Substituting these values in Eq. 3.13, we obtain
V
2
¼
a
2
ab cos c ac cos b
ba cos c
b
2
bc cos a
ca cos b cb cos a
c
2
ð3:15Þ
Further, simplifying the determinant, we obtain
V
2
¼ a
2 b
2 c
2 1 À cos
2
a À cos
2
b À cos
2
c þ 2cosacosbcosc
À
Á
V ¼ abc 1 À cos
2
a À cos
2
b À cos
2
c + 2cosacosbcosc
À
Á 1=2
ð3:16Þ
This is a general equation expressing the volume of a triclinic unit cell.
Equations for other unit cells can be obtained by substituting the values of
respective axial parameters (axes and angles) in Eq. 3.16. Unit cell volumes of
different lattice types are provided in Table 3.2.
3.4 Angle Between Two Crystallographic Directions
The scalar product (also called the dot product) of two vectors ~ a and ~ b denoted by ~ a.
~ b is defined as a scalar quantity which is equal to the product of the magnitudes of
two vectors and the cosine of angle between their directions (Fig. 3.9), that is,
~ a: ~ b ¼ a
j j b
j j cos h
or cos h ¼
~ a: ~ b
a
j j b
j j
Here; ~ b:~ a ¼ b
j j a
j j cos Àh
ð Þ ¼ a
j j b
j j cos h
Table. 3.2 Unit cell volume of different lattice types
Lattice type
Volume
Cubic
a
3
Orthorhombic
abc
Tetragonal
a
2 c
Hexagonal
ffiffi
3
p a
2 c
2
Rhombohedral
a
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À 3cos
2 a þ 2cos
3 a
p
Monoclinic
abc sinb
Triclinic
abc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À cos 2 a À cos 2 b À cos 2 c þ 2 cos a cos b cos c
p
110
3 Unit Cell Calculations
V
2
¼
a
2
ab cos c ac cos b
ba cos c
b
2
bc cos a
ca cos b cb cos a
c
2
ð3:15Þ
Further, simplifying the determinant, we obtain
V
2
¼ a
2 b
2 c
2 1 À cos
2
a À cos
2
b À cos
2
c þ 2cosacosbcosc
À
Á
V ¼ abc 1 À cos
2
a À cos
2
b À cos
2
c + 2cosacosbcosc
À
Á 1=2
ð3:16Þ
This is a general equation expressing the volume of a triclinic unit cell.
Equations for other unit cells can be obtained by substituting the values of
respective axial parameters (axes and angles) in Eq. 3.16. Unit cell volumes of
different lattice types are provided in Table 3.2.
3.4 Angle Between Two Crystallographic Directions
The scalar product (also called the dot product) of two vectors ~ a and ~ b denoted by ~ a.
~ b is defined as a scalar quantity which is equal to the product of the magnitudes of
two vectors and the cosine of angle between their directions (Fig. 3.9), that is,
~ a: ~ b ¼ a
j j b
j j cos h
or cos h ¼
~ a: ~ b
a
j j b
j j
Here; ~ b:~ a ¼ b
j j a
j j cos Àh
ð Þ ¼ a
j j b
j j cos h
Table. 3.2 Unit cell volume of different lattice types
Lattice type
Volume
Cubic
a
3
Orthorhombic
abc
Tetragonal
a
2 c
Hexagonal
ffiffi
3
p a
2 c
2
Rhombohedral
a
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À 3cos
2 a þ 2cos
3 a
p
Monoclinic
abc sinb
Triclinic
abc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À cos 2 a À cos 2 b À cos 2 c þ 2 cos a cos b cos c
p
110
3 Unit Cell Calculations