Formula & Calculator
Interplanar Spacing for Cubic Crystals
Calculates the spacing between parallel lattice planes in a cubic crystal from the lattice parameter and Miller indices.
Interpretation
d_hkl = a / √(h² + k² + l²). Distance between parallel planes in a cubic lattice. Used in XRD to identify crystal structure and lattice parameter. Essential for crystallography.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| d_hkl | Interplanar spacing | angstrom |
| a | Cubic lattice parameter | angstrom |
| h, k, l | Miller indices of the crystal plane |
What it means
In a cubic crystal system, the interplanar spacing d_hkl for the plane with Miller indices (hkl) is given by d_hkl = a / √(h² + k² + l²), where a is the lattice constant. This formula is derived from the geometry of the cubic unit cell. It is used in X‑ray diffraction (Bragg’s law) to determine the positions of diffraction peaks, enabling identification of crystal structure and accurate lattice parameters. By measuring the angles of diffraction peaks, one can index the pattern and determine the crystal system. This is fundamental in materials characterisation, mineralogy, and solid‑state chemistry. Understanding this spacing is crucial for interpreting diffraction patterns and for analysing polycrystalline materials.
Worked example
Interplanar Spacing – Two Examples
Real‑World| Parameter | Value |
|---|---|
| a | 3.615 Å |
| (h,k,l) | (1,1,1) |
| Parameter | Value |
|---|---|
| a | 5.43 Å |
| (h,k,l) | (2,2,0) |
Common mistakes
- Interplanar spacing for cubic crystals: d_hkl = a / √(h²+k²+l²).
- Lattice parameter a: The length of the cubic unit cell edge – in m or Å.
- Miller indices h, k, l: Integers – do not reduce fractions incorrectly.
- Assumes orthogonal axes: Valid only for cubic systems – for other crystal systems, use the more general formula.
- Forbidden reflections: Some combinations may have zero intensity (e.g., FCC has systematic absences).
Applications
The interplanar spacing for cubic crystals, d_hkl = a / √(h² + k² + l²), gives the distance between parallel planes with Miller indices (hkl). It is used in X‑ray diffraction to identify peaks, to index patterns, and to determine crystal structure. Materials scientists rely on this to analyse polycrystalline and single‑crystal materials, to measure lattice parameters, and to study phase transformations. By calculating d_hkl, they can assign reflections in diffraction patterns, enabling phase identification and microstructural characterisation. This formula is a workhorse of crystallographic analysis.
- Indexing of X‑ray and electron diffraction patterns
- Phase identification in metals, ceramics, and minerals
- Lattice parameter determination from diffraction data
- Study of epitaxial relationships and texture
- Residual stress measurement via peak shift
Frequently Asked Questions
The interplanar spacing d_hkl for a cubic crystal with lattice parameter a is d_hkl = a / √(h² + k² + l²), where (hkl) are the Miller indices of the plane.
Using the cubic‑only formula for a non‑cubic crystal system (tetragonal, orthorhombic, etc.), where a different, more complex spacing equation is required.
The denominator √(h² + k² + l²) is the length of the reciprocal lattice vector. The interplanar spacing is inversely proportional to this length.
Using Bragg's law: nλ = 2d sinθ. For a known peak (hkl), d = a / √(h²+k²+l²). Measure θ, then solve for a.
For a simple cubic (a=1): d_100 = 1; d_110 = 1/√2 ≈ 0.707; d_111 = 1/√3 ≈ 0.577. These ratios are characteristic of the structure.
The formula gives the spacing for the actual lattice planes. For a given plane, higher order diffraction (n>1) corresponds to the same d but different path difference.
- Only for cubic systems.
- Assumes perfect crystals.
- Does not account for thermal expansion (a changes with temperature).
Measure the d‑spacings of the peaks. For a cubic crystal, the ratio of d² values (or sin²θ) corresponds to reciprocal of (h²+k²+l²). Compare with allowed reflections for the structure (FCC, BCC, etc.).
The interplanar spacing is the reciprocal of the magnitude of the reciprocal lattice vector G_hkl: d = 2π / |G| (in some conventions).
The formula uses the squares of the indices, so the sign does not matter. For example, d_1̄10 = d_110.