Interplanar Distance In Hexagonal Crystal Lattice Calculator

Hexagonal Crystal Lattice Formula:

\[ d = \sqrt{\frac{1}{\frac{\frac{4}{3}(h^2 + hk + k^2)}{a^2} + \frac{l^2}{c^2}}} \]

Miller Index along x-axis (h):

Miller Index along y-axis (k):

Miller Index along z-axis (l):

Lattice Constant a:

Lattice Constant c:

Interplanar Spacing (d):

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1. What is Interplanar Spacing in Hexagonal Crystals?

Interplanar spacing (d-spacing) is the distance between adjacent parallel planes of atoms in a crystal lattice. In hexagonal crystal systems, this distance is calculated using a specific formula that accounts for the unique symmetry and lattice parameters of hexagonal structures.

2. How Does the Calculator Work?

The calculator uses the hexagonal crystal lattice formula:

\[ d = \sqrt{\frac{1}{\frac{\frac{4}{3}(h^2 + hk + k^2)}{a^2} + \frac{l^2}{c^2}}} \]

Where:

\( d \) — Interplanar spacing (m)
\( h, k, l \) — Miller indices along x, y, z axes
\( a \) — Lattice constant along x-axis (m)
\( c \) — Lattice constant along z-axis (m)

Explanation: The formula accounts for the hexagonal symmetry where the x and y axes are equivalent and at 120° to each other, while the z-axis is perpendicular to the basal plane.

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 constants in meters. All values must be valid (lattice constants > 0). The calculator handles both positive and negative Miller indices.

5. Frequently Asked Questions (FAQ)

Q1: What are Miller indices?
A: Miller indices are a notation system in crystallography for planes in crystal lattices, represented by three integers (hkl) that represent the reciprocal of the intercepts with the crystallographic axes.

Q2: Why is the formula different for hexagonal crystals?
A: Hexagonal crystals have unique symmetry with two equivalent axes at 120° and a third perpendicular axis, requiring a specialized formula that accounts for this symmetry.

Q3: What are typical values for lattice constants?
A: For hexagonal crystals like zinc or magnesium, typical lattice constants range from 2-6 Å (2-6 × 10⁻¹⁰ m) for the a parameter and 3-10 Å for the c parameter.

Q4: Can negative Miller indices be used?
A: Yes, negative Miller indices are valid and represent planes that intersect the negative direction of the crystallographic axes.

Q5: What happens if the denominator becomes zero?
A: If the denominator calculation results in zero, the interplanar spacing becomes undefined as it would require division by zero, which is mathematically impossible.