Fundamentals of the Physics of Solids / 04-Bonding in Solids

than the vertices of a tetrahedron. We shall revisit this question at the end of Section 7.6 on the relationship between crystal structure and bonding.

4.4.8 Covalent Bonds in Solids

Covalent solids are held together by networks of directional covalent bonds between neighboring atoms, where the wavefunctions of covalent bonds are constructed from pairs of atomic states. If each bonding state is occupied by two electrons of opposite spins – i.e., each bond is saturated – then the electrons that participate in bonding are localized in space and cannot contribute to electrical conductivity. That is why covalent materials are usually insulators or semiconductors. This is the case for diamond and for two further elements of the carbon group: germanium and silicon, which have a diamond structure. Here each atom has four sp3 hybrid states, and the bonds formed by them make up a tetrahedral network, as shown in Fig. 7.16(a). The same structure is seen in semiconducting compounds formed by elements in groups 13 (IIIA) and 15 (VA) of the periodic table.

As it was mentioned in the previous subsection, besides sp3 wavefunctions, other hybrid states may also give rise to covalent bonding – however, their spatial directionality depends on which states are hybridized. The orientation of the bonds plays a crucial role in determining the crystal structure. The development of short-range order in the amorphous state of covalently bonded solids is also related to the directionality of the bonds.

The cohesive energy of covalent crystals is given by the sum of the binding energies of individual bonds. The binding energies of some typical covalent bonds are listed in Table 4.5. Much larger than their counterparts for molecular crystals, these values are comparable to the energies of ionic crystals.

Table 4.5. Binding energy of some typical covalent bonds (in units of eV and kJ/mol)

106 4 Bonding in Solids

4.5 Metallic Bond

A great part of the chemical elements have fewer electrons on the incomplete shells than what is necessary for having saturated covalent bonds between each pair of neighboring atoms in the solid state. Electrons participating in unsaturated bonds are not localized. One may also say that the atoms lose these outermost (in general, s or p) electrons. While the positively charged ions left behind are arranged in a more or less regular pattern, the freed electrons fill the region among the ions almost evenly. This moving cloud of electrons gives rise to metallic bonding. In transition metals, where incomplete d-shells are also found under the outermost shell, further electrons may participate in metallic bonding. The same kind of bonding may appear in materials built up of molecules with incomplete shells.

The wavefunction of such an electron system cannot be written as the product of the wavefunctions of pairs of electrons forming bonds. It has to be chosen in such a way that it should show explicitly the antisymmetry with respect to the interchange of the coordinates of any two electrons. This can be done through the generalization of the formulas in (4.4.43) or (4.4.47), using functions of the Slater determinant form. The analysis of such systems, the determination of electronic energies and states, and the study of those properties of solids that are due to electrons will be among our most important tasks. Volumes 2 and 3 are devoted almost exclusively to these issues. To a large extent, the present volume serves as preparation for this.

The determination of the total energy of metals is a di cult problem of solid-state physics. We shall take a closer look at it in Chapter 30, after the study of electron states. Here we only mention that the cohesive energy per atom is usually 1–5 eV.

4.6 The Hydrogen Bond

The hydrogen atom has a single electron on the 1s shell – lacking another one to have the shell closed. That is why covalent bonding would permit hydrogen to be linked to a single other ion. Due to its small size, even with ionic bonding only two ions can be tightly packed around the proton. However, because of its high ionization energy (13.6 eV) hydrogen is not easily ionized. Instead of participating in such bonds, hydrogen can create a special bond between highly electronegative atoms like fluorine, oxygen, or nitrogen. This is the hydrogen bond (or, as it is called in several languages, the “hydrogen bridge”).

In this type of bonding the hydrogen atom is not located at the midpoint between the F, O, N atoms, but has two symmetrical equilibrium positions between which it oscillates.

Crystalline ice is held together by such bonds, as shown in Fig. 4.12(a) – but hydrogen bonds play an important role in water, too. The distance between oxygen atoms is about 2.75 Å in the ground state of ice and about 2.9 Å

(a) (b)

Fig. 4.12. Hydrogen bonds in (a) ice; and (b) the alpha helix structure of a polypeptide

in water. The two equilibrium positions of the hydrogen atom are located at a distance of about 1 Å from either of them. The bond between the hydrogen and the nearby oxygen can be considered as covalent, although the huge di erence in electronegativity makes it highly polar. That is why the hydrogen can be linked to another oxygen atom on the other side, at a distance of 1.8–2 Å. The hydrogen bond is stronger on one side and weaker on the other; this is reflected in the notation O–H · · · O. The bond is relatively weak; breaking it requires some 0.2 eV of energy. Between two fluorine atoms the binding energy is 0.29 eV.

In crystalline ice each oxygen atom is connected to four others through hydrogen bonds. If a snapshot were made of the structure, in two of these bonds the hydrogen would be in its equilibrium position closer to the oxygen, and in the farther one in the other two. It can be shown that there are exponentially many states that satisfy this condition. This high degeneracy exists in the ground state, too – giving rise to a finite entropy at zero temperature. That is why ice is a model of choice in statistical physics.

Even more important is the fact that the structure of proteins is determined by hydrogen bonds (with binding energies of 8 to 40 kJ/mol) between the CO and NH groups of the polypeptide chains. The blueprint of life, the double helix of DNA is also held together by hydrogen bonds between base pairs. The schematic structure of an alpha helix is shown in Fig. 4.12(b).

108 4 Bonding in Solids

Hydrogen bonding is therefore essential for biological materials – however, substances held together by hydrogen bonds do not play an important role in solid-state physics. That is why we shall not pursue the study of this type of bond any further.

Further Reading

1.J. K. Burdett, Chemical Bonding in Solids, Oxford University Press, Oxford (1995).

2.R. McWeeny, Coulson’s Valence, Third Edition, Oxford University Press, Oxford (1979).

3.J. N. Murrell, S. F. A. Kettle, and J. M. Tedder, The Chemical Bond, Second Edition, John Wiley and Sons Ltd., Chichester (1985).

4.L. C. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, 3rd edition, Cornell University Press, Ithaka, N.Y. (1960).

5.D. Pettifor, Bonding and Structure of Molecules and Solids, Clarendon Press, Oxford (1995).

6.G. S. Rohrer, Structure and Bonding in Crystalline Materials, Cambridge University Press, Cambridge (2001).