Metallic and Ionic Structures and Bonding Ionic compounds are formed between elements having an electronegativity difference of about 2.0 or greater. Simple ionic compounds are characterized by high melting points and very low conductivities in their solid forms. When molten, however, conductivities are very high. Ionic compounds formed between s- and p-block elements involve ions with rare gas configurations. Both experimental (x-ray diffraction) and computational determinations of electron densities in these structures indicate that the electron density is very low at some point between the nuclei of the cation and anion. From this it is inferred that there is minimal overlap of the cation and anion valence orbitals. The notion that the ions can be viewed as spheres in close contact is derived in part from these observations. The figure shows an electron density map for a portion of the face of the unit cell of sodium chloride. Covalent compounds have highly directional interactions with a (relatively) small number of adjacent (overlap of orbitals with defined directions). Furthermore interactions with adjacent molecules in solids are weak as they involve only dipole-dipole and/or Van der Waals interactions. In metals each atom tends to have a much larger number of nearest neighbor atoms, determined only by the way in which the atoms are packed in the array. In many cases the arrangement is either cubic closest packed (ccp) or hexagonal closest packed (hcp), both of which result in twelve nearest neighbors. The unit cell of the former is termed face-centered cubic. Some structures are based upon less efficient packing (more free space) such as the body centered cubic structure. Drawings of these are shown below. spheres. Two layers of closest packed
Face centered cubic cell Body centered cell Regardless of solid-state structure the electronic structures of metals involve molecular orbitals that extend throughout the solid. The number of orbitals is so large that it is useful to view metals have having a band structure. Because there are typically either vacancies in the lower energy band or overlap of filled and vacant bands metals are good conductors. Many simple ionic compounds can be viewed as having structures determined by ccp or hcp packing of large spheres (anions) with voids in the structure occupied by smaller spheres (cations). Other structures are related to the body centered cubic structure in which the atoms at the corner of the cube are replaced by anions and the atom at the center is replaced by a cation. Because there is no significant overlap of orbitals between anions, or between anion and cation, ionic substances do not conduct. The unit cell is the smallest unit that repeated in each direction will make up the infinite array of a solid. For the ccp arrangement the unit cell is known as the face-centered cubic (fcc) cell. It is shown above in exploded form. In the actual fcc cell comprised of a single type of atom, the spheres will be in contact with one another and the cell dimension will be determined by the diagonal which will be four times the radius of the atom. This only one of three cubic lattice structures and one of fourteen basic lattice shapes. The entire group of fourteen is shown on the next page. In a cubic salt, which can be considered as a ccp arrangement of large spheres (usually anions) with smaller spheres (usually cations) occupying voids, the cell dimension will frequently be determined by the contact between the anion and cation along the edge of the cell, which is then equal to the sum of the diameters of the anion and the cation. Note that the empirical formula of a salt can be determined by examining the contents of the unit cell and its density can be calculated based upon the contents of the unit cell and its dimensions. For a cubic cell the all sides are equivalent. In considering the contents of a unit cell it is important to recognize that only those atoms (ions) that are fully in the cell contribute totally to the cell. Those sitting in the sides contribute only one half to the cell NaCl structure (and one half to the adjacent cell). Atoms or ions residing on edges contribute only one fourth to the cell under consideration (and one fourth each to three contiguous cells. Finally, atoms or ions residing at the corners of a cell contribute only one eighth to the cell (and one eighth to each of seven other contiguous cells that meet at the corner). This can be illustrated by the fcc structure adopted by sodium chloride and many other
The fourteen Bravais lattices divided into six systems according to their macroscopic symmetry. The symbols at the top of the page refer to primitive (P), body-centered (I), centered on one face (C), face-centered (F) and rhombohedral (R).
1:1 salts. Consideration of the anions (large spheres) indicates that there are six in faces, which contribute one half each for a contribution of three, and that there are eight on the corners, which contribute one eighth each for a contribution of 4 anions to the cell. Alternatively, using the cations one observes that there is one cation that is entirely in the cell and twelve on edges, which contribute one quarter each for a contribution of three cations. Thus the total cations equals the total anions, as it must, and there are four formula units per cell. In the ccp arrangement of spheres there are three types of voids (holes), trigonal (three nearest neighbors), tetrahedral (four nearest neighbors) and octahedral (six nearest neighbors). The latter two are most commonly occupied and it is important to know that there are two tetrahedral holes per sphere and one octahedral hole per sphere making up the ccp array. This can be determined by examination of the fcc unit cell, which contains a total of four spheres. As the top drawing to the right indicates there is one complete octahedral hole in the center of the unit cell and 1/4 of an octahedral hole along each edge of the cell (1 + 12/4 = 4 holes/cell or one hole per sphere. The bottom drawing shows one of the eight tetrahedral holes that are fully contained within the unit cell (each tetrahedral hole is formed by a corner sphere and the three adjacent spheres that sit on the faces. In the simple cubic arrangement the cation occupying the cubic hole would have eight neighbors. For the larger spheres making up the ccp array to remain just in contact the maximum radius that the smaller sphere can have will be 0.155 of the large radius of the large sphere for the trigonal hole, 0.225 for the tetrahedral hole and 0.414 for the octahedral hole. This leads to the definition of a radius ratio r + /r - for ions. In the simple cubic structure there is one cubic hole per unit cell or one hole per sphere. The radius ratio is 0.73. Now realize that the maximum electrostatic energy results when two opposite charges interact through the shortest distance and when a charge of one type interacts with the greatest possible number of opposite charges. Therefore, under ideal circumstances cations will occupy holes that minimize the distance between the cation and its anion and that maximize the number of nearest neighbors. In general a cation will not occupy a hole that is too small since the ion will not be centered in the hole. On the other hand a cation that is too large for a hole will only result in separation of the anions. Based upon this we can conclude that combinations of anions and cations with r + /r - of <0.22 should have the cation in the trigonal holes, those with ratios in the range 0.22-0.4 should have the cations in tetrahedral holes and those with ratios 0.4-0.73 should have the cations in octahedral holes. Those with r + /r - >0.73 should adopt the simple cubic structure in which the ions would have eight neighbors. Some examples of structures that result from occupancy of tetrahedral holes the ccp lattice are shown below.
Zinc sulfide (zinc blende) Platinum sulfide Calcium fluoride (fluorite) Lead oxide Note the different ways in which partial occupancy of tetrahedral holes occurs in the ZnS, PtS and PbO lattices, whereas all of the tetrahedral holes are occupied in the fluorite structure. In representing these structures the cations make up the fcc lattice in the fluorite, PbO and PtS structures. A non-closest packed structure is found in cesium chloride and related salts. Cesium chloride The energy of interaction between an anion and cation (ion pair) can be calculated on the basis of an electrostatic interaction between charges operating through a distance equal to the sum of the ionic radii. Electron-electron repulsions between the valence electrons of the cation and anion help to maintain an equilibrium internuclear distance. In the electrostatic model, the energy is proportional to the product of the ion charges Z + Z - and inversely proportional to the sum of the ionic radii, r + + r -. When this energy is calculated for a mole of sodium chloride ion pairs, the energy calculated is 589 kj mol -1. This is only about 75% of the
experimentally determined lattice energy for sodium chloride, which is 770 kj mol -1. The difference results from extended interactions between cations and anions that are not nearest neighbors. Summation of the interactions, nearest neighbors and beyond, results in an infinite series that converges to a value of 1.74756. There are similar constants for other structures. The simple cubic structure, which is found for cesium chloride has a constant of 1.76267 and that for ZnS (zinc blende) of 1.63805. Lattice energies can be estimated based upon consideration of several factors. The most important are: the product of the charge on the ions, the type of structure, and the internuclear distance between the cation and anion (sum of the ion radii). The energy is proportional to the first two and inversely proportional to the latter. So for a given structure, the energy goes up as the charge on the ions increase and their size decreases. The effects ion size and charge on properties is well illustrated by the data in the following table. Salt m.p./k density r o unit cell/å U/kJ mol -1 NaF 1268 2.558 2.31 4.63 928 NaCl 1073 2.165 2.76 5.64 770 NaBr 1020 3.20 2.94 5.94 736 MgO 3073 3.58 2.1 4.21 3791 MgCl 2 981 2.32 2.46 * 2526.*does not have fcc (sodium chloride) structure