Crystals, covalent

If electron-pair, or covalent, bonding is periodic in two or three dimensions, crystals result. The most important case is the carbon-carbon bond. If it is extended periodically in two-dimensions the result is graphite in three-dimensions it is diamond. Other elements that form electron-pair bonds are Si, Ge, and a-Sn. Some binary compounds are A1P (isoelectronic with Si), 

Except in ice, molecules in molecular crystals are generally packed together as closely as their size and shape allow. Because van der Waals forces and hydrogen bonding are usually quite weak compared with covalent and ionic bonds, molecular crystals are more easily broken apart than ionic and covalent crystals. Indeed, most molecular crystals melt at temperatures below 100°C. 

A large number of binary AB compounds formed by elements of groups IIIA and VA or IIA and VIA (the so-called III-V and II-VI compounds) also fcrystallize in diamond-like structures. Among the I-VII compounds, copper (I) halides and Agl crystallize in this structure. Unlike in diamond, the bonds in such binary compounds are not entirely covalent because of the difference in electronegativity between the constituent atoms. This can be understood in terms of the fractional ionic character or ionicity of bonds in these crystals. 

Pauling (1960) expressed the ionicity of a covalent bond A-B in terms of the difference between the electronegativities of the two atoms  

Here/j is the fractional ionic character of the bond while Xf and X are the Pauling electronegativities of the elements A and B. It is seen that/- = 0 when = Xg and fi= I whenX — Vg 1. In crystals, this expression is modified to account for the fact that an atom forms more than one bond and the number of bonds formed is not always equal to its formal valence. For crystals, the expression for ionicity reads as 

Phillips (1973a) has proposed a new scale of ionicity for AB-type crystals based on the assumption that the average band gap. Eg, of these crystals consists of both covalent and ionic contributions as expressed by Eg = + Ef, where and , are the 

The parameter is obtained by relating the static dielectric constant to Eg and taking in such crystals to be proportional to a - where a is the lattice constant. Phillips parameters for a few crystals are listed in Table 1.4. Phillips has shown that all crystals with a/ below the critical value of0.785 possess the tetrahedral diamond (or wurtzite) structure when f 0.785, six-fold coordination (rocksalt structure) is favoured. Pauling s ionicity scale also makes such structural predictions, but Phillips scale is more universal. Accordingly, MgS (f = 0.786) shows a borderline behaviour. Cohesive energies of tetrahedrally coordinated semiconductors have been calculated making use 

CHAPTER 12 Intermolecular Forces and the Physical Properties of Liquids and Solids 

The extreme case of a giant 3D coAraJent polymer is realized in the case of diamond [C] (see Fig. 2.22)  

2 Bonding aspects Prom atoms to solid state 

Roughly speaking, such a value can be justified by the experimental bond energies of aromatic hydrocarbons. 

As in the case of metals and in contrast to ionic and molecular crystals, covalent crystals yield extended bands in the band model, because of the significant overlap. 

As expected the bandgap drastically decreases from diamond to a-Sn. Diamond is a typical insulator with a band gap of 5eV in siUcon the band gap is still leV in germanium only 0.7eV a-Sn is already a semimetal (as is graphite). In the case of a-Sn the covalency is not very pronounced and the s-p splitting (metaUicity) predominates (see Section 2.2.1 and Fig. 2.14). 

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Of the three principal classes of crystals, ionic crystals, crystals containing electron-pair bonds (covalent crystals), and metallic crystals, we feel that a good understanding of the first class has resulted from the work done in the last few years. Interionic distances can be reliably predicted with the aid of the tables of ionic radii obtained by Goldschmidt1) by the analysis of the empirical data and by Pauling2) by a treatment based on modem theories of atomic structure. The stability,... 

Conclusion. There is no need to multiply examples of the type considered, especially since our knowledge of the principles determining the structures of covalent crystals is still so incomplete that we can offer no explanation for the anomalous manganese radius indeed, even the observed arrangement1) of the bonds formed by sulfur in sulvanite, Om3FS4, was entirely unexpected and has been given only an ad hoc explanation. 

Goldschmidt (1926) grouped metals and covalent crystals together, and Bernal (1929) pointed out that many properties of metals indicate that metallic bonds are closely similar to covalent bonds. I developed this idea further (Pauling, 1938), and formulated a set of metallic radii in 1947, with use of the empirical equa-... 

I feel now that I was influenced to some extent by my knowledge that in 1926 Goldschmidt had formulated a set of atomic radii that represented reasonably well the interatomic distances in both covalent crystals and metals (5). I was also impressed by a discussion of the properties of metals by Bernal, who, however, rejected the idea that covalent bonds are present in metals (4). Bragg also rejected this idea (5). 

When crystals with covalent bonds (e.g., AICI3 or TiCy melt, the melt conductivity remains low (e.g., below 0.1 S/m), which implies that the degree of dissociation of the covalent bonds after melting is low. The covalent crystals also differ from the ionic crystals by their much lower melting points. The differences between these two types of crystal are rather pronounced, whereas there are few crystalline solids with intermediate properties. 

However, It has been found that in many cases, simple models of the properties of atomic aggregates (monocrystals, poly crystals, and glasses) can account quantitatively for hardnesses. These models need not contain disposable parameters, but they must be tailored to take into account particular types of chemical bonding. That is, metals differ from covalent crystals which differ from ionic crystals which differ from molecular crystals, including polymers. Elaborate numerical computations are not necessary. 

A measure of shear strength is the shear modulus. For covalent crystals this correlates quite well with hardness (Gilman, 1973). It also correlates with the hardnesses of metals (Pugh, 1954), as well as with ionic crystals (Chin, 1975). Chin has pointed out that the proportionality number (VHN/C44) depends on the bonding type. This parameter has become known as the Chin-Gilman parameter. 

Gerk showed that Equation (2.1) is followed not only for metals, but also for ionic and covalent crystals if two adjustments are made. For covalent crystals, the temperature must be raised to a level where dislocations glide readily, but below the level where they climb readily. For ionic crystals, G (an average shear modulus) must be adjusted for elastic anisotropy. Thus it becomes ... 

A key feature of the motion of dislocation lines is that the motion is rarely concerted. One consequence is that the lines tend not to be straight, or smoothly curved. They contain perturbations ranging from small curvatures to cusps, and kinks. In covalent crystals where there are distinct bonds between the top... 

Figure 4.3 Glide activation energies for various covalent crystals versus their minimum energy gaps. The correlation coefficient is 0.95. Without the point for GaP, it would be much higher. 

Inelastic shearing of atoms relative to one another is the mechanism that determines hardness. The shearing is localized at dislocation lines and at kinks along these lines. The kinks are very sharp in covalent crystals where they encompass only individual chemical bonds. On the other hand, in metal crystals they are often very extended. In metallic glasses they are localized in configurations that have a variety of shapes. In ionic crystals the kinks are localized in order to minimize the electrostatic energy. 

The differences just outlined cannot be explained by means of a classical mechanical model. However, they can be explained by considering chemical bonding. In particular, hardness depends in covalent crystals on the fact that the valence (bonding) electrons are highly localized as shown by electron-diffraction studies which can provide maps of electron densities (bonds). 

Dislocation motion in covalent crystals is thermally activated at temperatures above the Einstein (Debye) temperature. The activation energies are well-defined, and the velocities are approximately proportional to the applied stresses (Sumino, 1989). These facts indicate that the rate determining process is localized to atomic dimensions. Dislocation lines do not move concertedly. Instead, sharp kinks form along their lengths, and as these kinks move so do the lines. The kinks are localized at individual chemical bonds that cross the glide plane (Figure 5.8). 

For covalent crystals temperature has little effect on hardness (except for the relatively small effect of decreasing the elastic shear stiffness) until the Debye temperature is reached (Gilman, 1995). Then the hardness begins to decrease exponentially (Figure 5.14). Since the Debye temperature is related to the shear stiffness (Ledbetter, 1982) this softening temperature is proportional to C44 (Feltham and Banerjee, 1992). 

In the high temperature regime for covalent crystals where the hardness drops rapidly, its values are affected by impurities (dopants). Both hardening and softening occur depending on whether the dopant is is a donor, or an accepter... 

A shear modulus of about 1 GPa has been measured for wet lysozyme. Thus its Chin-Gilman parameter is about 0.02 which is large compared with metals and small compared with covalent crystals. 

The concept of chemical hardness was originally developed as a measure of the stability of molecules. Its relationship to physical hardness and to solids is discussed here. Also, it is pointed out that shear moduli and polarizabilitites, as well as band gaps in covalent crystals, are related to it. 

Physical hardness can be defined to be proportional, and sometimes equal, to the chemical hardness (Parr and Yang, 1989). The relationship between the two types of hardness depends on the type of chemical bonding. For simple metals, where the bonding is nonlocal, the bulk modulus is proportional to the chemical hardness density. The same is true for non-local ionic bonding. However, for covalent crystals, where the bonding is local, the bulk moduli may be less appropriate measures of stability than the octahedral shear moduli. In this case, it is also found that the indentation hardness—and therefore the Mohs scratch hardness—are monotonic functions of the chemical hardness density. 

One implication of these findings is that chemical hardness is related to the band gaps of covalent crystals, consistent with its being related to the LUMO-HOMO gaps of molecules. Data indicate that this is indeed the case. Another... 

In relatively recent years, it has been found that that indentations made in covalent crystals at temperatures below their Debye temperatures often result from crystal structure changes, as well as from plastic deformation via dislocation activity. Thus, indentation hardness numbers may provide better critical parameters for structural stability than pressure cell studies because indentation involves a combination of shear and hydrostatic compression and a phase transformation involves both of these quantities. 

There is disagreement in the literature about the role of friction. Compare, for example, Cai (1993) with Ishikawa et al. (2000) This has arisen in various ways. In the case of metals, where the Chin-Gilman parameter is small, friction is not important for relatively large indents. However, as the C-G parameter becomes much larger for covalent crystals, and as the indent size decreases friction becomes more important. Also, environmental factors, such as humidity, affect friction coefficients. In the regime of superhardness with dry specimens and small indents friction becomes very important. 

The principal intention of the present book is to connect mechanical hardness numbers with the physics of chemical bonds in simple, but definite (quantitative) ways. This has not been done very effectively in the past because the atomic processes involved had not been fully identified. In some cases, where the atomic structures are complex, this is still true, but the author believes that the simpler prototype cases are now understood. However, the mechanisms change from one type of chemical bonding to another. Therefore, metals, covalent crystals, ionic crystals, and molecular crystals must be considered separately. There is no universal chemical mechanism that determines mechanical hardness. 

With the atom C strongly bound not only to B but also to the other atoms of a solid-state matrix (i.e., when C fB) the above ratio is small in the parameter mc/mB 1, so that the dominant contribution to the interaction with phonons is provided by the deformation potential. Reorientation probabilities were calculated, with the deformation term only taken into consideration, in Refs. 209, 210. For a diatomic group BC, c A Uv 0.1 eV, whereas eb 10 eV (a typical bond energy for ionic and covalent crystals). A strong binding of the atom C only to the atom B results in the dominant contribution from inertial forces.211 For OH groups, as an example, the second term in Eq. (A2.13) is more than 6 times as large as the first one. 

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