1. What Are Hexagonal Miller Indices?

Hexagonal Miller indices [hkil] are a four-index notation system used to describe crystallographic planes in hexagonal crystal systems. The fourth index (i) is derived from the first two indices (h and k) to maintain symmetry in the hexagonal coordinate system.

2. How Does the Calculator Work?

The calculator uses the fundamental relationship for hexagonal systems:

Where:

• \( h \) — First Miller index
• \( k \) — Second Miller index
• \( i \) — Third index (calculated)
• \( l \) — Fourth index (perpendicular to basal plane)

Explanation: The equation ensures that the three basal plane indices sum to zero, maintaining the symmetry of the hexagonal system.

3. Importance of Hexagonal Indices

Details: The four-index notation is essential for correctly identifying equivalent planes in hexagonal crystals and for understanding their symmetry properties in materials science and crystallography.

4. Using the Calculator

Tips: Enter integer values for h, k, and l indices. The calculator will automatically compute the i index. All values must be integers.

5. Frequently Asked Questions (FAQ)

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

Q2: What's the difference between [hkil] and (hkil)?
A: Square brackets denote specific planes, while parentheses denote families of equivalent planes.

Q3: Can I use three indices for hexagonal systems?
A: While possible, three-index notation obscures the hexagonal symmetry and makes equivalent planes less obvious.

Q4: What are common hexagonal crystal structures?
A: Examples include graphite, zinc, magnesium, and many important semiconductor materials.

Q5: How do I visualize hexagonal planes?
A: Specialized crystallographic software can help visualize planes in hexagonal systems.