TOPICS
Group 18 elements in the solid state
5.17 Defects in solid state lattices:
an introduction
So far in this chapter, we have assumed implicitly that all the pure substances considered have ideal lattices in which every site is occupied by the correct type of atom or ion. This state appertains only at 0 K, and above this temperature, lattice defects are always present; the energy required to create a defect is more than compensated for by the resulting increase in entropy of the structure. There are various types of lattice defects, but we shall introduce only the Schottky and Frenkel defects. Solid state defects are discussed further in Chapter 27. Spinels and defect spinels are introduced inBox 12.6.
Schottky defect
A Schottky defect consists of an atom or ion vacancy in a crystal lattice, but the stoichiometry of a compound (and thus electrical neutrality) must be retained. In a metal lattice, a vacant atom site may be present. Examples of Schottky defects in ionic lattices are a vacant cation and a vacant anion site in an MX salt, or a vacant cation and two vacant anion sites in an MX2 salt. Figure 5.26 illustrates a Schottky defect in an NaCl lattice; holes are present (Figure 5.26b) where ions are expected on the basis of the ideal lattice (Figure 5.26a).
Frenkel defect
In a Frenkel defect, an atom or ion occupies a normally vacant site, leaving its ‘own’ lattice site vacant. Figure 5.27 Fig. 5.25 The trend in the values of the first ionization
energies of the noble gases (group 18).
illustrates this for AgBr, which adopts an NaCl lattice. In Figure 5.27a, the central Agþ ion is in an octahedral hole with respect to the fcc arrangement of Brions. Migration of the Agþ ion to one of the previously unoccupied tetra-hedral holes (Figure 5.27b) generates a Frenkel defect in the lattice. This type of defect is possible if there is a relatively large difference in size between cation and anion; in AgBr, the cation must be accommodated in a tetrahedral hole which is significantly smaller than the octahedral site.
More generally, Frenkel defects are observed in lattices which are relatively open and in which the coordination number is low.
Experimental observation of Schottky and Frenkel defects
There are several methods that may be used to study the occurrence of Schottky and Frenkel defects in stoichiometric crystals, but the simplest, in principle, is to measure the
density of the crystal extremely accurately. Low concen-trations of Schottky defects lead to the observed density of a crystal being lower than that calculated from X-ray diffraction and data based on the size and structure of the unit cell. On the other hand, since the Frenkel defect does not involve a change in the number of atoms or ions present, no such density differences will be observed.
Glossary
The following terms were introduced in this chapter.
Do you know what they mean?
q close-packing (of spheres or atoms) q cubic close-packed (ccp) lattice q hexagonal close-packed (hcp) lattice q face-centred cubic (fcc) lattice q simple cubic lattice
q body-centred cubic (bcc) lattice q coordination number (in a lattice) q unit cell
q interstitial hole q polymorph q phase diagram q metallic radius q alloy
q electrical resistivity q band theory q band gap q insulator q semiconductor
q intrinsic and extrinsic semiconductors q n- and p-type semiconductors q doping (a semiconductor) q ionic radius
q NaCl lattice q CsCl lattice
q CaF2(fluorite) lattice q Antifluorite lattice q Zinc blende lattice q Diamond network q Wurtzite lattice q b-Cristobalite lattice q TiO2(rutile) lattice
q CdI2and CdCl2(layer) lattices q Perovskite lattice
q Lattice energy q Born–Lande´ equation q Madelung constant q Born exponent q Born–Haber cycle q Disproportionation q Kapustinskii equation q Schottky defect q Frenkel defect Fig. 5.26 (a) Part of one face of an ideal NaCl lattice;
compare this with Figure 5.15. (b) A Schottky defect involves vacant cation and anion sites; equal numbers of cations and anions must be absent to maintain electrical neutrality. Colour code: Na, purple; Cl, green.
Fig. 5.27 Silver bromide adopts an NaCl lattice. (a) An ideal lattice can be described in terms of Agþions occupying octahedral holes in a cubic close-packed array of bromide ions. (b) A Frenkel defect in AgBr involves the migration of Agþions into tetrahedral holes; in the diagram, one Agþion occupies a tetrahedral hole which was originally vacant in (a), leaving the central octahedral hole empty. Colour code: Ag, pale grey; Br, gold.
Chapter 5 . Glossary 159
Further reading
Packing of spheres and structures of ionic lattices
C.E. Housecroft and E.C. Constable (2002) Chemistry, Prentice Hall, Harlow – Chapters 7 and 8 give detailed accounts at an introductory level.
A.F. Wells (1984) Structural Inorganic Chemistry, 5th edn, Clarendon Press, Oxford – Chapters 4 and 6 present careful descriptions, ranging from basic to more advanced material.
Dictionary of Inorganic Compounds(1992), Chapman and Hall, London – The introduction to Vol. 4 gives a useful summary of structure types.
Structure determination SeeBox 5.5 Further reading.
Alloys
B.C. Giessen (1994) in Encyclopedia of Inorganic Chemistry, ed. R.B. King, Wiley, Chichester, Vol. 1, p. 90 – A detailed overview of alloys with further references.
A.F. Wells (1984) Structural Inorganic Chemistry, 5th edn, Clarendon Press, Oxford – Chapter 29 provides excellent coverage of metal and alloy lattice types.
Semiconductors
M. Hammonds (1998) Chemistry & Industry, p. 219 – ‘Getting power from the sun’ illustrates the application of the semicon-ducting properties of Si.
C.E. Stanton, S.T. Nguyen, J.M. Kesselman, P.E. Laaibinis and N.S. Lewis (1994) in Encyclopedia of Inorganic Chemistry, ed.
R.B. King, Wiley, Chichester, vol. 7, p. 3725 – An up-to-date general survey of semiconductors which defines pertinent terminology and gives pointers for further reading.
J. Wolfe (1998) Chemistry & Industry, p. 224 – ‘Capitalising on the sun’ describes the applications of Si and other materials in solar cells.
Solid state: for more general information
A.K. Cheetham and P. Day (1992) Solid State Chemistry, Clarendon Press, Oxford.
M. Ladd (1994) Chemical Bonding in Solids and Fluids, Ellis Horwood, Chichester.
M. Ladd (1999) Crystal Structures: Lattices and Solids in Stereo-view, Ellis Horwood, Chichester.
L. Smart and E. Moore (1992) Solid State Chemistry: An Introduction, Chapman and Hall, London.
A.R. West (1999) Basic Solid State Chemistry, 2nd edn, Wiley-VCH, Weinheim.
Problems
5.1 Outline the similarities and differences between cubic and hexagonal close-packed arrangements of spheres, paying particular attention to (a) coordination numbers, (b) interstitial holes and (c) unit cells.
5.2 State the coordination number of a sphere in each of the following arrangements: (a) ccp; (b) hcp; (c) bcc; (d) fcc;
(e) simple cubic.
5.3 (a) Lithium metal undergoes a phase change at 80 K (1 bar pressure) from the a- to b-form; one form is bcc and the other is a close-packed lattice. Suggest, with reasons, which form is which. What name is given to this type of structural change? (b) Suggest why tin buttons on nineteenth-century military uniforms crumbled in exceptionally cold winters.
5.4 Refer to Table 5.2. (a) Write an equation for the process for which the standard enthalpy of atomization of cobalt is defined. (b) Suggest reasons for the trend in standard enthalpies of atomization on descending group 1. (c) Outline possible reasons for the trend in values of aHoon going from Cs to Bi.
5.5 ‘Titanium dissolves nitrogen to give a solid solution of composition TiN0:2; the metal lattice defines an hcp arrangement.’ Explain what is meant by this statement, and suggest whether, on the basis of this evidence, TiN0:2is likely to be an interstitial or substitutional alloy. Relevant data may be found in Appendix 6 and Table 5.2.
5.6 What do you understand by the ‘band theory of metals’?
5.7 (a) Draw a representation of the structure of diamond and give a description of the bonding. (b) Is the same picture of the bonding appropriate for silicon, which is isostructural with diamond? If not, suggest an alternative picture of the bonding.
5.8 (a) Give a definition of electrical resistivity and state how it is related to electrical conductivity. (b) At 273–290 K, the electrical resistivities of diamond, Si, Ge and a-Sn are approximately 1 1011, 1 103, 0.46 and 11 108m.
Rationalize this trend in values. (c) How does the change in electrical resistivity with temperature vary for a typical metal and for a semiconductor?
5.9 Distinguish between an intrinsic and extrinsic
semiconductor, giving examples of materials that fall into these classes, and further classifying the types of extrinsic semiconductors.
5.10 The metallic, covalent and ionic radii of Al are 143, 130 and 54 pm respectively; the value of rionis for a 6-coordinate ion. (a) How is each of these quantities defined?
(b) Suggest reasons for the trend in values.
5.11 With reference to the NaCl, CsCl and TiO2lattice types, explain what is meant by (a) coordination number, (b) unit cell, (c) ion sharing between unit cells, and (d) determination of the formula of an ionic salt from the unit cell.
5.12 Determine the number of formula units of (a) CaF2in a unit cell of fluorite, and (b) TiO2in a unit cell of rutile.
5.13 (a) Confirm that the unit cell for perovskite shown in Figure 5.23a is consistent with the stoichiometry CaTiO3.
(b) A second unit cell can be drawn for perovskite; this has Ti(IV) at the centre of a cubic cell; Ti(IV) is in an octahedral environment with respect to the O2ions. In what sites must the Ca2þlie in order that the unit cell depicts the correct compound stoichiometry? Draw a diagram to illustrate this unit cell.
5.14 (a) Give a definition of lattice energy. Does your definition mean that the associated enthalpy of reaction will be positive or negative? (b) Use the Born–Lande´ equation to calculate a value for the lattice energy of KBr, for which r0¼ 328 pm. KBr adopts an NaCl lattice; other data may be found in Tables 5.3 and 5.4.
5.15 Using data from the Appendices and the fact that
fHoð298 KÞ ¼ 859 kJ mol1, calculate a value for the lattice energy of BaCl2. Outline any assumptions that you have made.
5.16 (a) Given that U(0 K) and fHo(298 K) for MgO are
3795 and 602 kJ mol1respectively, derive a value for
EAHo(298 K) for the reaction:
OðgÞ þ 2e"O2ðgÞ
Other data may be found in the Appendices. (b) Compare the calculated value with that obtained using electron affinity data from Appendix 9, and suggest reasons for any differences.
5.17 Discuss the interpretation of the following:
(a) fHo(298 K) becomes less negative along the series LiF, NaF, KF, RbF, CsF, but more negative along the series LiI, NaI, KI, RbI, CsI.
(b) The thermal stability of the isomorphous sulfates of Ca, Sr and Ba with respect to decomposition into the metal oxide (MO) and SO3increases in the sequence CaSO4<SrSO4<BaSO4.
5.18 Data from Tables 5.3 and 5.4 are needed for this question.
(a) Estimate the lattice energy of CsCl if the CsCl internuclear distance is 356.6 pm. (b) Now consider a polymorph of CsCl that crystallizes with an NaCl lattice;
estimate its lattice energy given that the CsCl distance is 347.4 pm. (c) What conclusions can you draw from your answers to parts (a) and (b)?
5.19 Which of the following processes are expected to be exothermic? Give reasons for your answers.
(a) Naþ(g) + Br(g)"NaBr(s) (b) Mg(g) "Mg2þ(g) + 2e (c) MgCl2(s)"Mg(s) + Cl2(g) (d) O(g) + 2e"O2(g) (e) Cu(l)"Cu(s) (f ) Cu(s)"Cu(g)
(g) KF(s)"Kþ(g) + F(g)
Overview problems
5.20 Give explanations for the following observations.
(a) Raising the temperature of a sample of a-Fe from 298 K to 1200 K (at 1 bar pressure) results in a change of coordination number of each Fe atom from 8 to 12.
(b) Although a non-metal, graphite is often used as an electrode material.
(c) The semiconducting properties of silicon are improved by adding minute amounts of boron.
5.21 ReO3is a structure-prototype. Each Re(VI) centre is octahedrally sited with respect to the O2centres. The unit cell can be described in terms of a cubic array of Re(VI) centres, with each O2centre at the centre of each edge of the unit cell. Draw a representation of the unit cell and use your diagram to confirm the stoichiometry of the compound.
5.22 Suggest an explanation for each of the following observations.
(a) The Cr and Ni content of stainless steels used to make knife blades is different from that used in the
manufacture of spoons.
(b) There is a poor match between experimental and calculated (Born–Lande´) values of the lattice energy for AgI, but a good match for NaI.
(c) ThI2has been formulated as the Th(IV) compound Th4þ(I)2(e)2. Comment on why this is consistent with the observation of ThI2having a low electrical resistivity.
5.23 The first list below contains words or phrases, each of which has a ‘partner’ in the second list, e.g. ‘sodium’ in the first list can be matched with ‘metal’ in the second list.
Match the ‘partners’; there is only one match for each pair of words or phrases.
List 1 List 2
Sodium Antifluorite structure
Cadmium iodide Extrinsic semiconductor Octahedral site Double oxide
Gallium-doped silicon Polymorphs Sodium sulfide Fluorite structure
Perovskite Metal
Calcium fluoride Intrinsic semiconductor Gallium arsenide Layered structure Wurtzite and zinc blende 6-Coordinate
Tin(IV) oxide Cassiterite
Chapter 5 . Problems 161