Interplanar Distance In Rhombohedral Crystal Lattice Calculator

Interplanar Spacing Formula:

\[ d = \sqrt{\frac{1}{\frac{((h^2 + k^2 + l^2)\sin^2\alpha + 2(hk + kl + hl)\cos^2\alpha - \cos\alpha)}{a^2(1 - 3\cos^2\alpha + 2\cos^3\alpha)}}} \]

Miller Index along x-axis (h):

Miller Index along y-axis (k):

Miller Index along z-axis (l):

Lattice Parameter Alpha (α):

radians

Lattice Constant a:

meters

Interplanar Spacing (d):

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1. What is Interplanar Spacing in Rhombohedral Lattices?

Interplanar spacing (d-spacing) is the distance between adjacent parallel planes with the same Miller indices in a crystal lattice. In rhombohedral crystal systems, this calculation is more complex due to the unique lattice parameters and angles.

2. How Does the Calculator Work?

The calculator uses the specialized formula for rhombohedral crystals:

\[ d = \sqrt{\frac{1}{\frac{((h^2 + k^2 + l^2)\sin^2\alpha + 2(hk + kl + hl)\cos^2\alpha - \cos\alpha)}{a^2(1 - 3\cos^2\alpha + 2\cos^3\alpha)}}} \]

Where:

\( h, k, l \) — Miller indices along x, y, and z axes
\( \alpha \) — Lattice parameter alpha (angle between b and c axes)
\( a \) — Lattice constant a

Explanation: This formula accounts for the unique geometry of rhombohedral crystal systems where all three axes are of equal length and all interaxial angles are equal but not 90°.

3. Importance of Interplanar Spacing Calculation

Details: Accurate calculation of interplanar spacing is crucial for X-ray diffraction analysis, crystal structure determination, materials characterization, and understanding material properties at the atomic level.

4. Using the Calculator

Tips: Enter Miller indices as integers, lattice parameter alpha in radians, and lattice constant a in meters. All values must be valid (Miller indices ≥ 0, alpha > 0, a > 0).

5. Frequently Asked Questions (FAQ)

Q1: What makes rhombohedral crystals different from other crystal systems?
A: Rhombohedral crystals have all three axes of equal length and all interaxial angles equal but not 90°, creating a unique symmetry that requires specialized calculations.

Q2: Why is the formula for rhombohedral crystals more complex?
A: The non-orthogonal nature of the lattice axes creates mathematical complexity that requires trigonometric functions to accurately calculate distances between crystal planes.

Q3: What are some common materials with rhombohedral crystal structure?
A: Examples include calcite, hematite, corundum (including ruby and sapphire), and various carbonate minerals.

Q4: How is this calculation used in materials science?
A: It's essential for interpreting X-ray diffraction patterns, determining crystal structures, analyzing strain in materials, and characterizing crystalline materials.

Q5: Can this calculator be used for other crystal systems?
A: No, this specific formula is designed only for rhombohedral crystal systems. Other crystal systems (cubic, tetragonal, etc.) have different formulas for calculating interplanar spacing.