the theoretical density
The theoretical density of a ceramic is the density it would have if it were a single, perfect, pore-free crystal, and it is calculated straight from the contents of one unit cell. It is also called the X-ray density, because the cell size comes from X-ray diffraction. It is the yardstick against which the real, always-lower density of a fired part is measured.
The formula is rho = (n times M) / (V_c times N_A), where n is the number of formula units in the cell, M is the mass of one formula unit (its molar mass), V_c is the cell volume, and N_A is Avogadro's number (6.022 times 10^23 per mole). Work it through for magnesium oxide: rock salt has n = 4 formula units, M = 40.30 grams per mole, and a cell edge of 0.4212 nm, so V_c = (4.212 times 10^-8 cm)^3 = 7.47 times 10^-23 cubic centimetres. Then rho = (4 times 40.30) / (7.47 times 10^-23 times 6.022 times 10^23) = 3.58 grams per cubic centimetre, exactly what is measured for pure dense MgO.
Theoretical density matters most as the target of sintering. A pressed powder (a green body) starts around 55 to 65 percent of theoretical density, full of pores; firing shrinks it and drives out those pores until a good engineering ceramic reaches over 98 percent. The ratio of measured to theoretical density, called relative density, is the single most-quoted number for how well a ceramic has densified. Honest caveat: theoretical density assumes a defect-free crystal; the density of any real bulk ceramic is lower because of residual porosity, and never higher.
For dense alumina the theoretical density is about 3.99 grams per cubic centimetre. If a fired alumina part measures 3.91, its relative density is 3.91/3.99 = 98 percent, meaning about 2 percent of its volume is still pores, a useful, quick read on sintering quality.
rho equals n times M divided by cell volume times Avogadro's number.
Theoretical density is the pore-free ceiling. A real ceramic never exceeds it and usually falls short; the gap is porosity, which is why relative density is the standard measure of how completely a ceramic has sintered.