Remember the purpose of this reading assignment is to prepare you for class. Reading for familiarity not mastery is expected.

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1 Remember the purpose of this reading assignment is to prepare you for class. Reading for familiarity not mastery is expected. After completing this reading assignment and reviewing the intro video you should be comfortable either 1) Describing the following key concepts in your own words, or 2) asking focused questions regarding the following key concepts. The definition of a crystalline solid Identifying and determining the values of the key features of a cubic crystal structure given a sketch Determining strongest directions, planes, and openness of cubic crystal structures.

2 Chapter I. Structure of Crystalline Solids Engineering materials are used in the solid state. A solid can be defined as a state of matter that does not flow when subjected to a non-uniform force, in contrast to liquids and gases which do flow. In solids the number of atoms in individual atoms varies dramatically. For example, an iron crystal (a single molecule of iron in the solid state) has the chemical formula Fe(s). This does neither means nor implies that a single molecule of iron in the solid state has only one atom. The same is true for materials such as table salt, sodium chloride. The formula is NaCl(s), this means that there is a one-to-one relationship between the number of sodium (Na) atoms and chlorine (Cl) atoms, but there are many more than two atoms per molecule. In fact, the correct answer to the question: How many iron atoms are there in a single molecule of Fe(s)? is, I don t know. A single grain of iron (which is a single crystal or molecule) can have a diameter of 1 :m. This is roughly 4000 atoms wide, meaning the overall molecule has approximately 65 billion atoms. In most solids the sizes of the individual grains varies, meaning that each molecule has a different number of atoms. The arrangement of atoms inside these molecules has an effect on the mechanical, electrical and chemical properties of the overall material. Therefore, it is important to have a basic understanding of this arrangement. Many metals, ceramics, and even some polymers are crystalline solids. This means that the material consists of individual molecules or crystals each containing many atoms. In a crystal the atoms have long-range order. This means that the atoms are arranged in a well defined, repeating pattern, over large (based on the size of the atom) distances in three dimensional space. A. Basic Terms and Definitions A crystal as a solid with long-range order. That is the atoms are arranged periodically in three dimensional space. To illustrate the structures of crystals, atoms are considered to be spheres with a definite radius. The electrons around an atomic nucleus are spread out in a cloud, although there is a most probable distance of each electron from the nucleus. In spite of this atomic structure, the hard sphere model for crystal structure is an accurate way to describe the structure. The nuclei of the atoms have fixed positions that can define the crystal structure. Having defined a crystal as a solid where the atoms (molecules or ions) are arranged periodically in three dimensional space, the next question to be answered is how are they arranged. A seventeenth century scientist named Bravais discovered that there were 14 unique ways to arrange solid spheres, with three dimensional periodicity. These are called the Bravais Lattices. The smallest complete lattice is called the unit cell. Thus, by knowing and understanding the unit cell, one can understand begin to predict the properties of a crystalline solid. In this chapter and book we will focus on cubic crystal structures. Other structures exist, it is important that you know they exist, but the details are beyond what is needed of materials for a general engineer. A cube can be completely defined by one measurement, the length of its edge. This is referred to as the lattice parameter, and is given the symbol a. Knowing a one can completely analyze the unit cell of a cubic structure. B. Cubic Crystal Structures (Metals) Metallic bonding, where electrons are delocalized, or shared among many atoms, is the most complicated. However, metals have the simplest crystal structures. There are three types of cuboc crystal structures observed in metals:

3 simple cubic, in which the atoms touch along the edge of the cube, body centered cubic, in which the atoms touch along the body diagonal of the cube, and face centered cubic, in which the atoms touch along the face diagonal of the cube. 1. The Simple Cubic Lattice Only one metal Po, above room temperature is known to have the simple cubic structure, however the structure is worth discussing, as it can be used to illustrate the principles of crystallography. The simple cubic unit cell is shown in Figure 1. The atoms touch along the cube edge. In this case the lattice parameter, or edge length, and is therefore twice the atomic radius. a=2r, therefore for Po, a = 2r = 2( nm) = nm Figure 1: Simple Cubic Unit Cell In the simple cubic lattice there is only one atom per unit cell. Each atom is represented by a sphere in a corner, only one-eighth of this sphere is a part of this cube and another way of looking at this is to say that each atom is in eight other unit cells. Therefore the number of atoms in the unit cell is, 8(1/8)=1. In this structure each atom touches six others. The number of atom an individual atom touches is referred to as the coordination number. The coordination number in a simple cubic unit cell is The Body-Centered Cubic Lattice (bcc) A common cubic structure is body-centered cubic, as shown in Figure 2 Here eight atom centers are on the corners of the unit cell cube, and one is at the center of the cube. Note the coordination number is thus 8. There are two atoms in 2 bcc cell, one from the corners and one in the center of the cell. The atoms touch along the diagonal of the cube, so that the relation between atomic radius and lattice parameter is, The lattice parameter of "-Fe is thus Figure 2: The Body-Centered Cubic (bcc) Unit Cell

4 4r a = 3 4( nm) = 3 = nm Metals which have the bcc structure include chromium( Cr ), vanadium( V ), niobium( Nb ), tungsten ( W ), molybdenum( Mo ), the alkali metals( Group I ), and iron below 910 o C("-Fe). 3. The Face-Centered Cubic Lattice (fcc) The most common metallic crystal structure is face centered cubic, shown in Figure 3. In the face-centered structure there are atom centers at each of the eight corners of the cube and at the centers of each of the six faces of the cube. The eight atoms at the cube corners, one-eighth of the unit cell and each of the six face centered atoms are half in the unit cell, making a total of four atoms in the unit cell. The coordination number in fcc is 12. The atoms in the face-centered cubic unit cell touch along the face diagonal in this structure, and therefore, a r = 4 2 Figure 3: Face-Centered Cubic Unit Cell (fcc) The lattice parameter for Cu is therefore, 4r 4( nm) a = = = nm Some metals that have face-centered cubic structure are aluminum(al), copper(cu), gold(au), lead(pb), nickel(nu), platinum(pr), and iron above 910 o C((-Fe). C. (Ionic Solids) Structures of ionic compounds such as MgO are more complicated than those of crystals containing only one kind of atom. Note, that as stated earlier, a single molecule of MgO(s) contains millions or billions of Mg ions and O ions. In ionic solids the ions arrange themselves in such a way that charge neutrality is maintained. Frequently this involves a long range three dimensional pattern and thus many ionic solids are crystalline. We will describe some of the most common compound structures CsCl (cesium chloride), NaCl (sodium chloride), ZnS (zinc blende), calcium fluorite (CaF 2 ), and barium titanate (BaTiO 3 ).

5 1. Coordination Number The coordination numbers of ions in ironically-bonded structures are related to the ionic radii of the ions. A positively charged ion will try to surround itself with as many negatively charged ions as possible. However the negatively charged ions repel one another. Thus there are two competing factors. The coordination number will depend upon the ratio of the two radii. The limiting case will be where the anions just touch one another. The ideal radius ratio for a particular coordination number can be calculated from a structure of closest packing for that radius ratio. These ideal ratios are given in Table 1. Consider 6-fold coordination, that is each cation (+ charge) is surrounded by six anions (- charge). This means that the cation must be large enough to just separate six negatively charged anions. Sketching this case, as in Figure 4, one can isolate a triangle as shown. Note that this is an isoceles right triangle. The length of the two equal sides must equal the sum of the radius of the cation and the radius of the anion. The length of the other side is twice the radius of the anion. Using the Pythagorean Theorem, 2( r + R) = ( 2R) 2( r + R) = 2R ( r + R) = 2R 2 2 ( 2 1) ( ) r = r R R = 2 1 or Figure 4: 6-fold coordination of ions. In order for the cation to have a coordination # of 6, the cation must be large enough to separate 6 anions or r R Thus the ratio at which all particles touch is If r/r is larger than 0.414, the ions can arrange themselves with 6 fold coordination and the anions will not touch. Table 1 shows the minimum radius ratio for a given coordination number in ionic solids. Table 1: Minimum Radius Ratios for Given Coordination Numbers in Ionic Crystals CN Min r/r

6 2. The Cesium Chloride Structure (CsCl) The coordination no. of CsCl can be found from the ratio of the ionic radii, r R + ( ) r Cs nm = = = r( Cr) nm Therefore the coordination number is 8, and each ion is surrounded by eight oppositely charged ions. The cesium chloride(cscl) unit cell is shown in Figure 5. It is inappropriate to call this bcc., because the central particle is not the same as those on the corners. In fact, this is a type of simple cubic structure. Like simple cubic there is only 1 CsCl molecule per unit cell. In this structure there is only one ion pair, for example chloride(cl)), is at the cube corners and the other (Cs+) at the center of the cube. Thus, the structure of the individual ions is simple cubic. The ions touch along the body diagonal l = a 3, one of each ion so the lattice Parameter is, 2 a = ( r + R) 3 Figure 5: Cesium Chloride Structure

7 3. The Sodium Chloride Structure (NaCl) In NaCl the ratio of ionic radii is, r R + + ( ) r Na nm = = = r( Cl) nm nm and thus the coordination number is 6. This sodium chloride structure is cubic and is shown in Figure 6. The unit cell can have either sodium or chloride ions at the corners of the cube; in Figure 6 the chloride ions are on the corners. In addition, chloride ions are at the face centers, so that the structure of the chloride ions by themselves is face-centered cubic. Sodium ions are between the corner chlorides; the sodium ions are on a face-centered cubic lattice. Therefore the Bravais lattice for this structure is considered to be face centered cubic, as the arrangement of both Na and Cl ions are face centered cubic. The ions touch along the edge of the cube, and therefore a = 2( r + R) Figure 6: Sodium Chloride Structure and the x-ray pattern will resemble that of an fcc metal. Like a lattice there are four NaCl molecules per unit cell.

8 4. Zinc Blende Structure (ZnS) In ZnS the ratio of the ionic radii is r R = + 2 ( ) 2 ( ) r Zn R S nm = = nm Therefore CN=4 and will the ions will have a tetrahedral arrangement. It is possible to arrange the tetrahedra into a cube as shown in Figure 7. Note that each ion pair touches along the body diagonal but, that this distance accounts for only 25%. of the length. Thus, where r represents the radius of the cation, and R - represents the radius of the anion. Figure 7: Zinc Blende (ZnS) structure. Each Zn ion is bonded to four S ions. The ions are shown to be the same size for clarity. 5. The Calcium Fluorite Structure (CaF 2 ) The calcium fluorite CaF 2 structure is shown in Figure 8. It resembles the zinc blende structure. The difference between the calcium fluorite structure and the zinc blende structure is the presence of additional ions. There are eight anions in the unit cell and four cations. Note that the stoichiometry of the unit cell is precisely the same as the stoichiometry of the chemical formula. There are four cations, (Ca +2 ), and eight anions (F - ) This must be the case, as the unit cell represents the compound. The geometric relationship between ion size and lattice parameter is the same. The ions still touch along the body diagonal, and this distance is one-quarter the length of the body diagonal. Thus, Figure 8:Calcium Fluorite (CaF 2 ) Structure, the F - ions are colored gray. For clarity all ions are shown to be the same size.

9 6. Perovskite (BaTiO 3 ) Structure The perovskite structure is a cubic structure based on three ions. The lattice is similar to both the fcc and NaCl lattices. O 2- ions occupy the face centered positions, Ba 2+ ions occupy the corner positions and a Ti 4+ ion occupies the body centered position. This means there are 3 oxygen anions, 1 barium cation and 1 titanium cation. Thus, the stoichiometry of the system is preserved. The center of this structure is shown in Figure 9. Note that the anions and the cations must touch, the question is which set of ions touch. By assuming the O 2- ions touch the Ba 2+ ions, it can be shown that the Ti 4+ ion is slightly too large to fit in the center of the cell. This means that the cell is slightly stretched. If Figure 9: Center of Perovskite Structure one applies a force to the cell, then its dimensions will change in one direction. Assume for example that one pressed on the top and bottom surface of the cell. The z-dimension would then decrease and the Ti 4+ ion would be pushed off-center. This will induce a charge imbalance, or electric dipole in the cell as shown in Figure 7. Thus by imposing a force on the cell, one causes an electric field to be induced. This is how a microphone works. One s voice (or the displacements in the air associated with sound waves) causes a force to be induced on the microphone, this generates an electric field which can be analyzed, manipulated and reproduced. Conversely, an electric field will cause the Ti 4+ ion to be displaced from the center of the cell. This will cause a dimensional change in the unit cell. The positive ion still attracts the negative ions. This is how a speaker works. An electric field causes the crystal to change dimension, which causes vibrations in the membrane of a speaker. These materials are called piezoelectrics.

10 D. Structure of Covalent Network Solids 1. Diamond Cubic Covalent network solid is a carbon (diamond), silicon, germanium, and SiO 2 are covalent network solids. That is the atoms are covalently bonded to one another in such a way that the molecular pattern repeats itself. This is most commonly seen where sp 3 hybridiztion occurs. Therefore the molecular geometry is tetrahedral. As with the Zinc blende (ZnS) Structure. The tetrahedral can be arranged in a cubic structure except that all atoms are identical. The diamond cubic structure is shown in Figure 10. Note there is only one interstitial atom per body diagonal. In the figure each interstitial atom is indicated by a four, and each atom is bonded to one corner atom and three face centered atoms. Figure 10: Diamond Cubic Unit Cell There are some differences between the fcc unit cell and the diamond cubic unit cell. The fcc unit cell has four atoms, the diamond cubic unit cell has eight. There are the four atoms from the fcc lattice and the four additional atoms internally, giving eight total. In addition each atom is bonded to four other atoms. The atoms touch along the body diagonal. This means the relationship between the lattice parameter and the radius of the atom will be different than for a fcc structure. As with ZnS the atoms touch along the body diagonal and this distance is only 1/4 of the length. However since the atoms, and thus their radii are identical the relationship between the lattice parameter and the atomic radius is, a 3 = 2r 4

11 2. Silica Almost all rocks and minerals are silicate compounds, which are compounds based on. Many silicates are important engineering materials. The simplest silicate is silicon dioxide, SiO 2. The Lewis structure of which is shown in Figure 11. Each oxygen ion is bonded to two silicon ions, and each silicon ion to four oxygen ions. The crystal structure is shown in Figure 12. Figure 12 Structure of $-cristobalite, the cubic form of silica. Figure 11: Lewis Structure of SiO 2 E. Crystal Features Within a given unit cell there are lattice positions, lattice directions, and lattice planes. Because, this text is focused on the general engineering student, not the materials engineering student, our focus will be limited to the most important directions and planes in cubic systems. 1. Lattice Positions Lattice positions are taken relative to an origin, usually fixed at the lower, left, rear corner of the unit cell; and measured in terms of the edge length of the cube. Thus, the position one unit in the x-direction from the origin is labeled 100 (note this is not pronounced one-hundred, but one-zero-zero ) and corresponds to the lower, left, front corner. The position which is one unit in the x-direction and one unit in the y-direction is labeled 110. The most important lattice positions for cubic crystal structures are shown in Figure 13. Figure 13: Lattice Positions in a Cubic Unit Cell

12 2. Lattice Directions In describing crystal structures is it useful to have a method of designating directions in the unit cell or crystal lattice. A direction in the crystal lattice is a vector from the origin of a coordinate system to a point in the lattice. A more formal way of defining a directions is: A direction [uvw] is the vector which extends from the origin to the point uvw. The mechanical and electrical properties of a crystal often depend on the crystals orientation. As the distance between atoms will vary from direction to direction, The [100] represents the edge, the vector from the origin to 100. The [110] represents the face diagonal, the vector from the origin to 110. The [111] the body diagonal, the vector from the origin to 111. Figure 14: Important Directions in Cubic Crystal Systems These directions are shown in Figure 14. This is most useful in determining which direction will be strongest in a unit cell. For example, the face diagonal will be the strongest direction in fcc as it is the direction in which the atoms touch. Comparison of Directions in fcc Crystals edge face diagonal body diagonal

13 4. Bulk Properties Density( also referred to as bulk or volume density), D, is defined as mass per unit volume. the density of a unit cell is therefore the mass of the unit cell divided by the volume of the unit cell. These quantities can be calculated for any unit cell from a knowledge of how many atoms are in the cell. To find the number of atoms in the unit cell, one must visualize how much of each atom is in the cell. For example, we can calculate the density of aluminum knowing only its atomic radius, nm, its crystal structure, fcc, and its atomic weight of 27.0 grams/mole. The unit cell contains four atoms, therefore the mass of the unit cell is the mass of four aluminum atoms. mass of the unit cell can be determined from, Since aluminum is fcc,. and The packing factor of a structure is defined as the ratio of the volume of the atoms in the unit cell to the volume of the unit cell itself. In the face-centered cubic structure, there are four atoms, so the volume occupied by the atoms is simply four times the volume of an atom, 4 4BR 3 /3 and the volume r a = 4 2 of the unit cell is a 3. However there is a relationship between a and r is The packing factor is: For the body-centered unit cell, the packing factor is Simple cubic 0.52 and diamond 0.34 For ionic solids this will depend on each system as the r/r will vary from compound to compound.

14 F. Non Cubic Structures The most important thing to remember about non-cubic crystal structures is that they exist and metals or ceramics with these structures are brittle. The most important non-cubic structures for the general engineer to be aware of are the hexagonal-close-packed (hcp) and body-centered-tetragonal (bct). Metals with the hcp structure include the rare-earths, magnesium and zinc. These metals are britlle. Certain steels, when cooled quickly from temperatures above 800 o C, will form a very strong, but brittle material called martensite. This has a bct structure. While a thorough understanding of these structures is necessary for materials engineers and metallurgists, they are beyond the scope of an introductory course for general engineers. G. Summary The arrangement of atoms in solids is the basis for all material properties. In this chapter, we have discussed cubic crystal systems which are the basis of many engineering materials.

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