Selected crystal structures

1.9 Selected crystal structures

1.9.1 Pure metals

We now examine the crystal structures of various elements (metallic and non-metallic) and com- pounds, using examples to illustrate important structure-building principles and structure–property

relations. 9 Most elements in the Periodic Table are metallic in character; accordingly, we commence with them. Metal ions are relatively small, with diameters in the order of 0.25 nm. A millimeter cube of metal therefore contains about 10 20 atoms. The like ions in pure solid metal are packed together in a highly regular manner and, in the majority of metals, are packed so that ions collectively occupy the minimum volume. Metals are normally crystalline and for all of them, irrespective of whether the packing of ions is close or open, it is possible to define and express atomic arrangements in terms of structure cells (Section 1.6). Furthermore, because of the non-directional nature of the metallic bond, it is also possible to simulate these arrangements by simple ‘hard-sphere’ modeling.

There are two ways of packing spheres of equal size together so that they occupy the minimum volume. The structure cells of the resulting arrangements, face-centered cubic (fcc) and close-packed hexagonal (cph), are shown in Figures 1.15a and b. The other structure cell (Figure 1.15c) has a body-centered cubic (bcc) arrangement; although more ‘open’ and not based on close packing, it is nevertheless adopted by many metals.

In order to specify the structure of a particular metal completely, it is necessary to give not only the type of crystal structure adopted by the metal, but also the dimensions of the structure cell. In cubic structure cells it is only necessary to give the length of an edge a, whereas in a hexagonal cell the two parameters a and c must be given, as indicated in Figures 1.15a–c. If a hexagonal structure is ideally close packed, the ratio c/a must be 1.633. In hexagonal metal structures, the axial ratio c/a is never exactly 1.633. These structures are therefore never quite ideally close packed, e.g. c/a(Zn) = 1.856, c/a(Ti) = 1.587. As the axial ratio approaches unity, the properties of cph metals begin to show similarities to fcc metals.

9 Where possible, compound structures of engineering importance have been selected as illustrative examples. Prototype structures, such as NaCl, ZnS, CaF 2 , etc., which appear in standard treatments elsewhere, are indicated as appropriate.

Atoms and atomic arrangements 25

Position of the center of the atom

(a) (b)

a (c)

Figure 1.15 Arrangement of atoms in: (a) face-centered cubic structure, (b) close-packed hexagonal structure and (c) body-centered cubic structure.

A knowledge of cell parameters permits the atomic radius r of the metal atoms to be calculated on the assumption that they are spherical and that they are in closest possible contact. The reader should verify √ √ that in the fcc structure r = (a 2)/4 and in the bcc structure r = (a 3)/4, where a is the cell parameter.

The coordination number (CN), an important concept in crystal analysis, is defined as the number of nearest equidistant neighboring atoms around any atom in the crystal structure. Thus, in the bcc structure shown in Figure 1.15c the atom at the center of the cube is surrounded by eight equidistant atoms, i.e. CN = 8. It is perhaps not so readily seen from Figure 1.15a that the coordination number for the fcc structure is 12. Perhaps the easiest method of visualizing this is to place two fcc cells side by side, and then count the neighbors of the common face-centering atom. In the cph structure with ideal packing (c/a = 1.633) the coordination number is again 12, as can be seen by once more considering two cells, one stacked on top of the other, and choosing the center atom of the common basal plane. This (0 0 0 1) basal plane has the densest packing of atoms and has the same atomic arrangement as the closest packed plane in the fcc structure. 10

The cph and fcc structures represent two effective methods of packing spheres closely; the dif- ference between them arises from the different way in which the close-packed planes are stacked. Figure 1.16a shows an arrangement of atoms in A-sites of a close-packed plane. When a second plane of close-packed atoms is laid down, its first atom may be placed in either a B-site or a C-site, which are entirely equivalent. However, once the first atom is placed in one of these two types of site, all other atoms in the second plane must be in similar sites. (This is because neighboring B- and C-sites are too close together for both to be occupied in the same layer.) At this stage there is no difference between the cph and fcc structure; the difference arises only when the third layer is put in position. In building up the third layer, assuming that sites of type B have been used to construct the second layer, as shown in Figure 1.16b, either A-sites or C-sites may be selected. If A-sites are chosen, then the atoms in the third layer will be directly above those in the first layer, and the structure will be cph, whereas if C-sites

10 The Miller indices for the closest packed (octahedral) planes of the fcc structure are {1 1 1}; these planes are best revealed by balancing a ball-and-stick model of the fcc cell on one corner.

26 Physical Metallurgy and Advanced Materials

A position

B position C position

Figure 1.16 (a) Arrangements of atoms in a close-packed plane. (b) Registry of two close-packed planes. (c) The stacking of successive planes.

Table 1.4 Crystal structures of some metals at room temperature. Element

Crystal structure Closest

Element

Crystal structure Closest

interatomic

interatomic

distance (nm) Aluminum

distance (nm)

cph (c/a = 1.568) 0.223

Potassium bcc

0.269 Chromium

cph (c/a = 1.886) 0.298

Rubidium bcc

0.289 Copper

cph (c/a = 1.623) 0.250

cph (c/a = 1.587) 0.299 Lithium

orthorhombic 0.275 Molybdenum bcc

cph (c/a = 1.623) 0.320

Vanadium bcc

cph (c/a = 1.856) 0.266 Niobium

Zirconium cph (c/a = 1.592) 0.318

are chosen this will not be the case and the structure will be fcc. Thus, a cph structure consists of layers of close-packed atoms stacked in the sequence of ABABAB or, of course, equally well, ACACAC. An fcc structure has the stacking sequence ABCABCABC so that the atoms in the fourth layer lie directly above those in the bottom layer. The density of packing within structures is sometimes expressed as an atomic packing fraction (APF), which is the fraction of the cell volume occupied by atoms. The

APF value for a bcc cell is 0.68; it rises to 0.74 for the more closely packed fcc and cph cells.

Table 1.4 gives the crystal structures adopted by some typical metals, the majority of which are either fcc or bcc. As indicated previously, an atom does not have precise dimensions; however, it is convenient to express atomic diameters as the closest distance of approach between atom centers.

Table 1.4 lists structures that are stable at room temperature; at other temperatures, some metals undergo transition and the atoms rearrange to form a different crystal structure, each structure being stable over a definite interval of temperature. This phenomenon is known as allotropy. The best-known commercially exploitable example is that of iron, which is bcc at temperatures below 910 ◦

C, fcc in the temperature range 910–1400 ◦

C and the melting point (1535 ◦ C). Other common examples include titanium and zirconium, which change from cph to bcc

C, and bcc at temperatures between 1400 ◦

Atoms and atomic arrangements 27 at temperatures of 882 ◦

C, respectively; tin, which changes from cubic (gray) to tetragonal (white) at 13.2 ◦ C; and the metals uranium and plutonium. Plutonium is particularly complex in that it has six different allotropes between room temperature and its melting point of 640 ◦ C.

C and 815 ◦

These transitions between allotropes are usually reversible and, because they necessitate rearrange- ment of atoms, are accompanied by volume changes and either the evolution or absorption of thermal energy. The transition can be abrupt but is often sluggish. Fortunately, tetragonal tin can persist in

a metastable state at temperatures below the nominal transition temperature. However, the eventual transition to the friable low-density cubic form can be very sudden. 11 Using the concept of a unit cell, together with data on the atomic mass of constituent atoms, it is possible to derive a theoretical value for the density of a pure single crystal. The param- eter a for the bcc cell of pure iron at room temperature is 0.28664 nm. Hence the volume of

the unit cell is 0.02355 nm 3 . Contrary to first impressions, the bcc cell contains two atoms, i.e. (8 1 × 8 atom) + 1 atom. Using the Avogadro constant N A , 12 we can calculate the mass of these two atoms as 2(55.85/N A ) or 185.46 × 10 −24 kg, where 55.85 is the relative atomic mass of iron. The theoretical density (mass/volume) is thus 7875 kg m −3 . The reason for the slight discrepancy between this value and the experimentally determined value of 7870 kg m −3 will become evident when we discuss crystal imperfections in Chapter 3.

Worked examples

(1) Calculate what fraction of the structure is occupied by atoms, i.e. atomic packing factor, in (a) fcc and (b) bcc, assuming the atoms behave as hard spheres. (2) What are the positions (coordinates) of the largest interstices in (a) fcc and (b) bcc? How many are there in each unit cell? Why does carbon not occupy the large interstice in iron? (3) Draw a cph structure cell and label the (0 0 0 1), (1 ¯1 0 0) and (2 ¯1 ¯1 1) planes. Mark in a 1 1 ¯2 0 and

1 0 ¯1 0 type direction. Magnesium has almost ideal c/a and has a density 1.74 Mg m −3 and relative atomic mass 24.31. Calculate the values of c and a.

Solutions

(1) In the fcc, volume of unit cell

=a 3 , occupied by four atoms.

Volume of atoms =4× π r 3

3 √ r = 2a/4

11 Historical examples of ‘tin plague’ abound (e.g. buttons, coins, organ pipes, statues). 12 The Avogadro constant N A is 0.602217 × 10 −24 mol −1 . The mole is a basic SI unit. It does not refer to mass and has

been likened to terms such as dozen, score, gross, etc. By definition, it is the amount of substance which contains as many elementary units as there are atoms in 0.012 kg of carbon-12. The elementary unit must be specified and may be an atom, a molecule, an ion, an electron, a photon, etc. or a group of such entities.

28 Physical Metallurgy and Advanced Materials

2 2a 3 π

Atomic packing factor =

3a 3 × 64 = √ = 0.74. 3 2

For bcc, volume of unit cell

=a 3 , occupied by two atoms.

Volume of atoms

, where r

Atomic packing factor =

3a 3 × 64 = 8 = 0.68.

(2) Fcc has a hole in the center of the cell [ 1 , 1 , 1 ] and at the middle of each edge [ 2 1 2 2 2 , 0, 0]. The number of interstices

For bcc, the coordinates are (0, 1 2 , 1 4 ) and there are four such sites on each face, which are

shared by two unit cells, so number of interstices 1 = 2 × 4 × 6 = 12.

Carbon does not occupy the tetrahedral site because it has to squeeze out four symmetrically arranged atoms. Instead, it occupies the octahedral site (0, 0, 1 2 ) because it pushes out only two atoms. (3)

(ii) Volume of cell

= 1.74 Mg m −3 , so a 2 c = 5.36 × 10 −29

Ideal c/a

a 3 = 5.36 × 10 −29 × 1.633, so a = 3.2 × 10 −10 m, c = 5.2 × 10 −10 m.

Atoms and atomic arrangements 29

A (a)

(b)

Figure 1.17 Two crystalline forms of carbon: (a) diamond and (b) graphite ( from Kingery, Bowen and Uhlmann, 1976; by permission of Wiley-Interscience).

1.9.2 Diamond and graphite

It is remarkable that a single element, carbon, can exist in numerous forms (see Section 10.5) including diamond, graphite and fullerene-based structures. Of these, diamond and graphite are most well known and are therefore described here. Diamond is transparent and one of the hardest materials known, finding wide use, notably as an abrasive and cutting medium. Graphite finds general use as a solid lubricant and writing medium (pencil ‘lead’). It is now often classed as a highly refractory ceramic because of its strength at high temperatures and excellent resistance to thermal shock.

We can now progress from the earlier representation of the diamond structure (Figure 1.3c) to

a more realistic version. Although the structure consists of two interpenetrating fcc substructures, in which one substructure is slightly displaced along the body diagonal of the other, it is sufficient for our purpose to concentrate on a representative structure cell (Figure 1.17a). Each carbon atom

is covalently bonded to four equidistant neighbors in regular tetrahedral 13 coordination (CN = 4). For instance, the atom marked X occupies a ‘hole’, or interstice, at the center of the group formed by atoms marked 1, 2, 3 and 4. There are eight equivalent tetrahedral sites of the X-type, arranged four-square within the fcc cell; however, in the case of diamond, only half of these sites are occupied.

Their disposition, which also forms a tetrahedron, maximizes the intervening distances between the four atoms. If the fcc structure of diamond depended solely upon packing efficiency, the coordination number would be 12; actually CN = 4, because only four covalent bonds can form. Silicon (Z = 14), germanium (Z = 32) and gray tin (Z = 50) are fellow members of Group IV in the Periodic Table and are therefore also tetravalent. Their crystal structures are identical in character, but obviously not in dimensions, to the diamond structure of Figure 1.17a.

Graphite is less dense and more stable than diamond. In direct contrast to the cross-braced structure of diamond, graphite has a highly anisotropic layer structure (Figure 1.17b). Adjacent layers in the

ABABAB sequence are staggered; the structure is not cph. A less stable rhombohedral ABCABC

sequence has been observed in natural graphite. Charcoal, soot and lampblack have been termed

13 The stability and strength of a tetrahedral form holds a perennial appeal for military engineers: spiked iron caltrops deterred attackers in the Middle Ages and concrete tetrahedra acted as obstacles on fortified Normandy beaches in World

War II.

30 Physical Metallurgy and Advanced Materials ‘amorphous carbon’; actually they are microcrystalline forms of graphite. Covalently bonded carbon

atoms, 0.1415 nm apart, are arranged in layers of hexagonal symmetry. These layers are approximately 0.335 nm apart. This distance is relatively large and the interlayer forces are therefore weak. Layers can

be readily sheared past each other, thus explaining the lubricity of graphitic carbon. (An alternative solid lubricant, molybdenum disulfide, MoS 2 , has a similar layered structure.) The ratio of property values parallel to the a-axis and the c-axis is known as the anisotropy ratio. (For cubic crystals, the ratio is unity.) Special synthesis techniques can produce near-ideal graphite 14 with an anisotropy ratio of thermal conductivity of 200.

Worked example

How many atoms are in the cubic unit cell of diamond and GaAs? How many are in their primitive cells?

Solution

Diamond is fcc with two atoms per lattice point, the two atoms being separated by 1 4 < 1 1 1>. There are four lattice points per cubic unit cell in fcc and therefore there are eight atoms per cubic unit cell in diamond. GaAs is similar to diamond, but the pair of atoms differ. There are therefore still eight atoms per cubic unit cell.

The primitive cell of fcc contains one lattice point and one atom. That of diamond therefore contains two atoms, as does that of GaAs.

1.9.3 Coordination in ionic crystals

We have seen in the case of diamond how the joining of four carbon atoms outlines a tetrahedron which is smaller than the structure cell (Figure 1.17a). Before examining some selected ionic compounds, it is necessary to develop this aspect of coordination more fully. This approach to structure building concerns packing and is essentially a geometrical exercise. It is subordinate to the more dominant demands of covalent bonding.

In the first of a set of conditional rules, assembled by Pauling, the relative radii of cation (r) and anion (R) are compared. When electrons are stripped from the outer valence shell during ion- ization, the remaining electrons are more strongly attracted to the nucleus; consequently, cations are usually smaller than anions. Rule I states that the coordination of anions around a reference cation is determined by the geometry necessary for the cation to remain in contact with each anion. For instance, in Figure 1.18a, a radius ratio r/R of 0.155 signifies touching contact when three anions are grouped about a cation. This critical value is readily derived by geometry. If the r/R ratio for threefold coordination is less than 0.155 then the cation ‘rattles’ in the central interstice, or ‘hole’, and the arrangement is unstable. As r/R exceeds 0.155 then structural distortion begins to develop.

In the next case, that of fourfold coordination, the ‘touching’ ratio has a value of 0.225 and joining of the anion centers defines a tetrahedron (Figure 1.18b). For example, silicon and oxygen ions have radii of 0.039 and 0.132 nm, respectively, hence r/R = 0.296. This value is slightly greater than the critical value of 0.225 and it follows that tetrahedral coordination gives a stable configuration;

indeed, the complex anion SiO 4 4 − is the key structural feature of silica, silicates and silica glasses. The quadruple negative charge is due to the four unsatisfied oxygen bonds which project from the group.

14 Applications range from rocket nozzles to bowl linings for tobacco pipes.

Atoms and atomic arrangements 31

r /R ⫽ 0.155

r /R ⫽ 0.414 CN ⫽ 3

r /R ⫽ 0.225

CN ⫽ 6 (Boric oxide B 2 O 3 )

CN ⫽ 4

(Periclase MgO) (a)

(Silica SiO 2 )

(c) Figure 1.18 Nesting of cations within anionic groups.

(b)

Table 1.5 Relation between radius ratio and coordination. r/R

Maximum coordination

Form of coordination

number (CN)

2 Linear 0.155–0.225

3 Equilateral triangle 0.225–0.414

4 Regular tetrahedron 0.414–0.732

6 Regular octahedron 0.732–1.0

8 Cube

1.00 12 Cuboctahedron

In a feature common to many structures, the tendency for anions to distance themselves from each other as much as possible is balanced by their attraction towards the central cation. Each of the four oxygen anions is only linked by one of its two bonds to the silicon cation, giving an effec- tive silicon/oxygen ratio of 1:2 and thus confirming the stoichiometric chemical formula for silica,

SiO 2 . Finally, as shown in Figure 1.18c, the next coordination polyhedron is an octahedron for which r/R = 0.414. It follows that each degree of coordination is associated with a nominal range of r/R values, as shown in Table 1.5. Caution is necessary in applying these ideas of geometrical packing because (1) range limits are approximative, (2) ionic radii are very dependent upon CN, (3) ions can be non-spherical in anisotropic crystals and (4) considerations of covalent or metallic bonding can be overriding. The other four Pauling rules are as follows:

• Rule II. In a stable coordinated structure the total valency of the anion equals the summated bond strengths of the valency bonds which extend to this anion from all neighboring cations. Bond strength is defined as the valency of an ion divided by the actual number of bonds; thus, for Si 4 + in tetrahedral coordination it is 4 4 = 1. This valuable rule, which expresses the tendency of each

ion to achieve localized neutrality by surrounding itself with ions of opposite charge, is useful in deciding the arrangement of cations around an anion. For instance, the important ceramic barium

titanate (BaTiO 3 ) has Ba 2 + and Ti 4 + cations bonded to a common O 2 − anion. Given that the coordination numbers of O 2 − polyhedra centered on Ba 2 + and Ti 4 + are 12 and 6, respectively, we calculate the corresponding strengths of the Ba–O and Ti–O bonds as 2 12 1 = 6 and 4 6 2 = 3 . The

32 Physical Metallurgy and Advanced Materials

B(Zn)

N(S)

Figure 1.19 Zinc blende (α-ZnS) structure, prototype for cubic boron nitride (BN) ( from Kingery, Bowen and Uhlmann, 1976; by permission of Wiley-Interscience).

valency of the shared anion is 2, which is numerically equal to (4

6 + (2 × 3 ). Accordingly, coordination of the common oxygen anion with four barium cations and two titanium cations is

a viable possibility. • Rule III. An ionic structure tends to have maximum stability when its coordination polyhedra share corners; edge- and face-sharing give less stability. Any arrangement which brings the mutually repelling central cations closer together tends to destabilize the structure. Cations of high valency (charge) and low CN (poor ‘shielding’ by surrounding anions) aggravate the destabilizing tendency.

• Rule IV. In crystals containing different types of cation, cations of high valency and low CN tend to limit the sharing of polyhedra elements; for instance, such cations favor corner sharing rather than edge sharing.

• Rule V. If several alternative forms of coordination are possible, one form usually applies throughout the structure. In this way, ions of a given type are more likely to have identical

surroundings. In conclusion, it is emphasized that the Pauling rules are only applicable to structures in which ionic

bonding predominates. Conversely, any structure which fails to comply with the rules is extremely unlikely to be ionic.

The structure of the mineral zinc blende (α-ZnS) shown in Figure 1.19 is often quoted as a prototype for other structures. In accord with the radius ratio r/R = 0.074/0.184 = 0.4, tetrahedral coordination is a feature of its structure. Coordination tetrahedra share only corners (vertices). Thus, one species of ion occupies four of the eight tetrahedral sites within the cell. These sites have been mentioned previously in connection with diamond (Section 1.9.2); in that case, the directional demands of the covalent bonds between like carbon atoms determined their location. In zinc sulfide, the position

of unlike ions is determined by geometrical packing. Replacement of the Zn 2 + and S 2 − ions in the prototype cell with boron and nitrogen atoms produces the structure cell of cubic boron nitride (BN). This compound is extremely hard and refractory and, because of the adjacency of boron (Z = 5)

and nitrogen (Z = 7) to carbon (Z = 6) in the Periodic Table, is more akin in character to diamond than to zinc sulfide. Its angular crystals serve as an excellent grinding abrasive for hardened steel.

Atoms and atomic arrangements 33

⫽ Mg 2⫹

⫽ Cu Magnesia

⫽ O 2⫺

⫽ Zn

b -Brass MgO

CuZn

Primitive cubic (CN ⫽ 6:6)

fcc

(CN ⫽ 8:8)

Figure 1.20 AB-type compounds ( from Kingery, Bowen and Uhlmann, 1976; by permission of Wiley-Interscience).

The precursor for cubic boron nitride is the more common and readily prepared form, hexagonal boron nitride. 15

This hexagonal form is obtained by replacing the carbon atoms in the layered graphite structure (Figure 1.17b) alternately with boron and nitrogen atoms, and also slightly altering the stacking registry of the layer planes. It feels slippery like graphite and is sometimes called ‘white graphite’. Unlike graphite, it is an insulator, having no free electrons.

Another abrasive medium, silicon carbide (SiC), can be represented in one of its several crys- talline forms by the zinc blende structure. Silicon and carbon are tetravalent and the coordination is tetrahedral, as would be expected.

1.9.4 AB-type compounds

An earlier diagram (Figure 1.3b) schematically portrayed the ionic bonding within magnesium oxide (periclase). We can now develop a more realistic model of its structure and also apply the ideas

of coordination. Generically, MgO is a sodium chloride-type structure (Figure 1.20a), with Mg 2 + cations and O 2 − anions occupying two interpenetrating 16 fcc sublattices. Many oxides and halides have this type of structure (e.g. CaO, SrO, BaO, VO, CdO, MnO, FeO, CoO, NiO; NaCl, NaBr, NaI, NaF, KCl, etc.). The ratio of ionic radii r/R = 0.065/0.140 = 0.46 and, as indicated by Table 1.5,

each Mg 2 + cation is octahedrally coordinated with six larger O 2 − anions, and vice versa (CN = 6:6).

15 The process for converting hexagonal BN to cubic BN (Borazon) involves very high temperature and pressure, and was developed by Dr R. H. Wentorf at the General Electric Company, USA (1957).

16 Sublattices can be discerned by concentrating on each array of like atoms (ions) in turn.

34 Physical Metallurgy and Advanced Materials Octahedra of a given type share edges. The ‘molecular’ formula MgO indicates that there is an exact

stoichiometric balance between the numbers of cations and anions; more specifically, the unit cell

2 ) = 4 cations and (12 × 4 ) + 1 = 4 anions. The second example of an AB-type compound is the hard intermetallic compound CuZn (β-brass)

depicted contains (8 1 × 1 8 )

shown in Figure 1.20b. It has a cesium chloride-type structure in which two simple cubic sublattices interpenetrate. Copper (Z = 29) and zinc (Z = 30) have similar atomic radii. Each copper atom is in eightfold coordination with zinc atoms, thus CN

1 cell contains (8 = 8:8. The coordination cubes share faces. Each unit × 8 ) = 1 corner atom and 1 central atom; hence the formula CuZn. In other words,

this compound contains 50 at.% copper and 50 at.% zinc.

Problems

1.1 (i) Write down the elements for which the energy of the 3d-level is higher than the 4s. (ii) Write down the electronic structure of potassium and bromine in the solid KBr.

1.2 For an Na + −Cl − ion pair with separation r nm, the potential energy is given by:

For solid NaCl, estimate the (i) equilibrium ion spacing, (ii) Young’s modulus and (iii) the bonding energy per ion pair.

1.3 In an ionically bonded ceramic the force between two ions is given by:

r 2 − r 10

(r is the distance between the ions, b and c are constants, b = 1.2 × 10 −28 Nm 2 and

c = 1.8 × 10 −105 Nm 10 ). Estimate in a simple way the equilibrium ion separation in the solid and the energy required to separate the ions in the solid to an infinite distance apart. What physical properties would you measure in order to check your answers and how would you measure them?

1.4 If the density of silicon is 2.55 Mg m −3 , what is its lattice parameter? (The relative atomic mass of silicon is 28.09 and Avogadro’s number is 6.02

1.5 If atoms in silicon are considered to be solid spheres which touch, what fraction of space is occupied by matter? If the lattice parameter is 0.543 nm, what is the atom radius?

1.6 For the atom at 1 1 4 1 , 4 , 4 in the unit cell of diamond, enumerate the vectors to its nearest neighbors.

1.7 (a) Taking the atomic radius of nickel to be 0.1246 nm, calculate the lattice parameter a. (b) From the relative atomic mass, i.e. atomic weight (58.7 g mol −1 ) and Avogadro’s constant

6.02 × 10 23 , calculate the theoretical density of nickel.

1.8 MgO is an ionic solid with Mg 2 + cations and O 2 − anions occupying two interpenetrating fcc sub-lattices. The ionic radius of Mg is 0.065 nm and that of oxygen 0.140 nm. Determine the coordination of Mg and O atoms and explain the value in terms of the ionic ratio.

Further reading

Barrett, C. S. and Massalski, T. B. (1966). Structure of Metals, 3rd edn. McGraw-Hill, New York. Brydson, J. A. (1989). Plastics Materials, 5th edn. Butterworths, London. Cottrell, A. H. (1975). Introduction to Metallurgy. Edward Arnold, London.

Atoms and atomic arrangements 35 Evans, R. C. (1966). An Introduction to Crystal Chemistry, 2nd edn. Cambridge University Press,

Cambridge. Huheey, J. E. (1983). Inorganic Chemistry, 3rd edn. Harper & Row, New York. Hume-Rothery, W., Smallman, R. E. and Haworth, C. W. (1975). The Structure of Metals and Alloys, 5th

edn (1988 reprint). Institute of Materials, London. Kelly, A. and Groves, G. W. (1973). Crystallography and Crystal Defects. Longmans, Harlow. Kingery, W. D., Bowen, H. K. and Uhlmann, D. R. (1976). Introduction to Ceramics, 2nd edn. John Wiley,

Chichester. Puddephatt, R. J. and Monaghan, P. K. (1986). The Periodic Table of the Elements. Clarendon Press, Oxford. van Vlack, L. H. (1985). Elements of Materials Science, 5th edn. Addison-Wesley, Reading, MA.

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Chapter 2

Phase equilibria and structure