d = a h
2
þ k
2
þ l
2
À
Á À1=2
or
d =
a
h
2
þ k
2
þ l
2
À
Á 1=2
2. Using General Method
Simple Cartesian geometry is used to determine the interplanar spacing in the unit
cells involving orthogonal coordinate system. However, for the unit cells involving
oblique system, simple Cartesian geometry is not very helpful. Therefore, let us use
the reciprocal lattice concept to get the general expression for interlayer spacing in
complex cases. Accordingly, let us write
1
d
2
hkl
¼ r hkl :r hkl ¼ ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ : ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ
¼ h
2 a
Ã
:a
Ã
þ hk a
Ã
:b
Ã
þ hl a
Ã
:c
Ã
þ kh b
Ã
:a
Ã
þ k
2 b
Ã
:b
Ã
þ kl b
Ã
:c
Ã
þ lh c
Ã
:a
Ã
þ lk c
Ã
:b
Ã
þ l
2 c
Ã
:c
Ã
ð4:5Þ
Making use of the identities a*.a* = a*
2 , etc. and a*.b* = b*.a* = a* b* cosc*,
etc. and collecting the like terms, Eq. 4.5 becomes
1
d
2
hkl
¼ h
2 a
Ã2
þ k
2 b
Ã2
þ l
2 c
Ã2
þ 2hk a
à b
à cos c
Ã
þ 2kl b
à c
à cos a
Ã
þ 2lh c
à a
à cos b
Ã
ð4:6Þ
Substituting the values of a
Ã
¼ bÂc
V , b*, c*, etc. and simplifying Eq. 4.6 will
become
1
d
2
hkl
¼
1
V
2
h
2 b
2 c
2 sin
2
a þ k
2 c
2 ac
2 sin
2
b þ l
2 a
2 b
2 sin
2
c þ
2hkabc
2 cos a cos b À cos c
ð
Þ þ
2kla
2 bc cos b cos c À cos a
ð
Þ þ
2lhkab
2 c cos c cos a À cos b
ð
Þ
2
6
6
4
3
7
7
5
ð4:7Þ
where V
2
¼ a
2 b
2 c
2 1 À cos
2
a À cos
2
b À cos
2
c þ 2 cos a cos b cos c
ð
Þ .
The general expression for interplanar spacing given by Eq. 4.6 is valid for
triclinic crystal system and the equations for other crystal systems can be obtained
by substituting respective axial parameters (i.e., axes and angles). They are provided in Table 4.4.
156
4 Unit Cell Representations of Miller Indices
2
þ k
2
þ l
2
À
Á À1=2
or
d =
a
h
2
þ k
2
þ l
2
À
Á 1=2
2. Using General Method
Simple Cartesian geometry is used to determine the interplanar spacing in the unit
cells involving orthogonal coordinate system. However, for the unit cells involving
oblique system, simple Cartesian geometry is not very helpful. Therefore, let us use
the reciprocal lattice concept to get the general expression for interlayer spacing in
complex cases. Accordingly, let us write
1
d
2
hkl
¼ r hkl :r hkl ¼ ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ : ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ
¼ h
2 a
Ã
:a
Ã
þ hk a
Ã
:b
Ã
þ hl a
Ã
:c
Ã
þ kh b
Ã
:a
Ã
þ k
2 b
Ã
:b
Ã
þ kl b
Ã
:c
Ã
þ lh c
Ã
:a
Ã
þ lk c
Ã
:b
Ã
þ l
2 c
Ã
:c
Ã
ð4:5Þ
Making use of the identities a*.a* = a*
2 , etc. and a*.b* = b*.a* = a* b* cosc*,
etc. and collecting the like terms, Eq. 4.5 becomes
1
d
2
hkl
¼ h
2 a
Ã2
þ k
2 b
Ã2
þ l
2 c
Ã2
þ 2hk a
à b
à cos c
Ã
þ 2kl b
à c
à cos a
Ã
þ 2lh c
à a
à cos b
Ã
ð4:6Þ
Substituting the values of a
Ã
¼ bÂc
V , b*, c*, etc. and simplifying Eq. 4.6 will
become
1
d
2
hkl
¼
1
V
2
h
2 b
2 c
2 sin
2
a þ k
2 c
2 ac
2 sin
2
b þ l
2 a
2 b
2 sin
2
c þ
2hkabc
2 cos a cos b À cos c
ð
Þ þ
2kla
2 bc cos b cos c À cos a
ð
Þ þ
2lhkab
2 c cos c cos a À cos b
ð
Þ
2
6
6
4
3
7
7
5
ð4:7Þ
where V
2
¼ a
2 b
2 c
2 1 À cos
2
a À cos
2
b À cos
2
c þ 2 cos a cos b cos c
ð
Þ .
The general expression for interplanar spacing given by Eq. 4.6 is valid for
triclinic crystal system and the equations for other crystal systems can be obtained
by substituting respective axial parameters (i.e., axes and angles). They are provided in Table 4.4.
156
4 Unit Cell Representations of Miller Indices