ATOMIC SCALE STRUCTURES
DESIGN EXAMPLE 2.5–2
2.6 ATOMIC PACKING AND COORDINATION NUMBERS
2.6 ATOMIC PACKING AND COORDINATION NUMBERS
As demonstrated above, several important macroscopic properties of materials can be estimated on the basis of only a knowledge of the atoms present, the type of bonding between atoms, and the shape of the bond-energy curve. Other properties, such as density, largely depend on the arrangement of the atoms in the solid. In this section we discuss the factors that influence the three-dimensional packing of atoms.
The arrangement of atoms within a solid can be principally characterized by the number of nearest neighbors, orcoordination number(CN), of each atom in the struc-ture. In turn, the coordination number is influenced primarily by the type of bonding present and by the relative sizes of the atoms or ions.
Consider the ionic compound CsCl. As discussed in Section 2.4.1, each ion can be approximated as a hard sphere. Because of the nature of the coulombic attractive force, we expect each Cs⫹cation to be surrounded by as many negatively charged Cl⫺anions as is geometrically possible. The total energy for the system is minimized when the number of oppositely charged nearest neighbors for each ion is maximized. The geometric factor responsible for determining the CN of ions is the ratio of the radii of the ions.
Using the procedure outlined in Section 2.4.1, the equilibrium separation distance for CsCl, x0(CsCl), can be found from the bond-energy curve. It is then possible to write
r共Cl⫺兲 ⫹ r共Cs⫹兲 ⫽ x0共CsCl兲 (2.6–1)
where the terms on the left are the radii of the ions. Similar equations may be written for a variety of compounds, and the resulting set of simultaneous equations may be solved for the respective radii. Using this procedure, a self-consistent set of ionic radii can be calculated. The result justifies the use of a rigid sphere model for describing ionic solids.
The atomic and ionic radii of the elements are given in Appendix C.
Let r represent the radius of the smaller ion, usually the cation, and let R represent the radius of the larger ion, usually the anion. The relationship between the ratio of the radii and the resulting CN can be determined using the following constraints: (1) cations
“touch” anions, (2) the number of anions surrounding a given cation will be as high as geometrically possible, and (3) the ions cannot overlap.
Consider the geometry when a small cation is bonded to a much larger anion. It will always be possible to place two anions in contact with the cation, but as shown in Figure 2.6–1, it may not be possible to place a third anion in contact with the cation
120° 30° r
R
R r
R (a)
(b) (c)
cos 30° = R/(r + R) r/R = 0.155 Region of
overlap
without forcing the anions to overlap. The repulsive force between the ions prevents this overlap. Since it is impossible for the smaller ion to have CN⫽ 3, this ratio of r兾R results in CNcation ⫽ 2.
If the radius of the cation is gradually increased while holding the radius of the anion constant, a value of r兾R is eventually reached for which CN ⫽ 3 is possible (see F ig-ure 2.6–1c). As r兾R continues to increase, CNs of 4, 6, 8, and eventually 12 become possible. The critical radius ratio for each coordination number is given in Table 2.6–1.
Note that these are minimum values for each CN. Although a given CN is geometrically possible for any r兾R ratio greater than the value given in Table 2.6–1, it will be energet-ically favorable only until the minimum value of the next highest CN is reached. Thus, the maximum value of r兾R for CN ⫽ 8 is the minimum r兾R value for CN ⫽ 12.
Examination of Appendix C shows that the radius of an anion is generally larger than the radius of the corresponding neutral atom, and the radius of a cation is usually smaller than that of the neutral atom. This behavior can be readily understood on the basis of electron-electron and electron-proton interactions. In cations, the magnitude of the elec-tronic repulsions decreases with the loss of electrons, and the positive charge in the nucleus is able to attract the remaining electrons more closely. The result is r⫹⬍ r0. Using the inverse argument, it can be shown that r⫺will be greater than r0.
Since r兾R must always be less than or equal to 1, and since for most ionically bonded compounds r (cation)⬍ r (anion), the appropriate radius ratio is usually r (cation) 兾R(anion). If, however, r (anion) ⬍ r (cation), then the r兾R ratio should be used to
FIGURE 2.6–1 Geometry for CN⫽ 2: (a) two anions “fit” but (b) a third would require overlap and (c) the mini-mum r兾R value for a CN ⫽ 3 is shown to be 0.155.
R r
r R
r R
r r R
R
2 0 0 < r/R < 0.155 Always possible
Coordination number
Critical (r/R) value
(r/R) Stability
range Geometry
3 0.155 0.155 ≤ r/R < 0.225
4 0.225 0.225 ≤ r/R < 0.414
6 0.414 0.414 ≤ r/R < 0.732
8 0.732 0.732 ≤ r/R < 1
12 1 r/R = 1
TABLE 2.6–1 The critical共r兾R兲 ratio for each coordination number. (Note that the drawings are not to scale.)
estimate the CN of the anion. Once the CN of the smaller ion is known, the CN of the larger ion can be determined based on the cation : anion ratio, or the stoichiometry of the compound.
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EXAMPLE 2.6–1
Table 2.6–1gives the ionic radius ratio range for CN⫽ 6 as 0.414 ⱕ 共r兾R兲 ⬍ 0.732. Derive these limiting values by investigating the critical geometry for CNs of 6 and 8.
Solution
The geometry for the critical (minimum) r兾R ratio for CN ⫽ 6 is shown in Table 2.6–1 . If a represents the length of the edge of the cube, then when all of the ions are just touching each other
r ⫹ R ⫽a
2 and R⫹ R ⫽ a
兹2 Dividing the first equation by the second equation yields
r ⫹ R
2R ⫽ 1
兹2
Solving for the desired quantity yields r
R ⫽ 兹2 ⫺ 1 ⫽ 0.414
Since the maximum value for CN⫽ 6 corresponds to the minimum value for CN ⫽ 8,we must repeat the procedure for the critical geometry for CN⫽ 8. In this case,we find
r⫹ R ⫽a兹3
2 and R⫹ R ⫽ a
Dividing the first equation by the second and solving for r兾R yields r
R ⫽ 兹3 ⫺ 1 ⫽ 0.732
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EXAMPLE 2.6–2
Calculate the CNs,assuming ionic bonding,for each of the elements in each of the following compounds:共a兲 MgO, 共b兲 Cr2O3,共c兲 K2O.
Solution
From Appendix C we find that the relevant ionic radii are r共Mg2⫹兲 ⫽ 0.078 nm, r共Cr3⫹兲 ⫽ 0.064 nm, r共K⫹兲 ⫽ 0.133 nm,and r共O2⫺兲 ⫽ 0.132 nm. The stable ranges of r兾R for each CN are given in Table 2.6–1.
a. For MgO we find r共Mg2⫹兲兾R共O2⫺兲 ⫽ 0.078兾0.132 ⫽ 0.59,which corresponds to CN共Mg2⫹兲 ⫽ 6. Since the anion : cation ratio is 1 : 1,each anion will have the same num-ber of nearest neighbors as each cation so that CN共O2⫺兲 ⫽ 6.
b. For Cr2O3we have r共Cr3⫹兲兾R共O2⫺兲 ⫽ 0.064兾0.132 ⫽ 0.485,which corresponds to CN共Cr3⫹兲 ⫽ 6. Since there are more anions than cations in the structure,CN共O2⫺兲 must be less than CN共Cr3⫹兲. We use the anion : cation ratio of 3 : 2 to find that
CN共O2⫺兲 ⫽ 共2兾3兲CN共Cr3⫹兲 ⫽ 4.
c. In the compound K2O the radius of the anion is less than the radius of the cation. There-fore, r兾R is used to predict the CN of the anion as follows:
r共O2⫺兲兾R共K⫹兲 ⫽ 0.132兾0.133 ⫽ 0.992,which corresponds to CN共O2⫺兲 ⫽ 8. Since there are more cations than anions in the structure,CN共K⫹兲 must be less than CN共O2⫺兲. We use the anion : cation ratio of 1 : 2 to find that CN共K⫹兲 ⫽ 共1兾2兲CN共O2⫺兲 ⫽ 4.
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In contrast to ionic materials,for which CN is determined by geometry,the number of nearest neighbors in a covalently bonded material is determined by the number of elec-trons in the valence shell of each atom. The governing equation for the electronegative elements in Groups IV through VII is
NB⫽ 共8 ⫺ Nv兲 (2.6–2)
where NB is the number of covalent bonds formed and Nv is the number of valence electrons in the neutral atom. In most covalent solids CN is equal to NB. The exceptions to this rule are the covalent compounds containing double or triple bonds. It should also be recognized that the monovalent electronegative element hydrogen often forms a single covalent bond.
Let us consider as examples several compounds involving H共1s1兲 and C 共1s22s22p2兲.
Since an H atom requires only one additional electron to achieve a full valence shell, each H atom bonds to one additional H atom to form diatomic hydrogen. Since both atoms in the H2structure have filled valence shells,this diatomic molecule is stable and will not form any additional primary bonds.
C
The bond structure for carbon is more complex. Since C has only four electrons in its outer shell, it must share four pairs of electrons in order to achieve a full complement of eight valence electrons. In the compound CH4, the C atom shares one electron with each H atom (see Figure 2.6–2a). The resulting methane molecule is stable. Next, consider the bonding arrangement for pure carbon. Each C atom may form a single covalent bond with four other C atoms, giving rise to the highly stable three-dimensional diamond structure shown in Figure 2.6–2b.
What is the bond structure in the molecule C2H4? Each H can satisfy its bonding requirements by forming a single covalent bond with one of the C atoms. In turn, each C atom will be covalently bonded to two H atoms. Each C atom, however, must still form two additional covalent bonds in order to achieve a filled valence shell. This can be accomplished if the C atoms share two pairs of electrons. The C2H4molecule is shown schematically in Figure 2.6–2c. This molecule is fundamentally different from either H2
or CH4in that it contains a double bond. C2H4is an unsaturated hydrocarbon. We will see later that the double bond allows many C2H4molecules to react to form the polymer polyethylene.
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EXAMPLE 2.6–3
Use either of the shorthand electron notations shown in Figure 2.6–2 to depict the covalent bonding arrangement in each of the following materials: (a) H2O, (b) C2H6, (c) C2H3Cl, and (d) Si.
Solution
The number of covalent bonds formed is directly related to the number of valence electrons共Nv兲 in an atom (Equation 2.6–2). From Appendix A, Nv共H兲 ⫽ 1, Nv共C兲 ⫽ 4, Nv共O兲 ⫽ 6, Nv共Cl兲 ⫽ 7, and Nv共Si兲 ⫽ 4.
a. In the H2O molecule the O atom will form one covalent bond with each of the H atoms.
This arrangement of electrons, shown in Figure 2.6–3a, allows all three atoms to obtain filled valence shells.
FIGURE 2.6–2 Schematic illustrations of the covalent bond structure in a series of compounds: (a) CH4, (b) pure carbon in the diamond structure, and (c) C2H4. The x’s represent the electrons from the H atoms and the •’s and䡩’s represent those from the C atoms. Note the double bond in the compound C2H4.
FIGURE 2.6–3 Schematic illustrations of the covalent bond structure in a series of compounds: (a) H2O, (b) C2H6, and (c) C2H3Cl. The x’s represent the electrons from the H and Cl atoms and the •’s and䡩’s represent those from the C and O atoms. Note the double bond in the compound C2H3Cl.
C C
C
C
C
109.5°
3-D structure, with all angles = 109.5°
(a)
2-D (planar) structure, with all angles = 120°
(b) H
H
C O
120°
b. In the C2H6molecule each H atom is bonded to one of the C atoms. Since each C atom must form four covalent bonds, there is a single covalent bond bridging the two C atoms (see Figure 2.6–3b).
c. In the compound C2H3Cl each H and Cl atom forms a single covalent bond with one of the C atoms. Each C atom must form four covalent bonds, so that there will be a double bond between the two C atoms (see Figure 2.6–3c).
d. In silicon, each atom must be bonded to four other Si atoms, and the resulting structure is similar to the diamond structure described previously (see Figure 2.6–2b).
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Covalent bonds are directional and are characterized by specific bond angles. The bond angles can be determined by the geometry of the structure or vice versa. Shared electrons, or bond pairs, and lone electron pairs constitute mutually repulsive negative-charge centers that tend to separate as much as possible. As shown in Figure 2.6–4a, the bond angle in a tetrahedral structure such as diamond is 109.5⬚, which places nearest-neighbor C atoms (and their associated shared electron pairs)as far apart as possible in space while satisfying the valency requirements. In contrast, when carbon is bonded to only three other atoms (one of which involves a double bond), the resulting structure is planar with a bond angle of about 120⬚, as shown in Figure 2.6–4b. The existence of specific bond angles in covalent molecules is important in understanding the properties of polymers.
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EXAMPLE 2.6–4
Sketch the three-dimensional arrangement of covalent bonds in the H2O molecule.
Solution
The geometry of the H2O molecule can be envisioned by placing the O atom at the center of an imaginary cube and noting that its four pairs of electrons, two bonding and two nonbonding electron pairs, must be spatially separated as much as possible. This separation, shown in Figure 2.6–5, is obtained by placing the electron pairs along directions pointing to an alternating set of four corners of the imaginary cube. The H atoms are positioned at two of the cube corners associated with the FIGURE 2.6–4 A schematic illustration of covalent bond angles in two compounds: (a) the bond angle in a tetrahe-dral structure such as diamond is 109.5⬚; (b) when the C is bonded to only three other atoms (one of which involves a double bond), the resulting structure is planar with a bond angle of⬃120⬚.
H
O H
104.5°
bonding electron pairs. The structure of H2O deviates slightly from this model, since nonbonding electron pairs repel each other slightly more than bonding electron pairs. The result is that the H @ O @ H bond angle is 104.5⬚—slightly less than the predicted 109.5⬚.
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The shared electrons in a metallic bond are delocalized. Thus, the CN of an atom in a metallic solid is determined primarily by geometrical considerations. Indeed, many pure metals (e.g., Al, Cu, and Ni), for which r兾R ⫽ 1, have structures with a CN of 12;
however, several common pure metals such as Fe, Cr, and W have CNs of only 8, even in their purest forms.
Coordination numbers are useful because they describe theshort-range order, defined as the number and type of nearest neighbors, associated with a particular solid struc-ture. All solids exhibit short-range order. As we expand the consideration to include second- and higher-order neighbors, we find that there are two distinct types of solids.
Those that exhibit both short-range order (SRO) andlong-range orderare called crys-talline materialswhile those with SRO only are termedamorphous, ornoncrystalline, materials.