Structure, Bonding, and Properties
1.5. STRUCTURE AND CHEMICAL BONDING
A phase is usually characterized by its chemical composition and crystal structure. A phase transition means that the crystallographic structure of the material is transformed to another one while the chemical composition is kept the same. A precise description of the phase transformation is the crystallographic system and the atom arrangement in the unit cell, and the stability of the transformed phase depends on its electronic structure (e.g., bonding).
Properties of materials are determined not only by the arrangement of atoms in the lattice but, more importantly, by the electronic structure of the valence electrons. The charge distribution between adjacent atoms determines the bonding, which depends on the symmetry of atom distribution and the electronic structure of the atoms. In general, the electronic structure and crystal structure are closely related and they are constrained with each other. The bonding in crystalline materials is generally classified as metallic, ionic, and covalent. In this section, the relationship between crystal structure and bonding is illustrated.
1.5.1. BONDING AND ION RADIUS
The electrons in a condensed matter occupy the energy states determined by the band structure of the solid. These states are expressed mathematically by the eigenvalues of the wave function '1'. The wave functions result theoretically as solutions of the Schriidinger equation for the complete set of all constituent atoms. Although the exact mathematical solution of this equation is quite difficult, we do have a well-founded knowledge about wave functions and thus about electrons in atomic systems. The knowledge is based on good mathematical approximations and experimental data.
Chemical bonding elucidates how atoms are accumulated by electrons and how an atom interacts with its neighbors.
When the wave function of an atom overlaps with those of its adjacent atoms, the electrons localized on the corresponding atoms strongly interact, resulting in a part of shared electrons between the atoms. It is called covalent bonding. The localized covalent bond is distinguished by its short range of action, which usually extends only from one atom to the adjacent one. Within this range it is strongly bonded due to shared electrons.
Depending on the tight interatomic bonds and the mutual repulsion of the valence electrons and on the space requirement of the bonded atoms, a certain symmetric order arises around an atom. When atoms are linked to assemble a crystal structure, the short- range order can result in a long-range order in a similar way. Therefore, in a nonpolymer molecule or molecular ion, a limited number of atoms is only linked by covalent bonds.
The covalent forces within the molecule are considerably stronger than all forces acting outward. Many properties of this kind of compound can be explained from the molecular structure without significant error. This is not equally valid for macromolecular compounds, in which a molecule consists of a nearly unlimited number of atoms.
Interactions with surrounding atoms and/or molecules cannot be neglected. Crystalline macromolecular substances are classified according to the kinds of conductivity of the covalently linked atoms as chain structures, namely layered structures, and three- dimensional network structures. The chain or layer may be electrically uncharged molecules that interact with each other only by van der Waals forces, or they can be polyanions or polycations combined together by intervening counterions. Network structures can also be modified with the counterions occupying cavities in the network.
The structure of the chain, layer, or network depends to a large extent on the covalent bonds and the resulting short-range order around each atom. Synthesizing new macromolecules and crystallites is an important field for developing functional materials, but this is beyond the scope of this book. We will use similar ideas to understand the structure evolution of inorganic compounds.
When an atom loses or gains an electron or several electrons in its outer shells to become ions to be assembled to form a crystal via the interaction of opposite charges, it is called ionic bonding. The crystal structures of ionic compounds with small molecular ions depend mainly on how space can be filled most effectively by ions, following a principle of cations around anions and anions around cations. Geometric factors such as the relative sizes of the ions and the shape of molecular ions are dominant.
Pure covalent and ionic bonding are two extreme cases. Most inorganic compounds have mixture of covalent and ionic bonding with a high fraction of ionic bonding. In contrast, most organic compounds are dominated by covalent bonding. Chemical bonding in a solid depends on the atoms, the lattice, the coordination situation, the valence value, and the environment of the crystal atoms.
Atom size is closely related to the chemical bonding of a solid. The size of an atom means that the electron density in an atom decreases asymptotically toward zero with increasing distance from the atomic center. An atom, in general, has no definite size.
When two atoms approach each other, interaction forces between them become more and more effective. Two types of forces have to find a balance condition: (1) attractive forces--electronic interactions of bonding molecular orbital; and electrostatic forces between the charges of ions or the partial charges of atoms having opposite signs. (2) repulsive forces-electrostatic forces between ions or partially charged atoms having charges of the same sign; the interpenetration of closed electron shells of atoms (resulting in antibonding states) and the electrostatic repulsion between their atomic nuclei when the atoms are too close to each another.
The effectiveness of these forces differs and they change with a different degree as a function of the interatomic distance. Repulsive force is the most effective at short distances, but its range is rather restrictive. At some definite interatomic distance attractive and repUlsive forces are balanced. This equilibrium distance corresponds to the minimum in the curve of potential energy against the interatomic distance. The
19
STRUCTURE, BONDING, AND PROPERTIES
20
CHAPTER 1
equilibrium distance that always occurs between atoms gives the impression of atoms being spheres of definite sizes. As a matter of fact, in many cases atoms can be treated as if they were more or less hard spheres. Due to the attractive forces between the atoms the difference in the radius of an atom depends on the type of bonding forces. For the same element several different sphere radii have to be assigned according to bonding type.
Based on experimental data for one specific kind of bonding the atomic radius of an element has a fairly constant value. However, in crystalline solids the value of an atomic radius depends on bonding type (van der Waals bonding, metallic bonding, ionic bonding, and covalence bonding) and coordination numbers. The larger the coordination number the larger the radius is.
The atom shape is not always spherical and the covalently bonded atoms are not exactly spherical; for example, hydrogen atom is not spherical in the hydrogen-carbon compound. If the covalent bond is more polar, the deviation from the spherical shape is less pronounced. The influence of bond polarity shows that the bond lengths depend on the oxidation states. If an atom is easily polarized the deviation of covalent bond lengths is large.
The radii of metallic bonding atoms depend on the bonding electron states, but the shape is almost spherical. However, when many bonding but few antibonding states are occupied, the resulting bond forces between the metal atoms are large. This is the case for the metals in the central part of the block of transition elements. The number of valence electrons and their occupation configuration in the electron energy bands are essential parameters for metal bonding. The ratio of the total number of available valence electrons to the number of atoms, or the "valence electron concentration," is a decisive factor affecting the effective atom size and its structure in intermetallic compounds, such as MgCu2 and SrSh.
The radii of ions in an ionic bonding compound are determined experimentally by diffraction techniques. The data recorded from x-ray diffraction can be used to calculate the electron density in the crystal. The point having the minimum electron density along the connection line between a cation and an adjacent anion can be taken as the contact point of the ions. The radius of the ion is defined to be the distance between the center of the highest electron density region and the point of minimum electron density. The crystal ions deviate from sphericity, indicating the electron shell polarity in the crystal, which means the presence of some degree of covalent bonding. It can be interpreted as a partial backflow of electron density from the anion to the cation. The electron density minimum does not necessarily represent the ideal place for the limit between cation and anion.
The commonly used values for ionic radii are based on an arbitrarily assigned standard radius for a specific ion. In this way, a consistent set of radii for other ions can be derived. The values of Shannon are based on a critical evaluation of experimentally determined interatomic distances and on the standard radius of 0.14 nm for the 02- ion with sixfold coordination. They are listed in Tables 1.1 and 1.2. The coordination number given is based on the cation as the center of a coordinated polyhedron.
1.5.2. LATIICE ENERGY OF AN IONIC COMPOUND
In an energy favorable packing of cations and anions, only amons are directly adjacent to a cation, and vice versa; thus, the attractive forces among ions of opposite charges outweigh the repulsive forces between ions of the same charge. The interaction
TABLE 1.1. IONIC RADII FOR MAIN GROUP ELEMENTS ACCORDING TO SHANNON (1976), BASED ON r(02-)= 140pm (lpm=O.OOlnmt
H Li Be B C N 0 F
-1 150 +1 76 +2 45 +3 27 +4 16 -3 146 -2 140 -1 133 -3 16
Na Mg Al Si P S Cl
+1 102 +2 72 +3 54 +4 40 +3 44 -2 184 -1 181 +5 38 +6 29
K Ca Ga Ge As Se Br
+1 138 +2 100 +3 62 +2 73 +3 58 -2 198 -1 196 +4 53 +5 46 +4 50
Rb Sr In Sn Sb Te
+1 152 +2 118 +3 80 +2 118 +3 76 -2 221 -1 220 +4 69 +5 60 +4 97 +5 95 +6 56 +7 53
Cs Ba Tl Pb Bi Po
+1 167 +2 135 +1 150 +2 119 +3 103 +4 94 +3 89 +4 78 +5 76 +6 67
a The corresponding oxidation states are also given. All values refer to coordination number 6 except c.n. 4 for N3- . The unit is pm.
between an anion and a cation has a short-range force and a long-range one. The short- range force is repulsive and the long-range force is attractive. The equilibrium positions are the sites at which the two forces are balanced, or the binding energy (or lattice energy) reaches a minimum point. The lattice energy is defined as the energy needed to disassemble one mole of a crystalline compound at temperature of 0 K in such a way that its components are moved to infinity. For ionic crystals, the long-range interaction between the ions is described by the Coulomb potential, which depends on the interatomic distance and the effective charge of each ion as well. Therefore, the lattice energy of an ionic compound is
(1.4) where NA is the number of anions, ql the effective charge of the anion, and qi the effective charge of cations located at rli from the first anion. The lattice energy is usually written in the convenient form
(1.5) where A is called the Madelung constant, which depends on the structure type of an ionic compound, and R is half of the mean interion distance. The Madelung constant is determined by the interatomic distance and the three-dimensional distribution configuration of the cations, but it is independent of the ionic charges and the lattice constants. If all the ions have the same absolute charge and the same size, then r
=
R, which means the radii of the ions can be used as R. If they are different R would be half of the interion distance. The lattice energy increases if R decreases. Table 1.3 lists the values of Madelung constants for some simple structures.21
STRUCTURE, BONDING, AND PROPERTIES
22