How to use this calculator
Choose the crystal structure and give either the atomic radius or the measured density, with the atomic weight in g/mol. For cubic crystals, pick a direction and plane, and Miller indices with an X-ray wavelength for the diffraction angle.
Formulas
Lattice parameterSC a = 2R, BCC a = 4R/√3, FCC a = 2R√2
Densityρ = nA/(VcNA)
Spacingdhkl = a/√(h² + k² + l²)
Braggnλ = 2d sin θ
Worked example
Copper: FCC, R = 0.1278 nm, A = 63.55 g/mol (the default; Callister Example 3.3)
- a = 2(0.1278)√2 = 0.3615 nm, so Vc = 4.723 × 10⁻²³ cm³.
- ρ = 4(63.55)/(4.723 × 10⁻²³ × 6.022 × 10²³) = 8.94 g/cm³ (measured 8.94).
- For (111) planes, d = 0.3615/√3 = 0.2087 nm; with Cu Kα (0.1542 nm), 2θ = 43.4°.
Frequently asked questions
Why do BCC and FCC skip some reflections?
Atoms in the extra positions scatter exactly out of phase for some planes: BCC needs h + k + l even, FCC needs h, k, l all odd or all even.
Why do HCP and FCC have the same APF?
Both stack close-packed planes; only the stacking sequence differs (ABAB vs ABCABC).