Density of Crystal Calculator

Theoretical crystal density, often called X-ray density, comes from weighing the contents of a single unit cell and dividing by its volume. The mass of one cell is Z times M divided by Avogadro number, where Z counts the formula units per cell and M is the molar mass in g/mol, so density equals ZM divided by Na times a3 for a cubic cell. Because lattice parameters are reported in angstroms, the edge must be multiplied by 1e-8 to reach centimetres before cubing, and forgetting that conversion shifts the answer by 24 orders of magnitude. Counting Z follows fixed sharing rules: a corner atom contributes one eighth, an edge atom one quarter, a face atom one half and a body atom one whole. Simple cubic therefore gives Z = 1, body-centred cubic Z = 2, face-centred cubic Z = 4, hexagonal close packed Z = 6 and diamond cubic Z = 8. Worked reference cases are useful checks. Copper, face-centred cubic with a = 3.615 angstrom and M = 63.55 g/mol, returns about 8.93 g/cm3 against a handbook value of 8.96. Body-centred iron with a = 2.866 angstrom gives 7.87 g/cm3, and rock salt with four NaCl units in a cell of 5.640 angstrom gives 2.16 g/cm3. Measured bulk densities are almost always slightly lower, since real samples contain porosity, grain boundaries and vacancies, and the ratio of pycnometric to X-ray density is used as a relative density figure in ceramics and powder metallurgy. Non-stoichiometric phases such as iron-deficient wustite break the assumption that Z is an exact integer.

The idea that a crystal density can be computed rather than measured dates to the first structure determinations. When William Henry Bragg and William Lawrence Bragg solved the sodium chloride and diamond structures in 1913, they immediately noticed that the cell dimension together with the known chemical formula and the accepted value of Avogadro number reproduced the observed density. That agreement was one of the strongest early arguments that X-ray diffraction was revealing real atomic positions, and it also worked in reverse, since a measured density plus a measured cell allowed Avogadro number itself to be estimated more accurately than by any competing method of the period. That reverse use grew into a precision metrology programme. From the 1960s onward national laboratories, including the US National Bureau of Standards and the German PTB, refined the X-ray crystal density technique using nearly perfect silicon single crystals, in which the lattice spacing, molar mass, sphere volume and mass could all be measured to extreme accuracy. The International Avogadro Coordination assembled spheres of isotopically enriched silicon-28 to remove molar mass uncertainty, reaching relative uncertainties of a few parts in 100 million by the mid 2010s. Those results, combined with watt balance measurements of the Planck constant, underpinned the decision of the General Conference on Weights and Measures in November 2018 to fix Avogadro number at exactly 6.02214076e23 and redefine the kilogram, effective 20 May 2019.