UNIT 1 Bonding in Solids & Crystal Structures and X-ray Diffraction 1.1 BONDING IN SOLIDS 1.1.1 INTRODUCTION The most convenient basis for solid state theory is a classification scheme based on the character of the interatomic bonding forces in various classes of crystalline materials. According to this scheme of classification, all solids fall into one of the five general categories: molecular, ionic, covalent, metallic and hydrogen bonded crystals. The distinction is not a sharp one, because some may belong to more than one class. It is a too fundamental fact that inert gases like helium, neon, argon, etc. exist in the atomic form and not easily combine with other atoms to form new compounds because of the magic number 2 in Helium and 8 in other cases. Interatomic force among the atoms in a solid is one of the tools to classify the solids and then to study their physical properties. 1.1.2 FORCES BETWEEN ATOMS AND BOND ENERGY The attractive electrostatic interaction between the electrons and positive charge of the nuclei is totally responsible to group them and then for holding together. Gravitational forces and magnetic forces are negligible compared with the said interaction. When two atoms (or ions) come close to one another, there will be repulsion between negatively charged electrons of both atoms. The repulsive force increases very rapidly as the distance of separation decreases. However when the distance of separation is large, there is attraction between positive nucleus and negative electrons. At some optimum distance (say r = ro), the attractive and repulsive forces just balance and hence the resultant force becomes 2 Applied Physics zero. Hence the potential energy at this equilibrium spacing, ro becomes minimum. This magnitude of minimum energy is called bond energy. It is usually expressed in kJ/kmol. The value for primary bond ranges from 100 × 103 kJ/kmol to 1000 kJ/kmol and that of secondary bond ranges from 1 × 103 to 60 × 103 kJ/kmol. The centre to centre distance of two bonding atoms is called the bond length. The bond length increases on heating. The bond energy and dissociation energy will be equal but opposite in sign. A crystal can only be stable if its total energy (K.E + P.E) is lower than the total energy of the atoms or molecules when they are free. The cohesive energy is defined as the difference of free atom energy and crystal energy. The variation of interatomic forces and potential energy vs distance of separation is shown in Fig. 1.1. The general equation that represents the force between two atoms or molecules or ions is A B F = − ...(1.1.1) r M rN A, B, M and N are constants; r is the interatomic distance. Fig. 1.1 Interatomic forces and potential energy vs distance of separation. From the above equation, one can arrive at the equation for the potential energy of the system: A 1 B 1 i.e., U(r) = F dr = − + + C ∫ M − 1 r M +1 N − 1 r N +1 a b = − + + C rm r n When U = 0 for r = ¥, C will be zero. Thus a b U(r) = − + ...(1.1.2) rm r n Bonding in Solids and Crystal Structures and X-ray Diffraction 3 Here a, b, m and n are different constants. The first term of the above equation refers to attraction and the second term repulsion. n is greater than m with n of the order of 9 and m = 2 for ionic structures. The forces of attraction result from interaction between outer electrons of two atoms. The forces of repulsion are from interpenetration of outer electronic shells. Only these forces decide the nature of bonds in solids. The equilibrium distance r0 may be determined as follows: dU am bn 0 = − = m+1 n+1 dr r=r 0 r0 r0 nb i.e., rn−m = 0 ma 1 nb nm− r = 0 ma Also m nb n r = r0 ...(1.1.3) 0 ma 1.1.3 COMPUTATION OF COHESIVE ENERGY The energy will be minimum at r = r0 a b − + i.e., []U(r) min = m n r0 r0 n m nb Using r0 = r0 , we get ma a mab − + Umin = m m r0 nbr0 a m i.e., U = − 1 − ...(1.1.4) min m r0 n Conclusion: All stable arrangements of atoms in solids are such that the potential energy is minimum. This is the only way of explaining the cohesion of atoms in solid aggregates. 4 Applied Physics 1.1.4 IONIC BONDING Ionic crystals are those compounds in which the valence electrons are completely transferred from one atom to the other; the final result being a crystal that is composed of positively and negatively charged ions. Atom whose outermost shell has only a few electrons given up these electrons so that it is left with completely filled outermost shell. This process is termed ionisation. On the other hand if the outermost shell is slightly short of eight electrons, then the atom has a tendency to receive the electrons and become an ion. An atom of sodium metal has one electron in its outermost shell and it can be easily released with a sodium positive ion left with. This electron can be easily added to the outermost shell of chlorine with seven electrons already. This chlorine atom becomes a negative ion. Because of the mutual attraction between positive and negative ions, a bond is developed between these two ions of opposite charges. The outermost electron of sodium is removed by supplying 495 × 103 kJ/kmol (called first ionisation potential of sodium). + 1 Na + E1 ® Na + e This released electron will now move to occupy the outermost shell of chlorine (with already seven electrons) and produce a negatively charged ion. Cl + e1 ® Cl The electron affinity of chlorine is 370 × 103 kJ/kmol. Thus net increase in potential energy for transferring an electron from sodium to chlorine is 125 × 103 kJ/kmol. The chemical reactions in the formation of NaCl at the equilibrium spacing are summarized below: Fig. 1.2 Schematic representation of the formation of ionic molecule of sodium chloride. Na + Cl ® Na+ + Cl ® NaCl Since chlorine exists as molecules, the chemical reaction must be written as 2Na + Cl ® 2Na+ + 2Cl ® 2NaCl 2 1.1.5 BOND ENERGY OF NaCl MOLECULE NaCl is one of the best examples of ionic compound and let the sodium and chlorine atoms be free at infinite distance of separation. The energy required to remove the outer electron from the Na atom (ionisation energy of sodium atom), leaving it a Na+ ion is about 5 eV; or Na + 5 eV ® Na+ + e Bonding in Solids and Crystal Structures and X-ray Diffraction 5 Similarly when the removed electron from sodium atom is added to chlorine atom, about 3.5 eV of energy (electron affinity of chlorine) is released. Cl + e ® Cl + 3.5 Thus a net energy of (5.0 3.5) = 1.5 eV is spent in creating a positive sodium ion and a negative chlorine ion at infinity. i.e., Na + Cl + 1.5 eV ® Na+ + Cl Now the electrostatic attraction between these ions brings them to the equilibrium spacing ro = 0.24 nm and the energy released in the formation of NaCl molecule is called the bond energy of the molecule and it is obtained as follows: e2 (1.6 ×10 −19 )2 − = − V = −12 −9 4π ∈0 r0 4π(8.85 × 10 )(0.24 ×10 ) 2.56 × 0.0096 ×10 −16 = 2.56 ´ 0.0096 ´ 1016 J = − eV 1.6 ×10 −19 V = 6 eV ...(1.1.5) Thus the energy released in the formation of NaCl molecule from the neutral Na and Cl atoms is (5.0 3.5 6) = 4.5 eV. Schematically, Na+ + Cl ® Na + Cl + 4.5 eV ® NaCl ¬ 0.24 nm ® This means that to dissociate one NaCl molecule 4.5 eV energy is required. 1.1.6 CALCULATION OF MADELUNG CONSTANT AND LATTICE ENERGY OF IONIC CRYSTALS The lattice energy of an ionic solid will differ from the bond energy of diatomic ionic molecules since, in the former case there will be interactions between more atoms. The cohesive energy of an ionic crystal is the energy that would be liberated by the formation of the crystal from individual neutral atoms. The arrangement of Na+ and Cl ions in a sodium chloride crystal is shown in Fig. 1.3. Each sodium ion in sodium chloride is subject to the attractive potential due to 6 chlorine ions each at a distance r. Thus the attractive potential at the sodium ion by the chlorine ion is e2 −6 U1 = 4π ∈0 r The next nearest neighbours are 12 of the same kind of ions at a distance r( 2). Thus e2 12 U2 = π ∈ and so on. 4 0 r()2 6 Applied Physics Fig. 1.3 Sodium chloride structure. Thus the total potential energy is 6e2 12e2 8e2 U = − + − + ... coub π∈ π∈ π∈ 4 0 r 4 0 r( 2) 4 0 r( 3) e2 12 8 6 = − 6 − + − + ... 4π ∈0 r 2 3 2 12 8 6 Let a = 6 − + − + ... 2 3 2 All these relations are applicable only to univalent alkali halides (i.e., Z1 = Z2 = 1). According to the scheme of Evjen, a = 1.75 which is close to the accurate value of 1.747558 obtained for NaCl structure by the Ewold method. This constant is called Madelung constant. Thus e2 − α Ucoub = 4π ∈0 r The potential energy contribution of the short-range repulsive forces can be expressed as B U = rep r n Bonding in Solids and Crystal Structures and X-ray Diffraction 7 where B is a constant and Urep is positive which increases rapidly with decreasing internuclear distance r.