🔷 Crystal Lattice & Unit Cell 3D

Build simple cubic, BCC, FCC, diamond, rock-salt, CsCl and HCP crystals in 3D, slice them with any Miller plane (hkl), and calculate theoretical density, packing fraction, coordination number, d-spacing and the X-ray Bragg angle for real materials.

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Material preset

Lattice constant a (pm)

Molar mass per formula unit (g/mol)

Cells per side 1

Miller h

Miller k

Miller l

X-ray λ (pm)

X-ray tube

Show (hkl) plane Nearest-neighbour bonds Space-filling atoms

Atoms per cell—

Coordination number—

Packing fraction—

Theoretical density—

Cell volume—

Nearest-neighbour distance—

d-spacing d(hkl)—

Bragg angle 2θ (n = 1)—

Atomic radius (hard sphere)—

First allowed XRD peaks (2θ)—

How this works — lattices, packing and X-ray diffraction

A crystal is a unit cell repeated in every direction. Atoms on corners are shared by 8 cells, on edges by 4 and on faces by 2 — so simple cubic holds 1 atom per cell, BCC 2, FCC 4 and diamond 8.

Density: ρ = Z · M / (N_A · V_cell), where Z is formula units per cell, M the molar mass and N_A = 6.02214076×10²³ mol⁻¹. Copper (FCC, a = 361.49 pm, M = 63.546) gives 8.93 g/cm³ — the measured value is 8.96.
Packing fraction: volume of touching hard spheres ÷ cell volume — SC 52.4 %, BCC 68.0 %, FCC and ideal HCP 74.0 % (the densest possible, Kepler's conjecture), diamond 34.0 %.
Miller plane (hkl): the plane cutting the axes at a/h, a/k, a/l. For cubic cells the spacing between parallel planes is d = a / √(h² + k² + l²).
Bragg's law: n λ = 2 d sin θ. The default λ is Cu Kα₁ (154.06 pm), the most common lab X-ray source. Not every plane reflects: FCC needs h, k, l all odd or all even; BCC needs h + k + l even; diamond also forbids all-even sums that are not multiples of 4 — the tool checks these selection rules for you.

Hexagonal close packing uses a and c = 1.633 a (ideal); its d-spacing and selection rule are not cubic, so the Miller and Bragg fields apply to the cubic structures only.