1. Weiss Parameters
Weiss parameters describe the intercepts of a crystal plane on the three crystallographic axes. They were introduced by Christian Samuel Weiss in 1815 as the first systematic method to describe crystal planes.
Definition
If a plane intercepts the crystallographic axes at distances ma, nb, and pc from the origin, where a, b, and c are the unit cell parameters, then the Weiss parameters are:
Weiss Parameters = m : n : p
Example 1: Weiss Parameters
A plane intercepts the X-axis at 2a, Y-axis at 3b, and Z-axis at 1c.
Weiss Parameters: 2 : 3 : 1
Limitations of Weiss Parameters
- Cannot handle planes parallel to axes (infinite intercepts)
- Ratios can become unwieldy with fractional values
- Not convenient for mathematical calculations
- Led to the development of Miller indices
2. Miller Indices
Miller indices, introduced by William Hallowes Miller in 1839, provide a more convenient notation for crystal planes. They are the reciprocals of Weiss parameters, cleared of fractions.
Definition and Method
Find the intercepts of the plane on the three axes in terms of lattice parameters a, b, c
Take reciprocals of these intercepts
Clear fractions by multiplying by the LCM
Reduce to smallest integers (if possible)
Enclose in parentheses: (hkl)
If intercepts are ma, nb, pc
Reciprocals: 1/m : 1/n : 1/p
Miller Indices: (hkl) where h:k:l = 1/m : 1/n : 1/p
Example 2: Converting Weiss to Miller
Given: Weiss parameters = 2 : 3 : 1
Step 1: Intercepts = 2a : 3b : 1c
Step 2: Reciprocals = 1/2 : 1/3 : 1/1
Step 3: LCM of denominators = 6
Step 4: Multiply by 6: 3 : 2 : 6
Miller Indices: (326)
Special Cases
| Plane Type |
Intercepts |
Miller Indices |
Example |
| Parallel to X-axis |
∞ : n : p |
(0kl) |
(011) |
| Parallel to Y-axis |
m : ∞ : p |
(h0l) |
(101) |
| Parallel to Z-axis |
m : n : ∞ |
(hk0) |
(110) |
| Parallel to XY-plane |
∞ : ∞ : p |
(001) |
(001) |
Notation:
• (hkl) - Single plane
• {hkl} - Family of equivalent planes
• [hkl] - Direction
• ⟨hkl⟩ - Family of equivalent directions
• (h̄kl) - Negative index (h̄ represents -h)
3. Interplanar Distance (d-spacing)
The interplanar distance dhkl is the perpendicular distance between parallel planes in a family (hkl). This parameter is crucial for X-ray diffraction analysis.
Formulas for Different Crystal Systems
1. Cubic System (a = b = c, α = β = γ = 90°)
dhkl = a/√(h² + k² + l²)
Example 3: Cubic d-spacing
Given: NaCl with a = 5.64 Å, find d200
Solution:
d200 = 5.64/√(2² + 0² + 0²) = 5.64/√4 = 5.64/2 = 2.82 Å
2. Tetragonal System (a = b ≠ c, α = β = γ = 90°)
1/d²hkl = (h² + k²)/a² + l²/c²
3. Orthorhombic System (a ≠ b ≠ c, α = β = γ = 90°)
1/d²hkl = h²/a² + k²/b² + l²/c²
4. Hexagonal System (a = b ≠ c, α = β = 90°, γ = 120°)
1/d²hkl = 4(h² + hk + k²)/(3a²) + l²/c²
5. Rhombohedral System (a = b = c, α = β = γ ≠ 90°)
1/d²hkl = (h² + k² + l²)sin²α + 2(hk + kl + hl)(cos²α - cosα) / [a²(1 + 2cos³α - 3cos²α)]
6. Monoclinic System (a ≠ b ≠ c, α = γ = 90°, β ≠ 90°)
1/d²hkl = 1/sin²β × [h²/a² + l²sin²β/c² + 2hlcosβ/(ac)] + k²/b²
7. Triclinic System (a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°)
Most complex formula involving all lattice parameters and angles
Physical Significance
- X-ray Diffraction: d-spacing determines diffraction angles via Bragg's law
- Crystal Structure: Reveals information about atomic arrangements
- Phase Identification: d-spacings are fingerprints for crystal phases
- Lattice Parameters: Can be calculated from measured d-spacings
4. Relationship Between Parameters
| Weiss Parameters |
Miller Indices |
d-spacing (Cubic) |
| 1 : 1 : 1 |
(111) |
a/√3 |
| 1 : 1 : ∞ |
(110) |
a/√2 |
| 1 : ∞ : ∞ |
(100) |
a |
| 2 : 2 : 1 |
(112) |
a/√6 |
| 1 : 1 : 1/2 |
(221) |
a/3 |
5. Applications in IIT JEE
Common Problem Types:
- Converting between Weiss parameters and Miller indices
- Calculating d-spacings for given Miller indices
- Using Bragg's law: nλ = 2d sinθ
- Determining crystal structure from d-spacing ratios
- Finding lattice parameters from diffraction data
Example 4: Complete Problem
Problem: In a cubic crystal with a = 4.0 Å, find the d-spacing for the plane with Weiss parameters 1 : 2 : 3.
Solution:
Step 1: Convert to Miller indices
Weiss: 1 : 2 : 3 → Reciprocals: 1 : 1/2 : 1/3
LCM = 6 → Miller indices: (623)
Step 2: Calculate d-spacing
d623 = 4.0/√(6² + 2² + 3²) = 4.0/√(36 + 4 + 9) = 4.0/√49 = 4.0/7 = 0.571 Å
Remember: Miller indices are always written as the smallest possible integers. When converting from Weiss parameters, always check if the final indices can be reduced further by dividing by their greatest common divisor.