Miller Indices (h k l) – Quick Notes
Definition
Miller indices are a set of three integers (h k l) used to represent the orientation of crystal planes in a crystal lattice.
They are written as:
👉 (h k l) → single plane
👉 {h k l} → family of equivalent planes
Why Miller Indices are used?
- To describe crystal planes in a standard way
- Easier than using intercept values
- Useful in X-ray diffraction, crystallography, solid state physics
Steps to Find Miller Indices
✔ Step 1: Find intercepts
Find where the plane cuts the axes (x, y, z)
Example: (2a, 3b, ∞c)
✔ Step 2: Take reciprocals
Reciprocals of intercepts:
- 1/2, 1/3, 0
✔ Step 3: Clear fractions
Multiply by LCM (6):
- (3, 2, 0)
✔ Step 4: Write Miller Index
👉 (3 2 0)
Important Rules
- If plane is parallel to an axis → intercept = ∞ → reciprocal = 0
- If plane cuts origin → shift origin first
- Always reduce to smallest integers
- Negative intercept → write with bar
Example: \bar{1} 0 1
Common Miller Indices
|
Plane |
Meaning |
|---|---|
|
(100) |
cuts x-axis only |
|
(010) |
cuts y-axis only |
|
(001) |
cuts z-axis only |
|
(110) |
cuts x and y |
|
(111) |
cuts all axes equally |
Family of Planes
- (100), (010), (001) → belong to {100} family
- (111), (1̅11), (11̅1) → symmetry equivalent planes
Important Special Cases
Parallel Plane
- Intercept = ∞
- Example: (100) → parallel to y & z axes
Negative Plane
- Represented with bar
- Example: (1̅ 0 0)
Interplanar Spacing (d-spacing)
Definition
Interplanar spacing is the perpendicular distance between two adjacent parallel crystal planes having the same Miller indices (h k l).
Formula (Cubic Crystal System)
d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}
Where:
- a = lattice constant (edge length of cube)
- (h k l) = Miller indices
- dâ‚•â‚–â‚— = interplanar spacing
Important Points
- Higher Miller index → smaller spacing
- (100) has largest spacing
- (111) has smallest spacing (in cubic crystals)
Special Cases
|
Plane |
d-spacing |
|---|---|
|
(100) |
a |
|
(110) |
a/√2 |
|
(111) |
a/√3 |
Exam Tip
👉 Always remember:
d \propto \frac{1}{\sqrt{h^2+k^2+l^2}}
2. Angle Between Two Planes
Formula (Cubic Crystal)
Angle between two planes (h₁k₁l₁) and (h₂k₂l₂):
\cos \theta = \frac{h_1 h_2 + k_1 k_2 + l_1 l_2} {\sqrt{h_1^2+k_1^2+l_1^2} \; \sqrt{h_2^2+k_2^2+l_2^2}}
Key Idea
👉 Angle between planes = angle between their normal vectors
So we treat:
- (h k l) as a vector
Example
Angle between (100) and (110):
\cos \theta = \frac{1\cdot1 + 0\cdot1 + 0\cdot0}{1 \cdot \sqrt{2}} = \frac{1}{\sqrt{2}}
\theta = 45^\circ
Important Results
|
Planes |
Angle |
|---|---|
|
(100) & (010) |
90° |
|
(100) & (110) |
45° |
|
(111) & (100) |
~54.7° |
3. Angle Between Planes & Directions (Important Concept)
- Plane normal = direction [h k l]
- So:
- Plane–plane angle = direction–direction angle
4. Exam Short Tricks
✔ Spacing
- Bigger indices → closer planes
- Use formula directly
✔ Angle
- Treat (h k l) as vector
- Use dot product formula
5. One-Line Revision
- Interplanar spacing: distance between parallel planes
- Formula: d = \frac{a}{\sqrt{h^2+k^2+l^2}}
- Angle: use dot product of (h k l) vectors
Applications
- X-ray diffraction analysis
- Crystal structure study
- Slip systems in metals (plastic deformation)
- Material science & engineering
Exam Tips
- Always write steps in order
- Show reciprocals clearly
- Reduce ratios before final answer
- Draw diagram if possible for full marks