Ionic crystals

In contrast with localized covalent bonding, ionic bonding is delocalized, and the structures are densely packed. That is, the bonding is distributed over distances that are large compared with the distance between any individual pair of atoms. The most common crystal structure is that of sodium chloride in which each negatively charged atom is surrounded by six posi-tively charged atoms (Figure 12.18(a)). A less common structure is that of cesium chloride in which eight ions of one sign surround each one of the opposite sign (Figure 12.17(b)).

The atoms on the left-hand side of the Periodic Table (the alkali metals) are relatively easily ionized, becoming positive ions. They are said to be electropositive. Those on the right-hand side (the halogen gases) have an affinity for charge, forming negative ions. They are said to be electronegative. The atoms of Group II (the alkaline earths) are less electropos-itive than the alkalis; and those of Group VI (the chalcogenides) are less electronegative than the halogens. Thus the atoms become less electropositive from one group (column) to another from left to right across the Periodic Table. They also become more electropositive as the rows of the table progress from I to VI.

In accordance with Coulomb’s Law, the ions with opposite charges attract one another, while those with like charges repel each other. The closeness of approach of the attractive

Figure 12.18 Structures of highly ionic crystals: (a) sodium chloride structure, coordination number 6; (b) cesium chloride structure, coordination number 8.

Figure 12.19 String of alternating+ and − ions with spacing d.

pairs is limited because the underlying ion cores repulse one another strongly as a result of the Pauli Exclusion Principle. This repulsion arises because of the overlap that occurs as electrons in the filled shells of one ion enter the space occupied by another. A strong resistance to the overlapping occurs because the Pauli Principle forbids more than one electron of each spin to occupy the same quantum state (the same amplitude function).

The Pauli overlap repulsion results from the additional constraint that the Pauli Principle imposes on the de Broglie effect.

The long-range nature of the bonding in ionic crystals can be demonstrated by considering a one-dimensional periodic string of positive and negative ions spaced a distance d apart (Figure 12.19). The coordinate origin is placed at one of the negative ions, and then the electrostatic energy of the infinite chain is calculated (Kittel, 1953).

The first interaction to the right of the origin is−q²/d, the next is +q²/2d, then −q²/3d, and so on. Since the same terms appear to the left of the origin, all of the terms must be doubled. Also note that all of the other interactions cancel one another out. Thus, the electrostatic energy is:

Ue= −(q²/d)2[1 − 1/2 + 1/3 − 1/4 + 1/5 · · ·] (12.19) The term in square brackets converges very slowly toward ln 2 (after ten terms the error is still about 12%) which means that full cohesion of the chain exists only if the chain is very long. Locally, the cohesion is weak or non-existent. In three dimensions the situation is similar. That is why, for example, the grain boundaries in ionic solids tend to be weak (Gilman, 1966). For an extended ionic crystal with nearest-neighbor interionic distances, r , the electrostatic energy is:

U_e(r )= −αq²/r (12.20)

whereα represents the three-dimensional summation series analogous to the one in Equa-tion (12.19). It is called the Madelung constant, and it has been evaluated for various crystal structures. A few values are given below:

Structure Madelung constant Sodium chloride 1.7476

Cesium chloride 1.7627

Zinc blende 1.6381

Wurtzite 1.641

Fluorite 2.519

Rutile 2.408

Figure 12.20 Dependence of the bulk moduli of some divalent ionic crystals (fluorite structure) on their crystallographic lattice parameters. Note that even for mixed crystals containing three different ions, the dependence expected from electrostatics is observed.

q is the magnitude of the ionic charge which depends on the valence of a particular ion. The minus sign means that this energy is attractive. The repulsions of the ions are sometimes represented by a power function, and sometimes by an exponential function. Since, over specific ranges of their arguments, these functions can approximate each other, this choice is somewhat arbitrary. Exponential repulsive functions are physically more consistent with quantum mechanics, but power functions are easier to manipulate. Using the latter, the repulsive energy term is:

U_r(r )= +A/rⁿ (12.21)

where A and n are constants. By combining Equations (12.19), and (12.20), the following expression for the bulk modulus can be obtained:

B= (n − 1)αq²/18(ro)⁴ (12.22)

where rois the equilibrium interionic distance, and n can be empirically determined to be 9–10 (a large exponent is needed if the ion cores are to repulse each other strongly; when n becomes very large the ions respond like rigid spheres). Note the presence of the Keyes parameter (q²/r_o⁴). The inverse fourth power dependence is followed quite well within each homologous series of ionic crystals (Gilman, 1969). Figure 12.20 shows this dependence for a variety of divalent ionic crystals.

The dependence on q is as expected. That is, for divalent ions (such as MgO), the bulk modulus is about four times as large as for monovalent ions of the same size.

BULK MODULUS (Kbar) 200 400 600 800

0

Li Na

K Rb

Cs I

Br CI

F 0

200 400 600 800

BULK MODULUS (Kbar)

Figure 12.21 Pattern of bulk moduli for the alkali halides. LiF has the maximum bulk modulus, and CsI the minimum.

The full set of alkali halides exhibits a definite pattern of bulk moduli as displayed in Figure 12.21 where the alkali metals are arrayed along one axis, and the halides along another. The largest value of B is for LiF, and the smallest for CsI. The modulus values are for the lowest reported temperatures, mostly 4.2 K. The pattern suggests that there might be a correlation with electronegativity, and indeed there is.

Using Mulliken’s definition (electronegativity= [ionization energy − affinity energy]/

2= µ), and forming the electronegativity difference for each compound, the data for each cation group are plotted (B versus electronegativity difference) in Figure 12.22(a). Each group has its own correlation line. However, if the data are expressed as electronegativ-ity densities (each value in Figure 12.22(a) being divided by the molecular volume), the entire set of data for 20 alkali halides follows a single regression line with r= 0.97, see Figure 12.22(b).

A graph similar to Figure 12.22(b) has been made using the average valence electron density as the parameter. The correlation coefficient is similar, indicating that the behavior

Figure 12.22 Connection between bulk stiffness and electronegativity difference density for the set of 20 alkali halides: (a) bulk modulus versus electronegativity difference (eV); (b) bulk modulus versus electronegativity difference density (eV/ų).

is similar to that expressed by Figure 12.4, further indicating that electronegativity is deter-mined by valence electron concentration for these crystals. This suggests that a theory of chemical reactivity might be based on selected mechanical parameters.

12.12 Fluorites

The bulk stiffnesses of the divalent alkaline earth compounds are proportional to the in-verse fourth powers of their interionic spacings as Figure 12.20 demonstrates. All of these compounds have the cubic fluorite structure (CaF2is the prototype for the series).

12.13 Chalcogenides (oxygen column of the Periodic Table)

The chalcogenide compounds (II–VI) are somewhat unusual in that those containing oxygen behave like ionic compounds, while the others behave like covalent compounds. This is illustrated in Figure 12.23 which shows data for both the divalent oxides and several other chalcogenide crystals. The difference in stability is striking. The divalent oxides follow the inverse fourth power rule, but the other chalcogenides follow a somewhat higher power rule. More striking is that the magnitudes of the stiffnesses for the oxides are about five times greater than those for the other chalcogenides.

The deformation resistances of atomic particles are closely related to their polarizabilities since both involve a change of shape. Figure 12.24 illustrates this for the case of the alkaline oxides where the correlation between the bulk stiffness and the inverse polarizability data of Duffy (1996) is excellent.

Figure 12.23 Differential behavior within the chalcogenide compounds. The oxides are much stiffer than the other chalcogenide compounds at the same molecular size.

Figure 12.24 Showing that the polarizability (deformability) of the oxygen ion determines the stiff-nesses of the alkaline oxides.

12.14 Silicates

As the following table indicates, the bulk moduli of some silicates are similar, suggesting that they are determined principally by the behavior of the silicate groups. The influence of the anion metals is small. The data are from Anderson and Isaak (1995).

Bulk modulus Silicate (Mbar) Co2SiO4 1.48 Fe2SiO4 1.38 Mg2SiO4 1.29 Mn2SiO4 1.29