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.

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3.1 Unit cells and the description of crystal structures

The ‘crystal lattice’ is the pattern formed by the points and used to represent the positions of these repeating structural elements.

A crystal of an element or compound can be regarded as constructed from regularly repeating structural elements, which may be atoms, molecules, or ions.

(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.

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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.

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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’)

All ordered structures adopted by compounds belong to one of the following seven crystal systems.

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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.

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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.

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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.

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The same process can be used to count the number of atoms, ions, or molecules that the unit cell contains

We use the following rules to work out the number of lattice points in a three-dimensional unit cell.

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 .

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For the body-centred cubic lattice depicted in Fig. 3.4, the number of lattice points is (1×1) + (8×1/8 ) = 2.

Thus, for the face-centred cubic lattice depicted in Fig. 3.5 the total number of lattice points in the unit cell is (8×1/8 ) +(6× 1/2) = 4.

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Structures may be drawn in projection, with atom positions denoted by fractional coordinates.

(b) Fractional atomic coordinates and projections

a) The structure of metallic tungsten and (b) its projection representation.

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The projection representation of an fcc unit cell.

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3.2 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

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The close packing of identical spheres can result in a variety of polytypes

hexagonally close-packed (hcp)

cubic close-packed (ccp)

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Because each ccp unit cell has a sphere at one corner and one at the centre of each face, a ccp unit cell is sometimes referred to as face-centred cubic (fcc).

The hexagonal close-packed (hcp) unit cell of the ABAB . . . polytype. The colours of the spheres correspond to the layers

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The cubic close-packed (fcc) unit cell of the ABC . . . polytype. The colours of the spheres correspond to the layers

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Calculating the unoccupied space in a close-packed array

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.

Calculate the percentage of unoccupied space in a close-packed arrangement of identical spheres.

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3.3 Holes in close-packed structures

The feature of a close-packed structure that enables us to extend the concept to describe structures more complicated than elemental metals is the existence of two types of hole, or unoccupied space between the spheres.

An octahedral hole lies between two triangles of spheres on adjoining layers

For a crystal consisting of N spheres in a close-packed structure, there are N octahedral holes.

(a) An octahedral hole and (b) a tetrahedral hole formed in an arrangement of close-packed spheres.

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(a) The location (represented by a hexagon) of the two octahedral holes in the hcp unit cell and

(b) the locations (represented by hexagons) of the octahedral holes in the ccp unit cell.

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Calculate the maximum radius of a sphere that may be accommodated in

an octahedral hole in a closepacked solid composed of spheres of radius r.

0.414r

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A tetrahedral hole, T, is formed by a planar triangle of touching spheres capped by a single sphere lying in the dip between them.

The tetrahedral holes in any closepacked solid can be divided into two sets: in one the apex of the tetrahedron is

directed up (T) and in the other the apex points down

(T’).

In an arrangement of N close-packed spheres there are N tetrahedral holes of each set and 2N tetrahedral holes in all.

The locations (represented by triangles) of the tetrahedral holes in the hcp unit cell

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The locations of the tetrahedral holes in the ccp unit cell

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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.

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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.

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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.

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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),

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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.

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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

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The structures of the metallic elements at room temperature. Elements with more complex structures are left blank.

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3.6 Polymorphism of metals

polymorphism, the ability to adopt different crystal forms under different conditions of pressure and temperature.

It is often, but not universally, found that

The less closely packed structures are favoured at high temperatures.

The most closely packed phases are thermodynamically favoured at low temperatures

Application of high pressure leads to structures with higher packing densities, such as ccp and hcp.

The polymorphs of metals are generally labelled , β, ,...with increasing temperature.

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3.8 Alloys

An alloy is a blend of metallic elements prepared by mixing the molten components and then cooling the mixture to produce a metallic solid.

Alloys typically form from two electropositive metals

(a) Substitutional solid solutions

Involves the replacement of one type of metal atom in a structure by another.

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Sodium and potassium are chemically similar and have bcc structures, the atomic radius of Na (191 pm) is 19 per cent smaller than that of K (235 pm) and the two metals do not form a solid solution.

Copper and nickel, have similar electropositive character, similar crystal structures (both ccp), and similar atomic radii (Ni 125 pm, Cu 128 pm, only 2.3 per cent different), and form a continuous series of solid solutions, ranging from pure nickel to pure copper.

Substitutional solid solutions are generally formed if three criteria are fulfilled:

1. The atomic radii of the elements are within about 15 per cent of each other. 2. The crystal structures of the two pure metals are the same. 3. The electropositive characters of the two components are similar.

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(b) Interstitial solid solutions of nonmetals

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) that can inhabit the interstices in the structure.

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.

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(c) Intermetallic compounds

Intermetallic compounds are alloys in which the structure adopted is different from the structures of either component metal.

when some liquid mixtures of metals are cooled, they form phases with definite structures that are often unrelated to the parent structure. These phases are called intermetallic compounds.

They include β-brass (CuZn) and compounds of composition MgZn2, Cu3Au, NaTl, and Na5Zn21.

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Composition, lattice type and unit cell content of iron and its alloys

What are the lattice types and unit cell contents of (a) iron metal (Fig. a) and (b) the iron/chromium alloy, FeCr

The structure type is the bcc

there are two Fe atoms in the unit cell

the lattice type is primitive, P.

There is one Cr atom and 1 Fe atom in the unit cell

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Ionic solids

3.9 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 tetrahedral holes in the lattice

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The relation of structure to the filling of holes

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(a) Binary phases, AXn

The simplest ionic compounds contain just one type of cation (A) and one type of anion (X) present in various ratios covering compositions such as AX and AX2.

Several different structures may exist for each of these compositions, depending on the relative sizes of the cations and anions and which holes are filled and to what degree in the close-packed array

The rock-salt structure is based on a ccp array of bulky anions with cations in all the octahedral holes.

Because each ion is surrounded by an octahedron of six counter-ions, the coordination number of each type of ion is 6 and the structure is said to have (6,6)-coordination.

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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.

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Similarly, compounds such as CaC2, CsO2, KCN, and FeS2 all adopt structures closely related to the rock-salt structure with alternating cations and complex anions

many 1:1 compounds in which the ions are complex units such as [Co(NH3)6][TlCl6].

The structure of this compound can be considered as an array of closepacked octahedral [TlCl6]3− ions with [Co(NH3)6]3+ ions in all the octahedral holes.

The structure of CaC2 is based on the rock-salt structure but is elongated in the direction parallel to the axes of the C2

2− ions.

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caesium-chloride structure

which is possessed by CsCl, CsBr, and CsI, as well as some other compounds formed of ions of similar radii to these.

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.

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The structure of ammonium chloride, NH4Cl, reflects the ability of the tetrahedral NH4

+ ion to form hydrogen bonds to the tetrahedral array of Cl− ions around it.

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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.

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This structure, which has (4,4)-coordination, is adopted by ZnO, AgI, and one polymorph of SiC, as well as several other compounds

The wurtzite structure polymorph of zinc sulfide

It derived from an expanded hcp anion array rather than a ccp array

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The fluorite (CaF2) lattice

Each cation is 8-coordinate and each anion 4-coordinate; six of the Ca2+ ions are shared between two unit cells and the 8-coordinate environment can be appreciated by envisaging two adjacent unit cells.

The unit cell of CaF2; the Ca2+ ions are shown in red and the F− ions in green.

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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.

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The rutile structure, a mineral form of titanium(IV) oxide, TiO2. The structure can also be considered an example of hole filling in an hcp anion arrangement, the cations occupy only half the octahedral holes.

Each Ti4 atom is surrounded by six O atoms and each O atom is surrounded by three Ti4 ions; hence the rutile structure has (6,3)-coordination.

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(b) Ternary phases AaBbXn

it is difficult to predict the most likely structure type based on the ion sizes and preferred coordination numbers.

The mineral perovskite, CaTiO3, is the structural prototype of many ABX3 solids

The perovskite structure is cubic with each A cation surrounded by 12 X anions and each B cation surrounded by six X anions

the coordination number of the Ti4+ ion in the perovskite CaTiO3 is 6

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