3 The structures of simple solids

a    b    c    

α    β    γ    None 

 Monoclinic 

  a  

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

 

a    b    c    

β  

 One two-fold rotation axis and

/or a mirror plane 

 Orthorhombic 

  a  

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

  a    b    c  

 Three perpendicular two-fold axes and

/or mirror 

planes 

 Rhombohedral    a  

=  b  =  c ,   α  =  β  =  γ  ≠ 90°  

 

a    

α  

 One three-fold rotation axis 

 Tetragonal 

  a  

=  b  ≠   c ,   α  =  β  =  γ  = 90° 

  a    c  

 One four-fold rotation axis 

 Hexagonal 

  a  

=  b  ≠   c ,   α  =  β  = 90°,  γ  = 120°    a    c  

 One six-fold rotation axis 

 Cubic 

  a  

=  b  =  c ,   α  =  β  =  γ  = 90° 

  a  

 Four three-fold rotation axes tetrahedrally 

arranged 

a

a

a

a

a

a

b

c

c

a

a

b

c

β

a

b

γ

α

β

c

α

a

a

c

a

a

α

α

Cubic

Triclinic

Hexagonal

Tetragonal

Orthorhombic

Monoclinic

Rhombohedral

(trigonal)

120° 

   Figure  3.2  

The seven crystal systems.    

a

b

c

(+1,0,0)

O

   Figure  3.3  

Lattice points describing the 

translational symmetry of a primitive cubic 

unit cell. The translational symmetry is just 

that of the unit cell; for example the a lattice 

point at the origin, O, translates by (

+1,0,0) 

to another corner of the unit cell.    

a

b

c

(+½,½,½)

O

   Figure  3.4  

Lattice points describing the 

translational symmetry of a body-centred 

cubic unit cell. The translational symmetry 

is that of the unit cell and (

+½,+½,+½), so 

a lattice point at the origin, O, translates to 

the body centre of the unit cell.    

a

b

c

O

(+½,0,½)

   Figure  3.5  

Lattice points describing the 

translational symmetry of a face-centred 

cubic unit cell. The translational symmetry 

is that of the unit cell and (

+½,+½,0) , 

(

+½,0,+½), and (0,+½,+½) so a lattice 

point at the origin, O, translates to points in 

the centres of each of the faces.    

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68

3  The structures of simple solids

 Thus, for the face-centred cubic lattice depicted in  Fig.  3.5   the total number of lattice 

points in the unit cell is  (

) (

)

8

6

4

1

8

1

2

×

×

+

=    . For the body-centred cubic lattice depicted in 

 Fig.  3.4  , the number of lattice points is  (

) (

)

1 1

8

2

1

8

×

×

+

=    .  

   E X A MPLE  3.1 

Identifying lattice types   

  Determine the translational symmetry present in the structure of cubic ZnS ( Fig.  3.6  ) and identify the 

lattice type to which this structure belongs.