74 Chapter 4 Crystalline Solid 2p个个 conduction band band structure of diamond Figure 4.4 Band structure associated with the diamond structure in Group 14 number of energy levels in each and with an energy gap(band gap) between the bands. The latter is the case for carbon with the diamond structure ig. 4.4); since four electrons are available per C atom, they fill the lower nd (the valence band) completely, so that electronic conduction within this band is not possible, while the upper band (the conduction band) con- ains no electrons. The fermi level is located midway between the two bands. For diamond, the band gap is wide enough to prevent any popula tion of the conduction band from the valence band by thermal excitation of electrons from the latter, and so diamond is an electrical insulator. As we descend the periodic table to Si and Ge. however, the band gap narrows and a small fraction of the electrons in the valence band can be thermally excited into the otherwise empty conduction band. giving rise to a lim ited degree of electrical conductivity that increases with rising temperature (the opposite of the temperature dependence of electrical conductivity in metals). Such materials are known as intrinsic semiconductors In certain solids such as titanium dioxide or cadmium sulfide. the en- ergy of the band gap corresponds to that. of light (visible, ultraviolet,or infrared ), with the result that the solid, when illuminated, may become elec- trically conducting or acquire potent chemical redox characteristics because of the promotion of electrons to the conduction band (which is normall unoccupied). These properties have obvious practical significance and are considered at length in Chapter 19 4.3 The Close Packing Concept Figure 4.5 shows the manner of close packing of identical atoms--assumed to be spheres of equal radii-in a single plane, A. If a second layer B of the same atoms is close packed on top of layer A(Fig. 4.6), it will be seen that each B atom rests on three a atoms that are in mutual contact. so enclosing a void. The centers of the four atoms describe a regular tetra- hedron about the void, which is therefore called a tetrahedral interstice or T-hole. A second kind of interstice, bounded by six atoms(three fron
74 Chapter 4 Crystalline Solids Figure 4.4 Band structure associated with the diamond structure in Group 14 elements. number of energy levels in each and with an energy gap (band gap) between the bands. The latter is the case for carbon with the diamond structure (Fig. 4.4); since four electrons are available per C atom, they fill the lower band (the valence band) completely, so that electronic conduction within this band is not possible, while the upper band (the conduction band) contains no electrons. The Fermi level is located midway between the two bands. For diamond, the band gap is wide enough to prevent any population of the conduction band from the valence band by thermal excitation of electrons from the latter, and so diamond is an electrical insulator. As we descend the periodic table to Si and Ge, however, the band gap narrows, and a small fraction of the electrons in the valence band can be thermally excited into the otherwise empty conduction band, giving rise to a limited degree of electrical conductivity that increases with rising temperature (the opposite of the temperature dependence of electrical conductivity in metals). Such materials are known as intrinsic semiconductors. In certain solids such as titanium dioxide or cadmium sulfide, the energy of the band gap corresponds to that of light (visible, ultraviolet, or infrared), with the result that the solid, when illuminated~ may become electrically conducting or acquire potent chemical redox characteristics because of the promotion of electrons to the conduction band (which is normally unoccupied). These properties have obvious practical significance and are considered at length in Chapter 19. 4.3 The Close Packing Concept Figure 4.5 shows the manner of close packing of identical atoms--assumed to be spheres of equal radii--in a single plane, A. If a second layer B of the same atoms is close packed on top of layer A (Fig. 4.6), it will be seen that each B atom rests on three A atoms that are in mutual contact, so enclosing a void. The centers of the four atoms describe a regular tetrahedron about the void, which is therefore called a tetrahedral interstice or T-hole. A second kind of interstice, bounded by six atoms (three from each
4.3 The Close Packing Concept 75 Figure 4.5 Close packing of spheres in a single layer. B Figure 4.6 Placement of a layer B of close-packed spheres on top of a layer A generating octahedral(O) and tetrahedral(T) interstices between the layers layer), is also generated, and these are called octahedral interstices or O holes(Fig. 4.6). A different(smaller) kind of atom could be accommodated in the T-holes between the a and B layers. As we shall see in Section 4.4, the crystal structures of many ionic compounds can be represented in terms of the systematic filling of O-and /or T-holes in a close-packed array of ions X by smaller ions M(usually cations). The X sublattice may be somewhat expanded from close packed to accommodate M, but the point is that the essential geometric features of closest packing are frequently present in ionic crystal structures When we add a third layer C of atoms on top of the two layers, we find there are two close packed possibilities; this can be tested with small disks on Fig. 4.6. Each atom of layer C must rest on three of layer B. One possibility is to place atoms of layer C directly above atoms of layer A Thus, we create a new layer just like a, and further layers are added to
4.3 The Close Packing Concept 75 Figure 4.5 Close packing of spheres in a single layer. Figure 4.6 Placement of a layer B of close-packed spheres on top of a layer A, generating octahedral (O) and tetrahedral (T) interstices between the layers. layer), is also generated, and these are called octahedral interstices or Oholes (Fig. 4.6). A different (smaller) kind of atom could be accommodated in the T-holes between the A and B layers. As we shall see in Section 4.4, the crystal structures of many ionic compounds can be represented in terms of the systematic filling of O- and/or T-holes in a close-packed array of ions X by smaller ions M (usually cations). The X sublattice may be somewhat expanded from close packed to accommodate M, but the point is that the essential geometric features of closest packing are frequently present in ionic crystal structures. When we add a third layer C of atoms on top of the two layers, we find there are two close packed possibilities; this can be tested with small disks on Fig. 4.6. Each atom of layer C must rest on three of layer B. One possibility is to place atoms of layer C directly above atoms of layer A. Thus, we create a new layer just like A, and further layers are added to
76 Chapter 4 Crystalline Solids Figure 4.7 Cubic close-packed layers A, B, and C within a face-centered cubic unit cell give a sequence ABABAB.,. This is known as hexagonal close packing (hcp). If, however, we place the C atoms in positions directly above the octahedral holes that exist between A and b(such as the one marked"O n Fig 4.6), we have a new arrangement. The fourth layer would go above the a atoms(if the same packing sequence is adhered to), so the layer order would be ABCABC.. This is called cubic close packing(ccp) because, as Fig. 4.7 shows, it generates a unit cell that has cubic symmetry. More specifically, it is a face-centered cubic(fcc)unit cell, so called because there is an atom at the center of each face of a cube in addition to one at every corner. In contrast, a simple cubic unit cell is one in which only the corner atoms are present. Simple cubic unit cells, however, are rarely encountered n practice 4.3.1 Structures of metals The concept of close packing is particularly useful in describing the crystal structures of metals, most of which fall into one of three classes: hexagonal close packed, cubic close packed (i.e, fcc), and body-centered cubic(bcc) The bcc unit cell is shown in Fig. 4.8; its structure is not close packed. The stablest structures of metals under ambient conditions are summarized in Table 4.1, Notable omissions from Table 4.1. such as aluminum, tin, and manganese, reflect structures that are not so conveniently classified. The artificially produced radioactive element americium is interesting in that the close-packed sequence is ABAC., while one form of polonium has
76 Chapter 4 Crystalline Solids Figure 4.7 Cubic close-packed layers A, B, and C within a face-centered cubic unit cell. give a sequence ABABAB .... This is known as hexagonal close packing (hcp). If, however, we place the C atoms in positions directly above the octahedral holes that exist between A and B (such as the one marked "0" in Fig. 4.6), we have a new arrangement. The fourth layer would go above the A atoms (if the same packing sequence is adhered to), so the layer order would be ABCABC .... This is called cubic close packing (ccp) because, as Fig. 4.7 shows, it generates a unit cell that has cubic symmetry. More specifically, it is a face-centered cubic (fcc) unit cell, so called because there is an atom at the center of each face of a cube in addition to one at every corner. In contrast, a simple cubic unit cell is one in which only the corner atoms are present. Simple cubic unit cells, however, are rarely encountered in practice. 4.3.1 Structures of Metals The concept of close packing is particularly useful in describing the crystal structures of metals, most of which fall into one of three classes: hexagonal close packed, cubic close packed (i.e., fcc), and body-centered cubic (bcc). The bcc unit cell is shown in Fig. 4.8; its structure is not close packed. The stablest structures of metals under ambient conditions are summarized in Table 4.1. Notable omissions from Table 4.1, such as aluminum, tin, and manganese, reflect structures that are not so conveniently classified. The artificially produced radioactive element americium is interesting in that the close-packed sequence is ABAC..., while one form of polonium has
4.3 The Close Packing Concept 7 TABLE 4.1 Structures of Some Metallic Elements at Ambient Temperature and Pressure Body-centered cubic Cubic close packed Hexagonal close packed alkali metals V. Nb. Ta Cr、Mo.W Ni. Pd. Pt Te Re U, Np Pb. Th most lanthanides α0C Figure 4.8 The body-centered cubic(bcc)unit cell the rare simple cubic structure. Cobalt has an essentially close-packed structure, but the layer sequence is not regular Metals are often polymorphic, that is, they may exhibit alternative struc tures, particularly at other temperatures and pressures. An important ex- ample is iron, which has a body-centered structure(a-iron) at room tem- perature, but, on heating, goes over to a face-centered cubic form (y-iron) at 906C and returns to another body-centered cubic structure(8-iron) bove 1401 C. The usual expectation, though, is that close-packed struc- tures will be favored by low temperatures and body-centered by high, since the increased lattice vibrations at high temperatures will work against close cking. The polymorphism of tin can be exasperating in cold climates The familiar "white tin"(B-Sn)has a dense, complicated structure that slowly goes over to "gray tin"(a-Sn)with the more open diamond struc ture(Fig 3.1)on prolonged exposure to temperatures significantly below the transition temperature of 14.2 C. Tin represents one of the relatively rare cases in which the low temperature form is the less dense(by 21%) The effect is to make tin sheets that have been exposed to the cold for extended periods appear to have contracted a terrible skin disease Alloys are metals made by combining two or more elements. Two struc- tural types may be identified: substitutional alloys, in which atoms of one
4.3 The Close Packing Concept 77 TABLE 4.1 Structures of Some Metallic Elements at Ambient Temperature and Pressure Body-centered cubic Cubic close packed Hexagonal close packed alkali metals Cu, Ag, Au Be, Mg V, Nb, Ta Rh, Ir Zn, Cd, In Cr, Mo, W Ni, Pd, Pt Tc, Re, Ru, Os U, Np Pb, Th most lanthanides Figure 4.8 The body-centered cubic (bcc) unit cell. the rare simple cubic structure. Cobalt has an essentially close-packed structure, but the layer sequence is not regular. Metals are often polymorphic, that is, they may exhibit alternative structures, particularly at other temperatures and pressures. An important example is iron, which has a body-centered structure (a-iron) at room temperature, but, on heating, goes over to a face-centered cubic form (v-iron) at 906~ and returns to another body-centered cubic structure (5-iron) above 1401 ~ The usual expectation, though, is that close-packed structures will be favored by low temperatures and body-centered by high, since the increased lattice vibrations at high temperatures will work against close packing. The polymorphism of tin can be exasperating in cold climates. The familiar "white tin" (/%Sn) has a dense, complicated structure that slowly goes over to "gray tin" (a-Sn) with the more open diamond structure (Fig. 3.1) on prolonged exposure to temperatures significantly below the transition temperature of 14.2 ~ Tin represents one of the relatively rare cases in which the low temperature form is the less dense (by 21%). The effect is to make tin sheets that have been exposed to the cold for extended periods appear to have contracted a terrible skin disease. Alloys are metals made by combining two or more elements. Two structural types may be identified: substitutional alloys, in which atoms of one