3 The structures of simple solids

---

66

3  The structures of simple solids

 In   ionic bonding  , ions of different elements are held together in rigid, symmetrical arrays 

as a result of the attraction between their opposite charges. Ionic bonding also depends on 

electron loss and gain, so it is found typically in compounds of metals with electronegative 

elements. However, there are plenty of exceptions: not all compounds of metals are ionic 

and some compounds of nonmetals (such as ammonium nitrate) contain features of ionic 

bonding as well as covalent interactions. There are also materials that exhibit features of 

both ionic and metallic bonding. 

 Both ionic and metallic bonding are nondirectional, so structures where these types of 

bonding occur are most easily understood in terms of space-fi lling models that maximize, 

for example, the number and strength of the electrostatic interactions between the ions. 

The regular arrays of atoms, ions, or molecules in solids that produce these structures are 

best represented using a repeating unit that is produced as a result of the effi cient methods 

of fi lling space, known as the unit cell. 

    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. The spheres used to describe metallic solids rep-

resent neutral atoms because each cation can still be considered as surrounded by its full 

complement of electrons. The spheres used to describe ionic solids represent the cations 

and anions because there has been a substantial transfer of electrons from one type of 

atom to the other. 

     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 geometric pattern formed by the points that represent the positions of these repeat-

ing structural elements. 

      (a)   Lattices  and  unit  cells   

  Key  points:  

The lattice defi nes a network of identical points that has the translational symmetry of 

a structure. A unit cell is a subdivision of a crystal that when stacked together following translations 

reproduces the crystal. 

 A   lattice   is a three-dimensional, infi nite array of points, the   lattice points  , each of which is 

surrounded in an identical way by neighbouring points. The lattice defi nes the repeating 

nature of the crystal. The   crystal structure   itself is obtained by associating one or more 

identical structural units, such as atoms, ions, or molecules, with each lattice point. In 

many cases the structural unit may be centred on the lattice point, but that is not necessary. 

 A   unit cell   of a three-dimensional crystal is an imaginary parallel-sided region (a ‘paral-

lelepiped’) from which the entire crystal can be built up by purely translational displace-

ments;   

1

    unit cells so generated fi t perfectly together with no space excluded. 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. Thus, in the two-dimensional pattern in  Fig.  3.1  , 

a variety of unit cells (a parallelogram in two dimensions) may be chosen, each of which 

repeats the contents of the box under translational displacements. Two possible choices 

of repeating unit are shown, but (b) would be preferred to (a) because it is smaller. The 

relationship between the lattice parameters in three dimensions as a result of the sym-

metry of the structure gives rise to the seven   crystal systems   ( Table  3.1   and  Fig.  3.2  ). All 

ordered structures adopted by compounds belong to one of these crystal systems; most 

of those described in this chapter, which deals with simple compositions and stoichio-

metries, belong to the higher symmetry cubic and hexagonal systems. The angles ( 

α ,  β ,  γ ) 

and lengths ( a ,  b ,  c ) used to defi ne the size and shape of a unit cell, relative to an origin, 

are the   unit cell parameters   (the ‘lattice parameters’); the angle between  a  and  b  is denoted 

  1 

  A translation exists where it is possible to move an original fi gure or motif in a defi ned direction by a 

certain distance to produce an exact image. In this case a unit cell reproduces itself exactly by translation 

parallel to a unit cell edge by a distance equal to the unit cell parameter. 

(a)

(b)

(c)

Possible unit cell

Preferred unit cell choice

Not a unit cell

   Figure  3.1  

A two-dimensional solid and 

two choices of a unit cell. The entire crystal 

is produced by translational displacements 

of either unit cell, but (b) is generally 

preferred to (a) because it is smaller.    

2523_Ch03.indd   66

2523_Ch03.indd   66

10/1/2013   7:03:46 PM

10/1/2013   7:03:46 PM

---

67

The description of the structures of solids

 

γ , that between  b  and  c  is  α , and that between  a  and  c  is  β  ;  a triclinic unit cell is illustrated 

in  Fig.  3.2  .    

 A   primitive   unit cell (denoted by the symbol P) has just one lattice point in the unit cell 

( Fig.  3.3  ) and the translational symmetry present is just that on the repeating unit cell. More 

complex lattice types are   body-centred   (I, from the German word  innenzentriert , referring to 

the lattice point at the unit cell centre) and   face-centred   (F) with two and four lattice points 

in each unit cell, respectively, and additional translational symmetry beyond that of the unit 

cell ( Figs  3.4   and   3.5  ). The additional translational symmetry in the   body-centred cubic   (bcc) 

lattice, equivalent to the displacement  (

,

,

+ + +

1

2

1

2

1

2

)     from the unit cell origin at (0,0,0), pro-

duces a lattice point at the unit cell centre; note that the surroundings of each lattice point 

are identical, consisting of eight other lattice points at the corners of a cube. Centred lattices 

are sometimes preferred to primitive (although it is always possible to use a primitive lattice 

for any structure), for with them the full structural symmetry of the cell is more apparent.    

 We use the following rules to work out the number of lattice points in a three-dimen-

sional unit cell. The same process can be used to count the number of atoms, ions, or 

molecules that the unit cell contains (Section 3.9). 

• 

    A lattice point in the body of—that is fully inside—a cell belongs entirely to that cell 

and counts as 1.  

• 

   A lattice point on a face is shared by two cells and contributes 

1

2

   to the cell.  

• 

   A lattice point on an edge is shared by four cells and hence contributes 

1

4

   .  

• 

   A lattice point at a corner is shared by eight cells that share the corner, and so contributes 

1

8

   .     

   Table  3.1  

The seven crystal systems  

 System 

 Relationships  between 

lattice parameters 

 Unit  cell 

defi ned by 

 Essential  symmetries 

 Triclinic 

  a  

≠  b  ≠   c ,   α  ≠   β  ≠   γ  ≠ 90°