C APTER1Bonding and Crystal A. BONDING IN SOLIDS BONDING IN SOLIDS AND CRYSTAL STRUCTURE The attractive electrostatic interaction between the negative charge of the electrons and the positive charge of the nuclei is totally responsible for holding the atoms together. Magnetic forces have only a weak effect on cohesion and gravitational forces are negligible. A crystal can be stable if its total energy (kinetic energy + potential energy) is lower than the total energy of the atoms or molecules when they are free. The cohesive energy is defined as the difference between free atom energy and crystal energy. QUESTIONS AND ANSWERS Q.1.1. Explain bond type and bond energy. Ans. Depending upon the strength and bond energy, the chemical bonds may broadly be classified as primary bonds and secondary bonds. The bond energies in primary bonds may vary in the range of 1 to 10 ev. The examples of primary bonds are ionic, covalent and metallic pulley. Metallic bonds are relatively weaker than the other two. Secondary bonds such as van der Waals bonds and hydrogen bonds may have energies ranging from 0.01 to 0.5 ev. Bond length is defined as the centre to centre distance of the atoms forming a molecule. The strong bond strength means the atoms are strongly pulled together and therefore, the bond length is shorter. Bond length is relatively larger in weaker bonds. The bond length in primary bonds may vary from 0.1 nm to 0.2 nm and that in secondary may range from 0.2 to 0.5 nm. Q.1.2. Discuss the forces between atoms. If α + β is the net potential energy n m r r of the system, get the expression for the net interatomic forces. Draw the graph connecting the interatomic separation and resultant force. Ans. Assuming that the atoms consist of moving electrical charges, one of two things can happen as the atoms approach each other. Either they attract or repel each other. The 1
2 Essentials of Engineering Physics potential energy due to the attraction is negative, since the atoms do the work of attraction. The repulsive energy is positive since external work must be done to bring the atoms together and it is inversely proportional to some power of the interatomic separation, r. The net potential energy is sum of both the terms : U = decrease in potential energy (due to attraction) + increase in potential energy (due to repulsion) U U total r o U rep U attr Fig. 1.1 (a). A plot of interaction energy against interatomic distance. r U = U attractive + U repulsive U = a r m + b n...(1.1) r where α is the proportionality constant of attraction and β of repulsion. The solid curve in Fig. 1.1 (a) shows the variation of potential energy. The force between the atoms can be derived from Eqn. 1.1 by recalling that the force is the derivative of potential energy. Thus F(r) = du dr mα nβ =...(1.2) m+ 1 n+ 1 r r The resultant force is given by the solid curve in Fig. 1.1 (b). At large separations, the atoms do not interact with each other, so that U = 0 and F = 0. As the atoms approach each other, they exert attractive forces on each other, primarily because of the positive and negative charges in the atoms. When the atoms are separated by only a few atomic diametres, the repulsive forces between the like charges of the nuclei, however, start the assert themselves although somewhat more slowly. At the separation called equilibrium spacing r 0 the forces of attraction just equal to that of repulsion, and the potential energy is at a minimum. i.e., stability is reached. Q. 1.3. Give an introduction to binding in solids. Describe how ionic bonds are formed in solids with suitable figures and illustrations. Ans. The individual atoms of a crystal are held in position by bonds which are fundamentally electric in origin but, (except for the ionic bonds) arise essentially through quantum-mechanical considerations. In discussing interatomic bonds, we shall treat bonding U 0 U total r o U rep U attr Fig. 1.1 (b). U att, U rep and U total as a function of r ; r 0 is the equilibrium interatomic distance. r
Bonding and Crystal 3 in terms of five idealized types: ionic, hydrogen, covalent, metallic, and molecular. It should be emphasized at the start that this is an arbitrary choice; many bonds take part in two or more of these idealized forms. One should remember that the valence electron is very easily removed from an alkali metal leaving behind a very stable structure resembling an inert gas but with an extra positive nuclear charge. On the other hand an element from Group VII (chlorine, bromine or iodine the halogens) is but one electron short of an inert structure. Since the inert gas structure is so stable we might expect that a halogen atom would readily accept an extra electron and might even be reluctant to lose it. Taking these two tendencies together we can understand what happens when sodium and chlorine atoms are brought together in equal numbers. It is easy for the valence electron of each sodium atom to be transferred to a chlorine atom, making both more stable. But now there must be an electrostatic attraction between the ions so formed, since each sodium ion carries a positive charge and each chlorine ion a negative one. The attraction brings them closer and closer together until the inner electron clouds begin to overlap. At this point a strong repulsive force is manifested just to push them apart till the two forces just balance one another at a particular distance of separation d called equilibrium spacing. In this way the attraction of unlike ions overcomes the repulsion of like ions and a stable structure is formed (Fig. 1.3). Sodium chloride and magnesium oxide are two substances formed by this type of binding mechanism. The examples and chemical equations are given below: Na Na + Na + + + Cl (c) Ionic molecule Cl (a) Atoms Cl (b) Ions Fig. 1.2. Schematic representation of the formation of an ionic molecule of sodium chloride. Fig. 1.3. Part of the crystal of NaCl modelled as a close packed arrangement of spheres. CAPTER 1 2Na + Cl 2 2Na + + 2Cl 2NaCl...(1.3) and 2Mg + O 2 2Mg + + + 2O 2MgO...(1.4) Q.1.4. Discuss briefly the binding mechanism in covalently bonded crystals. Ans. Covalent bond is also known as homopolar bond. The electronic structure of an atom is relatively stable if it has eight electrons in its outer valence shell. Sometimes an atom may acquire these electrons by sharing electrons with an adjacent similar atom. Thus, when electrons are shared (and not transferred) between atoms; it gives rise to a covalent bond as shown in Fig. 1.4. The formation of a chlorine molecule is an excellent example of covalently bonded one. ere the outer shell of each atom possesses seven electrons ( ).
: : : : : : : : 4 Essentials of Engineering Physics Each chlorine atom would like to gain an electron, and thus form a stable octet. This can be done by sharing of two electrons between pairs of chlorine atoms thereby producing stable diatomic molecules. In other words, each atom contributes one electron for the sharing process. : Cl. +. Cl : : Cl : Cl : = Cl 2 Fig. 1.4. Covalent bonding in chlorine molecule. Another example of covalent bonding is a molecule of methane, C 4 in which carbon atom and hydrogen atoms share the electrons. The carbon atom has four electrons in its outer shell, and these are joined by four more electrons, contributed singly by each of the four hydrogen atoms (Fig. 1.5). The number of covalent bonds formed by an element is equal to (8 N), where N is the number of electrons outside the full shell. Covalent bond provides strong attractive forces between atoms. The solid material Diamond, which is the hardest material found in nature formed under this kind of sharing of atoms is another example. The two semiconductors germanium and silicon are the other two examples. Q.1.5. Write a short note on metallic bond bringing in some of the properties of metallic crystals. Ans. Metallic bond arises from the sharing of outer electrons by all the atoms of a crystal. In copper, for example, each atom contributes roughly one electron to the crystal as a whole. These free electrons, leaving the metal atoms in place as ions, serve as pervasive glue, moving among the ions and binding them together. Because of the free electrons metallic crystals are opaque and strongly reflecting; they are excellent conductors with resistance decreasing as the temperature is lowered. Excellent thermal conductivity comes also from the free electrons. Typical metals have moderately high melting points; they are mechanically tough and ductile rather than brittle. The binding energy is usually a few electron volts per atom. Q.1.6. List out the properties of ionic crystals and covalent crystals. Ans. (1) Properties of ionic crystals (i) The ionic bond is fairly strong. (ii) They have high melting point and high boiling point due to high energies. (iii) The electrical conductivity is much lower than that of a metal and it increases with temperature. Ions themselves are responsible for charge transport. (iv) They are transparent for all frequencies up to the point called the fundamental absorption frequency. Above this frequency, they are opaque. (v) They usually crystallize in relatively closed packed structure. (2) Characteristics of covalent crystals (i) Covalent crystals are usually hard (diamond is the hardest substance known), brittle having high binding energies and thus have high melting and boiling points. (ii) Bonds are strongly directional. C Fig. 1.5. Covalent bonding in a molecule of methane (C 4 ).
Bonding and Crystal 5 (iii) Bonds are saturated. (iv) The conductivity of the crystal varies over a wide range. Some crystals are insulators (diamond), and some have even higher conductivities. The conductivity increases with increase of temperature. (v) They are transparent to long wavelength radiation but opaque to shorter wavelength. Q.1.7. Enuciate with suitable figures and examples the different types of intermolecular bonds (the van der Waals bond). Ans. Unlike the three preceeding classes of bonding, in which electrons are either exchanged or shared, molecular bonds involve no transfer or exchange of charge. Rather the bond arises from the van der Waals interaction between electrically neutral molecules. Even an atom such as neon or argon, with a spherically symmetric electron distribution has a small instantaneous electric dipole moment as a result of motions of the electrons. This produces at the location of a neighbouring molecule a small electric field which polarizes the latter. A weak attractive interaction results. The potential energy associated with the dipoles is proportional to the square of these moments and to the inverse sixth power of the separation. Because the dipole moments are very small an ion-core repulsion prevents the molecules from coming very close, molecular binding is weak of the order of a few hundredths of an electron volt per atom. This small binding leads to a very low melting and boiling points. The crystals are electrically insulating, insoluble, soft, and readily deformable. Besides the three primary bonds as discussed above, there exists various secondary bonds. Secondary bonds are also called molecular bonds and are secondary in the sense of being relatively weak in comparison with the three primary bonds. The bond energy of secondary bonds vary between 4 to 42 kj/g mole. Intermolecular bonds do not involve the transfer or sharing of electrons between atoms. The three types of intermolecular bonds are: (i) Dispersion bonds (ii) Dipole bonds (iii) ydrogen bonds (i) Dispersion bond: As electrons rotate around their nuclei, they tend to keep in phase; and since the electrons of adjacent atoms in a molecule tend to repel each other, the result is that the molecule has a small fluctuating net charge on each end. [The hydrogen molecule is instantly charged negatively on the right end and positively on the left] The fluctuating charge on one molecular tends to interact with the fluctuating charge on a neighbouring molecule, resulting in, a net attraction and thus the dispersion bond. CAPTER 1 + + + F + 105 O + O 105 Bond () i () ii ( iii) + Fig. 1.6. Intermolecular bonds. (i) Dispersion effect (electronic polarization), (ii) Dipole bond, and (iii) ydrogen bond.
6 Essentials of Engineering Physics Molecules of inert gases, which consist of single atoms, are held together by dispersion forces when the gases are solidified. (ii) Dipole bond: Consider a molecule of hydrogen fluoride. There are two electrons that surround the positive charges in the nucleus of the hydrogen and there are eight electrons that surround the nucleus of the fluorine atom i.e., electrons surround nucleus of fluorine more completely than that of hydrogen. This creates an electric imbalance. Consequently, the centre of positive charges and the centre of negative charges do not coincide, rather remain separated. This produces an electric dipole. An electric dipole provides a mechanism for molecular bonding. The presence of a permanent dipole moment increases the attractive forces between molecules and facilitates their closer approach. A dipole bond is much weaker than the ionic bond but it is considerably stronger than the dispersion bond. (Strong) Attractive forces between positive & negative ions Shared valence electrons (a) (Positive) Ion (b) (Positive) Metal ion (Weak) Attractive forces between polarized atoms (Negative) Electron cloud (c) Centres of positive & of negative charges separated in each atom (d) 1. Weak attractive forces between polarized atoms. 2. Centre of + and of - electricity separated in each atom. Fig. 1.7. The arrangement of atoms in. (a) Ionic bond, (b) Covalent bond, (c)metallic bond, and (d) Molecular bond.
Bonding and Crystal 7 (iii) ydrogen bond: This is a special or peculiar type (strong) dipole bond that occurs between the molecules in which one end is hydrogen atom. The one electron belonging to the hydrogen atom is fairly loosely held, and it may keep all the electrons around itself, leaving the hydrogen atom, in effect, as a positive ion. This tendency can produce a strong permanent dipole that can bond to other similar dipoles with a force nearly equal to that involved in the ionic bond. A good example of hydrogen bonding is water. Argon, calomel, ice, paraffins and solid carbondioxide possess molecular bond. CAPTER 1 Properties of Molecular Crystals 1. Molecular crystals have small binding energy. 2. Molecular structure can be both crystalline and non-crystalline. 3. They are usually transparent to light. 4. They are good insulators as no valence electrons are available. 5. They are soluble in both polar and non-polar solvents. 6. They have low melting points because of extremely weak van der Waals forces. Table 1.1. Bond energies of some selected materials S. Materials Type of bond Bond Energy No. (kj/g mol) 1. Iron Metallic 401.3 2. Sodium chloride Ionic 639.3 3. Silicon dioxide Covalent 1692.9 4. Nitrogen Inter molecular 7.8 (van der Waals) Q.1.8. Write a brief note on crystals with mixed bonding. Ans. The stability of interatomic bonds depends upon the extent of overlapping of orbitals. In the same shell, p orbitals tend to overlap more than the s orbitals when the two atoms come close together. Therefore, p orbitals are preferred for most stable interatomic bonds. The resulting bond angles are obviously close to 90 since this is the angle the p orbitals make with one another. In some cases the bonds are partially ionic in character and therefore large deviation in bond angles result. The bond angles in such cases may be of the order of 110 or so. The deviation in bond angles from 90 may be due to hybridisation of the s and p orbits (or the wave functions). A typical example of hybrid bonding is carbon-carbon bonds formed in diamond. Besides diamond, hybridisation occurs in organic molecules, such as methane. Each carbon atom has four valence electrons that can take part in covalent bonding. Covalent bonds are formed not only due to overlapping of pure s or pure p orbitals but also due to overlapping of s and p orbitals, called the hybrid bonding. The electronic configuration of normal carbon atom is 1s 2 2s 2 2p 2 with the electron spin paired in the 1s and 2s orbitals. Only the 2p electron spins are unpaired and are free to form bonds. Therefore, one can expect only two carbon bonds. But when the carbon atoms approach each other, the s and p orbits overlap.
8 Essentials of Engineering Physics As a consequence one s electron is excited to p orbital and aligns itself parallel to other two. This is in accordance with unds rule that p orbital is half filled now. The electronic configuration of a carbon atom now can be represented as 1s 2 2s 1 2p 3 and the electron spins are distributed as: 1s 2 ( ) 2s 1 ( ) 2p 3 ( ) This configuration results in four unpaired electron spins and these unpaired electrons interact rather strongly with one another and modify the shapes of s and p orbitals. The electrostatic repulsion is so great that each electrons cloud concentrates itself away from those of the other three (see Fig. 1.8) and the four arrange themselves with the largest possible angle 109.5, between each pair. The favourable bonding directions of these orbitals are disposed towards 4 corners of a regular tetrahedron as you see in the second figure (1.9). Such a rearrangement of Nucleus Valence electron clouds Core electron clouds Fig. 1.8. The lobe-shaped valence electron clouds in the carbon atom. orbitals is called hybridisation and the four orbitals are called sp 3 hybrids. The four bonds are of equal strength. (a) (b) Fig. 1.9. (a) The unit cell of the diamond structure; (b) the crystal of diamond is built by stacking many unit cells together. Because of the strong repulsion, this rearrangement is difficult to distort and the carbon atoms are joined up with the electron clouds of the other neighbouring atoms pointing towards one another. Thus each carbon atom is surrounded by 8 electrons which is in agreement with the 8 N rule. The structure so formed can be considered as interpenetrating two fcc lattices along the body diagonal by 1/4 cube edge as illustrated in Fig. 1.9. The structure is very strong and rigid. Covalent bonds similar to diamond are found in CCl 4, SiF 4, C 4, etc. The bond in graphite is sp 2 hybrid in which 3 bonds are formed in a plane with interbond angles of Fig. 1.10. The structure of graphite. 120 and the fourth bond is formed in a direction
Bonding and Crystal 9 perpendicular to sp 2 bonds and between p orbital and sp 2 bonds are shown in Fig. 1.10. The perpendicular bond is however a weak bond and hence graphite has very low yield strength in this direction. This is why graphite in flaky is nature. B. CRYSTAL STRUCTURE The conditions of temperature and pressure at the surface of the earth are such that a large number of materials exist in the solid state, in which they maintain both shape and volume. Ionic and covalent bindings are dominant for some solids, but a different variation of the coulomb interaction leads to metallic crystal with high strength, ductility, and conductivities, both electrical and thermal. Some scientists prefer to reserve the term solid for what we shall call a crystalline solid. Of a piece of glass, a bar of silver and a strip of wood, only the silver qualifies as crystalline. The distinction is made on the basis of the kind of order which exists in the material i.e, on the type of geometrical arrangement in which the atoms fall. By means of X-rays it is possible to determine the relative positions of various kinds of atoms in a material. Suppose for a moment that we could actually see the individual atoms in a piece of glass. If we locate a silicon atom and examine its neighbours, we find that they are not distributed in a random fashion. There is a definite geometrical order. owever, if we look at other silicon atom, we find that their neighbours form a somewhat different pattern. For glass there is no orderly arrangement which persists throughout the material; there is a short range order but not long range order. Such a material which retains its shape is sometimes called an amorphous solid or a supercooled liquid. Crystalline solids exhibit long range order; in a large crystal the ordering extends uninterrupted over distances of many million atomic diameters. CAPTER 1 QUESTIONS AND ANSWERS Q.1.9. Explain crystal lattice, Basis vectors and crystal structure. Ans. For discussing crystals it is convenient to introduce the concept a space lattice, a network of straight lines constructed in such a way that it divides space into identical volumes with no space excluded. The lattice points are at the intersections of these lines. Associated with each lattice point there is either a single atom or an identical complex of atoms. For every crystal there is thus a network of lattice points for each of which there is either a single atom or group of atoms called the basis of the crystal. The arrangement of atoms in a crystal used to be periodic earlier. For convenience (first) we may replace each atom by a geometrical point located at the equilibrium position of that atom. The resulting pattern of these imaginary special points is termed as the crystal lattice or simply lattice and the points are referred to as lattice points. In such an arrangement, any point has the same environment as every other point. Such arrays of lattice points in two or three dimensions are called plane lattice and space lattice respectively. There are two classes of b O a Fig. 1.11. Two-dimensional array of points (A plane lattice).
10 Essentials of Engineering Physics lattice, namely Bravais and non-bravais. In a Bravais lattice, all lattice points are equivalent. ence all the atoms in the crystal are of the same kind. But in a non-bravais lattice, some of the lattice points are not equivalent. Consider any lattice point O, in a plane lattice shown in Fig. 1.11. For any type of simplest lattice there exist two fundamental translation vectors Æ a and Æ b ; these are the shortest vectors which satisfy the condition that if we go from any initial point in a crystal any integer n Æ 1 a distances, n Æ 2 b distances, we arrive at a point which has identically the same environment as the initial point. Thus any two lattice points can be connected by a translation vector T = n Æ 1 a + n Æ 2 b. Similarly a space lattice can be constructed using a set of basis vectors Æ a, Æ b and Æ c along the three directions of the crystallographic axes. With respect to any point as origin, location of any other lattice point is written as T = n Æ 1 a + n Æ 2 b + n Æ 3 c...(1.5) where n 1, n 2 and n 3 are arbitrary integers and T is the translation vector in three dimensions. Q.1.10. Discuss unit cells, lattice parameters and crystal systems. Ans. As already explained, a space lattice can be considered as an infinite array of points in space, so arranged that it divides space into volumes with no space excluded. Every point, which is called a lattice point, has identical surroundings with every other point. The smallest volume that contains the full pattern of repetition is called a unit cell. Identical unit cell must completely fill the space when they are packed face to face, thus generating a space lattice. If a unit cell is so chosen that it contains lattice points only at its corners, it is called a primitive unit cell or simple unit cell. A primitive unit cell contains only one lattice point because each point at eight corners is shared equally with adjacent unit cells. The edge of the unit cell, called a lattice constant or a lattice parameter. Crystal structure is got by attaching an atom or a group of atoms/molecules to each lattice point. The group of these atoms/molecules is called basis as said earlier. Periodic repetition of the basis in space generates a crystal structure. Thus crystal structure = Lattice + Basis Lattice + Basis Structure + = Primitive lattice Basis Fig. 1.12. Crystal structure.