Interplanar Angle for Orthorhombic System Calculator

Interplanar Angle Formula for Orthorhombic System:

\[ \theta = \cos^{-1}\left(\frac{\frac{h_1h_2}{a^2} + \frac{k_1k_2}{b^2} + \frac{l_1l_2}{c^2}}{\sqrt{\left(\frac{h_1^2}{a^2} + \frac{k_1^2}{b^2} + \frac{l_1^2}{c^2}\right) \cdot \left(\frac{h_2^2}{a^2} + \frac{k_2^2}{b^2} + \frac{l_2^2}{c^2}\right)}}\right) \]

Miller Index h along plane 1:

Miller Index k along plane 1:

Miller Index l along plane 1:

Miller Index h along plane 2:

Miller Index k along plane 2:

Miller Index l along plane 2:

Lattice Constant a:

Lattice Constant b:

Lattice Constant c:

Interplanar Angle:

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1. What is Interplanar Angle?

The interplanar angle is the angle between two crystallographic planes in a crystal lattice. For orthorhombic crystal systems, this angle depends on the Miller indices of the planes and the lattice constants along the three axes.

2. How Does the Calculator Work?

The calculator uses the formula for orthorhombic systems:

Where:

\( h_1, k_1, l_1 \) — Miller indices for plane 1
\( h_2, k_2, l_2 \) — Miller indices for plane 2
\( a, b, c \) — Lattice constants along x, y, and z axes respectively
\( \theta \) — Interplanar angle in degrees

Explanation: The formula calculates the cosine of the angle between normals to the two planes, then takes the inverse cosine to find the angle itself.

3. Importance of Interplanar Angle Calculation

Details: Calculating interplanar angles is crucial in crystallography for understanding crystal structure, analyzing X-ray diffraction patterns, and determining crystal orientations in materials science.

4. Using the Calculator

Tips: Enter integer values for Miller indices and positive values for lattice constants. All lattice constants must be greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is an orthorhombic crystal system?
A: An orthorhombic crystal system has three mutually perpendicular axes of different lengths (a ≠ b ≠ c) with all angles equal to 90°.

Q2: What are Miller indices?
A: Miller indices are a notation system in crystallography for planes in crystal lattices, represented by three integers (h,k,l).

Q3: Can this calculator be used for other crystal systems?
A: No, this specific formula is designed for orthorhombic systems. Other crystal systems have different formulas for calculating interplanar angles.

Q4: What are typical values for lattice constants?
A: Lattice constants are typically in the range of 0.1-1.0 nm (1.0-10.0 × 10⁻¹⁰ m) for most crystalline materials.

Q5: Why is the interplanar angle important in X-ray diffraction?
A: The interplanar angle determines the diffraction pattern and helps identify crystal structures and orientations in X-ray crystallography.