1. What Are Hexagonal Miller Indices?

Hexagonal Miller indices [uvtw] are a four-index notation system used to describe crystallographic planes in hexagonal crystal systems. The fourth index (w) is derived from the first two indices (u and v) by the relation w = -u - v.

2. How Does the Calculator Work?

The calculator uses the fundamental relationship for hexagonal indices:

Where:

• \( u \) — First index (dimensionless)
• \( v \) — Second index (dimensionless)
• \( t \) — Third index (dimensionless)
• \( w \) — Fourth index (calculated)

Explanation: The four-index notation accounts for the symmetry of the hexagonal system and provides a more symmetrical representation of equivalent planes.

3. Importance of Miller Indices

Details: Miller indices are crucial for identifying crystal planes, understanding material properties, and interpreting diffraction patterns in hexagonal crystals like graphite, zinc, and many ceramics.

4. Using the Calculator

Tips: Enter integer values for u, v, and t indices. The calculator will automatically compute the w index to complete the four-index notation.

5. Frequently Asked Questions (FAQ)

Q1: Why use four indices for hexagonal systems?
A: The four-index notation better reflects the hexagonal symmetry and makes equivalent planes more obvious in the notation.

Q2: What's the difference between [uvw] and [uvtw]?
A: [uvw] is the three-index notation, while [uvtw] is the four-index notation specific to hexagonal systems with w = -u - v.

Q3: Can all three indices be zero?
A: No, all indices cannot be zero simultaneously as this would represent no specific plane.

Q4: How are negative indices represented?
A: Negative indices are typically written with a bar over the number (e.g., [1̄1̄2̄0]), though here we use the minus sign for simplicity.

Q5: What crystals use this notation?
A: Hexagonal close-packed (HCP) crystals like zinc, magnesium, titanium, and many ceramic materials use this notation.