Interplanar Distance in Triclinic Crystal Lattice Calculator

Interplanar Spacing Formula for Triclinic Lattice:

\[ d = \sqrt{\frac{1}{\frac{(b^2 c^2 \sin^2\alpha \cdot h^2) + (a^2 c^2 \sin^2\beta \cdot k^2) + (a^2 b^2 \sin^2\gamma \cdot l^2) + (2abc^2 (\cos\alpha \cos\beta - \cos\gamma) \cdot hk) + (2bca^2 (\cos\gamma \cos\beta - \cos\alpha) \cdot lk) + (2acb^2 (\cos\alpha \cos\gamma - \cos\beta) \cdot hl)}{V^2}}} \]

Lattice Constant a:

Lattice Constant b:

Lattice Constant c:

Lattice Parameter Alpha (α):

degrees

Lattice Parameter Beta (β):

degrees

Lattice Parameter Gamma (γ):

degrees

Miller Index h:

Miller Index k:

Miller Index l:

Interplanar Spacing (d):

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1. What is Interplanar Spacing?

Interplanar spacing (d-spacing) is the distance between adjacent parallel planes of atoms in a crystal lattice. In triclinic crystal systems, this calculation is complex due to the lack of symmetry constraints and requires consideration of all lattice parameters and Miller indices.

2. How Does the Calculator Work?

The calculator uses the triclinic interplanar spacing formula:

Where:

\( a, b, c \) — Lattice constants along x, y, z axes (m)
\( \alpha, \beta, \gamma \) — Angles between lattice vectors (degrees)
\( h, k, l \) — Miller indices of the crystal plane
\( V \) — Volume of the unit cell (m³)

Explanation: This comprehensive formula accounts for the asymmetric nature of triclinic crystals by considering all lattice parameters and their angular relationships.

3. Importance of Interplanar Spacing Calculation

Details: Accurate d-spacing calculation 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 lattice constants in meters, angles in degrees, and integer Miller indices. All values must be valid (positive lattice constants, angles between 0-180°, integer Miller indices).

5. Frequently Asked Questions (FAQ)

Q1: Why is the triclinic formula more complex than other crystal systems?
A: Triclinic crystals have the lowest symmetry with no equal lengths or right angles, requiring consideration of all six lattice parameters in the calculation.

Q2: What are typical values for interplanar spacing?
A: Interplanar spacing typically ranges from 0.1 to 10 Å (10⁻¹¹ to 10⁻⁹ m), depending on the crystal structure and Miller indices.

Q3: How does interplanar spacing relate to X-ray diffraction?
A: According to Bragg's law (nλ = 2d sinθ), the d-spacing determines the diffraction angles where constructive interference occurs.

Q4: Can this calculator be used for other crystal systems?
A: This specific formula is for triclinic systems. Other crystal systems have simplified formulas due to their higher symmetry.

Q5: What units should I use for lattice constants?
A: Lattice constants are typically in meters, but the calculator will work with any consistent unit system as long as all lengths use the same unit.