HKL Calculator
For a unit cell and a Miller index: the cell volume, the d-spacing of that reflection, the angle it appears at, the reflections symmetry makes equivalent to it, and the powder pattern the whole cell would give.
- You supply
- A crystal system, the cell parameters that system leaves free, a wavelength — or an anode preset — and h k l.
- Reading it
- Results update as you type. The equivalent reflections assume the holohedral Laue class of the system you chose; a structure in a lower class has fewer of them than shown here. Stick heights in the pattern are multiplicity, not intensity.
Worked examples: cubic – NaCl, Fm3m (200) · cubic – Cu, Fm3m (111) · cubic – α-Fe, Im3m (110) · cubic – CsCl, Pm3m (110) · hexagonal – quartz, P3221 (101) · tetragonal – rutile, P42/mnm (110) · orthorhombic – aragonite, Pmcn (111) · monoclinic – ZrO2, P21/c (111) · triclinic – albite, C1 (111)
Input
Results
| a = | 5.640 Å |
|---|---|
| b = | 5.640 Å |
| c = | 5.640 Å |
| α = | 90° |
| β = | 90° |
| γ = | 90° |
| V = | 179.4 Å3 |
| d = | 2.820 Å |
| 2θ = | 31.73° |
The cell, and the plane
200 cuts the axes at a at 0.5, parallel to b, parallel to c.
The spheres are lattice points, not atoms. A lattice point is a position the whole structure repeats from; how many atoms sit on each one, and where, is the basis and is not shown here. NaCl and copper have the same face‑centred lattice and quite different structures.
Powder pattern
23 reflections between 5° and 90° 2θ, lattice P (no reflection condition). Hover a stick to identify it.
200 is the highlighted stick.
| 2θ (°) | d (Å) | hkl | multiplicity |
|---|---|---|---|
| 15.71 | 5.6400 | 100 | 6 |
| 22.29 | 3.9881 | 110 | 12 |
| 27.39 | 3.2563 | 111 | 8 |
| 31.73 | 2.8200 | 200 | 6 |
| 35.59 | 2.5223 | 210 | 24 |
| 39.12 | 2.3025 | 211 | 24 |
| 45.49 | 1.9940 | 220 | 12 |
| 48.42 | 1.8800 | 300 (6) + 221 (24) | 30 |
| 51.22 | 1.7835 | 310 | 24 |
| 53.92 | 1.7005 | 311 | 24 |
| 56.52 | 1.6281 | 222 | 8 |
| 59.05 | 1.5643 | 320 | 24 |
Stick heights are multiplicity only — how many symmetry-equivalent reflections coincide at that angle. They are not intensities: there is no structure factor, no Lorentz–polarisation factor and no temperature factor here, so a measured pattern will have the peaks in these positions with quite different heights. Positions and absences are the part to read.
How this is calculated
For a cubic cell (a = b = c, α = β = γ = 90°) the general form reduces to
λ = 1.541838 Å is Cu Kα:
Significant figures follow the cell parameters you entered, so a cell given to four digits is answered to four. The equivalent reflections below assume the holohedral Laue class of this crystal system.
The general form, and what this cell strikes out
S11 = b2c2 sin2α, S22 = a2c2 sin2β, S33 = a2b2 sin2γ, S12 = abc2(cos α cos β − cos γ), S23 = a2bc(cos β cos γ − cos α), S13 = ab2c(cos γ cos α − cos β). This cell strikes out 3 of the six. A struck term is zero for every reflection in this crystal system, which is exactly what having fewer free cell parameters buys you.