The structures of simple solids

1. The structures of simple solids • The majority of inorganic compounds exist as solids and comprise ordered arrays of atoms, ions, or molecules. • Some of the simplest solids are the metals, the structures of which can be described in terms of regular, space-filling arrangements of the metal atoms. These metal centres interact through metallic bonding The description of the structures of solids The arrangement of atoms or ions in simple solid structures can often be represented by different arrangements of hard spheres.

2. 3.1 Unit cells and the description of crystal structures • A crystal of an element or compound can be regarded as constructed from regularly repeating structural elements, which may be atoms, molecules, or ions. The ‘crystal lattice’ is the pattern formed by the points and used to represent the positions of these repeating structural elements. (a) Lattices and unit cells A lattice is a three-dimensional, infinite array of points, the lattice points, each of which is surrounded in an identical way by neighbouring points, and which defines the basic repeating structure of the crystal.

3. The crystal structure itself is obtained by associating one or more identical structural units (such as molecules or ions) with each lattice point. A unit cell of the crystal is an imaginary parallel-sided region (a ‘parallelepiped’) from which the entire crystal can be built up by purely translational displacements Unit cells may be chosen in a variety of ways but it is generally preferable to choose the smallest cell that exhibits the greatest symmetry Two possible choices of repeating unit are shown but (b) would be preferred to (a) because it is smaller.

4. All ordered structures adopted by compounds belong to one of the following seven crystal systems. The angles (, β, ) and lengths (a, b, c) used to define the size and shape of a unit cell are the unit cell parameters (the ‘lattice parameters’)

5. A primitive unit cell (denoted by the symbol P) has just one lattice point in the unit cell, and the translational symmetry present is just that on the repeating unit cell. Lattice points describing the translational symmetry of a primitive cubic unit cell.

6. body-centred (I, from the German word innenzentriet, referring to the lattice point at the unit cell centre) with two lattice points in each unit cell, and additional translational symmetry beyond that of the unit cell Lattice points describing the translational symmetry of a body-centred cubic unit cell.

7. face-centred (F) with four lattice points in each unit cell, and additional translational symmetry beyond that ofthe unit cell Lattice points describing the translational symmetry of a face-centred cubic unit cell.

8. We use the following rules to work out the number of lattice points in a three-dimensional unit cell. The same process can be used to count the number of atoms, ions, or molecules that the unit cell contains 1. A lattice point in the body of, that is fully inside, a cell belongs entirely to that cell and counts as 1. 2. A lattice point on a face is shared by two cells and contributes 1/2 to the cell. 3. A lattice point on an edge is shared by four cells and hence contributes 1/4 . 4. A lattice point at a corner is shared by eight cells that share the corner, and so contributes 1/8 .

9. Thus, for the face-centred cubic lattice depicted in Fig. the total number of lattice points in the unit cell is (8×1/8 ) +(6× 1/2) = 4. For the body-centred cubic lattice depicted in Fig. , the number of lattice points is (1×1) + (8×1/8 ) = 2.

10. The close packing of identical spheres can result in a variety of polytypes cubic close-packed (ccp) hexagonally close-packed (hcp)

11. In both the (a) ABA and (b) ABC close-packed arrangements, the coordination number of each atom is 12. Dr. Said M. El-Kurdi

12. The close packing of spheres Many metallic and ionic solids can be regarded as constructed from entities, such as atoms and ions, represented as hard spheres. Close-packed structure, a structure in which there is least unfilled space. The coordination number (CN) of a sphere in a close-packed arrangement (the ‘number of nearest neighbours’) is 12, the greatest number that geometry allows A close-packed layer of hard spheres

13. Interstitial holes: hexagonal and cubic close-packing Close-packed structures contain octahedral and tetrahedral holes (or sites).

14. There is one octahedral hole per sphere, and there are twice as many tetrahedral as octahedral holes in a close-packed array Tetrahedral hole can accommodate a sphere of radius  0.23 times that of the close-packed spheres Octahedral hole can accommodate a sphere of radius 0.41 times that of the close-packed spheres

15. 3.5 Nonclose-packed structures Not all elemental metals have structure based on close-packing and some other packing patterns use space nearly as efficiently. Even metals that are close-packed may undergo a phase transition to a less closely packed structure when they are heated and their atoms undergo large-amplitude vibrations.

16. Non-close-packing: simple cubic and body centred cubic arrays Unit cells of (a) a simple cubic lattice and (b) a body-centred cubic lattice.

17. The least common metallic structure is the primitive cubic (cubic-P) structure , in which spheres are located at the lattice points of a primitive cubic lattice, taken as the corners of the cube. The coordination number of a cubic-P structure is 6. One form of polonium (-Po) is the only example of this structure among the elements under normal conditions.

18. Body-centred cubic structure (cubic-I or bcc) in which a sphere is at the centre of a cube with spheres at each corner Metals with this structure have a coordination number of 8 Although a bcc structure is less closely packed than the ccp and hcp structures (for which the coordination number is 12),

21. 6.4 Polymorphism in metals Polymorphism: phase changes in the solid state If a substance exists in more than one crystalline form, it is polymorphic. under different conditions of pressure and temperature The polymorphs of metals are generally labelled, β, ,...with increasing temperature.

22. Solid mercury (-Hg), however, has a closely related structure: it is obtained from the cubic-P arrangement by stretching the cube along one of its body diagonals A second form of solid mercury (β-Hg) has a structure based on the bcc arrangement but compressed along one cell direction

23. Phase diagrams A pressure–temperature phase diagram for iron

24. 6.5 Metallic radii The metallic radius is half of the distance between the nearest neighbor atoms in a solid state metal lattice, and is dependent upon coordination number.

28. 6.7 Alloys and intermetallic compounds compound of two or more metals, or metals and non-metals; alloying changes the physical properties and resistance to corrosion, heat etc. of the material. Alloys are manufactured by combining the component elements in the molten state followed by cooling.

29. Substitutional alloys In a substitutional alloy, atoms of the solute occupy sites in the lattice of the solvent metal similar size same coordination environment sterling silver which contains 92.5% Ag and 7.5% Cu

30. Interstitial alloys In an interstitial solid solution, additional small atoms occupy holes within the lattice of the original metal structure. Interstitial solid solutions are often formed between metals and small atoms (such as boron, carbon, and nitrogen) One important class of materials of this type consists of carbon steels in which C atoms occupy some of the octahedral holes in the Fe bcc lattice.

31. Intermetallic compounds When melts of some metal mixtures solidify, the alloy formed may possess a definite structure type that is different from those of the pure metals. e.g. b-brass, CuZn. At 298 K, Cu has a ccp lattice and Zn has a structure related to an hcp array, but b-brass adopts a bcc structure.

32. The structures of metals and alloys Many metallic elements have close-packed structures, One consequence of this close-packing is that metals often have high densities because the most mass is packed into the smallest volume. • Osmium has the highest density of all the elements at 22.61 g cm−3 and the density of tungsten, 19.25 g cm−3, which is almost twice that of lead (11.3 g cm−3) Calculate the density of gold, with a cubic close-packed array of atoms of molar mass M=196.97 g mol−1 and a cubic lattice parameter a = 409 pm.

33. Gold (Au) crystallizes in a cubic close-packed structure (the face-centered cube) and has a density of 19.3 g/cm3. Calculate the atomic radius of gold.

34. The unoccupied space in a close-packed structure amounts to 26 per cent of the total volume. However, this unoccupied space is not empty in a real solid because electron density of an atom does not end as abruptly as the hard-sphere model suggests. Calculating the unoccupied space in a close-packed array Calculate the percentage of unoccupied space in a close-packed arrangement of identical spheres.

35. 6.8 Bonding in metals and semiconductors Electrical conductivity and resistivity An electrical conductor offers a low resistance (measured in ohms, ) to the flow of an electrical current (measured in amperes, A).

36. The electrical conductivity of a metal decreases with temperature; that of a semiconductor increases with temperature.

37. Band theory of metals and insulators A band is a group of MOs, the energy differences between which are so small that the system behaves as if a continuous, non-quantized variation of energy within the band is possible.

38. The relative energies of occupied and empty bands in (a) an insulator, (b) a metal in which the lower band is only partially occupied, (c) a metal in which the occupied and empty bands overlap, and (d) a semiconductor.

39. A band gap occurs when there is a significant energy difference between two bands. 6.9 Semiconductors For C, Si, Ge and -Sn, the band gaps are 5.39, 1.10, 0.66 and 0.08 eV respectively. C being an insulator Each of Si, Ge and -Sn is classed as an intrinsic semiconductor

40. Electrons present in the upper conduction band act as charge carriers and result in the semiconductor being able to conduct electricity. • removal of electrons from the lower valence band creates positive holes into which electrons can move, again leading to the ability to conduct charge. A charge carrier in a semiconductor is either a positive hole or an electron that is able to conduct electricity.

41. Extrinsic (n- and p-type) semiconductors Extrinsic semiconductors contain dopants; a dopant is an impurity introduced into a semiconductor in minute amounts to enhance its electrical conductivity. In Ga-doped Si, the substitution of a Ga (group 13) for a Si (group 14) atom in the bulk solid produces an electron deficient site. In As-doped Si, replacing an Si (group 14) by an As (group 15) atom introduces an electron-rich site.

42. (a) In a p-type semiconductor (e.g. Ga-doped Si), electrical conductivity arises from thermal population of an acceptor level which leaves vacancies (positive holes) in the lower band. (b) In an n-type semiconductor (e.g. As-doped Si), a donor level is close in energy to the conduction band.

43. 6.10 Sizes of ions Ionic radii Values of the ionic radius (rion) may be derived from X-ray diffraction data. internuclear distance >>> we generally take this to be the sum of the ionic radii of the cation and anion

44. Ionic solids Characteristic structures of ionic solids Many of the structures can be regarded as derived from arrays in which the larger of the ions, usually the anions, stack together in ccp or hcp patterns and the smaller counter-ions (usually the cations) occupy the octahedral or tetrahedralholes in the lattice

45. The rock salt (NaCl) structure type In salts of formula MX, the coordination numbers of M and X must be equal. Na+ and Cl- ion is 6-coordinate in the crystal lattice

46. The number of formula units present in the unit cell is commonly denoted Z Show that the structure of the unit cell for sodium chloride (Figure) is consistent with the formula NaCl.

47. The caesium chloride (CsCl) structure type cubic unit cell with each corner occupied by an anion and a cation occupying the ‘cubic hole’ at the cell centre (or vice versa); as a result, Z =1. The coordination number of both types of ion is 8, so the structure is described as having (8,8)-coordination.

48. The fluorite (CaF2) structure type Ca ions are shown in red and the F ions in green In salts of formula MX2, the coordination number of X must be half that of M.

49. The antifluorite lattice The antifluorite structure is the inverse of the fluorite structure in the sense that the locations of cations and anions are reversed. The latter structure is shown by some alkali metal oxides, including Li2O. In it, the cations (which are twice as numerous as the anions) occupy all the tetrahedral holes of a ccp array of anions. The coordination is (4,8) rather than the (8,4) of fluorite itself.

50. The sphalerite structure, which is also known as the zinc-blende structure, it is based on an expanded ccp anion arrangement but now the cations occupy one type of tetrahedral hole, one half the tetrahedral holes present in a close-packed structure. Each ion is surrounded by four neighbours and so the structure has (4,4)-coordination and Z= 4.