Calculator
Example Data Table
| Crystal system | Inputs (a,b,c,α,β,γ) | Expected reciprocal lengths (with 2π) | Notes |
|---|---|---|---|
| Cubic | a=b=c=4 Å, α=β=γ=90° | a*=b*=c*=2π/4 ≈ 1.570796 1/Å | Reciprocal angles remain 90°. |
| Tetragonal | a=b=3 Å, c=5 Å, α=β=γ=90° | a*=b*=2π/3 ≈ 2.094395 1/Å; c*=2π/5 ≈ 1.256637 1/Å | Useful for indexing peaks. |
| Hexagonal | a=b=2.5 Å, c=4 Å, α=β=90°, γ=120° | a*=b*=4π/(√3 a) ≈ 2.902079 1/Å; c*=2π/4 ≈ 1.570796 1/Å | Non-orthogonal basal plane. |
Formula Used
The reciprocal basis vectors are computed from the real-space basis vectors a1, a2, a3 using the unit-cell volume V = a1 · (a2 × a3).
In cell-parameter mode, real-space vectors are built in a conventional Cartesian frame.
How to Use This Calculator
- Select an input mode: cell parameters or basis vectors.
- Choose a length unit and enable 2π if needed.
- Enter values carefully; avoid degenerate vectors.
- Press Submit to display results above the form.
- Download CSV or PDF for reports or sharing.
- Validate angles and lengths against symmetry expectations.
Professional Article
1) Why reciprocal space matters
Diffraction experiments, electronic band calculations, and phonon studies are naturally expressed in reciprocal space. A reciprocal basis converts real-space periodicity into wavevector coordinates, letting you index peaks, define Brillouin zones, and compare datasets across instruments. This calculator produces the full reciprocal basis consistently from either cell parameters or explicit basis vectors.
2) Real-space vectors and unit-cell volume
The core stability check is the real-space volume, V = a1 · (a2 × a3). For physically meaningful lattices, V must be nonzero and positive in magnitude. For example, a cubic cell with a = 4 Å yields V = 64 ų. Near-coplanar vectors reduce V and amplify numerical noise in reciprocal vectors.
3) Reciprocal basis construction
The reciprocal vectors are built using cross products: b1 ∝ (a2 × a3), b2 ∝ (a3 × a1), and b3 ∝ (a1 × a2), each divided by V. If you enable the 2π factor, magnitudes align with common scattering and band-structure conventions. With 2π enabled, cubic a = 4 Å gives |b1| = 2π/4 = 1.570796 1/Å.
4) Reciprocal lengths and angles
Reciprocal lengths a*, b*, c* are the norms of b1, b2, b3, while α*, β*, γ* come from dot-product angles. Orthogonal real cells keep reciprocal angles near 90°, but non-orthogonal inputs shift them predictably. For hexagonal a = 2.5 Å and γ = 120°, the basal reciprocal length becomes 4π/(√3 a) ≈ 2.902079 1/Å (with 2π).
5) Metric tensors as quality control
The real-space metric tensor G (Gij = ai · aj) captures lengths and inter-vector correlations in one matrix. The reciprocal metric tensor G* does the same in reciprocal space. Consistent lattices show symmetry patterns: cubic cells yield diagonal G and G* (within rounding), while triclinic cells produce fully populated symmetric matrices. These tensors help validate imported crystallography data.
6) Units, scaling, and practical ranges
Because reciprocal vectors scale as 1/length, switching from Å to nm changes magnitudes by a factor of 10. Typical crystalline lattice constants lie around 2–10 Å, giving reciprocal magnitudes on the order of 0.6–3 1/Å (or 4–20 1/nm), depending on whether the 2π factor is included. Always record the unit choice alongside exports.
7) Using results for indexing and k-space tasks
Once b1, b2, b3 are known, any reciprocal vector G = h b1 + k b2 + l b3 follows directly, supporting peak indexing, zone boundary estimates, and distance calculations in k-space. For quick checks, compare computed |G| values against experimental scattering vector magnitudes. The CSV/PDF exports are designed for lab notebooks and reports.
8) Common pitfalls and verification steps
Errors usually come from inconsistent angle definitions, mixing degrees and radians, or swapping axes in vector mode. Verify that α, β, γ are strictly between 0° and 180°, and confirm V is not close to zero. For known symmetries (cubic, tetragonal, hexagonal), compare a*, b*, c* against expected analytical forms to catch input mistakes early.
FAQs
Q1: Should I enable the 2π factor?
A: Enable it for scattering, diffraction, and many band-structure workflows where k-vectors use 2π/L. Disable it if you need the dual basis defined purely by bᵢ·aⱼ = δᵢⱼ.
Q2: What does a zero or tiny volume mean?
A: It means your real-space vectors are nearly coplanar or linearly dependent, so the unit cell is degenerate. Reciprocal vectors will blow up or become unreliable. Adjust inputs to restore a valid 3D cell.
Q3: Why are my reciprocal angles not 90 degrees?
A: Non-orthogonal real-space angles generally produce non-orthogonal reciprocal angles. This is expected in monoclinic and triclinic systems. Check that your α, β, γ definitions match standard conventions and your unit choice is consistent.
Q4: Can I use vector mode for arbitrary coordinate frames?
A: Yes. Vector mode accepts any Cartesian frame as long as a1, a2, a3 are expressed consistently in that frame. The calculator uses dot and cross products, so the geometry is frame-independent under rigid rotations.
Q5: How do I compute a specific (h k l) reciprocal vector?
A: Use G = h·b1 + k·b2 + l·b3. The output b-vectors can be combined with your integer indices to get the scattering vector direction and magnitude for peak indexing or k-space mapping.
Q6: Why do results change when I change units?
A: Reciprocal values scale inversely with length. If you convert Å to nm, real lengths increase by 10, so reciprocal magnitudes decrease by 10. The calculator keeps the geometry identical while rescaling outputs to 1/unit.
Q7: What is the purpose of the metric tensors?
A: Metric tensors summarize lattice geometry compactly. G encodes real-space lengths and angles, while G* encodes reciprocal-space geometry. They help verify symmetry, compare datasets, and compute distances without rebuilding vectors repeatedly.
Accurate reciprocal lattices sharpen every crystallographic interpretation you make.