Saved calculations
| Time | Method | Details | d (Å) | d (nm) | d (pm) |
|---|---|---|---|---|---|
| No results yet. Run a calculation to populate this table. | |||||
Example data table
| Scenario | Inputs | Output (d) |
|---|---|---|
| Bragg (solve d) | n=1, λ=1.5406 Å, 2θ=30° | d ≈ 2.98 Å |
| Cubic (hkl) | a=3.90 Å, (111) | d ≈ 2.25 Å |
| Hexagonal (hkl) | a=2.50 Å, c=4.10 Å, (100) | d ≈ 2.17 Å |
Formula used
This tool computes the interplanar spacing d using two common routes.
1) Bragg’s law (XRD)
nλ = 2d sin(θ)
- θ = (2θ)/2 when your instrument reports 2θ.
- Solve for d = nλ / (2 sin θ), or solve for 2θ when d is known.
2) Miller indices + lattice parameters
- Cubic: 1/d² = (h²+k²+l²)/a²
- Tetragonal: 1/d² = (h²+k²)/a² + l²/c²
- Orthorhombic: 1/d² = h²/a² + k²/b² + l²/c²
- Hexagonal: 1/d² = (4/3)(h²+hk+k²)/a² + l²/c²
- Monoclinic (β unique): 1/d² = (h²/a² + k² sin²β/b² + l²/c² − 2hl cosβ/(ac)) csc²β
- Triclinic: uses the full angle‑dependent expression.
How to use this calculator
- Select a method: Bragg’s law or Miller indices + lattice parameters.
- Choose your preferred output unit for d.
- Enter all required inputs, then click Calculate.
- Read the result shown above the form.
- Download your saved results as CSV or PDF.
1) What lattice spacing means
Lattice spacing (often written as d) is the distance between repeating atomic planes inside a crystal. Each set of planes is labeled by Miller indices (h k l). Smaller d values usually mean planes are closer together and diffraction peaks appear at higher angles.
2) Why diffraction peaks give d
X‑ray diffraction works because crystal planes reflect X‑rays like a mirror array. When the path difference equals an integer number of wavelengths, the reflected waves add up strongly and you see a peak. This is the physical reason Bragg’s law is so useful in lab and industrial measurements.
3) Bragg’s law inputs used here
The calculator uses nλ = 2d sin(θ), where instruments typically report 2θ. Common laboratory wavelengths include Cu Kα at about 1.5406 Å and Mo Kα near 0.7107 Å. Typical scans run from ~10° to 120° in 2θ, covering many practical spacings.
4) Interpreting practical d ranges
Many dense inorganic crystals produce d values around 0.8–3.5 Å for strong peaks, while layered or porous materials can show larger spacings, sometimes above 5 Å. If your computed d looks unusually large or tiny, double‑check the wavelength unit and confirm you used θ as half of 2θ.
5) Using Miller indices with cell parameters
When you already know the unit cell, d(hkl) can be computed directly from the crystal system relations. For cubic crystals, the result depends only on a and h²+k²+l². Hexagonal and tetragonal systems separate in‑plane and out‑of‑plane contributions, so a and c both matter.
6) Angle systems and when to enter β, α, γ
Lower‑symmetry cells need angles because axes are not perpendicular. In monoclinic cells, only β is non‑90° (with the common “β unique” convention). Triclinic cells may require all three angles. If angles create an invalid cell, the calculator warns you before returning a result.
7) Uncertainty and measurement quality
Small angle errors can noticeably change d at low 2θ. That is why the optional uncertainty fields are helpful: the tool applies basic propagation from ±λ and ±2θ (when solving for d). Use well‑calibrated standards for the most reliable spacing values.
8) Good reporting practices
For reproducible work, record the radiation source (λ), peak position (2θ), indexing (hkl), and the unit cell parameters used. Reporting d in Å is common, but nm and pm are useful for cross‑checking units. Saving multiple results and exporting them supports quick lab notes and clean documentation.
FAQs
1) Is d the same as the lattice constant a?
No. a is a unit‑cell edge length. d is the spacing between specific planes (hkl). In cubic crystals, some planes have d related to a, but they are not identical.
2) My instrument shows 2θ. What do I enter?
Enter the reported 2θ value. The calculator internally uses θ = (2θ)/2 for Bragg’s law, which matches standard XRD practice.
3) What order n should I use?
Most routine powder diffraction uses n=1. Higher orders can occur, but they often overlap with other peaks. Use higher n only if your analysis or instrument setup explicitly requires it.
4) Which wavelength is most common in labs?
Cu Kα (about 1.5406 Å) is very common for powder XRD. Other sources exist (like Mo Kα) and must be entered correctly, including the proper unit.
5) Why do Bragg and hkl methods give different answers?
If inputs are consistent, they should agree. Differences usually come from wrong indexing, incorrect unit cell parameters, mixed wavelengths (Kα1/Kα2), or peak position shifts from strain, calibration, or sample displacement.
6) Can this tool handle non‑90° unit cells?
Yes. Monoclinic uses β, and triclinic can use α, β, γ. If angles imply an impossible cell geometry, the calculator will stop and show an error.
7) What’s a quick way to sanity‑check results?
Check units first, then confirm θ is half of 2θ. Compare your d to typical ranges (often ~0.8–5 Å). Extremely large or tiny values usually indicate a unit or angle mistake.