Appendix 2A Indices of Planes and Directions in Hexagonal Crystals

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1 Appendix 2 Appendix 2A Indices of Planes and Directions in Hexagonal Crystals Figure 2A.1a shows a hexagonal unit cell defined by two equal unit cell lattice parameters a 1 and a 2 in the basal plane, and the lattice parameter c that is perpendicular to the basal plane. To produce the hexagonal crystal, the unit cell defined by a 1, and c is rotated about the c axis twice by 120 degrees. The crystal planes are expressed with three Miller indices (hkl ) within the unit cell defined by a 1, and c, using the procedures outlined in Sections and However, the rotation of a 2 by 120 degrees about the c axis produces a 3. A hexagonal crystal can be defined by four lattice parameters a 1, a 3, and c, and most crystallographers use the four-index system. However, a 3 is not an independent lattice parameter; it is geometrically related to a 1 and a 2. The four Miller indices (hkl ) of a plane for a hexagonal crystal are determined by first using the procedure for three Miller indices, outlined in Section 2.4.2, to determine (hkl ) with the unit cell defined by a 1, and c. Then the fourth index i is determined from Equation 2A.1: h 1 k 5 2i 2A.1 where i is the Miller index related to the a 3 lattice parameter. Figure 2A.1a shows a hexagonal crystal with some planes noted in the four-miller-index system. The Miller index i corresponds to the inverse of the plane intercepts along the a 3 axis. c (0001) c (11 00) (112 0) (101 0) [100] = [21 1 0] a 3 a 1 (112 1) a 3 a 2 a 1 (a) a 3 a 2 [110] = [112 0] a 1 1 (b) +2 [010] = [ ] a 2 Figure 2A.1 (a) A unit cell defined by two equal unit cell lattice parameters a 1 and a 2 in the basal plane, and the lattice parameter c that is perpendicular to the basal plane. To produce the hexagonal crystal, the unit cell defined by a 1, and c is rotated about the c axis twice by 120 degrees. (b) Some directions in the hexagonal unit cell. ((a) Adapted from Barrett, C., and Massalski, T. B., Structure of Metals, 3rd revised ed., Pergamon, New York (1980), p. 12. (b) Reprinted from Askeland, D. R., Fulay, P. P., and Wright, W,., The Science and Engineering of Materials, 6th ed., Cengage Learning, Stamford, CT (2011), p. 81.)

2 W-132 CHAPTER 2 Directions are not as easy to visualize and deduce in the four-index system as are planes. If the directional indices [u9 v9 w9 ] are determined in the three-index system using the cell defined by a 1, and c, then the four indices [uvtw] are defined by the following equations: u 5 n 3 s2u9 2 v9d, v 5 n s2v9 2 u9d, t 5 2su 1 vd, and w 5 nw9 3 2A.2 The factor n is a number that may be necessary to reduce the indices [uvtw] to the lowest possible integers. Example Problem 2A.1 Determine the four Miller indices for the direction [010] defined with three Miller indices. In the unit cell defined by a 1, and c, the direction [010] is along the a 2 axis. In the four-index system, the Miller indices are derived from Equation 2A.2. u 5 n 3 s2u9 2 v9 d s0 2 1d v 5 n 3 s2v9 2 u9 d s2d t 5 2su 1 vd w 5 0 If n 5 3, then the four-directional Miller indices are [1210]. Although it is difficult for anyone but crystallographers to visualize the [1210] direction in the hexagonal unit cell being parallel to the a 2 axis, the advantage of the four-index system is that equivalent directions have related indices. For example, the [1120] direction is parallel to the a 3 axis. However, in the three-index system the direction parallel to the a 3 axis is given by [110], and it is not obvious that this is equivalent to [010] along the a 2 axis. Appendix 2B The Lennard-ones Potential for Inert-Gas Atoms If m 5 12 and n 5 6 in Equation 2.9, this is called a Lennard-ones pair potential [V PL (r)], and it is appropriate for the inert-gas atoms. A convenient form of Equation 2.9 for a pair potential with m 5 12 and n 5 6 is V PL srd r r2 6 2B.1 4

3 Atoms, Chemical Bonding, Material Structure, and Physical Properties W-133 Table 2B.1 Values of the Atom Pair Bond Energy (2«) and for Inert Gas Atoms (1Å m ) Experiment Gordon/Kim Inert-Gas Atom in Å «in 10 3 ev in Å «in 10 3 ev He Ne Ar Kr Xe Experimental values of s and «are from Bernardes, N., Phys. Rev. 112 (1958) 1534, and the values of Gordon, R.G. and Kim, Y.S.,. Chem. Phys. 56 (1972) 3122 where 2 is the bond energy per atom pair, is equal to the value of interatomic separation (r) where V PL (r) is equal to 0, and the equilibrium interatomic separation (r 0 ) is equal to Values of «and for the inert-gas atoms are presented in Table 2B.1. From Equation 2.11, if the pair potential is given by Equation 2B.1, the applied force required to move the atom to the position r is given by F PL srd 5 dv PL srd dr r 21 r r 21 r2 6 4 In calculations of force from interatomic potentials, it is necessary to use a standard system of units, such as SI, and to convert the values of to joules and to meters, because of the differentiation. 2B.2 Example Problem 2B.1 With the potential for Group 0 atoms in Equation 2B.1, prove that the equilibrium interatomic separation (r 0 ) is equal to 1.12, and prove that the cohesive energy of a pair of atoms is 2. The equilibrium interatomic separation is found using Equation 2.12: dv p d F p d dr Substituting for V p from Equation 2B.1 into Equation 2.11 and setting r 5 r 0 results in dv PL dr Setting the terms inside the square brackets to 0 results in f 12 s212dr s26dr 0 27 g r s6dr 0 27

W-134 CHAPTER 2 Dividing each side of the equation by 6 s6dr 0 27 results in Solving this equation for r 0 results in 2 6 r 0 26 5 1 r 0 5 2 1y6 5 1.

4 W-134 CHAPTER 2 Dividing each side of the equation by 6 s6dr 0 27 results in Solving this equation for r 0 results in 2 6 r r y Substituting the value of r y6 into Equation 2B.1 to evaluate the pair bond energy at equilibrium results in V PL y y As was discussed in association with Equation 2B.1, is equal to r 0 /1.12, and 2 is equal to the pair bond energy at the equilibrium interatomic separation. Example Problem 2B.2 Determine the equilibrium interatomic separation for solid argon using the data for the Lennard- ones pair potential in Table 2B.1. From Table 2B.1 for argon, nm. To obtain the interatomic separation for an argon crystal we can use r y (0.340) nm. Example Problem 2B.3 (a) From the data in Table 2B.1, construct a Lennard-ones pair potential for a pair of argon atoms. (b) Construct an equation for the cohesive energy of solid argon as a function of interatomic separation. a) From Table 2B.1 the experimental values for argon are ev and nm. Inserting these values into Equation 2B.1 for the Lennard-ones pair potential gives us V PL srd 5 4s e Vd n m r n m r b) In the FCC structure of argon, each atom has twelve nearest neighbors that form six pair bonds. If only nearest-neighbor interactions are considered, then from Equations 2B.1 and Equation 2.13, the cohesive energy of the argon crystal as a function of interatomic separation is given by n m V c o h srd s e Vd 31 r n m r 2 6 4

5 Atoms, Chemical Bonding, Material Structure, and Physical Properties W-135 Appendix 2C Ionic-Crystal Separation Energy and Cohesive Energy An ion-pair potential is required to calculate the ionic-crystal cohesive energy. One approach discussed by Harrison for forming an interionic pair potential [V Pion (r)] that does not require the evaluation of any new parameters is to model the coulombic interaction of ions [V C (r)] with Equation 2.14 and to model the interaction of the ions that have inert-gas atom electron configurations with a Lennard-ones pair potential [V PL ( r)] using Equation 2B.1, as shown in Equation 2C.1. V Pion srd 5 V C srd 1 V PL srd 5 2 szed2 4 0 r r r C.1 In Equation 2C.1 the value of is the cohesive energy for the inert-gas atom configurations of the ions, not including the coulombic contribution [V C (r)], and likewise 1.12 is the equilibrium interatomic separation for the inert-gas atom configuration and not for the ionic solid. Values for and are taken from Table 2B.1 for ion-pair configurations, such as Ne Ne, that result from Na 1 and F 2 ions, and Ar Ar values are used for K 1 and Cl 2. The values of and for different Group 0 atom pairs are taken from Table 2C.1. For example, in NaCl the Na 1 ion has the neon electron configuration, and Cl 2 has the argon electron configuration; thus and are taken from the NeAr line in Table 2C.1. Two sets of data are presented: values by Gordon and Kim and average values based upon the Ne Ne and Ar Ar values in Table 2B.1. With this formalism it is possible to construct an interionic potential for materials such as MgO. For MgO the value of Z in Equation 2C.1 is 2; both Mg 12 and O 22 have the electron configuration of neon; thus the values of and for MgO are those for neon in Table 2B.1. Table 2C.1 Lennard-ones potential parameters and for mixed inert-gas atom bonds in Å «in ev Atoms Gordon/Kim Average Gordon/Kim Geometric mean HeAr NeAr NeKr ArKr Values are by Gordon, R. G., and Kim, Y. S.,. Chem. Phys., 56 (1972) 3122; and average values of and geometric-mean values of are based on values from Table 2B.1. Example Problem 2C.1 Develop a pair potential for K 1 and Cl 2 ions. K 1 and Cl 2 ions both have the electron configuration of argon; thus the Lennard ones part of the KCl potential is the same as that of argon. K 1 and Cl 2 are both singly charged ions; thus Z 5 1. Taking the experimental

W-136 CHAPTER 2 values in Table 2B.1 for argon: 5 10.4 3 10 23 ev and 5 0.340 nm. It is necessary to use a unified system of units to calculate the coulombic energy. In the SI system of units, 5 16.

6 W-136 CHAPTER 2 values in Table 2B.1 for argon: ev and nm. It is necessary to use a unified system of units to calculate the coulombic energy. In the SI system of units, per atom pair, and m. The total pair potential is V PK C L srd 5 V C srd 1 V PL srd 5 2 e2 4 0 r 1 4s d m r m r A component of the cohesive energy per atom of the ionic crystal is the sum of the pair-potential energies in Equation 2C.1 over the entire crystal, converted from energy per pair to energy per atom. In Equation 2C.1 the pair interactions associated with the Lennard-ones part of the potential are short range, and the potential energy is most significant for the nearest neighbors. Thus in a crystal with a coordination number of C N, there are C N /2 pair bonds to the nearest-neighbor ions, and the contribution to the energy per atom from the Lennard-ones part of the potential is approximated as V PL (r) (C N /2), where V PL (r) is the Lennard-ones pair potential energy in Equation 2B.1. However, the coulombic energy term extends over a long range, and it must be summed over the entire crystal. Sums of the coulombic potential over all ions in the crystal result in a coulombic energy per ion [ (r)] given by Equation 2C.2. srd 5 MV C srd MsZed2 8 0 r 2C.2 The Madelung constant (M) is the sum of the convergent series of the long-range coulombic interactions in Equation 2C.1 for each ion with all other ions in the crystal. The factor of 1 2 results from the change from energy per pair of ions to energy per ion. In the summation of the coulombic terms, the next nearest neighbors in the NaCl crystal structure, shown in Figure 2.26d, result in a repulsive interaction, because the charges are of the same sign. The Madelung constant has one value for a particular type of crystal structure, and some values of M are presented in Table 2C.2. All materials that form in the same crystal structure as NaCl have the same value of M For example, MgO forms in the same crystal structure as NaCl; thus MgO has the same value of M as NaCl does, even though the interionic separation in NaCl is different than in MgO. Other ionic crystal structures have their own unique values of M. Table 2C.2 The Madelung Constant (M) for Some Crystal Structures Structure Type M Rock salt (NaCl) CsCl Zinc blende (ZnS) Wurtzite (ZnS) Fluorite (CaF 2 ) Corundum (Al 2 O 3 ) Based on data from Chaing, Y-M., Birnie D. B., and Kingery, W. D., Physical Ceramics Principles for Ceramic Science and Engineering, ohn Wiley & Sons, New York (1997), p. 12.

7 Atoms, Chemical Bonding, Material Structure, and Physical Properties W-137 Example Problem 2C.2 Evaluate the first five terms of the Madelung constant for an infinite series of ions, with alternating unit charge, in a line with a spacing r 0. The series of ions appears as ` ` with a spacing of r 0 between ions. Pick one ion, such as the negative ion in the center of the sequence, and call this the origin. Then sequentially number all of the other ions in both directions. The coulombic potential per ion for the interaction of the ion at the origin with all the other ions ( ) in this one-dimensional crystal is as follows for an even number of ion pairs (N): 5 e r r r r r Nr e2 8 0 r N e2 8 0 r N2 2 For the first five pairs of ions of this one-dimensional crystal, M Including all other terms in the series the sum converges to M The sum is independent of the origin; starting at any ion yields the same result. Also, the value of M is independent of r 0 and of the lattice parameter. The crystal potential energy per ion [V cry (r)] as a function of interionic separation is then given by Equation 2C.3. V cry srd 5 1 M 2 2 V srd 1 C 1 C N 2 2 V PLsrd 5 2MsZed2 8 0 r C N r r C.3 This potential energy does not include the energy to create the ions. The crystal separation energy per ion [E sep (r 0 )] is the energy required to separate the ions from their equilibrium interionic positions in a crystal to infinity, divided by the total number of ions, and it is the negative of the crystal potential energy [V cry ( r 0 )] of the ions at their equilibrium interionic positions. Example Problem 2C.3 (a) Determine the equation for the crystal potential energy per ion as a function of interatomic separation for a KCl crystal. Assume that only nearest-neighbor interactions are necessary for the Lennard-ones portion of the potential. KCl has the same crystal structure as NaCl. (b) Use a spreadsheet program such as Microsoft Excel, or write a computer program to calculate the crystal potential energy per ion [V cry (r)] as a function of interatomic separation for KCl.

8 W-138 CHAPTER 2 a) In the structure of NaCl, shown Figure 2.26d, there are six nearest neighbors to the central atom. It is the same for every other ion in the NaCl structure. Each ion is surrounded by six ions of the opposite sign. In the Lennard-ones potential energy for the crystal, there are three nearest-neighbor pair bonds for each ion (C N /2). The Lennard-ones potential energy per ion in the crystal is then given by 1 C N 2 2 V srd 5 PL r r r 2 12 io n m m r The Madelung constant for the NaCl structure is 1.748, and Z 5 1 for KCl. The coulombic energy per ion in the crystal is calculated as follows. srd 5 2 MsZed2 8 0 r e2 8 0 r s C d C s F /m dr s F /m dr srd m 10 F s C 2 d 1 r ? m /io n r The total crystal potential energy per ion as a function of r is calculated by the following equation. V cry srd ? m /io n r io n m r m r b) Plotting this function using increments of 0.01 nm from 0.20 nm to 0.60 nm, and increments of 0.1 nm from 0.6 to 1.0 nm, results in Figure 2C Energy (ev) Interionic Separation (nm) Figure 2C.1 nanometers. The KCl interionic potential in energy electron-volts, as a function of interionic separation in

9 Atoms, Chemical Bonding, Material Structure, and Physical Properties W-139 The minimum in Figure 2C.1 corresponds to the negative of the separation energy of KCl, and the location of the minimum along the radius axis in nanometers corresponds to the equilibrium interionic separation. From Figure 2C.1, this appears to be at approximately 0.3 nm. The experimental value is nm, for excellent agreement. Example Problem 2C.4 Calculate the crystal potential energy per ion of KCl, using the experimental equilibrium interionic separation between nearest neighbors in KCl of nm at T 5 0 K. Substituting r m into the equation for V cry (r) developed in Example Problem 2C.3, V cry srd ? m /io n m V c ry d io n m m m m2 6 4 io n [s1.090d 12 2 s1.090d 6 ] io n V c ry d io n [ ] io n V c ry d io n [1.13] io n V c ry d io n io n io n e V io n The experimental value for the crystal potential energy of KCl at 0 kelvin (from Kittel, C., Introduction to Solid State Physics, 3rd ed., ohn Wiley, New York [1967], p. 80) is ev per ion. This is very good agreement, considering that no adjustable parameters are included in the potential and the Lennard-ones potential term is limited to nearest neighbors.

E12 UNDERSTANDING CRYSTAL STRUCTURES

E12 UNDERSTANDING CRYSTAL STRUCTURES E1 UNDERSTANDING CRYSTAL STRUCTURES 1 Introduction In this experiment, the structures of many elements and compounds are rationalized using simple packing models. The pre-work revises and extends the material