Note that there are actually seven different lattice systems, some of which have more than one type of lattice, for a total of 14 different types of unit cells. We leave the more complicated geometries for later in this module. Some metals crystallize in an arrangement that has a cubic unit cell with atoms at all of the corners and an atom in the center, as shown in Figure 6. This is called a body-centered cubic BCC solid. Atoms in the corners of a BCC unit cell do not contact each other but contact the atom in the center.
Any atom in this structure touches four atoms in the layer above it and four atoms in the layer below it.
Thus, an atom in a BCC structure has a coordination number of eight. Elements or compounds that crystallize with the same structure are said to be isomorphous. Many other metals, such as aluminum, copper, and lead, crystallize in an arrangement that has a cubic unit cell with atoms at all of the corners and at the centers of each face, as illustrated in Figure 7.
This arrangement is called a face-centered cubic FCC solid. The atoms at the corners touch the atoms in the centers of the adjacent faces along the face diagonals of the cube.
Because the atoms are on identical lattice points, they have identical environments. This structure is also called cubic closest packing CCP. In CCP, there are three repeating layers of hexagonally arranged atoms.
Each atom contacts six atoms in its own layer, three in the layer above, and three in the layer below. In this arrangement, each atom touches 12 near neighbors, and therefore has a coordination number of The fact that FCC and CCP arrangements are equivalent may not be immediately obvious, but why they are actually the same structure is illustrated in Figure 8.
Because closer packing maximizes the overall attractions between atoms and minimizes the total intermolecular energy, the atoms in most metals pack in this manner.
We find two types of closest packing in simple metallic crystalline structures: CCP, which we have already encountered, and hexagonal closest packing HCP shown in Figure 9. Both consist of repeating layers of hexagonally arranged atoms. In both types, a second layer B is placed on the first layer A so that each atom in the second layer is in contact with three atoms in the first layer.
The third layer is positioned in one of two ways. In HCP, atoms in the third layer are directly above atoms in the first layer i. In CCP, atoms in the third layer are not above atoms in either of the first two layers i. About two—thirds of all metals crystallize in closest-packed arrays with coordination numbers of The edge length of its unit cell is Two adjacent edges and the diagonal of the face form a right triangle, with the length of each side equal to The density of calcium can be found by determining the density of its unit cell: for example, the mass contained within a unit cell divided by the volume of the unit cell.
The edge length of its unit cell is pm. The axes are defined as being the lengths between points in the space lattice. Consequently, unit cell axes join points with identical environments. There are seven different lattice systems, some of which have more than one type of lattice, for a total of fourteen different unit cells, which have the shapes shown in Figure Ionic crystals consist of two or more different kinds of ions that usually have different sizes.
The packing of these ions into a crystal structure is more complex than the packing of metal atoms that are the same size. Most monatomic ions behave as charged spheres, and their attraction for ions of opposite charge is the same in every direction. Consequently, stable structures for ionic compounds result 1 when ions of one charge are surrounded by as many ions as possible of the opposite charge and 2 when the cations and anions are in contact with each other.
Structures are determined by two principal factors: the relative sizes of the ions and the ratio of the numbers of positive and negative ions in the compound. In simple ionic structures, we usually find the anions, which are normally larger than the cations, arranged in a closest-packed array.
As seen previously, additional electrons attracted to the same nucleus make anions larger and fewer electrons attracted to the same nucleus make cations smaller when compared to the atoms from which they are formed. The smaller cations commonly occupy one of two types of holes or interstices remaining between the anions. The smaller of the holes is found between three anions in one plane and one anion in an adjacent plane.
The four anions surrounding this hole are arranged at the corners of a tetrahedron, so the hole is called a tetrahedral hole. The larger type of hole is found at the center of six anions three in one layer and three in an adjacent layer located at the corners of an octahedron; this is called an octahedral hole.
Figure 12 illustrates both of these types of holes. Depending on the relative sizes of the cations and anions, the cations of an ionic compound may occupy tetrahedral or octahedral holes, as illustrated in Figure Relatively small cations occupy tetrahedral holes, and larger cations occupy octahedral holes.
If the cations are too large to fit into the octahedral holes, the anions may adopt a more open structure, such as a simple cubic array. The larger cations can then occupy the larger cubic holes made possible by the more open spacing. A compound that crystallizes in a closest-packed array of anions with cations in the tetrahedral holes can have a maximum cation:anion ratio of ; all of the tetrahedral holes are filled at this ratio.
Compounds with a ratio of less than may also crystallize in a closest-packed array of anions with cations in the tetrahedral holes, if the ionic sizes fit. In these compounds, however, some of the tetrahedral holes remain vacant. Occupancy of Tetrahedral Holes Zinc sulfide is an important industrial source of zinc and is also used as a white pigment in paint. Zinc sulfide crystallizes with zinc ions occupying one-half of the tetrahedral holes in a closest-packed array of sulfide ions.
What is the formula of zinc sulfide? Thus, the formula is ZnS. Check Your Learning Lithium selenide can be described as a closest-packed array of selenide ions with lithium ions in all of the tetrahedral holes.
What it the formula of lithium selenide? Thus, compounds with cations in octahedral holes in a closest-packed array of anions can have a maximum cation:anion ratio of Ratios of less than are observed when some of the octahedral holes remain empty. Stoichiometry of Ionic Compounds Sapphire is aluminum oxide. If we choose the first arrangement and repeat the pattern in succeeding layers, the positions of the atoms alternate from layer to layer in the pattern ABABAB…, resulting in a hexagonal close-packed hcp structure part a in Figure If we choose the second arrangement and repeat the pattern indefinitely, the positions of the atoms alternate as ABCABC…, giving a cubic close-packed ccp structure part b in Figure Because the ccp structure contains hexagonally packed layers, it does not look particularly cubic.
The hcp and ccp structures differ only in the way their layers are stacked. The illustrations in a show an exploded view, a side view, and a top view of the hcp structure.
The simple hexagonal unit cell is outlined in the side and top views. Note the similarity to the hexagonal unit cell shown in Figure The ccp structure in b is shown in an exploded view, a side view, and a rotated view.
The rotated view emphasizes the fcc nature of the unit cell outlined. The line that connects the atoms in the first and fourth layers of the ccp structure is the body diagonal of the cube. Table Most metals have hcp, ccp, or bcc structures, although several metals exhibit both hcp and ccp structures, depending on temperature and pressure. The smallest repeating unit of a crystal lattice is the unit cell. The simple cubic unit cell contains only eight atoms, molecules, or ions at the corners of a cube.
A body-centered cubic bcc unit cell contains one additional component in the center of the cube. A face-centered cubic fcc unit cell contains a component in the center of each face in addition to those at the corners of the cube. The simple cubic and bcc lattices have coordination numbers of 6 and 8, respectively. A crystalline solid can be represented by its unit cell, which is the smallest identical unit that when stacked together produces the characteristic three-dimensional structure.
Why is it valid to represent the structure of a crystalline solid by the structure of its unit cell? What are the most important constraints in selecting a unit cell? All unit cell structures have six sides. Can crystals of a solid have more than six sides? Explain your answer. Explain how the intensive properties of a material are reflected in the unit cell. Are all the properties of a bulk material the same as those of its unit cell?
The experimentally measured density of a bulk material is slightly higher than expected based on the structure of the pure material. Propose two explanations for this observation. The experimentally determined density of a material is lower than expected based on the arrangement of the atoms in the unit cell, the formula mass, and the size of the atoms. What conclusion s can you draw about the material?
Only one element polonium crystallizes with a simple cubic unit cell. Why is polonium the only example of an element with this structure? What is meant by the term coordination number in the structure of a solid? How does the coordination number depend on the structure of the metal? Arrange the three types of cubic unit cells in order of increasing packing efficiency. What is the difference in packing efficiency between the hcp structure and the ccp structure?
The structures of many metals depend on pressure and temperature. Which structure—bcc or hcp—would be more likely in a given metal at very high pressures?
Explain your reasoning. A metal has two crystalline phases. Sketch a phase diagram for this substance. As one example, the cubic crystal system is composed of three different types of unit cells: 1 simple cubic , 2 face-centered cubic , and 3 body-centered cubic. These are shown in three different ways in the Figure below. Three unit cells of the cubic crystal system. Each sphere represents an atom or an ion. In the simple cubic system, the atoms or ions are at the corners of the unit cell only.
In the face-centered unit cell, there are also atoms or ions in the center of each of the six faces of the unit cell. In the body-centered unit cell, there is one atom or ion in the center of the unit cell in addition to the corner atoms or ions. Pay special attention to the last diagram for each type of cell. You will notice that the atoms or ions at the edges of each face or at the corners are not complete spheres.
The atom in the face is shared with the adjacent cell. FCC unit cells consist of four atoms, eight eighths at the corners and six halves in the faces. Table 1 shows the stable room temperature crystal structures for several elemental metals.
As atoms of melted metal begin to pack together to form a crystal lattice at the freezing point, groups of these atoms form tiny crystals. These tiny crystals increase in size by the progressive addition of atoms. The resulting solid is not one crystal but actually many smaller crystals, called grains. These grains grow until they impinge upon adjacent growing crystals. The interface formed between them is called a grain boundary. Grains are sometimes large enough to be visible under an ordinary light microscope or even to the unaided eye.
The spangles that are seen on newly galvanized metals are grains. See A Particle Model of Metals Activity Figure 5 shows a typical view of a metal surface with many grains, or crystals.
Figure 5: Grains and Grain Boundaries for a Metal. Crystal Defects: Metallic crystals are not perfect. Sometimes there are empty spaces called vacancies, where an atom is missing.
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