Chapter 4 Crystalline solids IN A PERFECT crystal, the constituent atoms, ions, or molecules are packed ogether in a regular array (the crystal lattice), the pattern of which is repeated periodically ad infinitum. Thus, regularly repeating planes of atoms are formed. The smallest complete repeating three-dimensional unit is called the unit cell, and the crystallographer's primary objective is to determine the dimensions and geometry of the unit cell, as well as the precise deployment of the atoms within it. -6 4.1 Determination of Crystal Structure Just as the rulings on a diffraction grating create colored interference patterns in the reflected light, so layers of atoms in a crystal give rise to diffraction patterns in incident radiation of the appropriate wavelength-in this case, monochromatic (i.e. single wavelength)X-rays or beams of elec trons or neutrons(which also have wave like properties). X-Ray diffraction is caused by interaction of an incoming X-ray photon with the electron den- sity in the crystal. Since atoms of very low atomic number such as hydrogen have relatively low electron density, they do not show up strongly in X-ray diffraction patterns, and so h atoms in particular are often missing from crystal structures determined by X-ray diffraction. To locate the hydrogens in such cases, we can resort to diffraction of a beam of neutrons of a sin gth of a material proportional to its momentum When X-rays of wavelength A are reflected from parallel planes of atoms of spacing d, they will reinforce one another if rays from successive planes arrive at the detector a distance A apart (or nA, where n is a positive inte ger); otherwise, they will tend to cancel. As Fig. 4.1 shows, rays reflected from successive planes at an angle 8 will each travel 2d sin 8 further than
Chapter 4 Crystalline Solids IN A PERFECT crystal, the constituent atoms, ions, or molecules are packed together in a regular array (the crystal lattice), the pattern of which is repeated periodically ad infinitum. Thus, regularly repeating planes of atoms are formed. The smallest complete repeating three-dimensional unit is called the unit cell, and the crystallographer's primary objective is to determine the dimensions and geometry of the unit cell, as well as the precise deployment of the atoms within it. 1-6 4.1 Determination of Crystal Structure Just as the rulings on a diffraction grating create colored interference patterns in the reflected light, so layers of atoms in a crystal give rise to diffraction patterns in incident radiation of the appropriate wavelength--in this case, monochromatic (i.e., single wavelength) X-rays or beams of electrons or neutrons (which also have wave like properties). X-Ray diffraction is caused by interaction of an incoming X-ray photon with the electron density in the crystal. Since atoms of very low atomic number such as hydrogen have relatively low electron density, they do not show up strongly in X-ray diffraction patterns, and so H atoms in particular are often missing from crystal structures determined by X-ray diffraction. To locate the hydrogens in such cases, we can resort to diffraction of a beam of neutrons of a single, known velocity (since the wavelength of a material particle is inversely proportional to its momentum). When X-rays of wavelength A are reflected from parallel planes of atoms of spacing d, they will reinforce one another if rays from successive planes arrive at the detector a distance ~ apart (or n~, where n is a positive integer); otherwise, they will tend to cancel. As Fig. 4.1 shows, rays reflected from successive planes at an angle 0 will each travel 2dsin 0 further than 69
70 Chapter 4 Crystalline Solids monochromatic lllll Figure 4.1 Diffraction of X-rays by layers of atoms. The path of ray 2 to the detector is longer than the path of ray 1 by 2(d sin 8) (film strip or electronic X-ray counter) monochromatic X powdered crystalline diffracted sample Figure 4.2 Characterization of a powdered solid by its X-ray diffraction pattern their immediate predecessors to reach the detector. Thus, when reinforced X-rays are recorded at the detector, Eq 4. 1(the Bragg equation)must hold and, knowing A and measuring 0, we can obtain d 入=2dsin日 (4.1) If a single crystal is rotated in a monochromatic X-ray beam, a pattern of spots of reinforced X-rays can be recorded, traditionally on a photo graphic film placed behind the crystal perpendicular to the primary beam (giving the so-called Laue photographs). Nowadays, X-ray diffractometers use electronic photon counters as detectors. Since, as noted above differ ent atoms have different X-ray scattering powers, both the positions and
70 Chapter 4 Crystalline Solids monochromatic X-rays layers of atoms 1 i ' 2 i Figure 4.1 Diffraction of X-rays by layers of atoms. The path of ray 2 to the detector is longer than the path of ray 1 by 2(dsin 0). detector (film strip or electronic X-ray counter) monochromatic powdered crystalline diffracted sample rays Figure 4.2 Characterization of a powdered solid by its X-ray diffraction pattern. their immediate predecessors to reach the detector. Thus, when reinforced X-rays are recorded at the detector, Eq. 4.1 (the Bragg equation) must hold, and, knowing A and measuring 0, we can obtain d: nA = 2dsinO. (4.1) If a single crystal is rotated in a monochromatic X-ray beam, a pattern of spots of reinforced X-rays can be recorded, traditionally on a photographic film placed behind the crystal perpendicular to the primary beam (giving the so-called Laue photographs). Nowadays, X-ray diffractometers use electronic photon counters as detectors. Since, as noted above, different atoms have different X-ray scattering powers, both the positions and
4.2 Bonding in Solids 71 the intensities of these spots are important in working out the structure of the crystal in terms of the planes of atoms present. In the case of crys- tals containing molecular units, these molecular structures will also show up in the analysis of the diffraction pattern. With the advent of powerful digital computers, such determinations of structure have become routine in modern research in synthetic chemistry Alternatively, when a powdered crystalline solid diffracts monochromatic X-radiation, the diffraction pattern will be a series of concentric rings rather than spots, because of the random orientation of the crystals in the sample(Fig 4. 2). The structural information in this pattern is limited however, because even solid compounds that have the same structure bi different composition will almost inevitably have different d values, each individual solid chemical compound will have its own characteristic powder diffraction pattern X-Ray powder diffraction patterns are catalogued in the JCPDs data file, and can be used to identify crystalline solids, either as pure phases or mixtures. Again, both the positions and the relative intensities of the features are important in interpretation of powder diffraction patterns, al- hough it should be borne in mind that diffraction peak heights in the read out from the photon counter are somewhat dependent on particle size. For example, a solid deposit accumulating in a heat exchanger can be quickly identified from its X-ray powder diffraction pattern, and its source or mech- anism of formation may be deduced-for instance, is it a corrosion product. (if so, what is it, and where does it come from) or a contaminant introduced with the feedwater? 4.2 Bonding in Solids Bonding in solids takes several forms. Some elements such as carbon or com- pounds such as silica(SiO2 in its various forms--see Section 7.5) can form quasi-infinite networks of covalent bonds, as discussed in Section 3.2; such rystalline solids are typically very high melting (quartz has mp 1610C) On the other hand, small, discrete molecules like dihydrogen(H2)or sulfur (Sa, Section 3.4)interact only weakly with one another through van der Waals forces(owing to electric dipoles induced by the electrons and nuclei of one molecule in the electron cloud of a neighbor and vice versa)and form low melting crystals(H, has mp-259C; ar-S melts at 113C) netal M of low electronegativity (x) combines with a nonmetal X of high x, the product is likely to be a high-melting solid consisting of ions Im+ and X, held together in a regular pattern(the crystal lattice) by electrostatic forces rather than electron-sharing bonding (covalency) The nergy of these electrostatic interactions-called the lattice energy, U- makes formation of the ionic solid possible by compensating for the energy
4.2 Bonding in Solids 71 the intensities of these spots are important in working out the structure of the crystal in terms of the planes of atoms present. In the case of crystals containing molecular units, these molecular structures will also show up in the analysis of the diffraction pattern. With the advent of powerful digital computers, such determinations of structure have become routine in modern research in synthetic chemistry. Alternatively, when a powdered crystalline solid diffracts monochromatic X-radiation, the diffraction pattern will be a series of concentric rings, rather than spots, because of the random orientation of the crystals in the sample (Fig. 4.2). The structural information in this pattern is limited; however, because even solid compounds that have the same structure but different composition will almost inevitably have different d values, each individual solid chemical compound will have its own characteristic powder diffraction pattern. X-Ray powder diffraction patterns are catalogued in the JCPDS data file, 7 and can be used to identify crystalline solids, either as pure phases or as mixtures. Again, both the positions and the relative intensities of the features are important in interpretation of powder diffraction patterns, although it should be borne in mind that diffraction peak heights in the readout from the photon counter are somewhat dependent on particle size. For example, a solid deposit accumulating in a heat exchanger can be quickly identified from its X-ray powder diffraction pattern, and its source or mechanism of formation may be deducedmfor instance, is it a corrosion product (if so, what is it, and where does it come from) or a contaminant introduced with the feedwater? 4.2 Bonding in Solids Bonding in solids takes several forms. Some elements such as carbon or compounds such as silica (SiO2 in its various formsusee Section 7.5) can form quasi-infinite networks of covalent bonds, as discussed in Section 3.2; such crystalline solids are typically very high melting (quartz has mp 1610 ~ On the other hand, small, discrete molecules like dihydrogen (H2) or sulfur ($8, Section 3.4) interact only weakly with one another through van der Waals forces (owing to electric dipoles induced by the electrons and nuclei of one molecule in the electron cloud of a neighbor and vice versa) and form low melting crystals (H2 has mp -259 ~ a-S melts at 113 ~ When a metal M of low electronegativity (X) combines with a nonmetal X of high X, the product is likely to be a high-melting solid consisting of ions M m+ and X x-, held together in a regular pattern (the crystal lattice) by electrostatic forces rather than electron-sharing bonding (covalency). The energy of these electrostatic interactionswcalled the lattice energy, U-- makes formation of the ionic solid possible by compensating for the energy
72 Chapter 4 Crystalline Solids inputs, such as ionization potential needed to form the ions, and is clearly dependent to some degree on the structure of the crystal at the atomic or molecular level(Section 4.7) Bonding in metals involves delocalization of electrons over the whole metal crystal, rather like the T electrons in graphite(Section 3. 2)except that the delocalization, and hence also the high electrical conductivity, is hree dimensional rather than two dimensional. Metallic bonding is best de- scribed in terms of band theory, which is in essence an extension of molecular orbital(MO)theory(widely used to represent bonding in small molecules to arrays of atoms of quasi- infinite extent Molecular orbital theory is explained at length in almost all introduc tory chemistry textbooks, and only a brief summary is given here. As the simplest possible case, we consider interactions between two free hydro- gen atoms, A and B, each with a single electron, the time-average spatial distributions of which are described mathematically as wave functions or orbitals(ls orbitals, if the atoms are in their lowest energy states)oa and dB. When A and B approach one another closely enough to interact, the two atomic orbitals combine mathematically to give two molecular orbitals one(say, MO =da+B, if we assume the combination of atomic orbitals to be linear)lower in energy than the original atomic orbitals ne other (φMo=φA-φg) higher in energy. When the difference in energy between the two molecular orbitals is sufficient to overcome the spin pairing energy of the two electrons, the two electrons will both occupy Mo, with a net nergetic stabilization, At the A-b approach distance where the increas- ing mutual repulsion of the atomic nuclei balances this stabilization, we have a stable H2 molecule. The occupied molecular orbital Mo is there- fore known as a bonding orbital, and oMo (electronic occupation of which would destabilize the system)is known as an antibonding orbital If we now have, say, four H atoms in a row, there would be four molecular orbitals of different energies, two bonding and two antibonding; eight atoms give a stack of eight molecular orbitals; and so on(Fig. 4.3). As the number of participating atoms increases and we move from a one-dimensional row to a three-dimensional array of atoms, the range in energies between the lowest bonding and highest antibonding molecular orbitals levels out to asymptotic limits, giving eventually a band of bonding and antibonding orbitals, very closely spaced in energy. At the absolute zero of temperature only the lower half of this band(the bonding orbitals)would be filled with electrons; the highest occupied energy level is known as the Fermi level, after the Italian born U.S. physicist Enrico Fermi. For all accessible temperatures, a small fraction of the electrons will be excited thermally into energy levels higher than the Fermi level, leaving some depleted levels below it Because the band is partially filled and extends throughout the cryst electrons can move freely through it, with the number flowing in any
72 Chapter 4 Crystalline Solids inputs, such as ionization potential needed to form the ions, and is clearly dependent to some degree on the structure of the crystal at the atomic or molecular level (Section 4.7). Bonding in metals involves delocalization of electrons over the whole metal crystal, rather like the 7r electrons in graphite (Section 3.2) except that the delocalization, and hence also the high electrical conductivity, is three dimensional rather than two dimensional. Metallic bonding is best described in terms of band theory, which is in essence an extension of molecular orbital (MO) theory (widely used to represent bonding in small molecules) to arrays of atoms of quasi-infinite extent. Molecular orbital theory is explained at length in almost all introductory chemistry textbooks, and only a brief summary is given here. As the simplest possible case, we consider interactions between two free hydrogen atoms, A and B, each with a single electron, the time-average spatial distributions of which are described mathematically as wave functions or orbitals (ls orbitals, if the atoms are in their lowest energy states) CA and CS. When A and B approach one another closely enough to interact, the two atomic orbitals combine mathematically to give two molecular orbitals, one (say, CMO = CA + CB, if we assume the combination of atomic orbitals to be linear) lower in energy than the original atomic orbitals and the other (r -- CA- CB) higher in energy. When the difference in energy between the two molecular orbitals is sufficient to overcome the spin pairing energy of the two electrons, the two electrons will both occupy cPMO, with a net energetic stabilization. At the A- B approach distance where the increasing mutual repulsion of the atomic nuclei balances this stabilization, we have a stable H2 molecule. The occupied molecular orbital CMO is therefore known as a bonding orbital, and CMO (electronic occupation of which would destabilize the system) is known as an antibonding orbital. If we now have, say, four H atoms in a row, there would be four molecular orbitals of different energies, two bonding and two antibonding; eight atoms give a stack of eight molecular orbitals; and so on (Fig. 4.3). As the number of participating atoms increases and we move from a one-dimensional row to a three-dimensional array of atoms, the range in energies between the lowest bonding and highest antibonding molecular orbitals levels out to asymptotic limits, giving eventually a band of bonding and antibonding orbitals, very closely spaced in energy. At the absolute zero of temperature, only the lower half of this band (the bonding orbitals) would be filled with electrons; the highest occupied energy level is known as the Fermi level, after the Italianborn U. S. physicist Enrico Fermi. For all accessible temperatures, a small fraction of the electrons will be excited thermally into energy levels higher than the Fermi level, leaving some depleted levels below it. Because the band is partially filled and extends throughout the crystal, electrons can move freely through it, with the number flowing in any one
4.2 Bonding in Solids 73 energy (a)(b)(c)(d)(e)(f) Figure 4.3 Formation of electronic bands in a hypothetical array of hydrogen atoms.(a)Two H atoms, infinitely far apart. (b) Two H atoms interacting(as in the actual H, molecule).(c)Four, (d)eight, and(e) a very large number of H atoms interacting. The dots or (in the band) shading represent the occu pancy of the energy levels by the electrons, in the absence of thermal excitation (f) Percentage distribution of the electron population in a band at a nonzero direction being ordinarily balanced by an equal number coming the other way. If, however, an electric potential is applied across the solid, the band energy levels will be depressed in energy near the positive connection, while near the negative end they will be elevated. Consequently, there will be a net How of electrons through the band to occupy the lower energy levels preferentially. This amounts to conduction of electricity through a partly filled electronic band and is a characteristic property of metals. The free How of electrons is limited by scattering by the atoms, which are in effect present as cations. As the temperature increases, the amplitudes of the vibrations of the metal ions about their mean positions in the lattice in- crease, resulting in increased scattering of the electrons as they flow; thus, che electrical conductivity of a metal decreases with increasing temperature In our illustrative example, hydrogen was chosen for v. even though it is well known that solid hydrogen ordinarily consists of an array of discrete H2 molecules as in Fig. 4.3b and is therefore not normally a con ductor of electricity. It is expected, however, that pressures on the order of several hundred gigapascals will force the H atoms in solid hydrogen into sufficiently close proximity that a band structure will form, as in Fig. 4. 3f- exists in the core of the planet Jupiter, which is composed largely of bvd ly in other words, hydrogen will metallize. Such metallic hydrogen prol en. Demonstration of such metallization of hydrogen in the laboratory, however, has proved elusive(see Section 2.4) or hydrogen, only the 1 s orbital is energetically accessible for band for mation. For elements of lithium through fluorine, the 2s and, at somewhat higher energy, the three 2p orbitals are available, and, depending on the ways in which the atomic orbitals align with the crystal structure, these may form either a continuous s, p band or a pair of bands with the same
4.2 Bonding in Solids 73 Figure 4.3 Formation of electronic bands in a hypothetical array of hydrogen atoms. (a) Two H atoms, infinitely far apart. (b) Two H atoms interacting (as in the actual H2 molecule). (c) Four, (d) eight, and (e) a very large number of H atoms interacting. The dots or (in the band) shading represent the occupancy of the energy levels by the electrons, in the absence of thermal excitation. (f) Percentage distribution of the electron population in a band at a nonzero temperature. direction being ordinarily balanced by an equal number coming the other way. If, however, an electric potential is applied across the solid, the band energy levels will be depressed in energy near the positive connection, while near the negative end they will be elevated. Consequently, there will be a net flow of electrons through the band to occupy the lower energy levels preferentially. This amounts to conduction of electricity through a partly filled electronic band and is a characteristic property of metals. The free flow of electrons is limited by scattering by the atoms, which are in effect present as cations. As the temperature increases, the amplitudes of the vibrations of the metal ions about their mean positions in the lattice increase, resulting in increased scattering of the electrons as they flow; thus, the electrical conductivity of a metal decreases with increasing temperature. In our illustrative example, hydrogen was chosen for simplicity, even though it is well known that solid hydrogen ordinarily consists of an array of discrete H2 molecules as in Fig. 4.3b and is therefore not normally a conductor of electricity. It is expected, however, that pressures on the order of several hundred gigapascals will force the H atoms in solid hydrogen into sumciently close proximity that a band structure will form, as in Fig. 4.3fin other words, hydrogen will metallize. Such metallic hydrogen probably exists in the core of the planet Jupiter, which is composed largely of hydrogen. Demonstration of such metallization of hydrogen in the laboratory, however, has proved elusive (see Section 2.4). For hydrogen, only the ls orbital is energetically accessible for band formation. For elements of lithium through fluorine, the 2s and, at somewhat higher energy, the three 2p orbitals are available, and, depending on the ways in which the atomic orbitals align with the crystal structure, these may form either a continuous s, p band or a pair of bands with the same
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
78 Chapter 4 Crystalline Solid kind of metal partially replace those of another in the normal lattice sites and interstitial compounds, in which the intruding atoms(usually H, B, c, or N)reside in some, but usually not all, of the interstices(O-or T-holes) the lattice of a metal. For example, copper and gold form of stitutional alloys with the ccp structure, as expected from Table 4.1. The Cu and Au positions in the 1: l alloy CuAu can be ordered(with alternat ing layers of Cu and Au parallel to the foor of the unit cell of Fig. 4.7)or disordered(with random placement of Au and Cu throughout the lattice n the alloy Cu, Au, when ordered the Cu atoms occupy all the face-center sites and the au all the corners of the cubic unit cell but disordered struc tures are again possible. Interstitial compounds are not alloys in the usual sense of a combination of metals, but they do retain a metallic appearance and electrical conductivity. They are considered at length in Chapter 5 4.3.2 Metallic Glasses Metals are not necessarily crystalline, and there is much interest currently in the preparation and properties of metallic glasses. Glasses in general are solids that lack long-range structure at the atomic level; often, they are described as extremely viscous, supercooled liquids(i.e, liquids far below their normal freezing temperatures), but there is no evidence that they have any more tendency to flow than do crystalline solids. They are however, thermodynamically unstable with respect to slow crystallization devitrification; see Section 7.5). Metallic glasses are usually made by melt- spinning certain alloys at forced cooling rates high enough(10 to 106 K s-)to avert crystallization, but they can also be produced by vapor deposition or chemical precipitation. In the absence of any opportunity for crystal slip mechanisms such as characterize crystalline solids, metallic glasses often have very high tensile strength and wear resistance. Their properties are typically metallic, but, as might be inferred from Section 4.2, be relatively low. Examples of compositions of glass-forming alloys are Fe100--B2(a= 12-25), Al100-zLa(a =10 50-80), U100-=Cor (a 24-40), and Be4o Zr1oTi5o Paradoxically, one of the most technologically promising properties of etallic glasses is the partial recrystallization under controlled annealing to form metals with extremely fine, uniform, microcrystalline structures Such devitrification is extensively practiced with oxide glasses to produce tough microcrystalline ceramics(Section 7.5), and we now have the prospect of producing devitrified metal glasses with extraordinary mechanical prop- erties. For example, samples of devitrified Algs Nig Ce] Fe have been pre- ared with tensile strengths as high as 1560 MPa without brittleness, and devitrified Fe14Nd2B has found application as a very hard magnetic mate-
78 Chapter 4 Crystalline Solids kind of metal partially replace those of another in the normal lattice sites, and interstitial compounds, in which the intruding atoms (usually H, B, C, or N) reside in some, but usually not all, of the interstices (O- or T-holes) in the lattice of a metal. For example, copper and gold form a range of substitutional alloys with the ccp structure, as expected from Table 4.1. The Cu and Au positions in the 1:1 alloy CuAu can be ordered (with alternating layers of Cu and Au parallel to the floor of the unit cell of Fig. 4.7) or disordered (with random placement of Au and Cu throughout the lattice). In the alloy Cu3Au, when ordered, the Cu atoms occupy all the face-center sites and the Au all the corners of the cubic unit cell, but disordered structures are again possible. Interstitial compounds are not alloys in the usual sense of a combination of metals, but they do retain a metallic appearance and electrical conductivity. They are considered at length in Chapter 5. 4.3.2 Metallic Glasses Metals are not necessarily crystalline, and there is much interest currently in the preparation and properties of metallic glasses, s Glasses in general are solids that lack long-range structure at the atomic level; often, they are described as extremely viscous, supercooled liquids (i.e., liquids far below their normal freezing temperatures), but there is no evidence that they have any more tendency to flow than do crystalline solids. They are, however, thermodynamically unstable with respect to slow crystallization (devitrification; see Section 7.5). Metallic glasses are usually made by meltspinning certain alloys at forced cooling rates high enough (105 to 106 K s -1) to avert crystallization, but they can also be produced by vapor deposition or chemical precipitation. In the absence of any opportunity for crystal slip mechanisms such as characterize crystalline solids, metallic glasses often have very high tensile strength and wear resistance. Their properties are typically metallic, but, as might be inferred from Section 4.2, their electrical resistivities can be relatively low. Examples of compositions of glass-forming alloys are Fel00-xBx (x = 12-25), All00_xLa~ (x = 10, 50-80), Ulo0-~Co~ (x = 24-40), and Be40Zrl0Tis0. Paradoxically, one of the most technologically promising properties of metallic glasses is the partial recrystallization under controlled annealing to form metals with extremely fine, uniform, microcrystalline structures, s Such devitrification is extensively practiced with oxide glasses to produce tough microcrystalline ceramics (Section 7.5), and we now have the prospect of producing devitrified metal glasses with extraordinary mechanical properties. For example, samples of devitrified AlssNi9Ce2Fe have been prepared with tensile strengths as high as 1560 MPa without brittleness, and devitrified Fe14Nd2B has found application as a very hard magnetic material