About

Errors in crystallographic calculations propagate into diffraction analyses, phase identification, and mechanical property predictions. A misidentified lattice parameter by 0.01 Å shifts computed density by several percent, enough to misclassify an alloy phase. This tool computes unit cell volume V = a³, theoretical density ρ via Avogadro-based derivation, atomic packing factor (APF), coordination number, and interplanar spacing d_hkl for all three cubic Bravais lattices: Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC). All formulas follow standard solid-state physics conventions (Kittel, Callister). The tool assumes ideal hard-sphere models. Real crystals exhibit thermal expansion, vacancies, and anharmonic deviations not captured here.

Formulas

The volume of a cubic unit cell with lattice parameter a:

V = a³

Theoretical density ρ relates the number of atoms per cell Z, molar mass M, Avogadro's number N_A, and cell volume:

ρ = Z ⋅ MN_A ⋅ a³

Atomic packing factor (APF) measures the fraction of volume occupied by atoms, modeled as hard spheres of radius r:

APF = Z ⋅ 43 π r³a³

Lattice parameter to atomic radius relationships for each structure:

SC: a = 2r

BCC: a = 4r√3

FCC: a = 2√2 ⋅ r

Interplanar spacing for Miller indices (h, k, l):

d_hkl = a√h² + k² + l²

Where Z = atoms per unit cell (SC: 1, BCC: 2, FCC: 4), M = molar mass in g/mol, N_A = 6.02214076 × 10²³ mol⁻¹, a = lattice parameter in Å (converted to cm for density), r = atomic radius in Å.

Reference Data

Element	Symbol	Structure	Lattice Parameter a (Å)	Atomic Radius r (Å)	Molar Mass M (g/mol)	Density (g/cm³)
Polonium	Po	SC	3.359	1.680	209.00	9.20
Iron (α)	Fe	BCC	2.866	1.241	55.845	7.87
Chromium	Cr	BCC	2.884	1.249	51.996	7.19
Tungsten	W	BCC	3.165	1.371	183.84	19.25
Molybdenum	Mo	BCC	3.147	1.363	95.95	10.28
Vanadium	V	BCC	3.024	1.310	50.942	6.11
Niobium	Nb	BCC	3.300	1.430	92.906	8.57
Tantalum	Ta	BCC	3.303	1.430	180.95	16.69
Sodium	Na	BCC	4.291	1.858	22.990	0.97
Potassium	K	BCC	5.328	2.308	39.098	0.86
Lithium	Li	BCC	3.510	1.520	6.941	0.53
Barium	Ba	BCC	5.020	2.175	137.33	3.51
Aluminum	Al	FCC	4.050	1.432	26.982	2.70
Copper	Cu	FCC	3.615	1.278	63.546	8.96
Gold	Au	FCC	4.079	1.442	196.97	19.30
Silver	Ag	FCC	4.086	1.445	107.87	10.49
Platinum	Pt	FCC	3.924	1.387	195.08	21.45
Nickel	Ni	FCC	3.524	1.246	58.693	8.91
Lead	Pb	FCC	4.950	1.750	207.20	11.34
Palladium	Pd	FCC	3.890	1.376	106.42	12.02
Calcium	Ca	FCC	5.588	1.976	40.078	1.55
Strontium	Sr	FCC	6.084	2.151	87.62	2.63
Rhodium	Rh	FCC	3.804	1.345	102.91	12.41
Iridium	Ir	FCC	3.839	1.357	192.22	22.56

Frequently Asked Questions

The structure determines Z (atoms per unit cell): SC has 1, BCC has 2, FCC has 4. Since density ρ is directly proportional to Z, selecting the wrong structure type doubles or quadruples the density error. Always verify the crystal structure from diffraction data or phase diagrams before calculating.

The formula ρ = ZM / (N_Aa³) assumes a perfect crystal with no vacancies, interstitials, or thermal expansion. Real materials contain point defects (typically 10⁻⁴ vacancy fraction near melting), dislocations, and grain boundaries. Additionally, lattice parameters are temperature-dependent. Published values are usually at 20 - 25 °C. Expect discrepancies of 0.1 - 2% for well-annealed single crystals.

The hard-sphere model assumes atoms are rigid, non-overlapping spheres with fixed radius. In reality, electron clouds are diffuse and compressible. Metallic bonding radii differ from covalent or van der Waals radii. Under high pressure, atoms compress measurably. The model breaks down for alloys where atoms of different sizes create lattice strain, and for ionic crystals where cation/anion radius ratios govern stability rather than simple packing.

Selection rules depend on structure. For SC, all (hkl) reflections are allowed. For BCC, only planes where h + k + l = even are permitted. For FCC, h, k, l must be all odd or all even. Using forbidden indices in Bragg's law yields interplanar spacings that correspond to systematically absent reflections, leading to incorrect phase identification.

No. This tool is restricted to cubic Bravais lattices where all three axes are equal (a = b = c) and all angles are 90°. Hexagonal systems (HCP metals like Ti, Zr) require two independent parameters (a and c) and a different interplanar spacing formula involving four indices (hkil). Tetragonal systems need a and c with a ≠ c.

APF quantifies how efficiently atoms fill space. FCC and HCP both achieve the theoretical maximum of 0.7405 (74.05%), BCC reaches 0.6802, and SC only 0.5236. Higher APF generally correlates with higher density, greater ductility, and more close-packed slip planes. SC is extremely rare in nature (only α-Po) precisely because its low packing efficiency makes it energetically unfavorable.