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June 12, 2009
4.3. From Atoms to Solid Materials
Axes and interaxial angles
Bravais lattice cells
Three axes at right angles;
all equal:
a = b = c; α = β = γ =
Cubic F
Cubic l
Cubic P
Three axes at right angles;
two equal:
a = b =
c; α = β = γ =
90º
Tetragonal P
Tetragonal l
Orthorhombic
Three axes at right angles;
all unequal:
c; α = β = γ =
b =
a =
90º
C
P
F
I
Three axes, one pair not
at right angles, of any length:
90º
c; α = γ = = β
b =
a =
Monoclinic C
Monoclinic P
Three axes, not at right
angles, of any length:
a =
b =
=
=
90º
c; α β γ =
Triclinic P
Rhombohedral: three axes
equally inclined,
not at right angles; all equal:
a = b = c; α = β = γ 90º
Trigonal R (rhombohedral)
Hexagonal: three equal
axes coplanar at
120 , fourth axis at right
angles to these: = 90º 81 ch04
a = a = a = c
Trigonal and hexagonal C(or P) 1 2 3
α = β = 90º, γ = 120º
Figure 4.19. The 14 Bravais lattices. P: primitive cell; I: body-centred
cell; F: face-centred cell.
repeat unit that can generate the complete 3D crystal, and is cho-
sen as the sub-unit that has the highest symmetry and smallest
volume. There are 14 basic unit cells known as Bravais lattices
(Fig. 4.19).