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