3.3 Unit Cell Volume
In order to calculate the volume of a unit cell, let us make use of the vector algebra
to express the basis vectors [~ a, ~ b and ~ c] in terms of their components as
~ a ¼ a x b i þ a y b j þ a z b k
~ b ¼ b x b i þ b y b j þ b z b k
and ~ c ¼ c x b i þ c y b j þ c z b k
ð3:9Þ
From this, the volume of the unit cell can be written as
V ¼
a x a y a z
b x b y b z
c x c y c z
ð3:10Þ
From the properties of determinant, we know that the value of the determinant
remains unchanged when its rows and columns are interchanged. Consequently, it
follows that
V
2
¼
a x a y a z
b x b y b z
c x c y c z
Â
a x b x c x
a y b y c y
a z b z c z
ð3:11Þ
On multiplication, the right hand side of Eq. 3.11 becomes
V
2
¼
a x a x þ a y a y þ a z a z a x b x þ a y b y þ a z b z a x c x þ a y c y þ a z c z
b x a x þ b y a y þ b z a z b x b x þ b y b y þ b z b z b x c x þ b y c y þ b z c z
c x a x þ c y a y þ c z a z c x b x þ c y b y þ c z b z c x c x þ c y c y þ c z c z
ð3:12Þ
In the reduced form, Eq. 3.12 can be written as
V
2
¼
a.a a.b a.c
b.a b.b b.c
c.a c.b c.c
ð3:13Þ
where a.a = a
2 , b.b = b
2 , c.c = c
2
a Á b ¼ b Á a ¼ ab cos c ¼ ba cos c
b Á c ¼ c Á b ¼ bc cos a ¼ cb cos a
c Á a ¼ a Á c ¼ ca cos b ¼ ac cos b
ð3:14Þ
3.3 Unit Cell Volume
109