/face-centered locations, as do the cell corners. If, in fact, a different type of atom fills the center position than the cell corners in a body-centered cubic structure, the structure is correctly called a simple cubic (SC) structure. An alloy made up of about 50% copper and 50% zinc forms such a structure.

Ⅱ. Tetragonal unit cell

The tetragonal crystal system has similar unit cells to the cubic, but the sizes are not equal. As an example, the body-centered tetragonal (BCT) crystal lattice unit cell is shown in Figure 2.13. Tin forms a tetragonal unit cell. The tetragonal is similar because the axes are all normal to each other. The difference lies in the length of the intercepts. The x and y intercepts have the same magnitude. The z intercept is larger than the x or y intercept. Martensite, a combination of iron and carbon that is contained in a hard steel, has its atoms of iron and carbon in a tetragonal lattice structure.

Ⅲ. Hexagonal unit cell

The hexagonal crystal system (Figure 2.14a) can best be described using three axes (a1, a2, and a3) in the x-y plane 120° apart and a fourth axis(z) at 90° to the x-y plane. The intercepts along the three axes in the horizontal plane are equal in length (a = a = a), but the fourth intercept, labeled c, is of a different length. This unit cell is made up basically of two parallel planes (top and bottom basal) separated by a distance equal to the dimension c. The atoms shown in the figure trace out a right hexagonal prism. Each of these two planes can be divided into six equilateral triangles, with each side equal to the intercept a (Figure 2.14b).

Atoms of solid materials do not form the purely hexagonal unit cell as in Figure 2.14a because they cannot satisfy equilibrium conditions by being so far apart. In other words, they would be unstable. Consequently, they form the hexagonal unit cell, called close-packed hexagonal (CPH), as shown in Figure 2.14c, with its three mid-plane atoms at a distance of c/2. Zinc, titanium, and magnesium form CPH unit cells.

According to the French crystallographer, A Bravais, in 1848's statement, there are seven types of unit cells (Table 2.2), of which three were mentioned above; and atoms can form 14 patterns in space (Brevais lattices, see Figure 2.15), with which the earth's elements form their particular atomic structures.

As early as 1848, long before X-ray methods of crystallography were known, the French crystallographer A. Bravais stated that the atomic arrangements in all crystalline solids can be referred to 14 fundamental crystal classes (Brevais lattices, see Figure 2.15), consisting of four types of space lattice in combination with seven systems of unit cells (Table 2.2), of which three were discussed above. The earth's elements form their particular atomic structures.

2.2.3 Representative Parameters for Crystal Systems

Ⅰ. Miller indices for atomic planes (lattice planes)

In a crystal system with huge amount of periodically arranged lattice points, there are several different situations (Figure 2.16). For instance, the atom distance in different direction is generally different, also in lattice planes of different orientation, the in between distance is different. This anisotropic material structure leads to anisotropic material properties.

Therefore, it is appropriate to be able to describe the location of atoms in a unit cell as well as the direction of their movement. The sites or locations of atoms and/or points in a unit cell are described by atomic planes in unit-cell dimensions, which are designated by their Miller indices.

As shown in Figure 2.17, the Miller indices for atomic plane can be expressed:

where X, Y, Z = coordinates of the intersections between the plane and the coordinate axes; D = the smallest common multiple of X, Y and Z. The miller indices, written within curly brackets {}, mean that all permutations of the hkl indices and combination of signs occur. They define the directions of the normals to the type of planes in question. If one refers to a specific plane with the given directions, ordinary parentheses are used, (hkl). If X = Y = Z = 1, the Miller indices of the plane are (111). If the plane is parallel with one of the coordinate axes, the corresponding Miller index is zero as the intersection occurs at infinity. A bar over an index indicates a negative value of the intercept. Some examples of Miller indices for atomic planes are shown in Figure 2.18.

Ⅱ. Crystal directions (Lattice directions)

As with Miller indices for specifying atomic planes in a crystal-lattice system, there is Miller indices for directions, which are generally written as<uvw>. This means that all permutations of the uvw indices and combination of signs occur. If one refers to a specific line with the given direction, square brackets are used, [uvw].

To calculate the Miller indices for a direction, there are four steps: The first step is to determine the coordinates of two points that lie in the particular line of direction. The first point, sometimes called the "head point," is farthest from the origin. Using the origin as the second point simplifies the procedure. The second step is to subtract the second point from the first point. The third step calls for clearing of any fractions using the smallest common multiple to obtain indices with the lowest integer values. The fourth step is the writing of the results. Negative integer values are indicated by the use of a bar placed over the integer. In more complex crystal systems, the determination of directions requires temporary relocation of the origin to another point in the unit cell to simplify the procedure. If the properties of a crystal measured along two different directions are identical, the two directions are termed equivalent. Equivalent directions are referred to as a family of directions, which can be specified by using angle brackets: <110>. Figure 2.19 are examples of the Miller indices for directions.

Ⅲ. Coordination number

To describe how many atoms are touching each other in a group of coordinated atoms, the term coordination number (CN) is used. The CN is the number of neighboring atoms that are directly surrounding it. Note in Figure 2.14c that each upper and lower basal plane of a CPH unit cell contains an atom at its center. Each atom touches six atoms in its own plane, plus three atoms above and below in adjacent planes. Consequently, the CN for these atoms would be 12. The number of nearest atoms is dependent on two factors: (1) the type of bonding, and (2) the relative size of the atoms or ions involved. In our discussion of bonding, for example, we learned that valence electrons determine the type of bonding as well as the number of bonds an atom or ion can have. Carbon (C) in group IV has four covalent bonds and therefore a CN of 4. The group VII elements, such as chlorine(Cl), form only one bond (CN is 1). The relative size of the atoms determines how many neighboring atoms will touch another atom. Ionic bonding involves ions of different charges, hence, different sizes. The limiting factor in this case is the ratio of the size (radii) of the combined atoms of ions. The minimum ratios of atomic (ionic) radii produce various CNs. During ionization, atoms decrease in size when they change to cations and increase in size as they form anions.

Figure 2.20 represents the five ions occupying one of the six faces of the FCC unit cell for NaCl. The Na ion is just the right size to fit between the Cl ions at the corners of the unit cell. Thus, the ions are closely packed, with each cation separated from other cations by a layer of anions. Each cation and each anion are shared equally by six oppositely charged ions. Therefore, a CN of 6 describes this geometric arrangement. Actually, in the case of ionic crystal, the radius ratio between the anion and the cation determines the CN value. As the difference between r and R decreases, higher CNs are possible. A CN of 12 is the maximum, which occurs when the atoms (ions) have the same radius and the ratio becomes 1. In other words, as the r gets smaller than R (radius of surrounding atoms), the fewer neighboring atoms can make contact or touch the smaller atoms. Table 2.3 lists the minimum radii ratios for some common CNs.

With a CN of 12, each atom has contact with 12 other atoms. Each atom in an fcc or cph unit cell meets this description, provided that their radii are of similar size.

Ⅳ. Volume changes and packing factor

In discussing crystal structure changes, we mentioned that every change in atomic structure brings changes in properties of the solid. One of these changes is volume. When pure steel transforms from a BCC to FCC structure at 912°C, it decreases about 1.06% in volume. The explanation for this phenomenon involves the density of atoms in the various unit cells. Density is the ratio of the mass to the volume of a substance, which stays constant provided it is non-allotropic.

The atomic packing factor (APF), or packing factor (PF), is the ratio of the volume of atoms/molecules present in a crystal (unit cell) to the volume of the unit cell. In calculating the volume of an atom, we assume the atom is spherical. The difference between the PF and unity (1) is known as the void fraction, that is, the fraction of void (unoccupied or empty) space in the unit cell.

Using the simple cubic crystal unit cell sketched in Figure 2.21a, with atoms of equal radius (R) located at each corner, the volume of the cell occupied by the eight atoms is equivalent to one atom. Each corner atom contributes one-eighth of its volume. Therefore, the volume of one atom is(4/3)πR3. The (100) face plane of this simple cubic unit cell shows four corner atoms touching each other (see Figure 2.21b). The relationship of the radius R of an atom to the lattice parameter or edge length a of the unit cell is 2R. The volume of the unit cell is a3. The volume of atoms is (π/6)a3. Thus,

Solving for the PF in terms of the radius R produces the same results. The void factor is therefore 1 − 0.52 = 0.48. What the calculation tells us is that only about half (52%) of the space in the simple cubic unit cell is occupied by the atoms. This is too inefficient, so atoms of metals do not crystallize in this structure except Polonium. Remember, the closer the atoms come to each other, the less energy they have and the more stable is their structure.

The BCC unit cell is quite similar to the simple cubic unit cell with the addition of one atom in the very center of the unit cell. Therefore, the bcc unit cell contains the equivalent of two atoms.

For the FCC unit cell, there are four net atoms. Each of the eight corner atoms contributes one-eighth of an atom. Each of the six face atoms contributes one-half. The total is 1 + 3, or 4, atoms. Notice that the atomic radius (R), or a as used in the formula, cancels out in all these calculations, which tells us that the PF is not dependent on the radius of the spheres being packed if all the atoms are of the same size.

The FCC structure has the maximum PF for a pure metal. The CPH structure also has a PF of 0.74. Finally, we should note that the coordination number varies directly with the PF. As an example, the CNbcc = 8 and PFbcc =0.68; the CNfcc = 12 and PFfcc = 0.74.

In metallic materials, for example, the pure iron will change its structure on heating from BCC to FCC at 910℃. Knowing the PF for both these structures will lead us to the conclusion that iron will contract in volume as it is heated above 910℃. This change in structure forms the basis for the production of steel as well as the heat treatment of steel.

In organic materials, the concept of packing factor is the same as the above description, just that the atoms change to molecules. Since the unit is molecules that are hold together by weak van der Waals force and strong short-range repulsion, the molecules will adopt the densest possible packing with the least possible repulsion. The arrangement of the molecules will be determined by atom-atom potentials. The lattice energy is minimized when the number of van der Waals atom-atom contacts is as large as possible. The values of PF for the aromatic hydrocarbons lie between 0.68 (benzene) and 0.80 (perylene). For comparison: the packing coefficient of ice, which is bound through dipolar forces (hydrogen bonding) is only 0.38.

2.2.4 Crystal structures of Metals and Organic Materials

Ⅰ. Metal

Many metals have a close-packed structure, which means that the number of atoms per unit cell is greater than 1, the number of atoms in a simple crystal structure. The most common close packed crystal structures have already been given in Table 2.1.

• BCC structure

This structure is a simple cubic lattice with an additional atom in the centre of the cube (Figure 2.22). Number of atoms per unit cell = (8×1/8) +1 =2; Coordination Number = 8. Examples of metals with a BCC structure are Cr, Ti, W, Mo, V, and α-Fe etc.

• FCC structure

This structure is a simple cubic lattice with an additional atom in the centre of each of the six faces of the cube. Number of atoms per unit cell =(8×1/8) + (6×1/2) = 4.

Some {111} places of FCC structure are shown in Figure 2.23(b). As labeled, there are A, B and C planes. The pattern of these {111} planes are illustrated as following: Take the central atom in plane A as sample, besides the 6 peri-surrounding atoms, at top and bottom of plane A, there exist 3 surrounding atoms, respectively. Hence, the coordination number is 12. Examples of metals with a FCC structure are Al, Cu, Au, Ag, and γ–Fe etc.

• HCP structure

BCC and FCC structures are simple in the sense that atoms are directly placed in the corners of the Bravais point lattice. The hexagonal close-packed structure is more complicated as every lattice corner is occupied by an atom, along with three mid-plane atoms at a distance of c/2, like a dumb-bell. All dumb-bells are parallel in space. Number of atoms per unit cell = (8×1/8) + 1 = 2, Number of nearest neighbours = 12. The HCP structure can be regarded as three single unit cells connected rigidly with one another. Figure 2.24 is the unit cell of HCP cell. Examples of metals with a HCP structure are Mg, Zn, and Cd etc.

The stacking sequence of the atomic planes in an HCP structure is shown in Figure 2.25.

Ⅱ. Organics

In considering the packing modes within a crystal of organic materials, due to the relatively weak dispersive forces and strong short range repulsion, the trend is to pack as close as possible to maximize the van der Waals force in order to reduce lattice energy. And we also need to remember that even with planar molecules, the molecular surfaces are not structureless. The positions of the atoms correspond to 'hills', the positions in between to valleys in the molecular contour, as shown with anthracene molecule in Figure 2.26.

Therefore, an arrangement in which the hills of one molecule lie above the valleys of the neighboring molecules is generally energetically more favorable for nonpolar molecules than one in which the molecules lie directly above one another. This is most noticeable in the 'herringbone' pattern in many aromatic molecules crystalline (Figure 2.27).

In the case of long-chain alkanes, in particular the linear hydrocarbons with n carbon atoms, the preferential arrangements are the CH2 zig-zag chains of the individual molecules, which are parallel to each other, and thus form a layered structure (Figure 2.28). Such an arrangement guarantees the strongest dispersive interactions in view of the anisotropic polarizability of these molecules.